The Egyptian Mathematical Society

The Mathematical Sciences Research Centre (MSRC)

Technical Reports No. 1

Coalgebras and Bialgebras

by Robert Wisbauer Heinrich Heine Universität

Düsseldorf, Germany

Lectures given at

Cairo University and The American University in Cairo November - December 2004

http://msrc.etms-web.org 2

Contents

Preface iii

1 Modules and Algebras 1 1 Tensor product, tensor functor . . . . . . . . . . . . . . . . . . 1 2 Algebras and modules . . . . . . . . . . . . . . . . . . . . . . 15 3 The category σ[M] . . . . . . . . . . . . . . . . . . . . . . . . 22 4 External properties of σ[M] . . . . . . . . . . . . . . . . . . . 30

2 Coalgebras and comodules 38 5 Coalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 6 Coalgebra morphisms . . . . . . . . . . . . . . . . . . . . . . . 44 7 Comodules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 8 C-comodules and C^{∗}-modules . . . . . . . . . . . . . . . . . . 63 9 The rational functor . . . . . . . . . . . . . . . . . . . . . . . 74 10 Structure of comodules . . . . . . . . . . . . . . . . . . . . . . 81 11 Coalgebras over QF rings . . . . . . . . . . . . . . . . . . . . . 85 12 Bialgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 13 Antipodes and Hopf algebras . . . . . . . . . . . . . . . . . . . 108

3 Exercises 115

Bibliography 121

Index 122 ii

Preface

The notion of a coalgebra is dual to the notion of an algebra. Nevertheless both of them are related to similar objects in linear algebra. Coalgebras over fields are well-studied in the literature, e.g. in the texts of Sweedler [9], Abe [1], Montgomery [8], and Dǎscǎlescu, Nǎstǎsescu and Raianu [4].

To understand coalgebras over commutative rings is not only of interest for the sake of generalisation but also to provide techniques for the study of such structures over noncommutative rings - the corings. In these lectures we concentrate on coalgebras over commutative rings but the methods provided will readily apply to the more general case.

There are parts of module theory over algebras A that provide a useful framework for the theory of comodules. Given any left A-module M, denote by σ[M] the full subcategory of the category _{A}M of left A-modules that is subgenerated by M. This is the smallest Grothendieck subcategory of _{A}M containing M. Internal properties of σ[M] strongly depend on the module properties of M and this can be used as key for a homological classification of modules, that is, a characterisation of the structural properties of M by the properties of the category σ[M].

On the other hand, by definition, σ[M] is closed under direct sums, submodules and factor modules in_{A}M, and so it is a hereditary pretorsion class in AM. Hence torsion theory provides a setting for studying the outer properties of σ[M], that is, the behaviour of σ[M] as a subclass of _{A}M.

Both the inner and outer properties of the categories of type σ[M] are important in the study of coalgebras and comodules and it is the purpose of these lectures to make this clear. To begin with we present material from algebra and module theory in a form that is immediately applicable to comodules. Then a solid introduction to coalgebras and their comodules is given.

Portions of the notes are taken from the joint book with Tomasz Brzeziński [3] and I would like to use this opportunity to express my appreciation for his significant contributions to the subject. iv

Chapter 1

Modules and Algebras

1 Tensor product, tensor functor

The tensor product of modules is an important notion both for the theory of algebras and coalgebras. To ensure the generality desired for our purposes we give an account of the construction of the tensor product over noncommutative rings R. Notice that for some notions and results R need not have a unit.

1.1 Definition. Let M_{R} be a right module, _{R}N a left module over the ring R and G an abelian group.

A ℤ-bilinear map β : M × N → G is called R-balanced if, for all m ∈ M, n ∈ N and r ∈ R, we have: β(mr, n) = β(m, rn).

An abelian group T with an R-balanced map τ : M × N → T is called the tensor product of M and N if every R-balanced map

β : M × N → G, G an abelian group ,

can be uniquely factorized over τ, i.e., there is a unique ℤ-homomorphism γ : T → G which renders the following diagram commutative:

\xymatrix{M × N \ar[d]_{τ} \ar[r]^{β} & G . \\ T \ar[ur]_{γ}}

With standard arguments applied for universal constructions it is easily seen that the tensor product (T, τ) for a pair of modules M_{R}, _{R}N is uniquely determined up to isomorphism (of ℤ-modules).

1.2. Existence of tensor products. For the R-modules M_{R}, _{R}N, we form the direct sum of the family of ℤ-modules {ℤ_{(m,n)}}_{M×N} with ℤ_{(m,n)} ≃ ℤ, the free ℤ-module over M × N,

F = ⨁_{M×N}ℤ_{(m,n)} ≃ ℤ^{(M×N)}.

By construction, there is a (canonical) basis {f_{(m,n)}}_{M×N} in F. We simply write f_{(m,n)} = [m, n]. Let K denote the submodule of F generated by elements of the form

[m_{1} + m_{2}, n] − [m_{1}, n] − [m_{2}, n], [m, n_{1} + n_{2}] − [m, n_{1}] − [m, n_{2}],
[mr, n] − [m, rn], with m, m_{i} ∈ M, n, n_{i} ∈ N, r ∈ R.

Putting M ⊗_{R} N := F/K we define the map

τ : M × N → M ⊗_{R} N, (m, n) ↦ m ⊗ n := [m, n] + K .

By definition of K, the map τ is R-balanced. Observe that τ is not surjective but the image of τ, Im τ = {m ⊗ n | m ∈ M, n ∈ N}, is a generating set of M ⊗_{R} N as a ℤ-module.

If β : M × N → G is an R-balanced map we obtain a ℤ-homomorphism γ̃ : F → G, [m, n] ↦ β(m, n), and obviously K ⊂ Ke γ̃. Hence γ̃ factorizes over τ and we have the commutative diagram

\xymatrix{M × N \ar[d]_{τ} \ar[r]^{β} & G . \\ M ⊗_{R} N \ar@{-->}[ur]_{γ}}

γ is unique since its values on the generating set Im τ of T are uniquely determined.

Observe that every element in M ⊗_{R} N can be written as a finite sum

m_{1} ⊗ n_{1} + ⋯ + m_{k} ⊗ n_{k}.

However this presentation is not unique. m ⊗ n only represents a coset and m, n are not uniquely determined. Also a presentation of zero in M ⊗_{R} N is not unique. We may even have that M ⊗_{R} N is zero for non-zero M and N, e.g. ℤ_{2} ⊗_{ℤ} ℤ_{3} = 0.

1.3. Tensor product of homomorphisms. Consider two R-homomorphisms f : M_{R} → M′_{R} and g : _{R}N → _{R}N′.

(1) There is a unique ℤ-linear map

f ⊗ g : M ⊗_{R} N → M′ ⊗_{R} N′,
with f ⊗ g(m ⊗ n) = f(m) ⊗ g(n), m ∈ M, n ∈ N.

f ⊗ g is called the tensor product of the homomorphisms f and g.

(2) If f and g are surjective, then f ⊗ g is surjective and

Ke f ⊗ g = Ke f ⊗′ N + M ⊗′ Ke g,

where Ke f ⊗′ N denotes the submodule of M ⊗_{R} N generated by the elements u ⊗ n with u ∈ Ke f and n ∈ N, etc.

Proof. (1) Define a map

f × g : M × N → M′ ⊗_{R} N′, (m, n) ↦ f(m) ⊗ g(n).

It is ℤ-bilinear and R-balanced since f(mr) ⊗ g(n) = f(m) ⊗ g(rn). Hence the map f × g can be factorized over M ⊗_{R} N and we obtain the desired map f ⊗ g.

(2) If f and g are surjective, then obviously M′ ⊗_{R} N′ is generated as ℤ-module by the elements {f(m) ⊗ g(n) | m ∈ M, n ∈ N} and hence f ⊗ g is surjective.

It is clear that the subgroup H = Ke f ⊗′ N + M ⊗′ Ke g lies in Ke f ⊗ g and hence, with the canonical projection p, the map f ⊗ g factors into

M ⊗_{R} N ⟶^{p} M ⊗_{R} N/H ⟶^{ν} M′ ⊗_{R} N′.

Obvioulsy ν is surjective. To show that ν is an isomorphism we first consider a map

α : M′×N′ → M⊗_{R}N/H, (m′, n′) ↦ m⊗n+H, where m′ = f(m), n′ = g(n).

To see that the map is well-defined, i.e., that it is independent of the choice of m ∈ M with f(m) = m′, and similarly of n ∈ N, observe that for f(m_{1}) = m′ and g(n_{1}) = n′ we have

m_{1} ⊗ n_{1} = m ⊗ n + [(m_{1} − m) ⊗ n + m ⊗ (n_{1} − n) + (m_{1} − m) ⊗ (n_{1} − n)],

where the right summand […] lies in H. Clearly ν is ℤ-linear and R-balanced and hence induces a map ᾱ : M′ ⊗_{R} N′ → M ⊗_{R} N/H with ᾱ ∘ν = I, proving that ν is injective. ⊔ ⊓

1.4. Tensor product and direct sums. Let M_{R} be an R-module and RN = ⨁_{Λ}N_{λ}, with the canonical injections ε_{λ} : _{R}N_{λ} → _{R}N and projections π_{λ} : _{R}N → _{R}N_{λ}.

Then (M ⊗_{R} N, I_{M} ⊗ ε_{λ}) is a direct sum of {M ⊗_{R} N_{λ}}_{Λ}, i.e.,

M ⊗_{R} (⨁_{Λ}N_{λ}) ≃ ⨁_{Λ}(M ⊗_{R} N_{λ}).

We say the tensor product commutes with direct sums.

Proof. For the maps I_{M} ⊗ π_{λ} : M ⊗_{R} N → M ⊗_{R} N_{λ}, we derive from properties of tensor products of homomorphisms

(I_{M} ⊗ ε_{λ})(I_{M} ⊗ π_{μ}) = δ_{λμ}I_{M⊗_{R}N_{λ}}.

For a family {f_{λ} : M ⊗_{R} N_{λ} → X}_{Λ} of ℤ-linear maps, we define

f : M ⊗_{R} N → X, m ⊗ n ↦ ∑_{λ∈Λ}f_{λ}(I_{M} ⊗ π_{λ}(m ⊗ n)) ,

where the sum is always finite.

Obviously, f ∘ (I_{M} ⊗ ε_{λ}) = f_{λ} and (M ⊗_{R} N, I_{M} ⊗ ε_{λ}) is a direct sum of the {M ⊗_{R} N_{λ}}_{Λ}.

By symmetry, we obtain, for M_{R} = ⨁_{Λ′}M_{μ},

\begin{matrix}(⨁_{Λ′}M_{μ}) ⊗_{R} N ≃ ⨁_{Λ′}(M_{μ} ⊗_{R} N), \\ (⨁_{Λ′}M_{μ}) ⊗_{R} (⨁_{Λ}N_{λ}) ≃ ⨁_{Λ′×Λ}(M_{μ} ⊗ N_{λ}). \end{matrix}

⊔ ⊓

1.5. Module structure of tensor products. By construction, the tensor product M ⊗_{R} N of M_{R} and _{R}N is only an abelian group. However, if _{T} M_{R} or _{R}N_{S} are bimodules, then we may define module structures on M ⊗_{R} N:

If _{T} M_{R} is a (T, R)-bimodule, then the elements of T may be regarded as R-endomorphisms of M and the tensor product with I_{N} yields a map

T → End_{ℤ}(M ⊗_{R} N), t ↦ t ⊗ I_{N}.

From the properties of this construction noted in 1.3 we see that this is a ring homomorphism. Hence _{T} M ⊗_{R} N becomes a left T-module and the action of t ∈ T on ∑ m_{i} ⊗ n_{i} ∈ M ⊗_{R} N is given by

t(∑ m_{i} ⊗ n_{i}) = ∑(tm_{i}) ⊗ n_{i}.

For an (R, S)-bimodule _{R}N_{S}, we obtain in the same way that M ⊗_{R} N_{S} is a right S-module.

If _{T} M_{R} and _{R}N_{S} are bimodules, the structures defined above turn T M ⊗_{R} N_{S} into a (T, S)-bimodule since we have, for all t ∈ T, s ∈ S and m ⊗ n ∈ M ⊗_{R} N, that (t(m ⊗ n))s = (tm) ⊗ (ns) = t((m ⊗ n)s).

1.6. Tensor product with R. Regarding R as an (R, R)-bimodule, for every R-module _{R}N, there is an R-isomorphism

μ_{R} : R ⊗_{R} N → RN, ∑ r_{i} ⊗ n_{i} ↦ ∑ r_{i}n_{i}.

The map exists since the map R × N → RN, (r, n) ↦ rn is balanced, and obviously has the given properties.

Since the tensor product commutes with direct sums (see 1.4), we obtain, for every free right R-module F_{R} ≃ R_{R}^{(Λ)}, Λ an index set, a ℤ-isomorphism F ⊗_{R} N ≃ N^{(Λ)}.

1.7. Associativity of the tensor product. Assume three modules M_{R}, RN_{S} and _{S}L to be given. Then (M ⊗_{R} N) ⊗_{S} L and M ⊗_{R} (N ⊗_{S} L) can be formed and there is an isomorphism

σ : (M ⊗_{R} N) ⊗_{S} L → M ⊗_{R} (N ⊗_{S} L), (m ⊗ n) ⊗ l ↦ m ⊗ (n ⊗ l) .

Proof. We only have to show the existence of such a map σ. Then, by symmetry, we obtain a corresponding map in the other direction which is inverse to σ: We first define, for l ∈ L, a morphism f_{l} : N → N ⊗_{S} L, n ↦ n ⊗ l, then form the tensor product I_{M} ⊗ f_{l} : M ⊗_{R} N → M ⊗_{R} (N ⊗_{S} L) and obtain

β : (M ⊗_{R} N) × L → M ⊗_{R} (N ⊗_{S} L), (m ⊗ n, l) ↦ I_{M} ⊗ f_{l} (m ⊗ n).

It only remains to verify that β is balanced to obtain the desired map. ⊔ ⊓

1.8. Tensor functors. For an (S,R)-bimodule _{S}U_{R}, the assignments

SU ⊗_{R} − : Obj(R-Mod) ⟶ Obj(S-Mod), _{R}M ↦ _{S}U ⊗_{R} M,
Mor(R-Mod) ⟶ Mor(S-Mod),  f ↦ I_{U} ⊗ f,

yield a covariant functor _{S}U ⊗_{R} − : R-Mod → S-Mod with the properties (1) _{S}U ⊗_{R} − is additive and right exact;

(2) _{S}U ⊗_{R} − preserves direct sums.

Similarly we obtain a functor − ⊗_{S} U_{R} : Mod-S → Mod-R with the same properties.

Proof. Applying 1.3 it is easily checked that the given assignments define an additive functor. In 1.4 we have seen that it preserves direct sums. It remains to show that it is right exact. An exact sequence K →^{f} L →^{g} N → 0 in R-Mod yields the sequence

U ⊗_{R} K ⟶^{I⊗f} U ⊗_{R} L ⟶^{I⊗g} U ⊗_{R} N ⟶ 0 ,

where, by 1.3(2), I ⊗ g is surjective and

Ke I ⊗ g = U ⊗′ Ke g = U ⊗′ Im f = Im I ⊗ f,

proving the exactness of this sequence. ⊔ ⊓

Notice that the relation between tensor products and direct products is more complicated than that between tensor products and direct sums.

1.9. Tensor product and direct products. Let U_{R} be a right R-module and {L_{λ}}_{Λ} a family of left R-modules. With the canonical projections we have the maps

I_{U} ⊗ π_{μ} : U ⊗_{R} (∏_{Λ}L_{λ}) → U ⊗_{R} L_{μ}

and, by the universal property of the product,

φ_{U} : U ⊗_{R} (∏_{Λ}L_{λ}) → ∏_{Λ} U ⊗_{R} L_{λ}, u ⊗ (l_{λ})_{Λ} ↦ (u ⊗ l_{λ})_{Λ}.

It is easy to see that, for U = R, and hence also for U = R^{n}, φ_{U} is an isomorphism.

(1) The following assertions are equivalent:

(a) U is finitely generated;

(b) φ_{U} is surjective for every family {L_{λ}}_{Λ};

(c) φ̃_{U} : U ⊗ R^{Λ} → (U ⊗ R)^{Λ} ≃ U^{Λ} is surjective for any set Λ (or Λ = U).

(2) The following assertions are also equivalent:

(a) U is finitely presented in R-Mod;

(b) φ_{U} is bijective for every family {L_{λ}}_{Λ};

(c) φ̃_{U} : U ⊗_{R} R^{Λ} → U^{Λ} is bijective for every set Λ.

Proof. (1) (a) ⇒ (b) If U is finitely generated and R^{(A)} →^{f} R^{n} →^{g} U → 0 is exact, we can form the commutative diagram with exact rows:

\xymatrix{R^{(A)} ⊗ ∏_{Λ}L_{λ} \ar[d]^{φ_{R^{(A)}}} \ar[r]^{f⊗I} & R^{n} ⊗ ∏_{Λ}L_{λ} \ar[d]^{φ_{R^{n}}} \ar[r]^{g⊗I} & U ⊗ ∏_{Λ}L_{λ} \ar[d]^{φ_{U}} \ar[r] & 0 \\ ∏_{Λ}(R^{(A)} ⊗ L_{λ}) \ar[r]^{∏(f⊗I)} & ∏_{Λ}(R^{n} ⊗ L_{λ}) \ar[r]^{∏(g⊗I)} & ∏_{Λ}(U ⊗ L_{λ}) \ar[r] & 0.}

As pointed out above, φ_{R^{n}} is bijective and hence φ_{U} is surjective. (b) ⇒ (c) is obvious.

(c) ⇒ (a) Assume (c). Then, for Λ = U, the map φ̃ : U ⊗ R^{U} → U^{U} is surjective. For the element (u_{u})_{U} (= I_{U} in Map(U, U) = U^{U}), we choose ∑_{i≤k}m_{i} ⊗ (r_{u}^{i}) as a preimage under φ̃_{U}, with r_{u}^{i} ∈ R, m_{i} ∈ U, i.e.

(u_{u})_{U} = ∑_{i≤k}(m_{i}r_{u}^{i})_{U} = (∑_{i≤k}m_{i}r_{u}^{i})_{U}.

Hence, for every u ∈ U, we get u = ∑_{i≤k}m_{i}r_{u}^{i}, i.e. m_{1}, …, m_{k} is a generating set of U.

(2) (a) ⇒ (b) In the proof (1) (a) ⇒ (b) we can choose a finite index set A. Then φ_{R^{(A)}} is an isomorphism and hence also φ_{U}.

(b) ⇒ (c) is obvious.

(c) ⇒ (a) From (1) we already know that U is finitely generated. Hence there is an exact sequence 0 → K → R^{n} → U → 0, n ∈ ℕ. From this we obtain – for any set Λ – the following commutative diagram with exact rows

\xymatrix{ & K ⊗ R^{Λ} \ar[d]^{φ̃_{K}} \ar[r] & R^{n} ⊗ R^{Λ} \ar[d]^{φ_{R^{n}}} \ar[r] & U ⊗ R^{Λ} \ar[d]^{φ_{U}} \ar[r] & 0 \\ 0 \ar[r] & K^{Λ} \ar[r] & (R^{n})^{Λ} \ar[r] & U^{Λ} \ar[r] & 0 .}

Here φ̃_{R^{n}} is an isomorphism (see above) and φ̃_{U} is an isomorphism by (c). According to the Kernel Cokernel Lemma, φ̃_{K} is surjective and, by(1), K is finitely generated. Therefore, for some m ∈ ℕ, we get an exact sequence R^{m} → K → 0, and R^{m} → R^{n} → U → 0 is also exact. ⊔ ⊓

As a consequence of the right exactness of the tensor functor the following two results can be shown:

1.10. Zero in the tensor product. Let {n_{i}}_{i∈Λ} a generating set of the R-module _{R}N and {m_{i}}_{i∈Λ} a family of elements in the R-module M_{R} with only finitely many m_{i} ≠ 0.

Then ∑_{Λ}m_{i} ⊗ n_{i} = 0 in M ⊗_{R} N if and only if there are finitely many elements {a_{j}}_{j∈Λ′} in M and a family {r_{ji}}_{Λ′×Λ} of elements in R with the properties

(i) r_{ji} ≠ 0 for only finitely many pairs (j,i), (ii) ∑_{i∈Λ}r_{ji}n_{i} = 0 for every j ∈ Λ′, (iii) m_{i} = ∑_{j∈Λ′}a_{j}r_{ji}.

Proof. For elements with these properties we see

∑_{Λ}m_{i} ⊗ n_{i} = ∑_{Λ}∑_{Λ′}a_{j}r_{ji} ⊗ n_{i} = ∑_{Λ′}(a_{j} ⊗ ∑_{Λ}r_{ji}n_{i}) = 0.

Now assume ∑_{Λ}m_{i} ⊗ n_{i} = 0. With the canonical basis {f_{i}}_{i∈Λ} and the map g : R^{(Λ)} → _{R}N, f_{i} ↦ n_{i}, we obtain the exact sequence

0 ⟶ _{R}K ⟶^{ε} R^{(Λ)} ⟶^{g} _{R}N ⟶ 0.

Tensoring with M ⊗_{R} − yields the exact sequence

M ⊗_{R} K ⟶^{I⊗ε} M ⊗_{R} R^{(Λ)} ⟶^{I⊗g} M ⊗_{R} N ⟶ 0 .

By assumption, I ⊗ g(∑_{Λ}m_{i} ⊗ f_{i}) = ∑_{Λ}m_{i} ⊗ n_{i} = 0 and there is an element ∑_{j∈Λ′}a_{j} ⊗ k_{j} ∈ M ⊗_{R} K with I ⊗ ε(∑_{j∈Λ′}a_{j} ⊗ k_{j}) = ∑_{i∈Λ}m_{i} ⊗ f_{i}.

Every k_{j} ∈ K ⊂ R^{(Λ)} can be written as k_{j} = ∑_{i∈Λ}r_{ji}f_{i} with only finitely many r_{ji} ≠ 0. This implies 0 = (k_{j})εg = ∑_{i∈Λ}r_{ji}n_{i} for all j ∈ Λ′, and in M ⊗_{R} R^{(Λ)} we get

∑_{i∈Λ}m_{i} ⊗ f_{i} = ∑_{j∈Λ′}a_{j} ⊗ k_{j} = ∑_{i∈Λ}(∑_{j∈Λ′}a_{j}r_{ji}) ⊗ f_{i}.

From this the projections onto the components yield the desired condition m_{i} = ∑_{j∈Λ′}a_{j}r_{ji}. ⊔ ⊓ 1.11. Tensor product with cyclic modules. Let I be a right ideal of a ring R with many idempotents and _{R}M a left module. Then

R/I ⊗_{R} M ≃ RM/IM (≃ M/IM if 1 ∈ R).

Proof. From the exact sequence 0 → I → R → R/I → 0 we obtain the first row exact in the commutative diagram

I ⊗_{R} M ⟶ R ⊗_{R} M ⟶ R/I ⊗_{R} M ⟶ 0
↓ μ_{I} ↓ μ_{R} ↓ γ
0 ⟶ IM ⟶ RM ⟶ RM/IM ⟶ 0

with the map μ_{I} : I ⊗_{R} M → IM, i⊗m ↦ im. By 1.6, μ_{R} is an isomorphism and hence γ is an isomorphism by the Kernel Cokernel Lemma. ⊔ ⊓

An interesting connection between Hom- and tensor functors is derived from the definition of the tensor product:

1.12. Hom-tensor relation. Let U_{R} and _{R}M be R-modules, N a ℤ-module and denote by Ten(U ×M, N) the set of the R-balanced maps from U ×M into N. By the definition of U ⊗_{R} M (see 12.1), the canonical map τ : U × M → U ⊗_{R} M yields a ℤ-isomorphism

ψ_{1} : Hom_{ℤ}(U ⊗_{R} M, N) → Ten(U × M, N), α ↦ τα.

On the other hand, every β ∈ Ten(U × M, N) defines an R-homomorphism

h_{β} : M → Hom_{ℤ}(U, N), m ↦ β(−, m),

where Hom_{ℤ}(U, N) is regarded as a left R-module in the usual way. From this we obtain a ℤ-isomorphism

ψ_{2} : Ten(U × M, N) → Hom_{R}(M, Hom_{ℤ}(U, N)), β ↦ h_{β}.

Now every φ ∈ Hom_{R}(M, Hom_{ℤ}(U, N)) determines an R-balanced map

φ̃ : U × M → N, (u, m) ↦ φ(m)(u),

and the assignment φ ↦ φ̃ is a map inverse to ψ_{2}. The composition of ψ_{1} and ψ_{2} leads to the ℤ-isomorphism

ψ_{M} : Hom(U ⊗_{R} M, N) → Hom_{R}(M, Hom(U, N)), δ ↦ [m ↦ δ(− ⊗ m)],

with inverse map ψ_{M}^{−1} : φ ↦ [u ⊗ m ↦ φ(m)(u)].

If_{S}U_{R} is an (S, R)-bimodule and_{S}N an S-module, then_{S}U ⊗_{R} M is also a left S-module and with respect to this structure ψ_{M} becomes a ℤ-isomorphism

ψ_{M} : Hom_{S}(U ⊗_{R} M, N) → Hom_{R}(M, Hom_{S}(U, N)).

It is readily verified that, for every R-homomorphism g : _{R}M → _{R}M′, the following diagram is commutative:

[figure: Hom_{S}(U ⊗_{R} M′, N); Hom(I⊗g,N); ψ_{M′}; Hom_{S}(U ⊗_{R} M, N); ψ_{M}; Hom_{R}(M′, Hom_{S}(U, N)); Hom(g,Hom_{S}(U,N)); Hom_{R}(M, Hom_{S}(U, N)) .]

Similarly we obtain, for modules _{R}U_{S}, M_{R} and N_{S}, a ℤ-isomorpism

ψ′_{M} : Hom_{S}(M ⊗_{R} U, N) → Hom_{R}(M, Hom_{S}(U, N))

and a corresponding commutative diagram.

1.13. Definitions. Let M be a left R-module. A right R-module U_{R} is called M-flat if, for every monomorphism f : K → M in R-Mod, the map I_{U} ⊗ f : U ⊗_{R} K → U ⊗_{R} M is also a monomorphism.

U_{R} is said to be flat (with respect to R-Mod) if U is M-flat for every M ∈ R-Mod.

Since U ⊗_{R} − is always right exact, U_{R} is flat (with respect to R-Mod) if and only if the functor U ⊗_{R} − : R-Mod → ℤ-Mod is exact.

1.14. Direct sum of M-flat modules. Let {U_{λ}}_{Λ} be a family of right R-modules and _{R}M ∈ R-Mod. The direct sum ⨁_{Λ}U_{λ} is M-flat if and only if U_{λ} is M-flat for every λ ∈ Λ.

Proof. From the exact sequence 0 → K →^{f} M we form the commutative diagram

\xymatrix{(⨁_{Λ}U_{λ}) ⊗_{R} K \ar[r]^{I⊗f} \ar[d] & (⨁_{Λ}U_{λ}) ⊗_{R} M \ar[d] \\ ⨁_{Λ}(U_{λ} ⊗_{R} K) \ar[r]^{⊕(I_{λ}⊗f)} & ⨁_{Λ}(U_{λ} ⊗_{R} M),}

in which the vertical maps are the canonical isomorphisms (see 1.4). Hence I ⊗ f is monic if and only if all I_{λ} ⊗ f are monic. ⊔ ⊓

1.15. Properties of M-flat modules. Let U_{R} be a right R-module. Then:

(1) U_{R} is M-flat if and only if U ⊗_{R} − is exact with respect to every exact sequence 0 → K′ → M with K′ finitely generated.

(2) Let 0 → M′ → M → M′′ → 0 be an exact sequence in R-Mod. If U_{R} is M-flat, then U_{R} is also M′- and M′′-flat.

(3) Let {M_{λ}}_{Λ} be a family of left R-modules. If U_{R} is M_{λ}-flat for every λ ∈ Λ, then U_{R} is also ⨁_{Λ}M_{λ}-flat.

Proof. (1) Let 0 → K →^{ε} M be exact and ∑_{i≤n}u_{i} ⊗ k_{i} ∈ U ⊗_{R} K with (∑_{i≤n}u_{i} ⊗ k_{i})I ⊗ ε = 0 ∈ U ⊗_{R} M. Let K′ denote the submodule of K generated by k_{1}, …, k_{n}. Since the map

I ⊗ ε′ : U ⊗_{R} K′ → U ⊗_{R} K → U ⊗_{R} M

is monic by assumption, we get ∑_{i≤n}u_{i} ⊗ k_{i} = 0 in U ⊗_{R} K′. Then it also has to be zero in U ⊗_{R} K, i.e., I ⊗ ε is monic.

(2) Let U_{R} be M-flat. If 0 → K →^{ε′} M′ is exact, the canonical map

U ⊗_{R} K ⟶^{I⊗ε′} U ⊗_{R} M′ ⟶ U ⊗_{R} M is monic and U_{R} is M′-flat.

If 0 → L →^{f} M′′ is exact, we obtain, by a pullback, the commutative diagram with exact rows and columns

0  0
↓  ↓
0 ⟶ M′ ⟶ P ⟶ L ⟶ 0
∥ ↓ ↓ f
0 ⟶ M′ ⟶ M ⟶ M′′ ⟶ 0 .

Tensoring with U_{R} yields the following commutative diagram with exact rows and columns

0 ↓

U ⊗_{R} M′ ⟶ U ⊗_{R} P ⟶ U ⊗_{R} L ⟶ 0
∥ ↓ ↓ I ⊗ f
0 ⟶ U ⊗_{R} M′ ⟶ U ⊗_{R} M ⟶ U ⊗_{R} M′′ ⟶ 0 .

By the Kernel Cokernel Lemma, I ⊗ f has to be monic, i.e., U is M′′-flat.

(3) We show that U_{R} is M_{1} ⊕ M_{2}-flat if it is both M_{1}- and M_{2}-flat. Then we get assertion (3) for finite index sets Λ by induction. For arbitrary sets Λ we use (1): A finitely generated submodule K′ ⊂ ⨁_{Λ} M_{λ} is contained in a finite partial sum. Since the tensor product preserves direct summands, the assertion follows from the finite case.

Let U_{R} be M_{1}- and M_{2}-flat and 0 → K →^{f} M_{1} ⊕ M_{2} exact. Forming a pullback we obtain the commutative exact diagram

0  0  0
↓  ↓  ↓
0 ⟶ P ⟶  K  ⟶ L ⟶ 0
↓  ↓  ↓
0 ⟶ M_{1} ⟶ M_{1} ⊕ M_{2} ⟶ M_{2} ⟶ 0 .

Tensoring with U_{R} yields the commutative exact diagram

0  0
↓  ↓
U ⊗_{R} P ⟶  U ⊗_{R} K  ⟶ U ⊗_{R} L ⟶ 0
↓  ↓ I ⊗ f  ↓
0 ⟶ U ⊗_{R} M_{1} ⟶ U ⊗_{R} (M_{1} ⊕ M_{2}) ⟶ U ⊗_{R} M_{2} ⟶ 0 .

By the Kernel Cokernel Lemma, I ⊗ f has to be monic. ⊔ ⊓

1.16. Flat modules. Characterizations. For a right R-module U_{R}, the following assertions are equivalent:

(a) U_{R} is flat (with respect to R-Mod);

(b) U ⊗_{R} − is exact with respect to all exact sequences 0 → _{R}I → _{R}R (with _{R}I finitely generated);

(c) for every (finitely generated) left ideal _{R}I ⊂ R, the canonical map μ_{I} : U ⊗_{R} I → UI is monic (and hence an isomorphism).

Proof. The equivalence of (a) and (b) follows from 1.13.

(b) ⇔ (c) For every (finitely generated) left ideal I ⊂ R, we have the commutative diagram with exact rows (see 1.11)

U ⊗_{R} I ⟶^{I⊗ε} U ⊗_{R} R ⟶ U ⊗_{R} R/I ⟶ 0
↓ μ_{I}  ∥  ∥
0 ⟶ UI ⟶  U  ⟶ U/UI ⟶ 0 .

Hence μ_{I} is monic (an isomorphism) if and only if I ⊗ ε is monic. ⊔ ⊓

In a ring R with unit, for every left ideal I ⊂ R, we have R ⊗_{R} I ≃ I. Hence R_{R} is a flat module (with respect to R-Mod). Then, by 1.14, all free R-modules and their direct summands (= projective modules) are flat (with respect to R-Mod).

An R-module U_{R} is called faithfully flat (with respect to R-Mod) if U_{R} is flat (w.r. to R-Mod) and, for N ∈ R-Mod, the relation U ⊗_{R} N = 0 implies N = 0.

1.17. Faithfully flat modules. Characterizations. For a right R-module U_{R} the following assertions are equivalent:

(a) U_{R} is faithfully flat;

(b) U_{R} is flat and, for every (maximal) left ideal I ⊂ R, I ≠ R, we have U ⊗_{R} R/I ≠ 0 (i.e., UI ≠ U).

Proof. (a) ⇒ (b) Because of the isomorphism U ⊗_{R} R/I ≃ U/UI (see 1.11), U ⊗_{R} R/I ≠ 0 is equivalent to UI ≠ U. By (a), U ⊗_{R} R/I = 0 would imply I = R.

(b) ⇒ (a) If UI ≠ U for every maximal left ideal I ⊂ R, then this is also true for every proper left ideal I ⊂ R. Hence U ⊗_{R} K ≠ 0 for every cyclic R-module K. Since every R-module N contains a cyclic submodule and U_{R} is flat, we have U ⊗_{R} N ≠ 0. ⊔ ⊓

1.18. Pure morphisms. Related to any morphism f : M → M′ in R−Mod, there is an exact sequence

\xymatrix{0 \ar[r] & Ke f \ar[r] & M \ar[r]^{f} & M′ \ar[r] & Coke f \ar[r] & 0 .}

Given L ∈ Mod−R, we say the morphism f is L-pure if tensoring this sequence with L ⊗_{R} − yields an exact sequence. The morphism f is said to be pure if it is L-pure for every L ∈ Mod−R. Since the tensor functor is right exact, the following are equivalent:

(a) f is L-pure;

(b) \xymatrix{0 \ar[r] & L ⊗_{R} Ke f \ar[r] & L ⊗_{R} M \ar[r]^{I_{L}⊗f} & L ⊗_{R} M′} is exact;

(c) Ke f → M and Im f → M′ are L-pure (mono) morphisms.

For any inclusion i : N → M, the image of the map

I_{L} ⊗ i : L ⊗_{R} N → L ⊗_{R} M

is called the canonical image of L ⊗_{R} N in L ⊗_{R} M. If I_{L} ⊗ i is injective (i.e., i is an L-pure morphism), then N is said to be an L-pure submodule and we identify the canonical image of I_{L} ⊗ i with L ⊗_{R} N.

Obviously, any direct summand is a pure submodule, and if L is a flat right R-module, then every morphism f : M → M′ in R−Mod is L-pure.

1.19. Tensor product over commutative rings. Let M, N and L be modules over a commutative ring R.

An R-balanced map β : M × N → L is called R-bilinear if

β(rm, n) = rβ(m, n) for all r ∈ R, m ∈ M, n ∈ N.

By 1.5, _{R}M ⊗_{R} N is a left R-module with r(m ⊗ n) = (rm) ⊗ n and we see that the balanced map

τ : M × N → M ⊗_{R} N, (m, n) ↦ m ⊗ n,

is bilinear: τ(rm, n) = (rm)⊗n = rτ(m, n). Hence we have, for commutative rings R:

A map β : M × N → L is R-bilinear if and only if there is an R-linear map β̄ : M ⊗_{R} N → L with β = τβ̄.

With the notation Bil_{R}(M × N, L) = {β : M × N → L | β R-bilinear} we have an isomorphism of R-modules

Hom_{R}(M ⊗_{R} N, L) ≃ Bil_{R}(M × N, L).

If M and N are vector spaces over a field K, then, by the above considerations, M ⊗_{K} N is also a K-vector space and since the tensor product commutes with direct sums we find

dim_{K}(M ⊗_{K} N) = dim_{K}M · dim_{K}N.

Every free R-module_{R}F is isomorphic to R^{(Λ)} for a suitable index set Λ. Over non-commutative rings the cardinality of Λ need not be uniquely determined. However, over a commutative ring with unit we have:

If R^{(Λ)} ≃ R^{(Λ′)}, then Λ and Λ′ have the same cardinality.

Proof. For a maximal ideal m of R, tensoring with − ⊗_{R} R/m yields (R/m)^{(Λ)} ≃ (R/m)^{(Λ′)}. For vector spaces over a field (= R/m) it is known that the cardinality of a basis is uniquely determined. ⊔ ⊓

1.20. Twist map. For modules M, N over a commutative ring R, there is an R-isomorphism

tw : M ⊗_{R} N → N ⊗_{R} M, m ⊗ n ↦ n ⊗ m,

called the twist map.

The purity condition on the submodules imply the important

1.21. Intersection property.

(1) Let M′ ⊂ M and K′ ⊂ K be pure R-submodules, or assume K and K/K′ to be (M-) flat. Then the canonical image of M′ ⊗_{R} K′ in M ⊗_{R} K is equal to the intersection of the canonical images of M′ ⊗_{R} K and M ⊗_{R} K′ in M ⊗_{R} K, i.e.,

M′ ⊗_{R} K′ = (M′ ⊗_{R} K) ∩ (M ⊗_{R} K′).

(2) Let U, V ⊂ M be R-submodules and K a flat R-module. Then

(U ⊗_{R} K) ∩ (V ⊗_{R} K) = (U ∩ V) ⊗_{R} K.

Proof. (1) Under the given conditions we have the exact commutative diagram

0  0  0
↓  ↓  ↓
0 → M′ ⊗_{R} K′ → M′ ⊗_{R} K → M′ ⊗_{R} K/K′ → 0
↓ ↓ ↓
0 → M ⊗_{R} K′ → M ⊗_{R} K → M ⊗_{R} K/K′ → 0 ,

where the left square is a pullback (e.g., [10, 10.3]), and hence we can make the identification stated.

(2) This is shown with a similar argument. ⊔ ⊓

2 Algebras and modules

From now on R will usually denote an associative commutative ring with unit. We recall basic definitions for algebras in a form which is suitable for dualising to coalgebras. The unadorned symbol ⊗ will always stand for ⊗_{R}.

2.1. Algebras. An R-module A is said to be an R-algebra if there exists an R-linear map

μ : A ⊗_{R} A → A,

called the multiplication of A. For a, b ∈ A, we write μ(a ⊗ b) = a · b (or ab).

2.2. Unit element. An element e ∈ A is called a unit if a · e = a = e · a, for all a ∈ A. This yields an R-homomorphism ι : R → A, r ↦ re, with

μ ∘ (ι ⊗ I_{A}) = μ ∘ (I_{A} ⊗ ι) = I_{A},

(putting A ⊗_{R} R = A = R ⊗_{R} A) which correponds to the commutativity of the diagram

\xymatrix{R ⊗_{R} A \ar[r]^{ι⊗I_{A}} \ar[dr]_{≃} & A ⊗_{R} A \ar[d]^{μ} & A ⊗_{R} R \ar[dl]^{≃} \ar[l]_{I_{A}⊗ι} \\  & A}

It is easy to check that for any R-linear map ι : R → A with these properties, ι(1) is an identity element in A.

2.3. Associativity and commutativity An R-algebra A is said to be associative, if a(bc) = (ab)c, for all a, b, c ∈ A. For the defining map μ : A ⊗_{R} A → A, this corresponds to the commutativity of the diagram

\xymatrix{A ⊗_{R} A ⊗_{R} A \ar[r]^{I_{A}⊗μ} \ar[d]_{μ⊗I_{A}} & A ⊗_{R} A \ar[d]^{μ} \\ A ⊗_{R} A \ar[r]^{μ} & A .}

The commutativity of A is expressed by the commutativity of the following diagram, where tw denotes the twist map,

\xymatrix{A ⊗_{R} A \ar[dr]_{μ} \ar[rr]^{tw} &  & A ⊗_{R} A \ar[dl]^{μ} \\  & A & .}

2.4. Algebra morphisms. Given two R-algebras μ_{A} : A ⊗ A → A and μ_{B} : B ⊗_{R} B → B, an R-linear map f : A → B is said to be an algebra morphism, if f(ab) = f(a)f(b), for all a, b ∈ A, i.e., we have the commutative diagram

\xymatrix{A ⊗_{R} A \ar[r]^{μ_{A}} \ar[d]_{f⊗f} & A \ar[d]^{f} \\ B ⊗_{R} B \ar[r]^{μ_{B}} & B .}

Clearly f preserves the units ι_{A} : R → A, ι_{B} : R → B, if and only if we have a commutative diagram

\xymatrix{R \ar[d]_{ι_{A}} \ar[r]^{=} & R \ar[d]^{ι_{B}} \\ A \ar[r]^{f} & B.}

Obviously the identity I_{A} : A → A is an algebra morphism and the composition of algebra morphisms is again an algebra morphism.

2.5. Tensor product of algebras. Let μ_{A} : A⊗_{R}A → A and μ_{B} : B⊗_{R}B → B define R-algebras. Then A ⊗_{R} B is an R-algebra by the R-linear map

μ_{A⊗B} : (A ⊗_{R} B) ⊗_{R} (A ⊗ B) ⟶^{I⊗tw⊗I} (A ⊗_{R} A) ⊗_{R} (B ⊗_{R} B) ⟶^{μ_{A}⊗μ_{B}} A ⊗_{R} B .

It is straightforward to show that the algebra A ⊗_{R} B is associative (commutative) if both A and B are associative (commutative).

If e_{A} and e_{B} are the units in A and B, then e_{A} ⊗ e_{B} is the unit in A ⊗_{R} B. In particular, if B is an associative and commutative algebra with unit,

then it is easy to verify that μ_{A⊗B} is in fact B-linear, yielding a map

μ_{A⊗B} : (A ⊗_{R} B) ⊗_{B} (A ⊗_{R} B) → A ⊗_{R} B,

showing that A ⊗_{R} B is a B-algebra (scalar extension of A by B).

2.6. Tensor product of algebra morphisms. Let f : A → A_{1} and g : B → B_{1} be R-algebra morphisms. Then:

(1) There is an algebra morphism

h : A ⊗_{R} B → A_{1} ⊗_{R} B_{1}, a ⊗ b ↦ f(a) ⊗ g(b).

(2) If f and g are surjective, then h is also surjective and (see 1.3)

A_{1} ⊗_{R} B_{1} ≃ (A ⊗_{R} B)/(Ke f ⊗′ B + A ⊗′ Ke g).

Proof. (1) Put h = f ⊗ g, the tensor product of f and g as R-module morphism. It remains to show that h is a ring morphism:

h((a_{1} ⊗ b_{1})(a_{2} ⊗ b_{2})) = f(a_{1}a_{2}) ⊗ g(b_{1}b_{2})
= (f(a_{1}) ⊗ g(b_{1}))(f(a_{2}) ⊗ g(b_{2}))
= h(a_{1} ⊗ b_{1})h(a_{2} ⊗ b_{2}).

(2) This is shown in 1.3. ⊔ ⊓

2.7. Universal property of the tensor product. Let f : A → C and g : B → C be unital algebra morphisms such that

[f(A), g(B)] = 0,

where [−, −] denotes the commutator. Then there exists a unique algebra morphism h : A ⊗_{R} B → C, satisfying

h(a ⊗ b) = f(a)g(b) , for all a ∈ A, b ∈ B.

Proof. Since f and g are R-module morphisms, there is an R-module morphism

h = f ⊗ g : A ⊗_{R} B → C, (a, b) ↦ f(a)g(b).

It remains to verify that h is an algebra morphism. By our assumptions on the commutators of f(A) and g(B), we have

h((a_{1} ⊗ b_{1})(a_{2} ⊗ b_{2})) = f(a_{1}a_{2})g(b_{1}b_{2})
= f(a_{1})f(a_{2})g(b_{1})g(b_{2})
= (f(a_{1})g(b_{1}))(f(a_{2})g(b_{2}))
= h(a_{1} ⊗ b_{1})h(a_{2} ⊗ b_{2}).
⊔ ⊓

As a special case, 2.7 implies that in the category of commutative associative unital R-algebras, the tensor product yields the coproduct of two algebras.

2.8. A-modules and homomorphisms. Let μ : A ⊗_{R} A → A define an associative R-algebra with unit ι : R → A. Then an R-module M with an R-linear map ϱ_{M} : A ⊗_{R} M → M is called a (unital) left A-module if the following diagrams are commutative:

\xymatrix{A ⊗_{R} A ⊗_{R} M \ar[r]^{I_{A}⊗ϱ_{M}} \ar[d]_{μ⊗I} & A ⊗_{R} M \ar[d]^{ϱ_{M}} \\ A ⊗_{R} M \ar[r]^{ϱ}_{M} & M} , \xymatrix{R ⊗_{R} M \ar[r]^{ι⊗I_{M}} \ar[dr]_{≃} & A ⊗_{R} M \ar[d]^{ϱ_{M}} \\ & M .}

Again we will often write ϱ_{M}(a ⊗ m) = am and then the commutativity of the diagram can be rephrased by the familiar conditions

(ab)m = a(bm), 1_{A}m = m, for all a, b ∈ A, m ∈ M.

An R-linear map g : M → N between A-modules is called an A-morphism or A-homomorphism provided the following diagram commutes:

\xymatrix{A ⊗_{R} M \ar[r]^{I_{A}⊗g} \ar[d]_{ϱ_{M}} & A ⊗_{R} N \ar[d]^{ϱ_{N}} \\ M \ar[r]^{g} & N .}

Using the standard notation mentioned above, g : M → N is an A-morphism if and only if

g(am) = ag(m), for all a ∈ A, m ∈ M.

Notice that the A-morphisms between two A-modules M, N are characterized by an exact sequence

0 ⟶ Hom_{A}(M, N) ⟶ Hom_{R}(M, N) ⟶^{β} Hom_{R}(A ⊗_{R} M, N),

where β(f) = ϱ_{N} ∘ (id ⊗ f) − f ∘ ϱ_{M}.

The A-modules together with the A-morphisms form a category which we denote by A-Mod.

It is easily checked that here the defining map ϱ_{M} : A ⊗_{R} M → M is an A-morphism and this implies that A is a generator in A-Mod, i.e., every left A-module is a homomorphic image of a direct sum of copies of A (= A^{(Λ)}).

There are well-known relations between R-morphisms and A-morphisms which are readily derived from the basic relation 1.12.

2.9. Hom-tensor relations. For any R-module X, consider the R-linear map α : X → A ⊗_{R} X, x ↦ 1_{A} ⊗ x.

(1) For each A-module ϱ_{M} : A ⊗_{R} M → M, the map

Hom_{A}(A ⊗_{R} X, M) → Hom_{R}(X, M), f ↦ f ∘ α,

is an R-isomorphism with inverse map h ↦ ϱ_{M} ∘(id⊗h). So the functor

A ⊗_{R} − : R−Mod → A−Mod, X ↦ A ⊗_{R} X,

is left adjoint to the forgetful functor A−Mod → R−Mod.

(2) For any left A-module N, the R-linear map

Hom_{A}(M ⊗X, N) → Hom_{R}(X, Hom_{A}(M, N)), g ↦ [x ↦ g ∘(−⊗x)],

is an R-isomorphism with inverse map h ↦ [m ⊗ x ↦ h(x)(m)]. So

M ⊗_{R} − : R−Mod → A−Mod, X ↦ M ⊗_{R} X,

is left adjoint to the functor

Hom_{A}(M, −) : A−Mod → R−Mod, N ↦ Hom_{A}(M, N).

2.10. Tensor product with modules. Let M and M′ be left A-modules and Q an R-module. Consider the R-linear map

ν_{M} : Hom_{A}(M, M′) ⊗_{R} Q → Hom_{A}(M, M′ ⊗_{R} Q), [h ⊗ q ↦ (−)h ⊗ q].

(1) If Q is a flat R-module and M a finitely generated (finitely presented) A-module, then ν_{M} is injective (an isomorphism).

(2) ν_{M} is also an isomorphism in the following cases:

(i) M is a finitely generated, M′-projective A-module, or
(ii) M is M′-projective and Q is a finitely presented R-module, or
(iii) Q is a finitely generated projective R-module.

Proof. (1) It is easy to check that ν_{M} is an isomorphism for M = A and M = A^{k}, k ∈ ℕ. Since _{A}M is finitely generated, there exists an exact sequence of A-modules A^{(Λ)} → A^{n} → M → 0, with Λ an index set, n ∈ ℕ.

The functors Hom_{A}(−, M′) ⊗_{R} Q and Hom_{A}(−, M′ ⊗_{R} Q) yield the exact commutative diagram

0 → Hom_{A}(M, M′) ⊗_{R} Q → Hom_{A}(A^{n}, M′) ⊗_{R} Q → Hom_{A}(A^{(Λ)}, M′) ⊗_{R} Q
↓ ν_{M} ↓ ν_{A^{n}} ↓ ν_{A^{(Λ)}}
0 → Hom_{A}(M, M′ ⊗_{R} Q) → Hom_{A}(A^{n}, M′ ⊗_{R} Q) → Hom_{A}(A^{(Λ)}, M′ ⊗_{R} Q).

Since ν_{A^{n}} is an isomorphism, ν_{M} has to be injective.

If M is finitely presented we can choose Λ to be finite. Then also ν_{A^{(Λ)}} and ν_{M} are isomorphisms.

(2)(i) From the exact sequence of R-modules 0 → K → R^{(Λ)} → Q → 0, we construct the commutative diagram with the upper line exact,

Hom_{A}(M, M′) ⊗_{R} K → Hom_{A}(M, M′) ⊗_{R} R^{(Λ)} → Hom_{A}(M, M′) ⊗_{R} Q → 0
↓_{ν} ↓_{≃} ↓_{ν_{M}}
Hom_{A}(M, M′ ⊗_{R} K) → Hom_{A}(M, M′ ⊗_{R} R^{(Λ)}) → Hom_{A}(M, M′ ⊗_{R} Q) → 0 ,

where ν is defined as above replacing Q by K.

Since M is M′ ⊗_{R} R^{(Λ)}-projective, the lower sequence is also exact and hence ν_{M} is surjective. By the same argument we obtain that ν is surjective. Now it follows from the Kernel Cokernel Lemma that ν_{M} is injective.

(ii) This statement is obtained from the proof of (1), with Λ a finite set and K a finitely generated R-module.

(iii) The assertion is obvious for Q = R and is easily extended to finitely generated free (projective) modules Q. ⊔ ⊓

Combining the preceding observations we get

2.11. Tensor product with an algebra. Let A, B be R-algebras and M, M′ left A-modules. Consider the map

Hom_{A}(M, M′)⊗_{R}B → Hom_{A⊗B}(M⊗_{R}B, M′⊗_{R}B), f⊗b ↦ [m⊗b′ ↦ (m)f⊗b′b].

(1) If B is a flat R-module and M is a finitely generated (finitely presented) A-module, then the map is injective (an isomorphism).

(2) The map is also an isomorphism if

(i) M is a finitely generated, M′-projective A-module, or
(ii) M is M′-projective and B is a finitely presented R-module, or
(iii) B is a finitely generated projective R-module.

Proof. The map is the composition of maps

Hom_{A}(M, M′) ⊗_{R} B → Hom_{A}(M, M′ ⊗_{R} B) and
Hom_{A}(M, M′ ⊗_{R} B) → Hom_{A⊗_{R}B}(M ⊗_{R} B, M′ ⊗_{R} B).
⊔ ⊓

2.12. Tensor product of morphisms of modules. Let A, B be associative unital R-algebras, M, M′ left A-modules and N, N′ left B-modules.

(1) For f ∈ Hom_{A}(M, M′) and g ∈ Hom_{B}(N, N′),

f ⊗ g ∈ Hom_{A⊗B}(M ⊗_{R} N, M′ ⊗_{R} N′).

(2) The mapping (f, g) ↦ f ⊗ g induces an R-module morphism

ψ : Hom_{A}(M, M′) ⊗_{R} Hom_{B}(N, N′) → Hom_{A⊗B}(M ⊗_{R} N, M′ ⊗_{R} N′).

Assume M and N are finitely generated. Then ψ is an isomorphism if

(i) M is M′-projective and N is N′-projective, or
(ii) M and N are projective as A-, resp. B-modules, or
(iii) M is a finitely presented A-module, N and N′ are finitely generated,

projective B-modules, and B is a flat R-module.

(3) ψ : End_{A}(M) ⊗_{R} End_{B}(N) → End_{A⊗_{R}B}(M ⊗_{R} N) is an algebra morphism.

Proof. (1) Just verify that f ⊗ g is in fact an A ⊗ B-module morphism.

(2) ψ is well-defined since (f, g) ↦ f ⊗ g yields an R-bilinear map.

(i) We have isomorphisms

Hom_{A⊗_{R}B}(M ⊗_{R} N, M′ ⊗_{R} N′) ≃ Hom_{B}(N, Hom_{A}(M, M′ ⊗_{R} N′))
≃ Hom_{B}(N, Hom_{A}(M, M′) ⊗_{R} N′)
≃ Hom_{A}(M, M′) ⊗ Hom_{B}(N, N′) .

(ii) is a special case of (i). (iii) Since B ≃ Hom_{B}(B, B) we know that ψ is an isomorphism for N =

N′ = B. Similar to the above argument, this isomorphism can be extended to finitely generated free and projective modules N and N′.

(3) is easily verified. ⊔ ⊓

3 The category σ[M]

Throughout A will denote an R-algebra and M a left A-module. The module structure of M is reflected by the smallest Grothendieck category of Amodules containing M, which we briefly describe in this section.

An A-module N is called M-generated if there exists an epimorphism M^{(Λ)} → N for some set Λ.

3.1. The category σ[M]. An A-module N is called M-subgenerated if it is (isomorphic to) a submodule of an M-generated module. By σ[M] we denote the full subcategory of _{A}M whose objects are all M-subgenerated modules. Obviously the finitely generated (cyclic) submodules of M^{(ℕ)} form a set of generators in σ[M].

The trace functor T^{M} : _{A}M → σ[M], which sends any X ∈ _{A}M to

T^{M}(X) := ∑{f(N) | N ∈ σ[M], f ∈ _{A}Hom(N, X)},

is right adjoint to the inclusion functor σ[M] → _{A}M. For any family {N_{λ}}_{Λ} of modules in σ[M], the product in σ[M] is

∏_{Λ}^{M}N_{λ} = T^{M}(∏_{Λ}N_{λ}),

where the unadorned ∏ denotes the usual (Cartesian) product of A-modules, since, for any P ∈ σ[M],

AHom(P, T^{M}(∏_{Λ}^{M}N_{λ})) ≃ ∏_{Λ}_{A}Hom(P, N_{λ}).

Moreover, for any injective A-module Q, T^{M}(Q) is an injective object in the category σ[M].

N ∈ σ[M] is said to be a generator in σ[M] if it generates all modules in σ[M], and M is called a self-generator if it generates all its own submodules.

3.2. Injective modules. Let U and M be A-modules. U is said to be M-injective if every diagram in _{A}M with exact row

0 → K → M
↓
U

can be extended commutatively by some morphism M → U. This holds if AHom(−, U) is exact with respect to all exact sequences of the form 0 → K → M → N → 0 (in σ[M]). U is injective in σ[M] (in _{A}M) if it is N-injective, for every N ∈ σ[M] (N ∈ _{A}M, resp.).

3. The category σ[M] 23

3.3. Injectives in σ[M]. (Cf. [10, 16.3, 16.11, 17.9].)

(1) For Q ∈ σ[M] the following are equivalent:

(a) Q is injective in σ[M];

(b) the functor _{A}Hom(−, Q) : σ[M] → M_{R} is exact;

(c) Q is M-injective;

(d) Q is N-injective for every (finitely generated) submodule N ⊂ M;

(e) every exact sequence 0 → Q → N → L → 0 in σ[M] splits.

(2) Every M-injective object in σ[M] is M-generated.

(3) Every object in σ[M] has an injective hull.

3.4. Projectivity. Let M and P be A-modules. P is said to be M-projective if the functor _{A}Hom(P, −) is exact on all exact sequences of the form 0 → K → M → N → 0 in _{A}M. P is called projective in σ[M] (in _{A}M) if it is N-projective, for every N ∈ σ[M] (N ∈ _{A}M, repectively).

3.5. Projectives in σ[M]. (Cf. [10, 18.3].)

For P ∈ σ[M] the following are equivalent:

(a) P is projective in σ[M];

(b) the functor _{A}Hom(P, −) : σ[M] → M_{R} is exact;

(c) P is M^{(Λ)}-projective, for any index set Λ;

(d) every exact sequence 0 → K → N → P → 0 in σ[M] splits. If P is finitely generated, then (a)–(d) are equivalent to:

(e) P is M-projective.

A module P ∈ σ[M] is called a progenerator in σ[M] if it is finitely generated, projective and a generator in σ[M]. Notice that there may be no projective objects in σ[M]. A module N ∈ σ[M] is a subgenerator in σ[M] if σ[N] = σ[M].

3.6. Subgenerators. (Cf. [10].)

(1) For an A-module M the following are equivalent:

(a) M is a subgenerator in _{A}M (that is, σ[M] = _{A}M);

(b) M generates all injective modules in _{A}M;

(c) there is a monomorphism A → M^{k}, for some k ∈ ℕ.

(2) A faithful module _{A}M is a subgenerator in _{A}M provided (i) _{A}M is finitely generated over End_{A}(M), or (ii) _{A}A is finitely cogenerated, or (iii) σ[M] is closed under products in _{A}M.

3.7. Semisimple modules.

(1) The following are equivalent:

(a) M is a (direct) sum of simple modules;

(b) every submodule of M is a direct summand;

(c) every module (in σ[M]) is M-projective (or M-injective);

(d) every simple module (in σ[M]) is M-projective;

(e) every cyclic module (in σ[M]) is M-injective.

Modules M with these properties are called semisimple modules.

(2) Assume M to be semisimple.

(i) There exists a fully invariant decomposition

M = ⨁_{Λ}Tr(E_{λ}, M),

where {E_{λ}}_{Λ} is a minimal representing set of simple submodules of M and the Tr(E_{λ}, M) are minimal fully invariant submodules. (ii) The ring S = _{A}End(M) is von Neumann regular and M is semi-

simple as a right S-module. (iii) If all simple submodules of _{A}M are isomorphic, then all simple

submodules of M_{S} are isomorphic.

Proof. The first parts are shown in [10, 20.2–20.6].

(2)(ii) Let Am ⊂ M be a simple submodule. We show that mS ⊂ M is a simple S-submodule. For any t ∈ S with mt ≠ 0, Am ≃ Amt. Since these are direct summands in M, there exists some ϕ ∈ S with mtϕ = m and hence mS = mtS, implying that mS has no nontrivial S-submodules. As a semisimple module, M = ∑_{Λ} Am_{λ} with Am_{λ} simple. Now M = A(∑_{Λ} m_{λ}S), showing that M is a sum of simple S-modules am_{λ}S, where a ∈ A.

(2)(iii) It is straightforward to show that, for any m, n ∈ M, Am ≃ An implies mS ≃ nS. ⊔ ⊓

Definitions. A module M has finite length if it is Noetherian and Artinian. M is called locally Noetherian (Artinian, of finite length) provided every finitely generated submodule of M is Noetherian (Artinian, of finite length). M is called semi-Artinian if every factor module of M has a nonzero socle.

3.8. Local finiteness conditions. (Cf. [10, 27.5, 32.5].)

(1) The following are equivalent for a left A-module M:

(a) M is locally Noetherian;

(b) every finitely generated module in σ[M] is Noetherian; (c) any direct sum of M-injective modules is M-injective;

(d) every injective module in σ[M] is a direct sum of uniform modules.

(2) The following are equivalent for a left A-module M:

(a) M is locally of finite length;

(b) every finitely generated module in σ[M] has finite length;

(c) every injective module in σ[M] is a direct sum of M-injective hulls of simple modules.

(3) A module M is locally Artinian if and only if every finitely generated module in σ[M] is Artinian.

(4) A module M is semi-Artinian if and only if every module in σ[M] has a nonzero socle.

Definitions. A submodule K of M is said to be superfluous or small in M if, for every submodule L ⊂ M, K +L = M implies L = M. A small submodule is denoted by K << M. An epimorphism π : P → N with P projective in σ[M] and Ke π << P is said to be a projective cover of N in σ[M]. A module is called local if it has a largest proper submodule.

3.9. Local modules. (Cf. [10, 19.7].) For a projective module P ∈ σ[M], the following are equivalent:

(a) P is local;

(b) P is a projective cover of a simple module in σ[M];

(c) End(_{A}P) is a local ring.

Definitions. Let U be a submodule of the A-module M. A submodule V ⊂ M is called a supplement of U in M if V is minimal with the property U + V = M. It is easy to see that V is a supplement of U if and only if U + V = M and U ∩ V << V. Notice that supplements need not exist in general. M is said to be supplemented provided each of its submodules has a supplement. Definitions. A module P ∈ σ[M] is said to be semiperfect in σ[M] if every factor module of N has a projective cover in σ[M]. P is perfect in σ[M] if any direct sum P^{(Λ)} is semiperfect in σ[M].

3.10. Semiperfect modules. (Cf. [10, 42.5, 42.12].) For a projective module P in σ[M], the following are equivalent:

(a) P is semiperfect in σ[M];

(b) P is supplemented;

(c) every finitely P -generated module has a projective cover in σ[M];

(d) (i) P/Rad(P) is semisimple and Rad(P) << P, and (ii) decompositions of P/Rad(P) can be lifted to P; (e) every proper submodule is contained in a maximal submodule of P, and

every simple factor module of P has a projective cover in σ[M];

(f) P is a direct sum of local modules and Rad(P) << P.

3.11. Perfect modules. (Cf. [10, 43.2].) For a projective module P in σ[M], the following are equivalent:

(a) P is perfect in σ[M];

(b) P is semiperfect and, for any set Λ, Rad(P^{(Λ)}) << P^{(Λ)};

(c) every P -generated module has a projective cover in σ[M].

Definition. We call σ[M] a (semi)perfect category if every (simple) module in σ[M] has a projective cover in σ[M].

3.12. Semiperfect and perfect categories.

(1) For an A-module M the following are equivalent:

(a) σ[M] is semiperfect;

(b) σ[M] has a generating set of local projective modules;

(c) in σ[M] every finitely generated module has a projective cover.

(2) For M the following are equivalent:

(a) σ[M] is perfect;

(b) σ[M] has a projective generator that is perfect in σ[M].

Proof. (1) (a) ⇒ (b) The projective covers of all simple objects in σ[M] are local and form a generating set of σ[M] (by [10, 18.5]). Notice that local modules are supplemented.

(b) ⇒ (c) Any finite direct sum of supplemented modules is supplemented. Hence, for every finitely generated N ∈ σ[M], there exists an epimorphism P → N with some supplemented projective module P ∈ σ[M]. By 3.10, every factor module of P has a projective cover in σ[M], and so does N.

(c) ⇒ (a) is trivial.

(2) (a) ⇒ (b) Let P be the direct sum of projective covers of a representative set of the simple modules in σ[M]. Then P is a projective generator and every factor module of P^{(Λ)} has a projective cover, and hence P is perfect.

(b) ⇒ (a) is obvious. ⊔ ⊓

3.13. Left perfect rings. (Cf. [10, 43.9].) For A the following are equivalent:

(a) A is a perfect module in _{A}M;

(b) A/Jac(A) is left semisimple and Jac(A) is right t-nilpotent;

(c) every left A-module has a projective cover; (d) A satisfies the descending chain condition (dcc) on cyclic right ideals.

3.14. (f-)semiperfect rings. A ring A is said to be semiperfect if A is semiperfect as a left A-module or – equivalently – as a right A-module. More generally, A is called f-semiperfect (or semiregular) if A/Jac(A) is von Neumann regular and idempotents lift modulo Jac(A). Note that A is semiperfect if and only if finitely generated left and right A-modules have projective covers, and A is f-semiperfect if and only if every finitely presented left (and right) A-module has a projective cover (see [10, 42.11]). From [10, 42.12, 22.1] we recall:

3.15. (f-)semiperfect endomorphism rings. Put S = _{A}End(M).

(1) Assume M to be projective in σ[M]. Then:

(i) S is semiperfect if and only if M is finitely generated and semiperfect.

(ii) If M is semiperfect, then S is f-semiperfect.

(iii) If S is f-semiperfect, then Rad(M) << M and M is a direct sum of cyclic modules.

(2) If M is self-injective, then S is f-semiperfect.

(3) If M is self-injective and Soc(M) ⊴ M, then

Jac(S) = _{A}Hom(M/Soc(M), M).

3.16. Weak QF modules. If _{A}M is faithful, the following are equivalent:

(a) _{A}M is a weak QF module;

(b) (i) _{A}M is weakly _{A}M-injective and M_{S} is weakly M_{S}-injective, and (ii) A is dense in End_{S}(M);

(c) M_{S} is a weak QF module and A is dense in End_{S}(M);

(d) _{A}M and M_{S} are weak cogenerators in σ[_{A}M] and σ[M_{S}], respectively. For any weak QF module _{A}M, Soc_{A}M = Soc M_{S}.

A locally Noetherian weak QF module M is an injective cogenerator in σ[M] (by [10, 16.5]). A Noetherian weak QF module is called a quasi-Frobenius or QF module. A ring is a QF ring if it is QF as a left (or right) module.

Caution: “quasi Frobenius” is used in different ways in the literature. Let σ_{f}[M] denote the full subcategory of σ[M] whose objects are sub-

modules of finitely M-generated modules. With this notation σ_{f}[S_{S}] is the category of submodules of finitely generated right S-modules. This type of category is of particular interest in studying dualities.

3.17. Morita dualities.

(1) The following are equivalent:

(a) _{A}Hom(−, M): σ_{f}[M] → σ_{f}[S_{S}] is a duality;

(b) _{A}M is an injective cogenerator in σ[M], and M_{S} is an injective cogenerator in M_{S};

(c) _{A}M is linearly compact, finitely cogenerated, and an injective cogenerator in σ[M].

(2) If M is an injective cogenerator in σ[M], the following are equivalent:

(a) _{A}M is Artinian;

(b) _{A}M is semi-Artinian and M_{S} is S-injective;

(c) M_{S} is a Σ-injective cogenerator in M_{S};

(d) S is right Noetherian.

Next we consider relative properties of A-modules related to a fixed ring morphism ϕ : B → A. In this case, any left A-module M is naturally a left B-module and there is an interplay between the properties of M as an A-module and those of M as a B-module.

3.18. (A, B)-finite modules. The module M is said to be (A, B)-finite if every finitely generated A-submodule of M is finitely generated as a Bmodule. σ[M] is said to be (A, B)-finite if every module in σ[M] is (A, B)- finite.

Let σ[M] be (A, B)-finite.

(i) If B is a right perfect ring, then every module in σ[M] has the dcc on finitely generated A-submodules.

(ii) If B is left Noetherian, then every module in σ[M] is locally Noetherian. (iii) If B is left Artinian, then every module in σ[M] has locally finite length.

For the following observations we refer to [11, Section 20].

3.19. Relative notions. An exact sequence K →^{f} M →^{g} N in _{A}M is called (A, B)-exact if Im f is a direct summand of M as a left B-module.

Let M, P, Q be left A-modules. P is called (M, B)-projective if_{A}Hom(P,−) is exact with respect to all (A, B)-exact sequences in σ[M]. This is the case if and only if every (A, B)-exact sequence L → P → 0 in σ[M] splits.

Q is called (M, B)-injective if _{A}Hom(−, Q) is exact with respect to all (A, B)-exact sequences in σ[M]. This happens if and only if every (A, B)- exact sequence 0 → Q → L in σ[M] splits.

Over a semisimple ring B, (M, B)-projective and (M, B)-injective are synonymous to projective and injective in σ[M], respectively.

3.20. (A, B)-projectives and (A, B)-injectives.

(1) For any B-module X, A ⊗_{B} X is (A, B)-projective.

(2) P ∈ _{A}M is (A, B)-projective if and only if the map A ⊗_{B} P → P, a ⊗ p ↦ ap, splits in _{A}M.

(3) For any B-module Y, Hom_{B}(A, Y) is (A, B)-injective.

(4) Q ∈ _{A}M is (A, B)-injective if and only if the map Q → Hom_{B}(A, Q), q ↦ [a ↦ aq], splits in _{A}M.

The module M is called (A, B)-semisimple if every (A, B)-exact sequence in σ[M] splits. The ring A is said to be left (A, B)-semisimple if A is (A, B)- semisimple as a left A-module.

3.21. (A, B)-semisimple modules. The following are equivalent:

(a) M is (A, B)-semisimple;

(b) every A-module (in σ[M]) is (M, B)-projective;

(c) every A-module (in σ[M]) is (M, B)-injective.

4 External properties of σ[M]

So far we dealt with the internal structure of the category σ[M]. It is also of interest to look at the properties of σ[M] as a class of modules in _{A}M.

Let T be any associative ring (without a unit). A left T -module N is called s-unital if u ∈ T u for every u ∈ N. T itself is called left s-unital if it is s-unital as a left T -module. For an ideal T ⊂ A, every A-module is a T -module and we observe the elementary properties:

4.1. s-unital T -modules. For any subring T ⊂ A the following assertions are equivalent:

(a) M is an s-unital T -module;

(b) for any m_{1}, …, m_{k} ∈ N, there exists t ∈ T with m_{i} = tm_{i} for all i ≤ k;

(c) for any set Λ, N^{(Λ)} is an s-unital T -module.

Proof. (a) ⇒ (b) We proceed by induction. Assume the assertion holds for k−1 elements. Choose t_{k} ∈ T such that t_{k}n_{k} = n_{k} and put a_{i} = m_{i}−m_{k}n_{i}, for all i ≤ k. By assumption there exists t′ ∈ T satisfying a_{i} = t′a_{i}, for all i ≤ k − 1. Then t := t′ + t_{k} − t′t_{k} ∈ T is an element satisfying the condition in (b). The remaining assertions are easily verified. ⊔ ⊓

4.2. Flat factor rings. For an ideal T ⊂ A the following are equivalent:

(a) A/T is a flat right A-module;

(b) for every left ideal I of A, T I = T ∩ I;

(c) every injective left A/T -module is A-injective;

(d) for every A-module _{A}L ⊂ _{A}N, T L = T N ∩ L;

(e) T is left s-unital. Under these conditions T is a flat right A-module, and, for any N ∈_{A} M, the canonical map T ⊗_{A} N → T N is an isomorphism.

Proof. The equivalence of (a) and (b) is shown in [10, 36.6].

(a) ⇒ (c) Put D := A/T. Let N be an injective D-module and L ⊂ A a left ideal. By (a), the sequence 0 → D ⊗_{A} L → D ⊗_{A} A is exact in _{D}M and there is a commutative diagram with exact rows and canonical isomorphisms,

\xymatrix{_{D}Hom(D ⊗_{A} A, N) \ar[r] \ar[d]_{≃} & _{D}Hom(D ⊗_{A} L, N) \ar[d]^{≃} \ar[r] & 0 \\ _{A}Hom(A,_{D}Hom(D, N)) \ar[r] \ar[d]_{≃} & _{A}Hom(L,_{D}Hom(D, N)) \ar[r] \ar[d]^{≃} & 0 \\ _{A}Hom(A, N) \ar[r] & _{A}Hom(L, M) \ar[r] & 0 .}

4. External properties of σ[M] 31

Since N is an injective D-module, the first row is exact and so are the others, that is, N is injective as an A-module.

(c) ⇒ (a) Let N be a cogenerator in _{D}M that is A-injective. For a left ideal L ⊂ A there is an exact sequence 0 → K → D ⊗_{A} L → D ⊗_{A} A in _{D}M, and we want to prove K = 0. Consider the exact sequence

DHom(D ⊗_{A} A, N) → _{D}Hom(D ⊗_{A} L, N) → _{D}Hom(K, N) → 0.

Now in the above diagram the bottom row is exact (N is A-injective). This implies that the top row is also exact, that is, _{D}Hom(K, N) = 0. Since N is a cogenerator in _{D}M, we conclude K = 0.

The remaining implications are straightforward to verify. ⊔ ⊓

4.3. s-unital modules over ideals. Let _{A}M be faithful. For an ideal T ⊂ A the following are equivalent:

(a) M is an s-unital T -module;

(b) for every N ∈ σ[M], N = T N;

(c) for every N ∈ σ[M], the canonical map φ_{N} : T ⊗_{A} N → N is an isomorphism.

If T ∈ σ[M], then (a)–(c) are equivalent to:

(d) T^{2} = T and T is a generator in σ[M].

Proof. The implications follow from 4.1 and 4.2. ⊔ ⊓

4.4. Trace ideals. The trace of M in A, Tr(M, A), is called the trace ideal of M, and the trace of σ[M] in A, T^{M}(A) = Tr(σ[M], A), is called the trace ideal of σ[M]. Clearly Tr(M, A) ⊂ T^{M}(A), and equality holds if M is a generator in σ[M], or else if A is a left self-injective algebra.

4.5. Trace ideals of M. Denote ^{∗}M = _{A}Hom(M, A), and T = Tr(M, A) = ∗M(M). Any f ∈ ^{∗}M defines an A-linear map

ϕ_{f} : M → S, m ↦ f(−)m.

Δ = ∑_{f∈^{∗}M}Im ϕ_{f} is an ideal in S and MΔ ⊂ T M.

The following are equivalent:

(a) M = T M;

(b) M = MΔ;

(c) for any L ∈ _{A}M, Tr(M, L) = T L.

If this holds, T and Δ are idempotent ideals and Δ = Tr(M_{S}, S).

Proof. (a) ⇔ (b) are obvious from the definitions.

(a) ⇔ (c) Clearly T is M-generated, and M = T M implies that Mgenerated A-modules are T -generated.

Assume the conditions hold. By definition, Δ ⊂ Tr(M_{S}, S). For any Slinear map g : M → S and m ∈ M, write m = ∑_{i} m_{i}δ_{i}, where m_{i} ∈ M and δ_{i} ∈ Δ, to obtain

g(m) = g(∑_{i}m_{i}δ_{i}) = ∑_{i}g(m_{i})δ_{i} ∈ Δ,

thus showing Tr(M_{S}, S) ⊂ Δ. ⊔ ⊓

4.6. Canonical map. For any N ∈ M_{A} there is a map

α_{N,M} : N ⊗_{A} M → Hom_{A}(^{∗}M, N), n ⊗ m ↦ [f ↦ nf(m)],

which is injective if and only if

for any u ∈ N ⊗_{A} M, (I_{N} ⊗ f)(u) = 0 for all f ∈ ^{∗}M, implies u = 0.

A module M is said to be locally projective if, for any diagram of left A-modules with exact rows,

\xymatrix{0 \ar[r] & F \ar[r]^{i} & M \ar[d]^{g} \\  & L \ar[r]^{f} & N \ar[r] & 0,}

where F is finitely generated, there exists h : M → L such that g∘i = f ∘h∘i.

4.7. Locally projective modules. With the notation from 4.5, the following are equivalent:

(a) M is locally projective;

(b) α_{N,M} is injective, for any (cyclic) right A-module N;

(c) for each m ∈ M, m ∈ ^{∗}M(m)M;

(d) for any m_{1}, …, m_{k} ∈ M there exist x_{1}, …, x_{n} ∈ M, f_{1}, …, f_{n} ∈ ^{∗}M,

such that

m_{j} = ∑_{i}f_{i}(m_{j})x_{i}, for j = 1, …, k;

(e) M = T M, and M is an s-unital right Δ-module.

Proof. (a) ⇒ (d) Put N = M and L = A^{(Λ)} in the defining diagram.

(d) ⇒ (a) follows by the fact that A is projective as left A-module.

(b) ⇒ (c) Assume α_{N,M} to be injective for cyclic right A-modules N. For any m ∈ M put J = ^{∗}M(m) and consider the monomorphism

ϕ : M/JM ≃ A/J ⊗_{A} M ⟶^{α_{N,M}} Hom_{A}(^{∗}M, A/J).

For x ∈ M and f ∈ ^{∗}M, ϕ(x+JM)(f) = f(x)+J, and hence ϕ(m+JM) = 0. By injectivity of ϕ this implies m ∈ JM.

(d) ⇒ (b) Let N ∈ M_{A} and let v = ∑_{j=1}^{r} n_{j} ⊗ m_{j} ∈ N ⊗_{A} M. Choose x_{1}, …, x_{n} ∈ M and f_{1}, …, f_{n} ∈ ^{∗}M such that m_{j} = ∑_{i} f_{i}(m_{j})x_{i}, for j = 1, …, r. Then

v = ∑_{i,j} n_{j} ⊗ f_{i}(m_{j})x_{i} = ∑_{i} α_{N,M}(v)(f_{i}) ⊗ x_{i},

and hence v = 0 if α_{N,M}(v) = 0, that is, α_{N,M} is injective.

(c) ⇔ (d) ⇔ (e) For m ∈ M, m ∈ ^{∗}M(m)M means that there are x_{1}, …, x_{n} ∈ M and f_{1}, …, f_{n} ∈ ^{∗}M such that

m = ∑_{i} f_{i}(m)x_{i} = m[∑_{i} f_{i}(−)x_{i}] ∈ mΔ,

showing that M is an s-unital right Δ-module (see 4.1). ⊔ ⊓

4.8. T^{M} as an exact functor. Putting T̃ = T^{M}(A), the following assertions are equivalent:

(a) the functor T^{M} : _{A}M → σ[M] is exact;

(b) σ[M] is closed under extensions and the class {X ∈ _{A}M | T^{M}(X) = 0} is closed under factor modules;

(c) for every N ∈ σ[M], T̃N = N;

(d) M is an s-unital T̃-module.

Proof. (a) ⇒ (b) Let T^{M} be exact. For any exact sequence in _{A}M as a bottom row, there is a commutative diagram with exact rows,

\xymatrix{0 \ar[r] & T^{M}(K) \ar[r] \ar[d] & T^{M}(L) \ar[r] \ar[d] & T^{M}(N) \ar[d] \ar[r] & 0 \\ 0 \ar[r] & K \ar[r] & L \ar[r] & N \ar[r] & 0 .}

In case T^{M}(K) = K and T^{M}(N) = N this implies T^{M}(L) = L, showing that σ[M] is closed under extensions. Moreover T^{M}(L) = 0 implies T^{M}(N) = 0 as required.

(b) ⇒ (c) Since σ[M] is closed under extensions, T^{M}(A/T) = 0. For any N ∈ σ[M], N/T̃N is generated by A/T̃, and so by (b), T^{M}(N/T̃N) = 0 and hence N = T̃N.

(c) ⇒ (a) First observe that the hypothesis implies T^{M}(X) = T X, for any X ∈ _{A}M. Consider an exact sequence in _{A}M, 0 → K → L → N → 0. Since T_{A} is flat (see 4.2), tensoring with T̃ ⊗_{A} − yields an exact sequence 0 → T̃K → T̃L → T̃N → 0.

(c) ⇔ (d) is shown in 4.3. ⊔ ⊓

We say σ[M] is closed under small epimorphisms if, for any epimorphism f : P → N in _{A}M, where Ke f << P and N ∈ σ[M], we obtain P ∈ σ[M].

4.9. Corollary. Assume that the functor T^{M} : _{A}M → σ[M] is exact.

(1) σ[M] is closed under small epimorphisms.

(2) If P is finitely presented in σ[M], then P is finitely presented in _{A}M.

(3) If P is projective in σ[M], then P is projective in _{A}M.

Proof. (1) Put T̃ = T^{M}(A). Consider an exact sequence 0 → K → P → N → 0 in _{A}M, where K << P and N ∈ σ[M]. From this we obtain the following commutative diagram with exact rows:

\xymatrix{0 \ar[r] & K \ar[r] \ar[d] & P \ar[r] \ar[d]^{=} & N \ar[r] \ar[d] & 0 \\ 0 \ar[r] & K + T̃P \ar[r] & P \ar[r] & P/(K + T̃P) \ar[r] & 0.}

Clearly P/(K + T̃P) ∈ σ[M] and by condition 4.8(b), T̃(P/(K + T̃P)) = 0. This implies P = K + T̃P, that is, P ∈ σ[M].

(2) It is enough to show this for any cyclic module P ∈ σ[M] that is finitely presented in σ[M]. For this we construct the following commutative diagram with exact rows (applying T^{M}):

\xymatrix{0 \ar[r] & L_{0} \ar[r] \ar[d] & L_{1} \ar[r] \ar[d] & P \ar[r] \ar[d]^{=} & 0 \\ 0 \ar[r] & I \ar[r] \ar[d] & A \ar[r] \ar[d] & P \ar[r] & 0 \\  & I/L_{0} \ar[r]^{≃} & A/L_{1},}

where L_{0} and L_{1} are suitable finitely generated modules in σ[M]. So I/L_{0} is finitely generated, and hence so is I and P is finitely presented in _{A}M.

(3) This is shown with a similar diagram as in the proof of (2). ⊔ ⊓

4.10. Corollary. Suppose that σ[M] has a generator that is locally projective in _{A}M. Then T^{M} : _{A}M → σ[M] is an exact functor.

Proof. Let P ∈ σ[M] be a locally projective generator. Then clearly σ[M] = σ[P] and T̃ = T^{M}(A) = Tr(P, A). By 4.5 and 4.7, T̃^{2} = T̃ and T̃P = P. So T̃ generates P and 4.8 applies. ⊔ ⊓

4.11. Projective covers in σ[M]. Let σ[M] be locally Noetherian and suppose that A is f-semiperfect. Then the following are equivalent:

(a) the functor T^{M} : _{A}M → σ[M] is exact;

(b) σ[M] has a generator that is (locally) projective in _{A}M; (c) there are idempotents {e_{λ}}_{Λ} in A such that the Ae_{λ} are in σ[M] and

form a generating set in σ[M];

(d) σ[M] is a semiperfect category.

Proof. (a) ⇒ (c) Let S be any simple module in σ[M]. S is finitely presented in σ[M] and hence in _{A}M (by 4.9(2)). Since A is f-semiperfect, S has a projective cover P in _{A}M (see 3.14). By 4.9(1), P ∈ σ[M] and clearly P ≃ Ae for some idempotent e ∈ A. Now a representing set of simple modules in σ[M] yields the required family of idempotents.

(c) ⇒ (b) is obvious and (b) ⇒ (a) follows from 4.10.

(c) ⇔ (d) This is clear by 3.12. ⊔ ⊓

A submodule N ⊂ M is said to be fully invariant if it is invariant under endomorphisms of M, that is, N is an (A, S)-submodule. The ring of (A, S)- endomorphisms of M is the centre of S (e.g., [11, 4.2]).

4.12. Big cogenerators. An M-injective module Q ∈ σ[M] is said to be a big injective cogenerator in σ[M] if every cyclic module in σ[M] is isomorphic to a submodule of Q^{(ℕ)}. Clearly every big injective cogenerator in σ[M] is a cogenerator as well as a subgenerator in σ[M]. Such modules always exist:

Let {N_{λ}}_{Λ} be a representing set of the cyclic modules in σ[M]. Then the M-injective hull of ⨁_{Λ} N_{λ} is a big injective cogenerator in σ[M].

If M is locally of finite length, every injective cogenerator in σ[M] is big.

4.13. Correspondence relations. Let Q be a big injective cogenerator in the category σ[M].

(1) For every N ∈ σ[M], σ[N] = σ[Tr(N, Q)].

(2) The assignment σ[N] ↦ Tr(N, Q) yields a bijective correspondence between the subcategories of type σ[N] of σ[M] and the fully invariant submodules of Q.

(3) If σ[N] is closed under essential extensions (injective hulls) in σ[M], then Tr(N, Q) is an A-direct summand of Q.

(4) If M is locally Noetherian and Tr(N, Q) is an A-direct summand of Q, then σ[N] is closed under essential extensions in σ[M].

(5) N ∈ σ[M] is semisimple if and only if Tr(N, Q) ⊂ Soc(_{A}Q).

Proof. Since Q is M-injective, Tr(σ[N], Q) = Tr(N, Q).

(1) Tr(N, Q) is a fully invariant submodule that, by definition, belongs to σ[N]. Consider any finitely generated L ∈ σ[N]. Then, by assumption, L ⊂ Q^{k}, for some k ∈ ℕ, and hence L ⊂ Tr(L, Q)^{k} ⊂ Tr(N, Q)^{k}. This implies N ∈ σ[Tr(N, Q)].

Parts (2) and (5) are immediate consequences of (1).

(3) If σ[N] is closed under essential extensions in σ[M], then Tr(N, Q) is an A-direct sumand in Q (and hence is injective in σ[M]).

(4) Let M be locally Noetherian and Tr(N, Q) a direct summand of Q. Consider any N-injective module L in σ[N]. Then L is a direct sum of Ninjective uniform modules U ∈ σ[M]. Clearly U is (isomorphic to) a direct summand of Tr(N, Q) and hence of Q; that is, U is M-injective and so L is M-injective, too. ⊔ ⊓

4.14. Sum and decomposition of subcategories. For any K, L ∈ σ[M] we write σ[K] ∩ σ[L] = 0, provided σ[K] and σ[L] have no nonzero module in common. Given a family {N_{λ}}_{Λ} of modules in σ[M], define

∑_{Λ}σ[N_{λ}] := σ[⨁_{Λ}N_{λ}].

Moreover, we write

σ[M] = ⨁_{Λ}σ[N_{λ}],

provided, for every module L ∈ σ[M], L = ⨁_{Λ}T^{N_{λ}}(L) (internal direct sum). This decomposition of σ[M] is known as a σ-decomposition. The category σ[M] is σ-indecomposable provided it has no nontrivial σ-decomposition.

4.15. σ-decomposition of modules. For a decomposition M = ⨁_{Λ} M_{λ}, the following are equivalent (cf. [12]):

(a) for any distinct λ, μ ∈ Λ, M_{λ} and M_{μ} have no nonzero isomorphic subfactors;

(b) for any distinct λ, μ ∈ Λ, σ[M_{λ}] ∩ σ[M_{μ}] = 0;

(c) for any L ∈ σ[M], L = ⨁_{Λ} T^{N_{λ}}(L). If these conditions hold, we call M = ⨁_{Λ} M_{λ} a σ-decomposition and in this case

σ[M] = ⨁_{Λ}σ[M_{λ}].

4.16. Corollary. Let σ[M] = ⨁_{Λ} σ[N_{λ}] be a σ-decomposition of σ[M]. Then the trace functor T^{M} is exact if and only if the trace functors T^{N_{λ}} are exact, for all λ ∈ Λ.

4.17. Corollary. If M is a projective generator or an injective cogenerator in σ[M], then any fully invariant decomposition of M is a σ-decomposition.

Proof. Let M = ⨁_{Λ} M_{λ} be a fully invariant decomposition. If M is a projective generator in σ[M], then every submodule of M_{λ} is generated by M_{λ}. Since the M_{λ} are projective in σ[M], any nonzero (iso)morphism between (sub)factors of M_{λ} and M_{μ} yields a nonzero morphism between M_{λ} and M_{μ}. So the assertion follows from 4.15.

Now suppose that M is an injective cogenerator in σ[M]. Then every subfactor of M_{λ} must be cogenerated by M_{λ}. From this it follows that for λ ≠ μ, there are no nonzero maps between subfactors of M_{λ} and M_{μ} and so 4.15 applies. ⊔ ⊓

As an example, consider the ℤ-module ℚ/ℤ = ⨁_{p prime}ℤ_{p^{∞}} and the decomposition of the category of torsion Abelian groups as a direct sum of the categories of p-groups,

σ[ℚ/ℤ] = ⨁_{p prime}σ[ℤ_{p^{∞}}].

Notice that, although ℚ/ℤ is an injective cogenerator in M_{ℤ} with a nontrivial σ-decomposition, M_{ℤ} is σ-indecomposable. This is possible since ℚ/ℤ is not a subgenerator in M_{ℤ}.

Coalgebras and comodules

Coalgebras and comodules are obtained by dualising the notions of algebras and modules. Throughout, R denotes a commutative and associative ring with a unit.

5 Coalgebras

The main aim of this section is to introduce and give examples of coalgebras and explain the (dual) relationship between algebras and coalgebras.

5.1. Coalgebras. An R-coalgebra is an R-module C with R-linear maps

Δ : C → C ⊗_{R} C and ε : C → R,

called (coassociative) coproduct and counit, respectively, with the properties

(I_{C} ⊗ Δ) ∘ Δ = (Δ ⊗ I_{C}) ∘ Δ, and (I_{C} ⊗ ε) ∘ Δ = I_{C} = (ε ⊗ I_{C}) ∘ Δ,

which can be expressed by commutativity of the diagrams

\xymatrix{C \ar[r]^{Δ} \ar[d]_{Δ} & C ⊗_{R} C \ar[d]^{I_{C}⊗Δ} & C \ar[r]^{Δ} \ar[dr]^{I_{C}} \ar[d]_{Δ} & C ⊗_{R} C \ar[d]^{ε⊗I_{C}} \\ C ⊗_{R} C \ar[r]^{Δ⊗I_{C}} & C ⊗_{R} C ⊗_{R} C & C ⊗_{R} C \ar[r]_{I_{C}⊗ε} & C .}

A coalgebra (C, Δ, ε) is said to be cocommutative if Δ = tw ∘ Δ, where

tw : C ⊗_{R} C → C ⊗_{R} C, a ⊗ b ↦ b ⊗ a,

is the twist map.

5.2. Sweedler’s Σ-notation. For an elementwise description of the maps we use the Σ-notation, writing for c ∈ C

Δ(c) = ∑_{i=1}^{k} c_{i} ⊗ c̃_{i} = ∑ c\underline{1} ⊗ c\underline{2}.

The first version is more precise; the second version, introduced by Sweedler, is very handy in explicit calculations. Notice that c\underline{1} and c\underline{2} do not represent single elements but families c_{1}, …, c_{k} and c̃_{1}, …, c̃_{k} of elements of C that are by no means uniquely determined. Properties of c\underline{1} can only be considered in context with c\underline{2}. With this notation, the coassociativity of Δ is expressed by

∑ Δ(c\underline{1}) ⊗ c\underline{2} = ∑ c_{\underline{1} \underline{1}} ⊗ c_{\underline{1} \underline{2}} ⊗ c\underline{2} = ∑ c\underline{1} ⊗ c_{\underline{2} \underline{1}} ⊗ c_{\underline{2} \underline{2}} = ∑ c\underline{1} ⊗ Δ(c\underline{2}),

and, hence, it is possible and convenient to shorten the notation by writing

\begin{aligned}(Δ ⊗ I_{C})Δ(c) = (I_{C} ⊗ Δ)Δ(c) & = & ∑ c\underline{1} ⊗ c\underline{2} ⊗ c\underline{3}, \\ (I_{C} ⊗ I_{C} ⊗ Δ)(I_{C} ⊗ Δ)Δ(c) & = & ∑ c\underline{1} ⊗ c\underline{2} ⊗ c\underline{3} ⊗ c\underline{4}, \end{aligned}

and so on. The conditions for the counit are described by

∑ ε(c\underline{1})c\underline{2} = c = ∑ c\underline{1}ε(c\underline{2}).

Cocommutativity is equivalent to ∑ c\underline{1} ⊗ c\underline{2} = ∑ c\underline{2} ⊗ c\underline{1}.

R-coalgebras are closely related or dual to algebras. Indeed, the module of R-linear maps from a coalgebra C to any R-algebra is an R-algebra.

5.3. The algebra Hom_{R}(C, A). For any R-linear map Δ : C → C ⊗_{R} C and an R-algebra A, Hom_{R}(C, A) is an R-algebra by the convolution product

f ∗ g = μ ∘ (f ⊗ g) ∘ Δ, i.e., f ∗ g(c) = ∑ f(c\underline{1})g(c\underline{2}),

for f, g ∈ Hom_{R}(C, A) and c ∈ C. Furthermore,

(1) Δ is coassociative if and only if Hom_{R}(C, A) is an associative R-algebra, for any R-algebra A.

(2) C is cocommutative if and only if Hom_{R}(C, A) is a commutative Ralgebra, for any commutative R-algebra A.

(3) C has a counit if and only if Hom_{R}(C, A) has a unit, for all R-algebras A with a unit.

Proof. (1) Let f, g, h ∈ Hom_{R}(C, A) and consider the R-linear map

μ̃ : A ⊗_{R} A ⊗_{R} A → A, a_{1} ⊗ a_{2} ⊗ a_{3} ↦ a_{1}a_{2}a_{3}.

By definition, the products (f ∗ g) ∗ h and f ∗ (g ∗ h) in Hom_{R}(C, A) are the compositions of the maps

\xymatrix{ & C ⊗_{R} C \ar[dr]^{Δ⊗I_{C}} \\ C \ar[ur]^{Δ} \ar[dr]_{Δ} &  & C ⊗_{R} C ⊗_{R} C \ar[r]^{f⊗g⊗h} & A ⊗_{R} A ⊗_{R} A \ar[r]^{μ̃} & A . \\  & C ⊗_{R} C \ar@{-->}[ur]_{I_{C}⊗Δ}}

It is obvious that coassociativity of Δ yields associativity of Hom_{R}(C, A).

To show the converse, we see from the above diagram that it suffices to prove that, (at least) for one associative algebra A and suitable f, g, h ∈ Hom_{R}(C, A), the composition μ̃ ∘ (f ⊗ g ⊗ h) is a monomorphism. So let A = T (C), the tensor algebra of the R-module C, and f = g = h, the canonical mapping C → T (C). Then μ̃ ∘ (f ⊗ g ⊗ h) is just the embedding C ⊗ C ⊗ C = T_{3}(C) → T (C).

(2) If C is cocommutative and A is commutative,

f ∗ g (c) = ∑ f(c\underline{1})g(c\underline{2}) = ∑ g(c\underline{1})f(c\underline{2}) = g ∗ f (c),

so that Hom_{R}(C, A) is commutative. Conversely, assume that Hom_{R}(C, A) is commutative for any commutative A. Then

μ ∘ (f ⊗ g)(Δ(c)) = μ ∘ (f ⊗ g)(tw ∘ Δ(c)).

This implies Δ = tw ∘ Δ provided we can find a commutative algebra A and f, g ∈ Hom_{R}(C, A) such that μ ∘ (f ⊗ g) : C ⊗_{R} C → A is injective. For this take A to be the symmetric algebra S(C ⊕ C). For f and g we choose the mappings

C → C ⊕ C, x ↦ (x, 0), C → C ⊕ C, x ↦ (0, x),

composed with the canonical embedding C ⊕ C → S(C ⊕ C).

With the canonical isomorphism h : S(C) ⊗ S(C) → S(C ⊕ C) and the embedding λ : C → S(C), we form h^{−1} ∘ μ ∘ (f ⊗ g) = λ ⊗ λ. Since λ(C) is a direct summand of S(C), we obtain that λ ⊗ λ is injective and so μ ∘ (f ⊗ g) is injective.

(3) It is easy to check that the unit in Hom_{R}(C, A) is

C ⟶^{ε} R ⟶^{ι} A, c ↦ ε(c)1_{A}.

For the converse, consider the R-module A = R ⊕ C and define a unital R-algebra

μ : A ⊗_{R} A → A, (r, a) ⊗ (s, b) ↦ (rs, rb + as).

Suppose there is a unit element in Hom_{R}(C, A),

e : C → A = R ⊕ C, c ↦ (ε(c), λ(c)),

with R-linear maps ε : C → R, λ : C → C. Then, for f : C → A, c ↦ (0, c), multiplication in Hom_{R}(C, A) yields

f ∗ e : C → A, c ↦ (0, (I_{C} ⊗ ε) ∘ Δ(c)).

By assumption, f = f ∗ e and hence I_{C} = (I_{C} ⊗ ε) ∘ Δ, one of the conditions for ε to be a counit. Similarly, the other condition is derived from f = e ∗ f.

Clearly ε is the unit in Hom_{R}(C, R), showing the uniqueness of a counit for C. ⊔ ⊓

Note in particular that C^{∗} = Hom_{R}(C, R) is an algebra with the convolution product known as the dual or convolution algebra of C.

Notation. From now on, C (usually) will denote a coassociative R-coalgebra (C, Δ, ε), and A will stand for an associative R-algebra with unit (A, μ, ι).

Many properties of coalgebras depend on properties of the base ring R. The base ring can be changed in the following way.

5.4. Scalar extension. Let C be an R-coalgebra and S an associative commutative R-algebra with unit. Then C ⊗_{R} S is an S-coalgebra with the coproduct

\xymatrix{Δ̃ : C ⊗_{R} S \ar[r]^{Δ⊗I_{S}} & (C ⊗_{R} C) ⊗_{R} S \ar[r]^{≃} & (C ⊗_{R} S) ⊗_{S} (C ⊗_{R} S)}

and the counit ε ⊗ I_{S} : C ⊗_{R} S → S. If C is cocommutative, then C ⊗_{R} S is cocommutative.

Proof. By definition, for any c ⊗ s ∈ C ⊗_{R} S,

Δ̃(c ⊗ s) = ∑(c\underline{1} ⊗ 1_{S}) ⊗_{S} (c\underline{2} ⊗ s).

It is easily checked that Δ̃ is coassociative. Moreover,

(ε ⊗ I_{S} ⊗ I_{C⊗_{R}S}) ∘ Δ̃(c ⊗ s) = ∑ ε(c\underline{1})c\underline{2} ⊗ s = c ⊗ s,

and similarly (I_{C⊗_{R}S} ⊗ ε ⊗ I_{S}) ∘ Δ̃ = I_{C⊗_{R}S} is shown. Obviously cocommutativity of Δ implies cocommutativity of Δ̃. ⊔ ⊓

To illustrate the notions introduced above we consider some examples.

5.5. R as a coalgebra. The ring R is (trivially) a coassociative, cocommutative coalgebra with the canonical isomorphism R → R ⊗_{R} R as coproduct and the identity map R → R as counit.

5.6. Free modules as coalgebras. Let F be a free R-module with basis (f_{λ})_{Λ}, Λ any set. Then there is a unique R-linear map

Δ : F → F ⊗_{R} F, f_{λ} ↦ f_{λ} ⊗ f_{λ},

defining a coassociative and cocommutative coproduct on F. The counit is provided by the linear map ε : F → R, f_{λ} ⟼ 1.

5.7. Semigroup coalgebra. Let G be a semigroup. A coproduct and counit on the semigroup ring R[G] can be defined by

Δ_{1} : R[G] → R[G] ⊗_{R} R[G], g ↦ g ⊗ g, ε_{1} : R[G] → R, g ↦ 1.

If G has a unit e, then another possibility is

Δ_{2} : R[G] → R[G] ⊗_{R} R[G], g ↦ \{ \begin{matrix}e ⊗ e & if g = e, \\ g ⊗ e + e ⊗ g & if g ≠ e. \end{matrix}

ε_{2} : R[G] → R, g ↦ \{ \begin{matrix}1 & if g = e, \\ 0 & if g ≠ e. \end{matrix}

Both Δ_{1} and Δ_{2} are coassociative and cocommutative.

5.8. Polynomial coalgebra. A coproduct and counit on the polynomial ring R[X] can be defined as algebra homomorphisms by

\begin{matrix}Δ_{1} : R[X] → R[X] ⊗_{R} R[X], & X^{i} ↦ X^{i} ⊗ X^{i}, \\ ε_{1} : R[X] → R, & X^{i} ↦ 1, i = 0, 1, 2, …. \end{matrix}

or else by

\begin{matrix}Δ_{2} : R[X] → R[X] ⊗_{R} R[X], & 1 ↦ 1, X^{i} ↦ (X ⊗ 1 + 1 ⊗ X)^{i}, \\ ε_{2} : R[X] → R, & 1 ↦ 1, X^{i} ↦ 0, i = 1, 2, …. \end{matrix}

Again, both Δ_{1} and Δ_{2} are coassociative and cocommutative.

5.9. Coalgebra of a projective module. Let P be a finitely generated projective R-module with dual basis p_{1}, …, p_{n} ∈ P and π_{1}, …, π_{n} ∈ P^{∗}. There is an isomorphism

P ⊗_{R} P^{∗} → End_{R}(P), p ⊗ f ↦ [a ↦ f(a)p],

and on P^{∗} ⊗_{R} P the coproduct and counit are defined by

Δ : P^{∗} ⊗_{R} P → (P^{∗} ⊗_{R} P) ⊗_{R} (P^{∗} ⊗_{R} P), f ⊗ p ↦ ∑_{i} f ⊗ p_{i} ⊗ π_{i} ⊗ p,

ε : P^{∗} ⊗_{R} P → R, f ⊗ p ↦ f(p).

By properties of the dual basis,

(I_{P ⊗_{R}P^{∗}} ⊗ ε)Δ(f ⊗ p) = ∑_{i} f ⊗ p_{i}π_{i}(p) = f ⊗ p,

showing that ε is a counit, and coassociativity of Δ is proved by the equality

(I_{P ⊗_{R}P^{∗}} ⊗Δ)Δ(f ⊗p) = ∑_{i,j}f ⊗p_{i} ⊗π_{i} ⊗p_{j} ⊗π_{j} ⊗p = (Δ⊗I_{P ⊗_{R}P^{∗}})Δ(f ⊗p).

The dual algebra of P^{∗} ⊗_{R} P is (anti)isomorphic to End_{R}(P) by the bijective maps

(P^{∗} ⊗_{R} P)^{∗} = Hom_{R}(P^{∗} ⊗_{R} P, R) ≃ Hom_{R}(P, P^{∗∗}) ≃ End_{R}(P),

which yield a ring isomorphism or anti-isomorphism, depending from which side the morphisms are acting.

For P = R we obtain R = R^{∗}, and R^{∗} ⊗_{R} R ≃ R is the trivial coalgebra. As a more interesting special case we may consider P = R^{n}. Then P^{∗} ⊗_{R} P can be identified with the matrix ring M_{n}(R), and this leads to the

5.10. Matrix coalgebra. Let {e_{ij}}_{1≤i,j≤n} be the canonical R-basis for M_{n}(R), and define the coproduct and counit

Δ : M_{n}(R) → M_{n}(R) ⊗_{R} M_{n}(R), e_{ij} ↦ ∑_{k}e_{ik} ⊗ e_{kj},
ε : M_{n}(R) → R, e_{ij} ↦ δ_{ij} .

The resulting coalgebra is called the (n, n)-matrix coalgebra over R, and we denote it by M_{n}^{c}(R).

Notice that the matrix coalgebra may also be considered as a special case of a semigroup coalgebra in 5.7.

From a given coalgebra one can construct the

5.11. Opposite coalgebra. Let Δ : C → C ⊗_{R} C define a coalgebra. Then

Δ^{tw} : C ⟶^{Δ} C ⊗_{R} C ⟶^{tw} C ⊗_{R} C, c ↦ ∑ c\underline{2} ⊗ c\underline{1},

where tw is the twist map, defines a new coalgebra structure on C known as the opposite coalgebra with the same counit. The opposite coalgebra is denoted by C^{cop}. Note that a coalgebra C is cocommutative if and only if C coincides with its opposite coalgebra (i.e., Δ = Δ^{tw}).

5.12. Duals of algebras. Let (A, μ, ι) be an R-algebra and assume _{R}A to be finitely generated and projective. Then there is an isomorphism

A^{∗} ⊗_{R} A^{∗} → (A ⊗_{R} A)^{∗}, f ⊗ g ↦ [a ⊗ b ↦ f(a)g(b)],

and the functor Hom_{R}(−, R) = (−)^{∗} yields a coproduct

μ^{∗} : A^{∗} → (A ⊗_{R} A)^{∗} ≃ A^{∗} ⊗_{R} A^{∗}

and a counit (as the dual of the unit of A)

ε := ι^{∗} : A^{∗} → R, f ↦ f(1_{A}).

This makes A^{∗} an R-coalgebra that is cocommutative provided μ is commutative. If _{R}A is not finitely generated and projective, the above construction does not work. However, under certain conditions the finite dual of A has a coalgebra structure.

6 Coalgebra morphisms

Morphisms are defined as R-linear map between coalgebras that respect the coalgebra structures (coproducts and counits).

6.1. Coalgebra morphisms. Given R-coalgebras C and C′, an R-linear map f : C → C′ is said to be a coalgebra morphism provided the diagrams

[figure: C; f; Δ; C′; Δ′; C ⊗_{R} C; f⊗f; C′ ⊗_{R} C′ ,; C; f; ε; C′; ε′; R]

are commutative. Explicitly, this means that

Δ′ ∘ f = (f ⊗ f) ∘ Δ, and ε′ ∘ f = ε,

that is, for all c ∈ C,

∑ f(c\underline{1}) ⊗ f(c\underline{2}) = ∑ f(c)\underline{1} ⊗ f(c)\underline{2}, and ε′(f(c)) = ε(c).

Given an R-coalgebra C and an S-coalgebra D, where S is a commutative ring, a coalgebra morphism between C and D is defined as a pair (α, γ) consisting of a ring morphism α : R → S and an R-linear map γ : C → D such that

γ′ : C ⊗_{R} S → D, c ⊗ s ↦ γ(c)s,

is an S-coalgebra morphism. Here we consider D as an R-module (induced by α) and C ⊗_{R} S is the scalar extension of C.

As shown in 5.3, for an R-algebra A, the contravariant functor Hom_{R}(−, A) turns coalgebras to algebras. It also turns coalgebra morphisms into algebra morphisms.

6.2. Duals of coalgebra morphisms. For R-coalgebras C and C′, an Rlinear map f : C → C′ is a coalgebra morphism if and only if

Hom(f, A) : Hom_{R}(C′, A) → Hom_{R}(C, A)

is an algebra morphism, for any R-algebra A.

Proof. Let f be a coalgebra morphism. Putting f^{∗} = Hom_{R}(f, A), we compute for g, h ∈ Hom_{R}(C′, A)

\begin{aligned}f^{∗}(g ∗ h) & = & μ ∘ (g ⊗ h) ∘ Δ′ ∘ f = μ ∘ (g ⊗ h) ∘ (f ⊗ f) ∘ Δ \\ & = & (g ∘ f) ∗ (h ∘ f) = f^{∗}(g) ∗ f^{∗}(h). \end{aligned}

To show the converse, assume that f^{∗} is an algebra morphism, that is,

μ ∘ (g ⊗ h) ∘ Δ′ ∘ f = μ ∘ (g ⊗ h) ∘ (f ⊗ f) ∘ Δ,

for any R-algebra A and g, h ∈ Hom_{R}(C′, A). Choose A to be the tensor algebra T (C) of the R-module C and choose g, h to be the canonical embedding C → T (C). Then μ∘(g ⊗h) is just the embedding C ⊗_{R} C → T_{2}(C) → T (C), and the above equality implies

Δ′ ∘ f = (f ⊗ f) ∘ Δ,

showing that f is a coalgebra morphism. ⊔ ⊓

6.3. Coideals. The problem of determining which R-submodules of C are kernels of a coalgebra map f : C → C′ is related to the problem of describing the kernel of f ⊗ f (in the category of R-modules M_{R}). If f is surjective, we know that Ke (f ⊗ f) is the sum of the canonical images of Ke f ⊗_{R} C and C ⊗_{R} Ke f in C ⊗_{R} C. This suggests the following definition.

The kernel of a surjective coalgebra morphism f : C → C′ is called a coideal of C.

6.4. Properties of coideals. For an R-submodule K ⊂ C and the canonical projection p : C → C/K, the following are equivalent:

(a) K is a coideal;

(b) C/K is a coalgebra and p is a coalgebra morphism;

(c) Δ(K) ⊂ Ke (p ⊗ p) and ε(K) = 0. If K ⊂ C is C-pure, then (c) is equivalent to:

(d) Δ(K) ⊂ C ⊗_{R} K + K ⊗_{R} C and ε(K) = 0. If (a) holds, then C/K is cocommutative provided C is also.

Proof. (a) ⇔ (b) is obvious.

(b) ⇒ (c) There is a commutative exact diagram

\xymatrix{0 \ar[r] & K \ar[r] \ar[d] & C \ar[r]^{p} \ar[d]^{Δ} & C/K \ar[d]^{Δ̄} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & Ke (p ⊗ p) \ar[r] & C ⊗_{R} C \ar[r]^{p⊗p} & C/K ⊗_{R} C/K \ar[r] & 0,}

where commutativity of the right square implies the existence of a morphism K → Ke (p ⊗ p), thus showing Δ(K) ⊂ Ke (p ⊗ p). For the counit ε̄: C/K → R of C/K, ε̄∘ p = ε and hence ε(K) = 0

(c) ⇒ (b) Under the given conditions, the left-hand square in the above diagram is commutative and the cokernel property of p implies the existence of Δ̄. This makes C/K a coalgebra with the properties required.

(c) ⇔ (d) If K ⊂ C is C-pure, Ke (p ⊗ p) = C ⊗_{R} K + K ⊗_{R} C. ⊔ ⊓ 6.5. Factorisation theorem. Let f : C → C′ be a morphism of Rcoalgebras. If K ⊂ C is a coideal and K ⊂ Ke f, then there is a commutative diagram of coalgebra morphisms

\xymatrix{C \ar[r]^{p} \ar[dr]_{f} & C/K \ar[d]^{f̄} \\  & C′ .}

Proof. Denote by f̄: C/K → C′ the R-module factorisation of f : C → C′. It is easy to show that the diagram

[figure: C/K; f̄; Δ̄; C′; Δ′; C/K ⊗_{R} C/K; f̄⊗f̄; C′ ⊗_{R} C′]

is commutative. This means that f̄ is a coalgebra morphism. ⊔ ⊓

6.6. The counit as a coalgebra morphism. View R as a trivial Rcoalgebra as in 5.5. Then, for any R-coalgebra C,

(1) ε is a coalgebra morphism;

(2) if ε is surjective, then Ke ε is a coideal.

Proof. (1) Consider the diagram

[figure: C; ε; Δ; R; ≃; c; ε(c); C ⊗_{R} C; ε⊗ε; R ⊗_{R} R; ∑ c\underline{1} ⊗ c\underline{2}; ∑ ε(c\underline{1}) ⊗ ε(c\underline{2}) .]

The properties of the counit yield

∑ ε(c\underline{1}) ⊗ ε(c\underline{2}) = ∑ ε(c\underline{1})ε(c\underline{2}) ⊗ 1 = ε(∑ c\underline{1}ε(c\underline{2})) ⊗ 1 = ε(c) ⊗ 1,

so the above diagram is commutative and ε is a coalgebra morphism. (2)This is clear by (1) and the definition of coideals. ⊔ ⊓

6.7. Subcoalgebras. An R-submodule D of a coalgebra C is called a subcoalgebra provided D has a coalgebra structure such that the inclusion map is a coalgebra morphism.

Notice that a pure R-submodule D ⊂ C is a subcoalgebra provided Δ_{D}(D) ⊂ D ⊗_{R} D ⊂ C ⊗_{R} C and ε|_{D} : D → R is a counit for D. Indeed, since D is a pure submodule of C, we obtain

Δ_{D}(D) = D ⊗_{R} C ∩ C ⊗_{R} D = D ⊗_{R} D ⊂ C ⊗_{R} C,

so that D has a coalgebra structure for which the inclusion is a coalgebra morphism, as required.

From the above observations we obtain:

6.8. Image of coalgebra morphisms. The image of any coalgebra map f : C → C′ is a subcoalgebra of C′.

6.9. Coproduct of coalgebras. For a family {C_{λ}}_{Λ} of R-coalgebras, put C = ⨁_{Λ} C_{λ}, the coproduct in M_{R}, i_{λ} : C_{λ} → C the canonical inclusions, and consider the R-linear maps

C_{λ} ⟶^{Δ_{λ}} C_{λ} ⊗ C_{λ} ⊂ C ⊗ C, ε : C_{λ} → R.

By the properties of coproducts of R-modules there exist unique maps

Δ : C → C ⊗_{R} C with Δ ∘ i_{λ} = Δ_{λ}, ε : C → R with ε ∘ i_{λ} = ε_{λ}.

(C, Δ, ε) is called the coproduct (or direct sum) of the coalgebras C_{λ}. It is obvious that the i_{λ} : C_{λ} → C are coalgebra morphisms.

C is coassociative (cocommutative) if and only if all the C_{λ} have the corresponding property. This follows – by 5.3 – from the ring isomorphism

Hom_{R}(C, A) = Hom_{R}(⨁_{Λ}C_{λ}, A) ≃ ∏_{Λ}Hom_{R}(C_{λ}, A),

for any R-algebra A, and the observation that the left-hand side is an associative (commutative) ring if and only if every component in the right-hand side has this property.

Universal property of C = ⨁_{Λ} C_{λ}. For a family {f_{λ} : C_{λ} → C′}_{Λ} of coalgebra morphisms there exists a unique coalgebra morphism f : C → C′ such that, for all λ ∈ Λ, there are commutative diagrams of coalgebra morphisms

\xymatrix{C_{λ} \ar[r]^{i_{λ}} \ar[dr]_{f_{λ}} & C \ar[d]^{f} \\  & C′ .}

Recall that for the definition of the tensor product of R-algebras A, B, the twist map tw : A ⊗_{R} B → B ⊗_{R} A, a ⊗ b ↦ b ⊗ a is needed. It also helps to define the 6.10. Tensor product of coalgebras. Let C and D be two R-coalgebras. Then the composite map

\xymatrix{C ⊗_{R} D \ar[r]^{Δ_{C}⊗Δ_{D}} & (C ⊗_{R} C) ⊗_{R} (D ⊗_{R} D) \ar[r]^{I_{C}⊗tw⊗I_{D}} & (C ⊗_{R} D) ⊗_{R} (C ⊗_{R} D)}

defines a coassociative coproduct on C ⊗_{R} D, and with the counits ε_{C} of C and ε_{D} of D the map ε_{C} ⊗ε_{D} : C ⊗_{R}D → R is a counit of C ⊗_{R}D. With these maps, C ⊗_{R} D is called the tensor product coalgebra of C and D. Obviously C ⊗_{R} D is cocommutative provided both C and D are cocommutative.

6.11. Tensor product of coalgebra morphisms. Let f : C → C′ and g : D → D′ be morphisms of R-coalgebras. The tensor product of f and g yields a coalgebra morphism

f ⊗ g : C ⊗_{R} D → C′ ⊗_{R} D′.

Proof. The fact that f and g are coalgebra morphisms implies commutativity of the top square in the diagram

\xymatrix{C ⊗_{R} D \ar[r]^{f⊗g} \ar[d]_{Δ_{C}⊗Δ_{D}} & C′ ⊗_{R} D′ \ar[d]^{Δ_{C′}⊗Δ_{D′}} \\ C ⊗_{R} C ⊗_{R} D ⊗_{R} D \ar[r]^{f⊗f⊗g⊗g} \ar[d]_{I_{C}⊗tw⊗I_{D}} & C′ ⊗_{R} C′ ⊗_{R} D′ ⊗_{R} D′ \ar[d]^{I_{C′}⊗tw⊗I_{D′}} \\ C ⊗_{R} D ⊗_{R} C ⊗_{R} D \ar[r]^{f⊗g⊗f⊗g} & C′ ⊗_{R} D′ ⊗_{R} C′ ⊗_{R} D′ ,}

while the bottom square obviously is commutative by the definitions. Commutativity of the outer rectangle means that f ⊗ g is a coalgebra morphism.

⊔ ⊓

To define the comultiplication for the tensor product of two R-coalgebras C, D in 6.10, the twist map tw : C ⊗_{R} D → D ⊗_{R} C was used. Notice that any such map yields a formal comultiplication on C ⊗_{R} D, whose properties strongly depend on the properties of the map chosen (see [3, 2.14]).

7 Comodules

As before, R denotes a commutative ring, M_{R} the category of R-modules, and C, more precisely (C, Δ, ε), stands for a (coassociative) R-coalgebra (with counit). We first introduce right comodules over C.

7.1. Right C-comodules. For M ∈ M_{R}, an R-linear map ϱ^{M} : M → M ⊗_{R} C is called a right coaction of C on M or simply a right C-coaction. To denote the action of ϱ^{M} on elements of M we write ϱ^{M}(m) = ∑ m\underline{0} ⊗ m\underline{1}.

A C-coaction ϱ^{M} is said to be coassociative and counital provided the diagrams

\xymatrix{M \ar[r]^{ϱ^{M}} \ar[d]_{ϱ^{M}} & M ⊗_{R} C \ar[d]^{I_{M} ⊗Δ} & M \ar[r]^{ϱ^{M}} \ar[dr]_{I_{M}} & M ⊗_{R} C \ar[d]^{I_{M} ⊗ε} \\ M ⊗_{R} C \ar[r]^{ϱ^{M} ⊗I_{C}} & M ⊗_{R} C ⊗_{R} C, &  & M}

are commutative. Explicitly, this means that, for all m ∈ M,

∑ ϱ^{M}(m\underline{0}) ⊗ m\underline{1} = ∑ m\underline{0} ⊗ Δ(m\underline{1}), m = ∑ m\underline{0}ε(m\underline{1}).

In view of the first of these equations we can shorten the notation and write

(I_{M} ⊗ Δ) ∘ ϱ^{M}(m) = ∑ m\underline{0} ⊗ m\underline{1} ⊗ m\underline{2},

and so on, in a way similar to the notation for a coproduct. Note that the elements with subscript 0 are in M while all the elements with positive subscripts are in C.

An R-module with a coassociative and counital right coaction is called a right C-comodule.

Recall that any semigroup induces a coalgebra (R[G], Δ_{1}, ε_{1}) (see 5.7) and for this the comodules have the following form.

7.2. Graded modules. Let G be a semigroup. Considering R with the trivial grading, an R-module M is G-graded if and only if it is an R[G]-comodule.

Proof. Let M = ⨁_{G}M_{g} be a G-graded module. Then a coaction of (R[G], Δ_{1}, ε_{1}) on M is defined by

ϱ^{M} : M ⟶ M ⊗_{R} R[G], m_{g} ↦ m ⊗ g.

It is easily seen that this coaction is coassociative and, for any m ∈ M,

(I_{M} ⊗ ε_{1})ϱ^{M}(m) = (I_{M} ⊗ ε_{1})(∑_{g∈G} m_{g} ⊗ g) = ∑_{g∈G} m_{g} = m .

Now assume that M is a right R[G]-comodule and for all m ∈ M write ϱ^{M}(m) = ∑_{g∈G} m_{g} ⊗ g. By coassociativity,

∑_{g∈G}(m_{g})_{h} ⊗ h ⊗ g = ∑_{g∈G} m_{g} ⊗ g ⊗ g,

which implies (m_{g})_{h} = δ_{g,h}m_{g} and also ϱ^{M}(m_{g}) = m_{g} ⊗ g. Then M_{g} = {m_{g} | m ∈ M} is an independent family of R-submodules of M. Now counitality of M implies

m = (I_{M} ⊗ ε_{1})(∑_{g∈G} m_{g} ⊗ g) = ∑_{g∈G} m_{g},

and hence M = ⨁_{G} M_{g}. ⊔ ⊓

7.3. Comodule morphisms. Let M, N be right C-comodules. An R-linear map f : M → N is called a comodule morphism (or (C-)colinear map) if and only if the diagram

\xymatrix{M \ar[r]^{f} \ar[d]_{ϱ^{M}} & N \ar[d]^{ϱ^{N}} \\ M ⊗_{R} C \ar[r]^{f⊗I_{C}} & N ⊗_{R} C}

is commutative. Explicitly, this means that ϱ^{N} ∘ f = (f ⊗ I_{C}) ∘ ϱ^{M}; that is, for all m ∈ M we require

∑ f(m)\underline{0} ⊗ f(m)\underline{1} = ∑ f(m\underline{0}) ⊗ m\underline{1}.

Clearly the sum of two C-morphisms is again a C-morphism and the set Hom^{C}(M, N) of C-morphisms from M to N is an R-module, which is determined by the exact sequence in M_{R},

0 → Hom^{C}(M, N) → Hom_{R}(M, N) ⟶^{γ} Hom_{R}(M, N ⊗_{R} C),

where γ(f) := ϱ^{N} ∘ f − (f ⊗ I_{C}) ∘ ϱ^{M}.

The class of right comodules over C together with the colinear maps form an additive category which we denote by M^{C}.

7.4. Left C-comodules and their morphisms are defined symmetrically by R-linear maps^{M}ϱ : M → C⊗_{R}M. For m ∈ M we write^{M}ϱ(m) = ∑ m\underline{−1}⊗m\underline{0}, and coassociativity is expressed as

∑ m\underline{−1} ⊗^{M}ϱ(m\underline{0}) = ∑ Δ(m\underline{−1}) ⊗ m\underline{0} = ∑ m\underline{−2} ⊗ m\underline{−1} ⊗ m\underline{0},

where the final expression is a notation. The condition for the counit reads m = ∑ ε(m\underline{−1})m\underline{0}.

The R-module of left C-morphisms is denoted by ^{C}Hom (M, N) and left C-comodules and their morphisms again form an additive category that is denoted by ^{C}M.

An example of a left and right C-comodule is provided by C itself. In both cases coaction is given by Δ (the regular coaction).

7.5. Kernels and cokernels in M^{C}. Let f : M → N be a morphism in M^{C}. The cokernel g of f in M_{R} yields the exact commutative diagram

\xymatrix{M \ar[r]^{f} \ar[d]_{ϱ^{M}} & N \ar[r]^{g} \ar[d]^{ϱ^{N}} & L \ar[r] & 0 \\ M ⊗_{R} C \ar[r]^{f⊗I_{C}} & N ⊗_{R} C \ar[r]^{g⊗I_{C}} & L ⊗_{R} C \ar[r] & 0,}

which can be completed commutatively in M_{R} by some ϱ^{L} : L → L ⊗_{R} C for which we obtain the diagram

\xymatrix{N \ar[r]^{ϱ^{N}} \ar[d]_{g} & N ⊗_{R} C \ar[r]^{ϱ^{N} ⊗I_{C}} \ar[r]_{I_{N} ⊗Δ} \ar[d]_{g⊗I_{C}} & N ⊗_{R} C ⊗_{R} C \ar[d]^{g⊗I_{C}⊗I_{C}} \\ L \ar[r]^{ϱ^{L}} & L ⊗_{R} C \ar[r]^{ϱ^{L}⊗I_{C}} \ar[r]_{I_{L}⊗Δ} & L ⊗_{R} C ⊗_{R} C .}

The outer rectangle is commutative for the upper as well as for the lower morphisms, and hence

(ϱ^{L} ⊗ I_{C}) ∘ ϱ^{L} ∘ g = (I_{L} ⊗ Δ) ∘ ϱ^{L} ∘ g .

Now, surjectivity of g implies (ϱ^{L} ⊗ I_{C}) ∘ ϱ^{L} = (I_{L} ⊗ Δ) ∘ ϱ^{L}, showing that ϱ^{L} is coassociative. Moreover,

(I_{L} ⊗ ε) ∘ ϱ^{L} ∘ g = (I_{L} ⊗ ε) ∘ (g ⊗ I_{C}) ∘ ϱ^{N} = g,

which shows that (I_{L} ⊗ ε) ∘ ϱ^{L} = I_{L}. Thus ϱ^{L} is counital, and so it makes L a comodule such that g is a C-morphism. This shows that cokernels exist in the category M^{C}.

Dually, for the kernel h of f in M_{R} there is a commutative diagram

\xymatrix{0 \ar[r] & K \ar[r]^{h} & M \ar[r]^{f} \ar[d]_{ϱ^{M}} & N \ar[d]^{ϱ^{N}} \\ 0 \ar[r] & K ⊗_{R} C \ar[r]^{h⊗I_{C}} & M ⊗_{R} C \ar[r]^{f⊗I_{C}} & N ⊗_{R} C ,}

where the top sequence is always exact while the bottom sequence is exact under special conditions. If this is the case, the diagram can be extended commutatively by a coaction ϱ^{K} : K → K ⊗_{R} C.

If moreover f is C⊗_{R}C-flat then - dual to the proof for cokernels - it can be shown that ϱ^{K} is coassociative and counital. Thus kernels of C-morphisms are induced from kernels in M_{R} provided certain additional conditions are imposed.

7.6. C-subcomodules. Let M be a right C-comodule. An R-submodule K ⊂ M is called a C-subcomodule of M provided K has a right comodule structure such that the inclusion is a comodule morphism.

If K is a C ⊗_{R} C-pure submodule of M, then K is a subcomodule of M provided ϱ^{M}(K) ⊂ K ⊗_{R} C ⊂ M ⊗_{R} C.

7.7. Coproducts in M^{C}. Let {M_{λ}, ϱ_{λ}^{M}}_{Λ} be a family of C-comodules. Put M = ⨁_{Λ} M_{λ}, the coproduct in M_{R}, i_{λ} : M_{λ} → M the canonical inclusions, and consider the linear maps

M_{λ} ⟶^{ϱ_{λ}^{M}} M_{λ} ⊗_{R} C ⊂ M ⊗_{R} C.

Note that the inclusions i_{λ} are R-splittings, so that M_{λ} ⊗_{R} C ⊂ M ⊗_{R} C is a pure submodule. By the properties of coproducts of R-modules there exists a unique coaction

ϱ^{M} : M → M ⊗_{R} C, such that ϱ^{M} ∘ i_{λ} = ϱ_{λ}^{M},

which is coassociative and counital since all the ϱ_{λ}^{M} are, and thus it makes M a C-comodule for which the i_{λ} : M_{λ} → M are C-morphisms with the following universal property:

Let {f_{λ} : M_{λ} → N}_{Λ} be a family of morphisms in M^{C}. Then there exists a unique C-morphism f : M → N such that, for each λ ∈ Λ, the following diagram of C-morphisms commutes:

\xymatrix{M_{λ} \ar[r]^{i_{λ}} \ar[dr]_{f_{λ}} & M \ar[d]^{f} \\  & N .}

Similarly to the coproduct, the direct limit of direct families of C-como-dules is derived from the direct limit in M_{R}.

7.8. Comodules and tensor products. Let M be in M^{C} and consider any morphism f : X → Y of R-modules. Then:

(1) X ⊗_{R} M is a right C-comodule with the coaction

I_{X} ⊗ ϱ^{M} : X ⊗_{R} M ⟶ X ⊗_{R} M ⊗_{R} C,

and the map f ⊗ I_{M} : X ⊗_{R} M → Y ⊗_{R} M is a C-morphism.

(2) In particular, X ⊗_{R} C has a right C-coaction

I_{X} ⊗ Δ : X ⊗_{R} C ⟶ X ⊗_{R} C ⊗_{R} C,

and the map f ⊗ I_{C} : X ⊗_{R} C → Y ⊗_{R} C is a C-morphism.

(3) For any index set Λ, R^{(Λ)} ⊗_{R} C ≃ C^{(Λ)} as comodules and there exists a surjective C-morphism

C^{(Λ′)} → M ⊗_{R} C, for some Λ′.

(4) The structure map ϱ^{M} : M → M ⊗_{R} C is a comodule morphism, and hence M is a subcomodule of a C-generated comodule.

Proof. (1) and (2) are easily verified from the definitions.

(3) Take a surjective R-linear map h : R^{(Λ′)} → M. Then, by (2),

h ⊗ I_{C} : R^{(Λ′)} ⊗_{R} C → M ⊗_{R} C

is a surjective comodule morphism.

(4) By coassociativity, ϱ^{M} is a comodule morphism (where M ⊗_{R} C has the comodule structure from (1)). Note that ρ^{M} is split by I_{M} ⊗ ε as an Rmodule; thus M is a pure submodule of M ⊗_{R} C and hence is a subcomodule.

⊔ ⊓

Similarly to the classical Hom-tensor relations (see 1.12) we obtain

7.9. Hom-tensor relations in M^{C}. Let X be any R-module.

(1) For any M ∈ M^{C}, the R-linear map

φ : Hom^{C}(M, X ⊗_{R} C) → Hom_{R}(M, X), f ↦ (I_{X} ⊗ ε) ∘ f,

is bijective, with inverse map h ↦ (h ⊗ I_{C}) ∘ ϱ^{M}.

(2) For any M, N ∈ M^{C}, the R-linear map

ψ : Hom^{C}(X⊗_{R}M, N) → Hom_{R}(X, Hom^{C}(M, N)), g ↦[x ↦ g(x⊗−)],

is bijective, with inverse map h ↦ [x ⊗ m ↦ h(x)(m)].

Proof. (1) For any f ∈ Hom^{C}(M, X ⊗_{R} C) the diagram

[figure: M; f; ϱ^{M}; X ⊗_{R} C; I_{X}⊗Δ; =; M ⊗_{R} C; f⊗I_{C}; X ⊗_{R} C ⊗_{R} C; I_{X}⊗ε⊗I_{C}; X ⊗_{R} C]

is commutative, that is,

f = (I_{X} ⊗ ε ⊗ I_{C}) ∘ (f ⊗ I_{C}) ∘ ϱ^{M} = (φ(f) ⊗ I_{C}) ∘ ϱ^{M}.

This implies that φ is injective.

Since ϱ^{M} is a C-morphism, so is (h ⊗ I_{C}) ∘ ϱ^{M}, for any h ∈ Hom_{R}(M, X). Therefore

φ((h ⊗ I_{C}) ∘ ρ^{M}) = (I_{X} ⊗ ε) ∘ (h ⊗ I_{C}) ∘ ϱ^{M} = h ∘ (I_{M} ⊗ ε) ∘ ϱ^{M} = h,

implying that φ is surjective.

(2) The Hom-tensor relations for modules provide one with an isomorphism of R-modules,

ψ : Hom_{R}(X ⊗_{R} M, N) → Hom_{R}(X, Hom_{R}(M, N)).  (∗)

For any x ∈ X, by commutativity of the diagram

[figure: M; x⊗−; ϱ^{M}; X ⊗_{R} M; I_{X}⊗ϱ^{M}; m; x ⊗ m; M ⊗_{R} C; (x⊗−)⊗I_{C}; X ⊗_{R} M ⊗_{R} C,; ϱ^{M}(m); x ⊗ ϱ^{M}(m) ,]

the map x ⊗ − is a C-morphism. Hence, for any g ∈ Hom^{C}(X ⊗_{R} M, N), the composition g ∘ (x ⊗ −) is a C-morphism. On the other hand, there is a commutative diagram, for all h ∈ Hom_{R}(X, Hom^{C}(M, N)),

[figure: X ⊗_{R} M; I_{X}⊗ϱ^{M}; N; ϱ^{N}; x⊗m; h(x)(m); X ⊗_{R} M ⊗_{R} C; N ⊗_{R} C ,; x⊗ϱ^{M}(m); (h(x)⊗I_{C}) ∘ ϱ^{M}(m) .]

This shows that ψ^{−1}(h) lies in Hom^{C}(X ⊗_{R} M, N) and therefore implies that ψ in (∗) restricts to the bijective map ψ : Hom^{C}(X ⊗_{R} M, N) → Hom_{R}(X, Hom^{C}(M, N)), as required. ⊔ ⊓

Unlike for A-modules, the R-dual of a right C-comodule need not be a left C-comodule unless additional conditions are imposed. To specify such sufficient conditions, first recall that, for a finitely presented R-module M and a flat R-module C, there is an isomorphism (compare 2.9)

ν_{M} : C ⊗_{R} Hom_{R}(M, R) → Hom_{R}(M, C), c ⊗ h ↦ c ⊗ h(−) .

7.10. Comodules finitely presented as R-modules. Let _{R}C be flat and M ∈ M^{C} such that _{R}M is finitely presented. Then M^{∗} = Hom_{R}(M, R) is a left C-comodule by the structure map

M^{∗}ϱ : M^{∗} → Hom_{R}(M, C) ≃ C ⊗_{R} M^{∗}, g ↦ (g ⊗ I_{C}) ∘ ϱ^{M} .

Proof. The comodule property of M^{∗} follows from the commutativity of the following diagram (with obvious maps), the central part of which arises from the coassociativity of C (tensor over R):

\xymatrix{M^{∗} \ar[r]^{≃} \ar[d] & Hom^{C}(M, C) \ar[r] \ar[d] & Hom^{C}(M, C⊗C) \ar[r]^{≃} \ar[d] & C⊗M^{∗} \ar[d]^{Δ⊗I_{M^{∗}}} \\ C⊗M^{∗} \ar[r]^{≃} & Hom^{C}(M, C⊗C) \ar[r] & Hom^{C}(M, C⊗C⊗C) \ar[r]^{≃} & C⊗C⊗M^{∗} .}

⊔ ⊓

For X = R and M = C, the isomorphism φ describes the comodule endomorphisms of C.

7.11. Comodule endomorphisms of C.

(1) There is an algebra anti-isomorphism φ : End^{C}(C) → C^{∗}, f ↦ ε ∘ f, with the inverse map h ↦ (h ⊗ I_{C}) ∘ Δ and so h ∈ C^{∗} acts on c ∈ C from the right by

c↼h = (h ⊗ I_{C})Δ(c) = ∑ h(c\underline{1})c\underline{2}.

(2) There is an algebra isomorphism φ′ : ^{C}End(C) → C^{∗}, f ↦ ε ∘ f, with the inverse map h ↦ (I_{C} ⊗ h) ∘ Δ and so h ∈ C^{∗} acts on c ∈ C from the left by

h⇀c = (I_{C} ⊗ h)Δ(c) = ∑ c\underline{1}h(c\underline{2}).

(3) For any f ∈ C^{∗} and c ∈ C,

\begin{aligned}Δ(f ⇀c) & = & ∑ c\underline{1} ⊗ (f ⇀c\underline{2}), \\ Δ(c↼f) & = & ∑(c\underline{1}↼f) ⊗ c\underline{2}, \\ Δ(f ⇀c↼g) & = & ∑(c\underline{1}↼g) ⊗ (f ⇀c\underline{2}), \\ ∑ c\underline{1} ⊗ (c\underline{2}↼f) & = & ∑(f ⇀c\underline{1}) ⊗ c\underline{2}. \end{aligned}

(4) The coproduct Δ yields the embedding

C^{∗} ≃ Hom^{C}(C, C) → Hom^{C}(C, C ⊗_{R} C) ≃ End_{R}(C).

Proof. (1) By 7.9(1), φ is R-linear and bijective. Take any f, g ∈ End^{C}(C), recall that (f ⊗ I_{C}) ∘ Δ = Δ ∘ f, and consider the convolution product applied to any c ∈ C,

(ε ∘ f) ∗ (ε ∘ g)(c) = ∑ ε(f(c\underline{1})) ε(g(c\underline{2}))
= ε ∘ g [(ε ⊗ I_{C}) ∘ (f ⊗ I_{C}) ∘ Δ(c)]
= ε ∘ g [(ε ⊗ I_{C}) ∘ Δ ∘ f(c)] = ε ∘ (g ∘ f)(c) .

This shows that φ is an anti-isomorphism.

(2) For all f, g ∈ ^{C}End (C), (I_{C} ⊗ g) ∘ Δ = Δ ∘ g, and hence

(ε ∘ f) ∗ (ε ∘ g)(c) = ∑ ε(f(c\underline{1})) ε(g(c\underline{2}))
= ε ∘ f [(I_{C} ⊗ ε) ∘ (I_{C} ⊗ g) ∘ Δ(c)]
= ε ∘ g [(I_{C} ⊗ ε) ∘ Δ ∘ g(c)] = ε ∘ (f ∘ g)(c) .

(3) By definition,

\begin{aligned}Δ(f ⇀c) & = & Δ(∑ c\underline{1}f(c\underline{2}) = ∑ c\underline{11} ⊗ c\underline{12}f(c\underline{2}) \\ & = & ∑ c\underline{1} ⊗ c\underline{21}f(c\underline{22}) = ∑ c\underline{1} ⊗ (f ⇀c\underline{2}). \end{aligned}

The remaining assertions are shown similarly.

(4) This follows from the Hom-tensor relations 7.9 for M = C = X . ⊔ ⊓

Notice that in 7.11(1) the comodule morphisms are written on the left of the argument. By writing morphisms of right comodules on the right side, we obtain an isomorphism between C^{∗} and the comodule endomorphism ring.

The next theorem summarises observations on the category of comodules.

7.12. The category M^{C}.

(1) The category M^{C} has direct sums and cokernels, and C is a subgenerator.

(2) M^{C} is a Grothendieck category provided that C is a flat R-module.

(3) The functor −⊗_{R}C : M_{R} → M^{C} is right adjoint to the forgetful functor (−)_{R} : M^{C} → M_{R}.

(4) For any monomorphism f : K → L of R-modules,

f ⊗ I_{C} : K ⊗_{R} C → L ⊗_{R} C

is a monomorphism in M^{C}.

(5) For any family {M_{λ}}_{Λ} of R-modules, (∏_{Λ} M_{λ}) ⊗_{R} C is the product of the M_{λ} ⊗_{R} C in M^{C}.

Proof. (1) The first assertions follow from 7.5 and 7.7. By 7.8(4), any comodule M is a subcomodule of the C-generated comodule M ⊗_{R} C.

(2) By 7.5, M^{C} has kernels provided C is a flat R-module. This implies that the intersection of two subcomodules and the preimage of a (sub)comodule is again a comodule. It remains to show that M^{C} has (a set of) generators. For any right C-comodule M, there exists a surjective comodule map g : C^{(Λ)} → M ⊗_{R} C (see 7.8). Then L := g^{−1}(M) ⊂ C^{(Λ)} is a subcomodule. Furthermore, for any m ∈ M there exist k ∈ ℕ and an element x in the comodule C^{k} ∩ L ⊂ C^{k} such that g(x) = m. Therefore m ∈ g(C^{k} ∩ L). This shows that M is generated by comodules of the form C^{k} ∩ L, k ∈ ℕ. Hence the subcomodules of C^{k}, k ∈ ℕ, form a set of generators of M^{C}.

(3) For all M ∈ M^{C} and X ∈ M_{R}, let φ_{M,X} denote the isomorphism constructed in 7.9(1). We need to show that φ_{M,X} is natural in M and X. First take any right C-comodule N and any g ∈ Hom^{C}(M, N). Then, for all f ∈ Hom^{C}(N, X ⊗_{R} C),

(φ_{M,X} ∘ Hom^{C}(g, X ⊗_{R} C))(f) = (I_{X} ⊗ ε) ∘ Hom^{C}(g, X ⊗_{R} C)(f)
= (I_{X} ⊗ ε) ∘ f ∘ g
= Hom_{R}(g, X)((I_{X} ⊗ ε) ∘ f)
= (Hom_{R}(g, X) ∘ φ_{N,X})(f).

Similarly, take any R-module Y and g ∈ Hom_{R}(X, Y). Then, for any map f ∈ Hom^{C}(M, X ⊗_{R} C),

(φ_{M,Y} ∘ Hom^{C}(M, g ⊗ I_{C}))(f) = (I_{Y} ⊗ ε) ∘ (Hom^{C}(M, g ⊗ I_{C})(f))
= (I_{Y} ⊗ ε) ∘ (g ⊗ I_{C}) ∘ f
= (g ⊗ ε) ∘ f = g ∘ (I_{X} ⊗ ε) ∘ f
= (Hom_{R}(M, g) ∘ φ_{M,X})(f).

This proves the naturality of φ and thus the adjointness property. Note that the unit of this adjunction is provided by the coaction ϱ^{M} : M → M ⊗_{R} C, while the counit is I_{X} ⊗ ε : X ⊗_{R} C → X.

(4) Any functor that has a left adjoint preserves monomorphisms (cf. [3, 38.21]). Note that monomorphisms in M^{C} need not be injective maps, unless RC is flat.

(5) By (3), for all X ∈ M^{C} there are isomorphisms

Hom^{C}(X, (∏_{Λ} M_{λ}) ⊗_{R} C) ≃ Hom_{R}(X, ∏_{Λ} M_{λ})
≃ ∏_{Λ} Hom_{R}(X, M_{λ})
≃ ∏_{Λ} Hom^{C}(X, M_{λ} ⊗_{R} C).

These isomorphisms characterise (∏_{Λ} M_{λ}) ⊗_{R} C as product of the M_{λ} ⊗_{R} C in M^{C}. ⊔ ⊓

7.13. C as a flat R-module. The following are equivalent:

(a) C is flat as an R-module;

(b) every monomorphism in M^{C} is injective;

(c) every monomorphism U → C in M^{C} is injective;

(d) the forgetful functor M^{C} → M_{R} respects monomorphisms.

Proof. (a) ⇒ (b) Consider a monomorphism f : M → N. Since RC is flat, the inclusion i : Ke f → M is a morphism in M^{C} (by 7.5) and f ∘ i = f ∘ 0 = 0 implies i = 0, that is, Ke f = 0.

(b) ⇒ (c) and (b) ⇔ (d) are obvious.

(c) ⇒ (a) For every ideal J ⊂ R, the canonical map J ⊗_{R} C → R ⊗_{R} C is a monomorphism in M^{C} by 7.12(4), and hence it is injective by assumption. This implies that _{R}C is flat (see 1.16). ⊔ ⊓

Recall that a monomorphism i : N → L in M_{R} is a coretraction provided there exists p : L → N in M_{R} with p ∘ i = I_{N}.

7.14. Relative injective comodules. A right C-comodule M is said to be relative injective or (C, R)-injective if, for every C-comodule map i : N → L that is an R-module coretraction, and for every morphism f : N → M in M^{C}, there exists a right C-comodule map g : L → M such that g ∘ i = f. In other words, we require that every diagram in M^{C}

\xymatrix{N \ar[rr]^{i} \ar[dr]_{f} &  & L \\  & M}

can be completed commutatively by some C-morphism g : L → M, provided there exists an R-module map p : L → N such that p ∘ i = I_{N}.

7.15. (C, R)-injectivity. Let M be a right C-comodule.

(1) The following are equivalent:

(a) M is (C, R)-injective;

(b) any C-comodule map i : M → L that is a coretraction in M_{R} is also a coretraction in M^{C};

(c) the coaction ϱ^{M} : M → M ⊗_{R} C is a coretraction in M^{C}.

(2) For any X ∈ M_{R}, X ⊗_{R} C is (C, R)-injective.

(3) If M is (C, R)-injective, then, for any L ∈ M^{C}, the canonical sequence

\xymatrix{0 \ar[r] & Hom^{C}(L, M) \ar[r]^{i} & Hom_{R}(L, M) \ar[r]^{γ} & Hom_{R}(L, M ⊗_{R} C)}

splits in M_{B}, where B = End^{C}(L) and γ(f) = ϱ^{M} ∘ f − (f ⊗ I_{C}) ∘ ϱ^{L} (see 7.3). In particular, End^{C}(C) ≃ C^{∗} is a C^{∗}-direct summand in End_{R}(C).

Proof. (1) (a) ⇒ (b) Suppose that M is (C, R)-injective and take N = M and f = I_{M} in 7.14 to obtain the assertion.

(b) ⇒ (c) View M ⊗_{R} C as a right C-comodule with the coaction I_{M} ⊗ Δ, and note that ϱ^{M} : M → M ⊗_{R} C is a right C-comodule map that has an R-linear retraction I_{M} ⊗ ε. Therefore ϱ^{M} is a coretraction in M^{C}.

(c) ⇒ (a) Suppose there exists a right C-comodule map h : M ⊗_{R} C → M such that h ∘ ρ^{M} = I_{M}, consider a diagram

\xymatrix{N \ar[rr]^{i} \ar[dr]_{f} &  & L \\  & M}

as in 7.14, and assume that there exists an R-module map p : L → N such that p ∘ i = I_{N}. Define an R-linear map g : L → M as a composition

\xymatrix{g : L \ar[r]^{ϱ^{L}} & L ⊗_{R} C \ar[r]^{f∘p⊗I_{C}} & M ⊗_{R} C \ar[r]^{h} & M} .

Clearly, g is a right C-comodule map as a composition of C-comodule maps. Furthermore,

\begin{aligned}g ∘ i & = & h ∘ (f ∘ p ⊗ I_{C}) ∘ ϱ^{L} ∘ i = h ∘ (f ∘ p ∘ i ⊗ I_{C}) ∘ ϱ^{N} \\ & = & h ∘ (f ⊗ I_{C}) ∘ ϱ^{N} = h ∘ ϱ^{M} ∘ f = f, \end{aligned}

where we used that both i and f are C-colinear. Thus the above diagram can be completed to a commutative diagram in M^{C}, and hence M is (C, R)- injective.

(2) The coaction for X ⊗_{R} C is given by ϱ^{X⊗_{R}C} = I_{X} ⊗ Δ, and it is split by a right C-comodule map I_{X} ⊗ ε ⊗ I_{C}. Thus X ⊗_{R} C is (C, R)-injective by part (1).

(3) Denote by h : M ⊗_{R} C → M the splitting map of ϱ^{M} in M^{C}. Then the map

Hom_{R}(L, M) ≃ Hom^{C}(L, M ⊗_{R} C) → Hom^{C}(L, M), f ↦ h ∘ (f ⊗ I_{C}) ∘ ϱ^{L},

splits the first inclusion in M_{B}, and the map

Hom_{R}(L, M ⊗_{R} C) → Hom_{R}(L, M), g ↦ h ∘ g

yields a splitting map Hom_{R}(L, M ⊗_{R}C) → Hom_{R}(L, M)/Hom^{C}(L, M), since for any f ∈ Hom_{R}(L, M),

h ∘ γ(f) = f − h ∘ (f ⊗ I_{C}) ∘ ϱ^{L} ∈ f + Hom^{C}(L, M).
⊔ ⊓

If _{R}C is flat, M^{C} is a Grothendieck category by 7.12, so exact sequences are defined in M^{C} and we can describe 7.16. Exactness of the Hom^{C}-functors. Assume _{R}C to be flat and let M ∈ M^{C}. Then:

(1) Hom^{C}(−, M) : M^{C} → M_{R} is a left exact functor.

(2) Hom^{C}(M, −) : M^{C} → M_{R} is a left exact functor.

Proof. (1) From any exact sequence X → Y → Z → 0 in M^{C} we derive the commutative diagram (tensor over R)

\xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & Hom^{C}(Z, M) \ar[r] \ar[d] & Hom^{C}(Y, M) \ar[r] \ar[d] & Hom^{C}(X, M) \ar[d] \\ 0 \ar[r] & Hom_{R}(Z, M) \ar[r] \ar[d] & Hom_{R}(Y, M) \ar[r] \ar[d] & Hom_{R}(X, M) \ar[d] \\ 0 \ar[r] & Hom_{R}(Z, M ⊗C) \ar[r] & Hom_{R}(Y, M ⊗C) \ar[r] & Hom_{R}(X, M ⊗C),}

where the columns are exact by the characterisation of comodule morphisms (in 7.3). The second and third rows are exact by exactness properties of the functors Hom_{R}. Now the diagram lemmata imply that the first row is exact, too.

Part (2) is shown with a similar diagram. ⊔ ⊓

An object Q ∈ M^{C} is injective in M^{C} if, for any monomorphism M → N in M^{C}, the canonical map Hom^{C}(N, Q) → Hom^{C}(M, Q) is surjective.

7.17. Injectives in M^{C}. Assume _{R}C to be flat.

(1) Q ∈ M^{C} is injective if and only if Hom^{C}(−, Q) : M^{C} → M_{R} is exact.

(2) If X ∈ M_{R} is injective in M_{R}, then X ⊗_{R} C is injective in M^{C}.

(3) If M ∈ M^{C} is (C, R)-injective and injective in M_{R}, then M is injective in M^{C}.

(4) C is (C, R)-injective, and it is injective in M^{C} provided that R is injective in M_{R}.

(5) If _{R}M is flat and N is injective in M^{C}, then Hom^{C}(M, N) is injective in M_{R}.

Proof. (1) The assertion follows by 7.16.

(2) This follows from the isomorphism in 7.9(1).

(3) Since M is R-injective, assertion (2) implies that M ⊗_{R} C is injective in M^{C}. Moreover, by 7.15, M is a direct summand of M ⊗_{R} C as a comodule, and hence it is also injective in M^{C}.

Part (4) is a special case of (2).

(5) This follows from the isomorphism in 7.9(2). ⊔ ⊓

An object P ∈ M^{C} is projective in M^{C} if, for any epimorphism M → N in M^{C}, the canonical map Hom^{C}(P, M) → Hom^{C}(P, N) is surjective.

7.18. Projectives in M^{C}. Consider any P ∈ M^{C}.

(1) If P is projective in M^{C}, then P is projective in M_{R}.

(2) If _{R}C is flat, the following are equivalent:

(a) P is projective in M^{C};

(b) Hom^{C}(P, −) : M^{C} → M_{R} is exact.

Proof. (1) For any epimorphism f : K → L in M_{R}, K ⊗_{R} C ⟶^{f⊗I_{C}} L⊗_{R} C is an epimorphism in M^{C} and the projectivity of P implies the exactness of the top row in the commutative diagram

\xymatrix{Hom^{C}(P, K ⊗_{R} C) \ar[r] \ar[d]^{≃} & Hom^{C}(P, L ⊗_{R} C) \ar[r] \ar[d]^{≃} & 0 \\ Hom_{R}(P, K) \ar[r]^{Hom(P,f)} & Hom_{R}(P, L) ,}

where the vertical maps are the functorial isomorphisms from 7.9(1). From this we see that Hom(P, f) is surjective, proving that P is projective as an R-module.

(2) This follows from left exactness of Hom^{C}(P, −) described in 7.16. ⊔ ⊓

Note that, although there are enough injectives in M^{C}, there are possibly no projective objects in M^{C}. This remains true even if R is a field.

7.19. Tensor product and Hom^{C}. Let _{R}C be flat, and consider M, N ∈ M^{C} and X ∈ M_{R} such that

(i) M_{R} is finitely generated and projective, and N is (C, R)-injective; or (ii) M_{R} is finitely presented and X is flat in M_{R}. Then there exists a canonical isomorphism

ν : X ⊗_{R} Hom^{C}(M, N) ⟶ Hom^{C}(M, X ⊗_{R} N), x ⊗ h ↦ x ⊗ h(−).

Proof. Consider the defining exact sequence for Hom^{C} (see 7.3),

(∗) \xymatrix{0 \ar[r] & Hom^{C}(M, N) \ar[r] & Hom_{R}(M, N) \ar[r] & Hom_{R}(M, N ⊗_{R} C).}

Tensoring with X_{R} yields the commutative diagram (tensor over R)

\xymatrix{0 \ar[r] & X ⊗Hom^{C}(M, N) \ar[r] \ar[d]^{ν} & X ⊗Hom_{R}(M,N) \ar[r] \ar[d]^{≃} & X ⊗Hom_{R}(M,N ⊗C) \ar[d]^{≃} \\ 0 \ar[r] & Hom^{C}(M,X ⊗N) \ar[r] & Hom_{R}(M, X⊗N) \ar[r] & Hom_{R}(M,X ⊗N ⊗C),}

where the bottom row is exact (again by 7.3) and the vertical isomorphisms follow from the finiteness assumptions in (i) and (ii) (cf. 2.9).

If X is flat, the top row is exact. On the other hand, if N is (C, R)- injective, the sequence (∗) splits by 7.15, and hence the top row is exact, too. Therefore, in either case, the exactness of the diagram implies that ν is an isomorphism, as required. ⊔ ⊓

8. C-comodules and C^{∗}-modules 63

8 C-comodules and C^{∗}-modules

For any R-coalgebra C the dual R-module C^{∗} = Hom_{R}(C, R) is an associative algebra. In this section we study the relationship between C-comodules and C^{∗}-modules.

8.1. C-comodules and C^{∗}-modules.

(1) Any M ∈ M^{C} is a (unital) left C^{∗}-module by

⇀ : C^{∗} ⊗_{R} M → M, f ⊗ m ↦ (I_{M} ⊗ f) ∘ ϱ^{M}(m) = ∑ m\underline{0}f(m\underline{1}).

(2) Any morphism h : M → N in M^{C} is a left C^{∗}-module morphism, that

is,

Hom^{C}(M, N) ⊂ _{C^{∗}}Hom (M, N).

(3) There is a faithful functor from M^{C} to σ[_{C^{∗}}C], the full subcategory of C^{∗}M consisting of all C^{∗}-modules subgenerated by C (cf. 3.1).

Proof. (1) By definition, for all f, g ∈ C^{∗} and m ∈ M, the actions f ⇀(g⇀m) and (f ∗ g)⇀m are the compositions of the maps in the top and bottom rows of the following commutative diagram:

\xymatrix{ & M ⊗_{R} C \ar[dr]^{ϱ^{M} ⊗I_{C}} \\ M \ar[ur]^{ϱ^{M}} \ar[dr]_{ϱ^{M}} &  & M ⊗_{R} C ⊗_{R} C \ar[r]^{I_{M} ⊗f⊗g} & M . \\  & M ⊗_{R} C \ar[ur]_{I_{M} ⊗Δ}}

Clearly, for each m ∈ M, ε⇀m = m, and thus M is a C^{∗}-module. (2) For any h : M → N in M^{C} and f ∈ C^{∗}, m ∈ M, consider

\begin{aligned}h(f ⇀m) & = & ∑ h(m\underline{0} f(m\underline{1})) = (I_{N} ⊗ f) ∘ (h ⊗ I_{C}) ∘ ϱ^{M}(m) \\ & = & (I_{N} ⊗ f) ∘ ϱ^{N} ∘ h(m) = f ⇀h(m). \end{aligned}

This shows that h is a C^{∗}-linear map.

(3) By 7.12, C is a subgenerator in M^{C} and hence all C-comodules are subgenerated by C as C^{∗}-modules (by (1),(2)); thus they are objects in σ[_{C^{∗}}C], and hence (1)–(2) define a faithful functor M^{C} → σ[_{C^{∗}}C]. ⊔ ⊓

Now, the question arises when M^{C} is a full subcategory of σ[_{C^{∗}}C] (or C^{∗}M), that is, when Hom^{C}(M, N) = Hom_{C^{∗}}(M, N), for any M, N ∈ M^{C}. In answering this question the following property plays a crucial role.

8.2. The α-condition. C is said to satisfy the α-condition if the map

α_{N} : N ⊗_{R} C → Hom_{R}(C^{∗}, N), n ⊗ c ↦ [f ↦ f(c)n],

is injective, for every N ∈ M_{R}. By 4.7, the following are equivalent:

(a) C satisfies the α-condition;

(b) for any N ∈ M_{R} and u ∈ N ⊗_{R} C, (I_{N} ⊗ f)(u) = 0 for all f ∈ C^{∗}, implies u = 0;

(c) C is locally projective as an R-module.

In particular, this implies that C is a flat R-module, and that it is cogenerated by R.

8.3. M^{C} as a full subcategory of _{C^{∗}}M. The following are equivalent:

(a) M^{C} = σ[_{C^{∗}}C];

(b) M^{C} is a full subcategory of _{C^{∗}}M;

(c) for all M, N ∈ M^{C}, Hom^{C}(M, N) = _{C^{∗}}Hom(M, N);

(d) _{R}C is locally projective;

(e) every left C^{∗}-submodule of C^{n}, n ∈ ℕ, is a subcomodule of C^{n}.

If these conditions are satisfied, the inclusion functor M^{C} → _{C^{∗}}M has a right adjoint, and for any family {M_{λ}}_{Λ} of R-modules,

(∏_{Λ}M_{λ}) ⊗_{R} C ≃ ∏_{Λ}^{C}(M_{λ} ⊗_{R} C) ⊂ ∏_{Λ}(M_{λ} ⊗_{R} C),

where ∏^{C} denotes the product in M^{C}.

Proof. (a) ⇔ (b) ⇔ (c) follow by the fact that C is always a subgenerator of ^{C}M (see 7.12) and the definition of the category σ[_{C^{∗}}C] (cf. 3.1).

(a) ⇒ (d) The equality obviously implies that monomorphisms in M^{C} are injective maps. Hence _{R}C is flat by 7.12(4). For any N ∈ M_{R} we prove the injectivity of the map α_{N} : N ⊗_{R} C → Hom_{R}(C^{∗}, N).

Hom_{R}(C^{∗}, N) is a left C^{∗}-module by

g · γ(f) = γ(f ∗ g), for γ ∈ Hom_{R}(C^{∗}, N), f, g ∈ C^{∗},

and considering N ⊗_{R} C as left C^{∗}-module in the canonical way we have

α_{N}(g⇀(n ⊗ c))(f) = ∑ n f(c\underline{1})g(c\underline{2}) = n f ∗ g(c) = [g · α_{N}(n ⊗ c)](f),

for all f, g ∈ C^{∗}, n ∈ N, and c ∈ C. So α_{N} is C^{∗}-linear, and for any right C-comodule L there is a commutative diagram

[figure: _{C^{∗}}Hom(L, N ⊗_{R} C); Hom(L,α_{N}); ≃; _{C^{∗}}Hom(L, Hom_{R}(C^{∗}, N)); ≃; Hom_{R}(L, N); =; Hom_{R}(L, N).]

The first vertical isomorphism is obtained by assumption and the Hom-tensor relations 7.9, explicitly,

C^{∗}Hom(L, N ⊗_{R} C) = Hom^{C}(L, N ⊗_{R} C) ≃ Hom_{R}(L, N).

The second vertical isomorphism results from the canonical isomorphisms

C^{∗}Hom(L, Hom_{R}(C^{∗}, N)) ≃ Hom_{R}(C^{∗} ⊗_{C^{∗}} L, N) ≃ Hom_{R}(L, N).

This shows that Hom(L, α_{N}) is injective for any L ∈ M^{C}, and so (the corestriction of) α_{N} is a monomorphism in M^{C}. Since _{R}C is flat, this implies that α_{N} is injective (by 7.13).

(d) ⇒ (e) We show that, for right C-comodules M, any C^{∗}-submodule N is a subcomodule. For this consider the map

ρ_{N} : N → Hom_{R}(C^{∗}, N), n ↦ [f ↦ f ⇀n].

The inclusion i : N → M yields the commutative diagram with exact rows

[figure: 0; N; i; M; p; ϱ^{M}; M/N; 0; 0; N ⊗_{R}C; i⊗I; α_{N,C}; M ⊗_{R}C; p⊗I; α_{M,C}; M/N ⊗_{R}C; α_{M/N,C}; 0; 0; Hom_{R}(C^{∗},N); Hom(C^{∗},i); Hom_{R}(C^{∗},M); Hom_{R}(C^{∗},M/N),]

where Hom(C^{∗}, i) ∘ ρ_{N} = α_{M,C} ∘ ϱ^{M} ∘ i. Injectivity of α_{M/N,C} implies (p ⊗ I) ∘ ϱ^{M} ∘ i = 0, and by the kernel property ϱ^{M} ∘ i factors through N → N ⊗_{R} C, thus yielding a C-coaction on N.

(e) ⇒ (a) First we show that every finitely generated C^{∗}-module N ∈ σ[_{C^{∗}}C] is a C-comodule. There exist a C^{∗}-submodule X ⊂ C^{n}, n ∈ ℕ, and an epimorphism h : X → N. By assumption, X and the kernel of h are comodules and hence N is a comodule (see 7.5). So, for any L ∈ σ[_{C^{∗}}C], finitely generated submodules are comodules and this obviously implies that L is a comodule.

It remains to prove that, for M, N ∈ M^{C}, any C^{∗}-morphism f : M → N is a comodule morphism. Im f ⊂ N and Ke f ⊂ M are C^{∗}-submodules and hence – as just shown – are subcomodules of N and M, respectively. Therefore the corestriction M → Im f and the inclusion Im f → N both are comodule morphisms and so is f (as the composition of two comodule maps).

For the final assertions, recall that the inclusion σ[_{C^{∗}}C] → _{C^{∗}}M has a right adjoint functor (trace functor, see 3.1) and this respects products. So the isomorphism follows from the characterisation of the products of the M_{λ}⊗_{R}C in M^{C} (see 7.12). ⊔ ⊓ 8.4. Coaction and C^{∗}-modules. Let _{R}C be locally projective. For any R-module M, consider an R-linear map ϱ : M → M ⊗_{R} C. Define a left C^{∗}-action on M by

⇀ : C^{∗} ⊗_{R} M → M, f ⊗ m ↦ (I_{M} ⊗ f) ∘ ϱ(m).

Then the following are equivalent:

(a) ϱ is coassociative and counital;

(b) M is a unital C^{∗}-module by ⇀.

Proof. The implication (a) ⇒ (b) is shown in 8.1. Conversely, suppose that M is a unital C^{∗}-module by ⇀, that is,

(f ∗ g)⇀m = f ⇀(g⇀m), for all f, g ∈ C^{∗}, m ∈ M.

By the definition of the action ⇀, this means that

(I_{M} ⊗ f ⊗ g) ∘ (I_{M} ⊗ Δ) ∘ ϱ(m) = (I_{M} ⊗ f ⊗ g) ∘ (ϱ ⊗ I_{C}) ∘ ϱ(m),

and from this _{R}C locally projective implies (I_{M} ⊗ Δ) ∘ ϱ(m) = (ϱ ⊗ I_{C}) ∘ ϱ(m) (see 8.2), showing that ϱ is coassociative. Moreover, for any m ∈ M, m = ε⇀m = (I_{M} ⊗ ε) ∘ ϱ(m). ⊔ ⊓

8.5. Left C-comodules and right C^{∗}-modules.

(1) Any M ∈ ^{C}M is a (unital) right C^{∗}-module by

↼ : M ⊗_{R} C^{∗} → M, m ⊗ f ↦ (f ⊗ I_{M}) ∘^{M}ϱ(m) = ∑ f(m\underline{−1})m\underline{0}.

(2) Any morphism h : M → N in ^{C}M is a right C^{∗}-module morphism, so

CHom (M, N) ⊂ Hom_{C^{∗}}(M, N)

and there is a faithful functor ^{C}M → σ[C_{C^{∗}}] ⊂ M_{C^{∗}}.

(3) _{R}C is locally projective if and only if ^{C}M = σ[C_{C^{∗}}].

Since C is a left and right C-comodule by the regular coaction, we can study the structure of C as a (C^{∗}, C^{∗})-bimodule.

8.6. C as a (C^{∗}, C^{∗})-bimodule. C is a (C^{∗}, C^{∗})-bimodule by

\begin{matrix}⇀ : C^{∗} ⊗ C → C, & f ⊗ c ↦ f ⇀c = (I_{C} ⊗ f) ∘ Δ(c), \\ ↼ : C ⊗ C^{∗} → C, & c ⊗ g ↦ c↼g = (g ⊗ I_{C}) ∘ Δ(c) . \end{matrix}

(1) For any f, g ∈ C^{∗}, c ∈ C,

f ∗ g (c) = f(g⇀c) = g(c↼f).

(2) C is faithful as a left and right C^{∗}-module.

(3) Assume C to be cogenerated by R. Then for any central element f ∈ C^{∗} and any c ∈ C, f ⇀c = c↼f.

(4) If C satisfies the α-condition, it is a balanced (C^{∗}, C^{∗})-bimodule, that is,

C^{∗}End(C) = End^{C}(C) ≃ C^{∗} ≃ ^{C}End(C) = End_{C^{∗}}(C) and
C^{∗}End_{C^{∗}}(C) = ^{C}End^{C}(C) ≃ Z(C^{∗}),

where morphisms are written opposite to scalars and Z(C^{∗}) denotes the centre of C^{∗}. In this case a pure R-submodule D ⊂ C is a subcoalgebra if and only if D is a left and right C^{∗}-submodule.

Proof. The bimodule property is shown by the equalities

\begin{aligned}(f ⇀c)↼g & = (g ⊗ I_{C} ⊗ f) ∘ ((Δ ⊗ I_{C}) ∘ Δ))(c) \\ & = (g ⊗ I_{C} ⊗ f) ∘ ((I_{C} ⊗ Δ) ∘ Δ))(c) & = f ⇀(c↼g). \end{aligned}

(1) From the definition it follows

\begin{aligned}f ∗ g (c) = (f ⊗ g) ∘ Δ(c) & = (f ⊗ I_{C}) ∘ (I_{C} ⊗ g) ∘ Δ(c) = f(g⇀c) \\ & = (I_{C} ⊗ g) ∘ (f ⊗ I_{C}) ∘ Δ(c) = g(c↼f). \end{aligned}

(2) For f ∈ C^{∗}, assume f ⇀c = 0 for each c ∈ C. Then applying (1) yields f(c) = ε(f ⇀c) = 0, and hence f = 0.

(3) For any central element f ∈ C^{∗}, by (1),

g(c↼f) = f ∗ g(c) = g ∗ f(c) = g(f ⇀c),

for all c ∈ C, g ∈ C^{∗}. Since C is cogenerated by R, this can only hold if, for all c ∈ C, c↼f = f ⇀c.

(4) The isomorphisms follow from 7.11, 8.3 and 8.5. Let D ⊂ C be a pure R-submodule. If D is a subcoalgebra of C, then it is a right and left subcomodule and hence a left and right C^{∗}-submodule. Conversely, suppose that D is a left and right C^{∗}-submodule. Then the restriction of Δ yields a left and right C-coaction on D and, by 1.21,

Δ(D) ⊂ D ⊗_{R} C ∩ C ⊗ D = D ⊗_{R} D ,

proving that D is a subcoalgebra. ⊔ ⊓ 8.7. When is M^{C} = _{C^{∗}}M? The following are equivalent:

(a) M^{C} = _{C^{∗}}M;

(b) the functor − ⊗_{R} C : M_{R} → _{C^{∗}}M has a left adjoint;

(c) _{R}C is finitely generated and projective;

(d) _{R}C is locally projective and C is finitely generated as right C^{∗}-module;

(e) ^{C}M = M_{C^{∗}}.

Proof. (a) ⇒ (b) is obvious (by 7.12(3)).

(b) ⇒ (c) Since − ⊗_{R} C is a right adjoint, it preserves monomorphisms (injective morphisms). Therefore, _{R}C is flat. Moreover − ⊗_{R} C preserves products, so for any family {M_{λ}}_{Λ} in M_{R} there is an isomorphism

(∏_{Λ}M_{λ}) ⊗_{R} C ≃ ∏_{Λ}(M_{λ} ⊗_{R} C),

which implies that _{R}C is finitely presented and hence projective.

(c) ⇒ (d) Clearly, projective modules are locally projective, and C finitely generated as an R-module implies that C is finitely generated as a right (and left) C^{∗}-module.

(d) ⇒ (a) By 8.6, C is a faithful left C^{∗}-module that is finitely generated as a module over its endomorphism ring C^{∗}. This implies that C is a subgenerator in _{C^{∗}}M, that is, M^{C} = σ[_{C^{∗}}C] = _{C^{∗}}M (see 3.6). ⊔ ⊓

The comodules of the coalgebra associated to any finitely generated projective R-module are of fundamental importance.

8.8. Projective modules as comodules. Let P be a finitely generated projective R-module with dual basis p_{1}, …, p_{n} ∈ P and π_{1}, …, π_{n} ∈ P^{∗}. Then P is a right P^{∗} ⊗_{R} P -comodule with the coaction

ϱ^{P} : P → P ⊗_{R} (P^{∗} ⊗_{R} P), p ↦ ∑_{i} p_{i} ⊗ π_{i} ⊗ p.

P is a subgenerator in M^{P^{∗}⊗_{R}P}, and there is a category isomorphism

M^{P^{∗}⊗_{R}P} ≃ M_{End_{R}(P)}.

The dual P^{∗} is a left P^{∗} ⊗_{R} P -comodule with the coaction

Pϱ : P → (P^{∗} ⊗_{R} P) ⊗_{R} P, f ↦ ∑_{i} f ⊗ p_{i} ⊗ π_{i}.

Proof. Coassociativity of ϱ^{P} follows from the equality

(I ⊗ Δ)ϱ^{P} (f ⊗ p) = ∑_{i,j} f ⊗ p_{i} ⊗ π_{i} ⊗ p_{j} ⊗ π_{j} ⊗ p = (ϱ^{P} ⊗ I)ϱ^{P} (f ⊗ p).

By properties of the dual basis, (I_{P} ⊗ ε)ϱ^{P} (p) = ∑_{i} p_{i}π_{i}(p) = p, so that P is indeed a right comodule over P^{∗} ⊗_{R} P. There exists a surjective R-linear map R^{n} → P^{∗} that yields an epimorphism P^{n} ≃ R^{n} ⊗ P → P^{∗} ⊗_{R} P in M^{P^{∗}⊗_{R}P}. So P generates P^{∗} ⊗_{R} P as a right comodule and hence is a subgenerator in M^{P^{∗}⊗_{R}P}. Since P^{∗} ⊗_{R} P is finitely generated and projective as an R-module, the category isomorphism follows by 8.7.

A simple computation shows that P^{∗} is a left comodule over P^{∗} ⊗_{R} P. ⊔ ⊓ As a special case, for any n ∈ ℕ, R^{n} may be considered as a right comodule

over the matrix coalgebra M_{n}^{c}(R) (cf. 5.10).

For an algebra A, any two elements a, b ∈ A define a subalgebra aAb ⊂ A, and for an idempotent e ∈ A, eAe is a subalgebra with a unit. Dually, one considers

8.9. Factor coalgebras. Let f, g, e ∈ C^{∗} with e ∗ e = e. Then:

(1) f ⇀C↼g is a coalgebra (without a counit) and there is a coalgebra mor-

phism

C → f ⇀C↼g, c ↦ f ⇀c↼g.

(2) e⇀C↼e is a coalgebra with counit e and coproduct

e⇀c↼e ↦ ∑ e⇀c\underline{1}↼e ⊗ e⇀c\underline{2}↼e.

The kernel of C → e⇀C↼e is equal to (ε − e)⇀C + C↼(ε − e).

(3) If C is R-cogenerated, and e is a central idempotent, then e⇀C is a subcoalgebra of C.

Proof. (1) For any f, g ∈ C^{∗} consider the left, respectively right, comodule maps L_{f} : C → C, c ↦ f ⇀c, and R_{g} : C → C, c ↦ c↼g. Construct the commutative diagram

\xymatrix{C \ar[r]^{L_{f}} \ar[d]_{Δ} & C \ar[r]^{R_{g}} \ar[d]^{Δ} & C \ar[d]^{Δ} \\ C ⊗_{R} C \ar[r]^{I_{C}⊗L_{f}} & C ⊗_{R} C \ar[r]^{R_{g}⊗I_{C}} & C ⊗_{R} C,}

which leads to the identity Δ∘R_{g} ∘L_{f} = (R_{g} ⊗L_{f})∘Δ. Putting δ := L_{f} ∘R_{g} = R_{g} ∘ L_{f}, we obtain the commutative diagram

\xymatrix{C \ar[r]^{Δ} \ar[d]_{δ} & C ⊗_{R} C \ar[d]^{R_{g}⊗L_{f}} \\ δ(C) \ar[r]^{Δ} & R_{g}(C) ⊗_{R} L_{f}(C) \ar[r]^{L_{f} ⊗R_{g}} & δ(C) ⊗_{R} δ(C).}

Thus Δ_{δ} = (L_{f} ⊗ R_{g}) ∘ Δ makes δ(C) a coalgebra. It is easily verified that

\xymatrix{C \ar[r]^{Δ} \ar[d]_{δ} & C ⊗_{R} C \ar[d]^{δ⊗δ} \\ δ(C) \ar[r]^{Δ_{δ}} & δ(C) ⊗_{R} δ(C)}

is a commutative diagram, and hence δ is a coalgebra morphism.

(2) The form of the coproduct follows from (1). For c ∈ C, ε(e⇀c↼e) = ε ∗ e(c↼e) = e(c) showing that e is the counit of e⇀C↼e.

For x ∈ C, e⇀x↼e = 0 implies x↼e = (ε − e)⇀(x↼e) ∈ (ε − e)⇀C, and so

x = x↼e + x↼(ε − e) ∈ (ε − e)⇀C + C↼(ε − e).

This proves the stated form of the kernel.

(3) By 8.6, for a central idempotent e and c ∈ C, e⇀c↼e = e⇀c. Putting f = g = e in (the proof of) (1) we obtain Δ_{e}(e⇀C) ⊂ e⇀C ⊗ e⇀C. ⊔ ⊓

8.10. Idempotents and comodules. Let e ∈ C^{∗} be an idempotent and consider the coalgebra e⇀C↼e (as in 8.9).

(1) For any M ∈ M^{C}, e⇀M is a right e⇀C↼e-comodule with the coaction

e⇀M → e⇀M ⊗_{R} e⇀C↼e, e⇀m ↦ ∑ e⇀m\underline{0} ⊗ e⇀m\underline{1}↼e.

(2) For any f : M → N ∈ M^{C}, f(e⇀M) = e⇀f(M), and so there is a covariant functor

e⇀− : M^{C} → M^{e⇀C↼e}, M ↦ e⇀M.

(3) For any M ∈ M^{C}, M^{∗} is a right C^{∗}-module canonically and

Hom_{R}(e⇀M, R) = (e⇀M)^{∗} ≃ M^{∗} · e.

(4) The map −↼e : e⇀C → e⇀C↼e is a surjective right e⇀C↼e-comodule morphism, and so e⇀C is a subgenerator in M^{e⇀C↼e}.

(5) (e⇀C↼e)^{∗} ≃ e∗C^{∗} ∗e, and hence there is a faithful functor M^{e⇀C↼e} → e∗C^{∗}∗eM.

(6) If _{R}C is locally projective, then e⇀C↼e is a locally projective R-module

and

M^{e⇀C↼e} = σ[_{e∗C^{∗}∗e} e⇀C] = σ[_{e∗C^{∗}∗e} e⇀C↼e].

Proof. (1), (3) and (4) are easily verified.

(2) By 8.1, right comodule morphisms are left C^{∗}-morphisms.

(5) The isomorphism in (3) holds similarly for the right action of e on C and from this the isomorphism in (5) follows.

(6) Clearly direct summands of locally projectives are locally projective, and hence the assertion follows from (3) and 8.3. ⊔ ⊓ 8.11. Finiteness Theorem. Assume _{R}C to be locally projective.

(1) Let M ∈ M^{C}. Every finite subset of M is contained in a subcomodule of M that is finitely generated as an R-module.

(2) Any finite subset of C is contained in a (C^{∗}, C^{∗})-sub-bimodule that is finitely generated as an R-module.

(3) Minimal C^{∗}-submodules and minimal (C^{∗}, C^{∗})-sub-bimodules of C are finitely generated as R-modules.

Proof. (1) Since any sum of subcomodules is again a subcomodule, it is enough to show that each m ∈ M lies in a subcomodule that is finitely generated as an R-module. Moreover, by the correspondence of subcomodules and C^{∗}-submodules, this amounts to proving that the submodule C^{∗}⇀m is finitely generated as an R-module. Writing ϱ^{M}(m) = ∑_{i=1}^{k} m_{i} ⊗ c_{i}, where m_{i} ∈ C^{∗}⇀m, c_{i} ∈ C, we compute for every f ∈ C^{∗}

f ⇀m = (I_{M} ⊗ f) ∘ ϱ^{M}(m) = ∑_{i=1}^{k}m_{i} f(c_{i}) .

Hence C^{∗}⇀m is finitely generated by m_{1}, …, m_{k} as an R-module.

(2) It is enough to prove the assertion for single elements c ∈ C. By (1), C^{∗}⇀c is generated as an R-module by some c_{1}, …, c_{k} ∈ C. By symmetry, each c_{i}↼C^{∗} is a finitely generated R-module. Hence C^{∗}⇀c↼C^{∗} is a finitely generated R-module.

(3) This is an obvious consequence of (1) and (2). ⊔ ⊓

A right C-comodule N is called semisimple (in M^{C}) if every C-monomor-phism U → N is a coretraction, and N is called simple if all these monomorphisms are isomorphisms. Semisimplicity of N is equivalent to the fact that every right C-comodule is N-injective. (Semi)simple left comodules and bicomodules are defined similarly.

The coalgebra C is said to be left (right) semisimple if it is semisimple as a left (right) comodule. C is called a simple coalgebra if it is simple as a (C, C)-bicomodule.

8.12. Semisimple comodules. Assume that _{R}C is flat.

(1) Any N ∈ M^{C} is simple if and only if N has no nontrivial subcomodules.

(2) For N ∈ M^{C} the following are equivalent:

(a) N is semisimple (as defined above);

(b) every subcomodule of N is a direct summand;

(c) N is a sum of simple subcomodules;

(d) N is a direct sum of simple subcomodules.

Proof. (1) By 7.13, any monomorphism U → N is injective, and hence it can be identified with a subcomodule. From this the assertion is clear.

(2) _{R}C flat implies that the intersection of any two subcomodules is again a subcomodule. Hence in this case the proof for modules (e.g., [10, 20.2]) can be transferred to comodules. ⊔ ⊓

8.13. Right semisimple coalgebras. For C the following are equivalent:

(a) C is a semisimple right C-comodule;

(b) _{R}C is flat and every right subcomodule of C is a direct summand;

(c) _{R}C is flat and C is a direct sum of simple right comodules;

(d) _{R}C is flat and every comodule in M^{C} is semisimple;

(e) _{R}C is flat and every short exact sequence in M^{C} splits;

(f) _{R}C is projective and C is a semisimple left C^{∗}-module;

(g) every comodule in M^{C} is (C-)injective;

(h) every comodule in M^{C} is projective;

(i) C is a direct sum of simple coalgebras that are right (left) semisimple;

(j) C is a semisimple left C-comodule.

Proof. (a) ⇒ (b) ⇒ (c) ⇒ (d) ⇒ (e) Assume every monomorphism i : U → C to be a coretraction. Then i is in particular an injective map, and hence, by 7.13, _{R}C is flat. Now the assertions follow by 8.12.

The implications (e) ⇒ (g) and (e) ⇒ (h) are obvious.

(h) ⇒ (f) By 7.18, any projective comodule is projective as an R-module. In particular, C is a projective R-module, and hence M^{C} = σ[_{C^{∗}}C] and all modules in σ[_{C^{∗}}C] are projective. This characterises C as a semisimple C^{∗}- module (see 3.7).

The implication (f) ⇒ (a) is obvious since M^{C} = σ[_{C^{∗}}C].

(g) ⇒ (a) This is obvious. Notice that, in view of (f), _{R}C is projective, and hence the C-injectivity of any comodule N implies that N is injective in M^{C} = σ[_{C^{∗}}C].

(f) ⇒ (i) Let C be a left semisimple C^{∗}-module. Let {E_{i}}_{I} be a minimal representative set of simple C^{∗}-submodules of C. Form the traces D_{i} := Tr_{C^{∗}}(E_{i}, C). By the structure theorem for semisimple modules (see 3.7),

C ≃ ⨁_{I}D_{i},

where the D_{i} are minimal fully invariant C^{∗}-submodules. Considering C^{∗} as an endomorphism ring acting from the right, this means that the D_{i} are minimal (C^{∗}, C^{∗})-submodules. By 8.6, each D_{i} is a minimal subcoalgebra of C and every subcoalgebra of D_{i} is a subcoalgebra of C. So every D_{i} is a right semisimple simple coalgebra.

(i) ⇒ (f) It follows from the proof (a) ⇒ (f) that all simple comodules of C are projective as R-modules and hence _{R}C is also projective. Now the assertion follows.

(f) ⇔ (j) By 3.7, the semisimple module _{C^{∗}}C is semisimple over its endomorphism ring, that is, C_{C^{∗}} is also semisimple. Since ^{C}M = σ[C_{C^{∗}}], the assertion follows from the preceding proof by symmetry. ⊔ ⊓

8.14. Simple coalgebras. For C the following are equivalent:

(a) C is a simple coalgebra that is right (left) semisimple;

(b) _{R}C is projective and C is a simple (C^{∗}, C^{∗})-bimodule containing a minimal left (right) C^{∗}-submodule;

(c) C is a simple coalgebra and a finite-dimensional vector space over R/m, for some maximal ideal m ⊂ R.

Proof. (a) ⇒ (b) We know from 8.13 that _{R}C is projective. Clearly a simple right subcomodule is a simple left C^{∗}-submodule. Let D ⊂ C be a (C^{∗}, C^{∗})-sub-bimodule. Then it is a direct summand as a left C^{∗}-module, and hence it is a subcoalgebra of C (by 8.6) and so D = C.

(b) ⇒ (c) Let D ⊂ C be a minimal left C^{∗}-submodule. For any maximal ideal m ⊂ R, mD ⊂ D is a C^{∗}-submodule and hence mD = 0 or mD = D. Since D is finitely generated as an R-module (by 8.11), mD = 0 for some maximal m ⊂ R. Moreover, mC = mD↼C^{∗} = 0, and so C is a finitedimensional R/m-algebra.

(c) ⇒ (a) is obvious. Notice that in this case M^{C} = _{C^{∗}}M (see 8.7). ⊔ ⊓

The Finiteness Theorem 8.11 and the Hom-tensor relations 7.9 indicate that properties of R have a strong influence on properties of C-comodules.

8.15. Coalgebras over special rings. Let _{R}C be locally projective.

(1) If R is Noetherian, then C is locally Noetherian as a right and left comodule, and in M^{C} and ^{C}M direct sums of injectives are injective.

(2) If R is perfect, then in M^{C} and ^{C}M any comodule satisfies the descending chain condition on finitely generated subcomodules.

(3) If R is Artinian, then in M^{C} and ^{C}M every finitely generated comodule has finite length.

Proof. All these assertions are special cases of 3.18. ⊔ ⊓

Notice that, over Artinian (perfect) rings R, _{R}C is locally projective if and only if _{R}C is projective (any flat R-module is projective).

9 The rational functor

We know from Section 8 that there is a faithful functor from the category of right C-comodules to the category of left C^{∗}-modules. Now we want to study an opposite problem provided the α-condition is satisfied: Suppose M is a left C^{∗}-module; does there exist a (maximal) part of M on which a right C-coaction can be defined?

Throughout we _{R}C to be locally projective.

9.1. Rational functor. For any left C^{∗}-module M, define the rational submodule

Rat^{C}(M) = T^{C}(M) = ∑ {Im f | f ∈ _{C^{∗}}Hom(U, M), U ∈ M^{C}},

where T^{C} is the trace functor _{C^{∗}}M → σ[C] (cf. 3.1). Clearly Rat^{C}(M) is the largest submodule of M that is subgenerated by C, and hence it is a right C-comodule. The induced functor (subfunctor of the identity)

Rat^{C} : _{C^{∗}}M → M^{C}, M ↦ Rat^{C}(M),

is called the rational functor. Since Rat^{C} is a trace functor, it is right adjoint to the inclusion M^{C} → _{C^{∗}}M (see 3.1) and its properties depend on (torsiontheoretic) properties of the class M^{C} in _{C^{∗}}M. Of course Rat^{C}(M) = M for M ∈ _{C^{∗}}M if and only if M ∈ M^{C}, and M^{C} = _{C^{∗}}M if and only if _{R}C is finitely generated and projective (see 8.7).

9.2. Rational elements. Let M be a left C^{∗}-module. An element k ∈ M is said to be rational if there exists an element ∑_{i} m_{i} ⊗ c_{i} ∈ M ⊗_{R} C, such that

fk = ∑_{i}m_{i}f(c_{i}), for all f ∈ C^{∗}.

This means that, from the diagram

\xymatrix{ & M \ar[d]^{ψ_{M}} &  &  & m \ar@{|->}[d] \\ M ⊗_{R} C \ar[r]^{α_{M}} & Hom_{R}(C^{∗}, M) & m ⊗ c \ar@{|->}[r] & [f ↦mf(c)], & [f ↦fm] ,}

we obtain ψ_{M}(k) = α_{M}(∑_{i} m_{i} ⊗ c_{i}) (see 8.2). Since it is assumed that α_{M} is injective, the element ∑_{i}m_{i} ⊗ c_{i} is uniquely determined.

9.3. Rational submodule. Let M be a left C^{∗}-module.

(1) An element k ∈ M is rational if and only if C^{∗}k is a right C-comodule with fk = f ⇀k, for all f ∈ C^{∗}.

(2) Rat^{C}(M) = {k ∈ M | k is rational}.

Proof. (1) Let k ∈ M be rational and ∑_{i} m_{i} ⊗ c_{i} ∈ M ⊗_{R} C such that fk = ∑_{i} m_{i}f(c_{i}) for all f ∈ C^{∗}. Put K := C^{∗}k and define a map

ϱ : K → M ⊗_{R} C, fk ↦ ∑_{i}m_{i} ⊗ f ⇀c_{i}.

For f, h ∈ C^{∗},

α_{M}(∑_{i}m_{i} ⊗ f ⇀c_{i})(h) = ∑_{i}m_{i}h(f ⇀c_{i}) = h ∗ f k = h · fk .

So the map ϱ is well defined since fk = 0 implies α_{M}(∑_{i} m_{i} ⊗ f ⇀c_{i}) = 0, and hence ∑_{i} m_{i} ⊗ f ⇀c_{i} = 0 by injectivity of α_{M}. Moreover, it implies that α_{M} ∘ ϱ(K) ⊂ Hom_{R}(C^{∗}, K), and we obtain the commutative diagram with exact rows

\xymatrix{ &  & K \ar[d]^{ϱ} \\ 0 \ar[r] & K ⊗_{R} C \ar[r] \ar[d]^{α_{K}} & M ⊗_{R} C \ar[r] \ar[d]^{α_{M}} & M/K ⊗_{R} C \ar[r] \ar[d]^{α_{M/K}} & 0 \\ 0 \ar[r] & Hom_{R}(C^{∗}, K) \ar[r] & Hom_{R}(C^{∗}, M) \ar[r] & Hom_{R}(C^{∗}, M/K) \ar[r] & 0,}

where all the α are injective. By the kernel property we conclude that ϱ factors through some ϱ^{K} : K → K ⊗_{R} C, and it follows by 8.4 that ϱ^{K} is coassociative and counital, thus making K a comodule. ⊔ ⊓

As a first application we consider the rational submodule of C^{∗∗}. The canonical map Φ_{C} : C → C^{∗∗} is a C^{∗}-morphism, since, for all c ∈ C, f, h ∈ C^{∗},

Φ_{C}(f ⇀c)(h) = h(f ⇀c) = ∑ h(c\underline{1})f(c\underline{2}) = Φ_{C}(c)(h ∗ f) = fΦ_{C}(c)(h).

Hence the image of Φ_{C} is a rational module. The next lemma shows that this is equal to the rational submodule of C^{∗∗}.

9.4. Rational submodule of C^{∗∗}. Φ_{C} : C → Rat^{C}(C^{∗∗}) is an isomorphism.

Proof. Local projectivity of _{R}C implies that Φ_{C} is injective. Let ϱ : Rat^{C}(C^{∗∗}) → Rat^{C}(C^{∗∗}) ⊗_{R} C denote the comodule structure map. For γ ∈ Rat^{C}(C^{∗∗}) write ϱ(γ) = ∑_{i}γ_{i} ⊗ c_{i}. Then, for any f ∈ C^{∗},

γ(f) = f · γ(ε) = ∑_{i}f(c_{i})γ_{i}(ε) = f(∑_{i}γ_{i}(ε)c_{i}),

where ∑_{i} γ_{i}(ε)c_{i} ∈ C. So γ ∈ Im Φ_{C}, proving that Φ_{C} is surjective. ⊔ ⊓

The rational submodule of _{C^{∗}}C^{∗} is a two-sided ideal in C^{∗} and is called the left trace ideal. From the above observations and the Finiteness Theorem it is clear that Rat^{C}(C^{∗}) = C^{∗} if and only if _{R}C is finitely generated.

Right rational C^{∗}-modules are defined in a symmetric way, yielding the right trace ideal ^{C}Rat(C^{∗}), which in general is different from Rat^{C}(C^{∗}).

9.5. Characterisation of the trace ideal. Let T = Rat^{C}(C^{∗}) be the left trace ideal.

(1) Let f ∈ C^{∗} and assume that f ⇀C is a finitely presented R-module. Then f ∈ T.

(2) If R is Noetherian, then T can be described as

T_{1} = {f ∈ C^{∗} | C^{∗} ∗ f is a finitely generated R-module};
T_{2} = {f ∈ C^{∗} | Ke f contains a right C^{∗}-submodule K, such that

C/K is a finitely generated R-module};

T_{3} = {f ∈ C^{∗} | f ⇀C is a finitely generated R-module }.

Proof. Assertion (1) and the inclusion T ⊂ T_{1} in (2) follow from the Finiteness Theorem 8.11.

[ T_{1} ⊂ T_{2} ]: For f ∈ T_{1}, let C^{∗}∗f be finitely R-generated by g_{1}, …, g_{k} ∈ C^{∗}. Consider the kernel of C^{∗} ∗ f,

K := ⋂ {Ke h | h ∈ C^{∗} ∗ f} = ⋂_{i=1}^{k} Ke g_{i} .

Clearly K is a right C^{∗}-submodule of C. Moreover, all the C/Ke g_{i} are finitely generated R-modules, and hence

C/K ⊂ ⨁_{i=1}^{k} C/Ke g_{i}

is a finitely generated R-module. This proves the inclusion T_{1} ⊂ T_{2}.

[ T_{2} ⊂ T_{3} ]: Let f ∈ T_{2}. Since Δ(K) ⊂ C ⊗_{R} K, f ⇀K = 0 and f ⇀C = f ⇀C/K is a finitely generated R-module, that is, f ∈ T_{3}.

[ T_{3} ⊂ T ]: For f ∈ T_{3}, the rational right C^{∗}-module f ⇀C is a finitely presented R-module. Then, by 7.10, (f ⇀C)^{∗} is a rational left C^{∗}-module. Since ε(f ⇀c) = f(c) for all c ∈ C, we conclude f ∈ (f ⇀C)^{∗} and hence f ∈ T.

⊔ ⊓

Before concentrating on properties of the trace ideal we consider density for any subalgebras of C^{∗}. From the Density Theorem we know that for any C-dense subalgebra T ⊂ C^{∗} the categories M^{C} and σ[_{T} C] can be identified.

9.6. Density in C^{∗}. For an R-submodule U ⊂ C^{∗}, the following assertions are equivalent:

(a) U is dense in C^{∗} in the finite topology (of R^{C});

(b) U is a C-dense subset of C^{∗} (in the finite topology of End_{R}(C)). If C is cogenerated by R, then (a), (b) imply:

(c) Ke U = {x ∈ C | u(x) = 0 for all u ∈ U} = 0. If R is a cogenerator in M_{R}, then (c) ⇒ (b).

Proof. (a) ⇔ (b) It can be derived from 7.11 that the finite topologies in C^{∗} and End^{C}(C) can be identified.

(a) ⇒ (c) Let C be cogenerated by R. Then, for any 0 ≠ x ∈ C, there exists f ∈ C^{∗} such that f(x) ≠ 0. Then, for some u ∈ U, u(x) = f(x) ≠ 0, that is, x ∉ Ke U, and hence Ke U = 0.

(c) ⇒ (b) Let R be a cogenerator in M_{R}. Let f ∈ C^{∗} and x_{1}, …, x_{n} ∈ C. Suppose that

f ⇀(x_{1}, …, x_{n}) ∉ U ⇀(x_{1}, …, x_{n}) ⊂ C^{n}.

Then there exists an R-linear map g : C^{n} → R such that

g(f ⇀(x_{1}, …, x_{n})) ≠ 0 and g(U ⇀(x_{1}, …, x_{n})) = 0.

For each u ∈ U (by 8.6),

0 = ∑_{i}g_{i}(u⇀x_{i}) = ∑_{i}u(x_{i}↼g_{i}) = u(∑_{i}x_{i}↼g_{i}),

where g_{i} : C → C^{n} ⟶^{g} R, and this implies ∑_{i} x_{i}↼g_{i} = 0 and

0 ≠ g(f ⇀(x_{1}, …, x_{n})) = ∑_{i}g_{i}(f ⇀x_{i}) = ∑_{i}f(x_{i}↼g_{i}) = f(∑_{i}x_{i}↼g_{i}) = 0,

contradicting the choice of g. ⊔ ⊓

9.7. Dense subalgebras of C^{∗}. For a subalgebra T ⊂ C^{∗} the following are equivalent:

(a) _{R}C is locally projective and T is dense in C^{∗};

(b) M^{C} = σ[_{T} C]. If T is an ideal in C^{∗}, then (a),(b) are equivalent to:

(c) C is an s-unital T -module and C satisfies the α-condition.

Proof. (a) ⇔ (b) There are embeddings M^{C} ⊂ σ[_{C^{∗}}C] ⊂ σ[_{T} C]. Now M^{C} = σ[_{C^{∗}}C] is equivalent to the α-condition while σ[_{T} C] = σ[_{C^{∗}}C] corresponds to the density property.

(a) ⇔ (c) For an ideal T the density property is equivalent to s-unitality of the T -module C. ⊔ ⊓

Combining the properties of the trace functor observed in 4.8 with the characterisation of dense ideals in 4.3, we obtain:

9.8. The rational functor exact. Let T = Rat^{C}(C^{∗}). The following statements are equivalent:

(a) the functor Rat^{C} : _{C^{∗}}M → M^{C} is exact;

(b) M^{C} is closed under extensions in _{C^{∗}}M and the class

{X ∈ _{C^{∗}}M | Rat^{C}(X) = 0}

is closed under factor modules;

(c) for every N ∈ M^{C} (with N ⊂ C), T N = N;

(d) for every N ∈ M^{C}, the canonical map T ⊗_{C^{∗}}N → N is an isomorphism;

(e) C is an s-unital T -module;

(f) T^{2} = T and T is a generator in M^{C};

(g) T C = C and C^{∗}/T is flat as a right C^{∗}-module;

(h) T is a left C-dense subring of C^{∗}.

9.9. Corollary. Assume that Rat^{C} is exact and let T = Rat^{C}(C^{∗}) ⊂ C^{∗}.

(1) M^{C} is closed under small epimorphisms in _{C^{∗}}M.

(2) If P is finitely presented in M^{C}, then P is finitely presented in _{C^{∗}}M.

(3) If P is projective in M^{C}, then P is projective in _{C^{∗}}M.

(4) For any M ∈ M^{C}, the canonical map _{C^{∗}}Hom(C^{∗}, M) → _{C^{∗}}Hom(T, M) is injective.

Proof. (1)–(3) follow from Corollary 4.9.

(4) By density, for every f ∈ _{C^{∗}}Hom(C^{∗}, M), f(ε) = tf(ε) = f(t) for some t ∈ T, and hence f(T) = 0 implies f(C^{∗}) = 0. ⊔ ⊓

Notice that the exactness of Rat^{C}, that is, the density of Rat^{C}(C^{∗}) in C^{∗}, also has some influence on left C-comodules.

9.10. Corollary. Assume that Rat^{C} is exact and let T = Rat^{C}(C^{∗}) ⊂ C^{∗}.

(1) For any N ∈ ^{C}M, the canonical map Hom_{C^{∗}}(C^{∗}, N) → Hom_{C^{∗}}(T, N) is injective.

(2) ^{C}Rat(C^{∗}) ⊂ T and equality holds if and only if T ∈ ^{C}M.

Proof. (1) By the preceding remark, C is also s-unital as a right T - module and hence the proof of Corollary 9.9(4) applies.

(2) By the density of T ⊂ C^{∗}, X↼T = X, for each X ∈ ^{C}M (see 4.3). This implies

Hom_{C^{∗}}(X, C^{∗}) = Hom_{C^{∗}}(X↼T, C^{∗}) = Hom_{C^{∗}}(X, C^{∗} ∗ T) = Hom_{C^{∗}}(X, T);

hence ^{C}Rat(C^{∗}) ⊂ T and ^{C}Rat(C^{∗}) = T provided T ∈ ^{C}M. ⊔ ⊓ The assertion in 4.10 yields here:

9.11. Corollary. Suppose that M^{C} has a generator that is (locally) projective in _{C^{∗}}M. Then Rat^{C} : _{C^{∗}}M → M^{C} is an exact functor.

Except when _{R}C is finitely generated (i.e., Rat^{C}(C^{∗}) = C^{∗}) the trace ideal does not contain a unit element. However, if C is a direct sum of finitely generated left (and right) C^{∗}-submodules, the trace ideal has particularly nice properties.

9.12. Trace ideal and decompositions. Let T := Rat^{C}(C^{∗}) and T′ := CRat(C^{∗}).

(1) If C is a direct sum of finitely generated right C^{∗}-modules, then T is C-dense in C^{∗} and there is an embedding

γ : T′ → ⨁_{Λ}T′ ∗ e_{λ} ⊂ T,

for a family of orthogonal idempotents {e_{λ}}_{Λ} in T.

(2) If C is a direct sum of finitely generated right C^{∗}-modules and of finitely generated left C^{∗}-modules, then T = T′ and T is a projective generator both in M^{C} and ^{C}M.

Proof. (1) Under the given conditions there exist orthogonal idempotents {e_{λ}}_{Λ} in C^{∗} with C = ⨁_{Λ} e_{λ}⇀C, where all e_{λ}⇀C are finitely generated right C^{∗}-modules. By the Finiteness Theorem 8.11, the e_{λ}⇀C are finitely generated as R-modules, and they are R-projective as direct summands of C. Now it follows from 9.5(1) that e_{λ} ∈ T. Clearly C is an s-unital left T -module and hence the density property follows (see 4.3).

Consider the assignment γ : T′ → ⨁_{Λ} T′ ∗ e_{λ}, t ↦ ∑_{Λ} t ∗ e_{λ}. For any t ∈ T′, t ∗ C^{∗} is finitely R-generated and so t ∗ e_{λ} = 0 for almost all λ ∈ Λ. Hence γ is a well-defined map. Assume γ(t) = 0. Then, for any c ∈ C, 0 = t ∗ e_{λ}(c) = t(e_{λ}⇀c), for all λ ∈ Λ, implying t = 0.

(2) By symmetry, (1) implies T = T′ and so T = ⨁_{Λ} T ∗ e_{λ} and T = ⨁_{Ω} f_{ω} ∗T, where the {f_{ω}}_{Ω} are orthogonal idempotents in C^{∗}, and the C↼f_{ω} are finitely R-generated (hence f_{ω} ∈ T′). Clearly each T ∗e_{λ} is a projective left T -module and f_{ω} ∗ T a projective right T -module. Now the density property implies that T is a projective generator both in M^{C} and in ^{C}M (see 9.8). ⊔ ⊓

Notice that, in 9.12, e_{λ} ∈ T′ need not imply that C↼e_{λ} is finitely Rgenerated, unless we know that R is Noetherian (see 9.5).

9.13. Decompositions over Noetherian rings. Let R be Noetherian, T = Rat^{C}(C^{∗}) and T′ = ^{C}Rat(C^{∗}). Then the following are equivalent:

(a) C_{C^{∗}} and _{C^{∗}}C are direct sums of finitely generated C^{∗}-modules;

(b) C_{C^{∗}} is a direct sum of finitely generated C^{∗}-modules and T = T′; (c) _{C^{∗}}C is a direct sum of finitely generated C^{∗}-modules and T = T′;

(d) C = T ⇀C and T = T′ and is a ring with enough idempotents. If these conditions hold, T is a projective generator both in M^{C} and in ^{C}M.

Proof. (a) ⇒ (b) follows by 9.12.

(b) ⇒ (d) Let C = ⨁_{Λ} e_{λ}⇀C, with orthogonal idempotents {e_{λ}}_{Λ} in C^{∗}, where all e_{λ}⇀C are finitely R-generated. Then e_{λ} ∈ T = T′ and T = ⨁_{Λ} T ∗ e_{λ}. For any t ∈ T, the module t⇀C is finitely R-generated (by 9.5) and so

t⇀C ⊂ e_{1}⇀C ⊕ ⋯ ⊕ e_{k}⇀C, for some idempotents e_{i} ∈ {e_{λ}}_{Λ}.

This implies t = (e_{1}+⋯+e_{k})∗t ∈ ⨁_{Λ} e_{λ}∗T. So ⨁_{Λ} T ∗e_{λ} = T = ⨁_{Λ} e_{λ}∗T, showing that T is a ring with enough idempotents.

(d) ⇒ (a) If T = ⨁_{Λ} e_{λ} ∗ T, then

C = T ⇀C = ⨁_{Λ}e_{λ}⇀C,

and e_{λ} ∈ T implies that e_{λ}⇀C is finitely R-generated. So, by 9.12, T is dense in C^{∗}, implying C↼T = C. Now symmetric arguments yield the decomposition of C as a direct sum of finitely R-generated left C^{∗}-modules.

(c) ⇔ (a) The statement is symmetric to (d) ⇔ (a).

If the conditions hold, the assertion follows by 9.12. ⊔ ⊓

Fully invariant submodules of C that are direct summands are precisely subcoalgebras that are direct summands, and they are of the form e⇀C, where e is a central idempotent in C^{∗}. Hence 9.13 yields:

9.14. Corollary. If R is Noetherian, the following are equivalent:

(a) C is a direct sum of finitely generated subcoalgebras;

(b) C is a direct sum of finitely generated (C^{∗}, C^{∗})-sub-bimodules;

(c) T ⇀C = C and T is a ring with enough central idempotents.

10 Structure of comodules

Throughout this section we assume that C is an R-coalgebra with _{R}C locally projective (see 8.2).

Let N be a right C-comodule. Then a C-comodule Q is said to be Ninjective provided Hom^{C}(−, Q) turns any monomorphism K → N in M^{C} into a surjective map. We recall characterisations from 3.3.

10.1. Injectives in M^{C}. (1) For Q ∈ M^{C} the following are equivalent:

(a) Q is injective in M^{C};

(b) the functor Hom^{C}(−, Q) : M^{C} → M_{R} is exact;

(c) Q is C-injective (as left C^{∗}-module);

(d) Q is N-injective for any (finitely generated) subcomodule N ⊂ C;

(e) every exact sequence 0 → Q → N → L → 0 in M^{C} splits.

(2) Every injective object in M^{C} is C-generated.

(3) Every object in M^{C} has an injective hull.

A C-comodule P is N-projective if Hom^{C}(P, −) turns any epimorphism N → L into a surjective map.

10.2. Projectives in M^{C}. (1) For P ∈ M^{C} the following are equivalent:

(a) P is projective in M^{C};

(b) the functor Hom^{C}(P, −) : M^{C} → M_{R} is exact;

(c) P is C^{(Λ)}-projective, for any set Λ;

(d) every exact sequence 0 → K → N → P → 0 in M^{C} splits.

(2) If P is finitely generated and C-projective, then P is projective in M^{C}.

Notice that projectives need not exist in M^{C}. As observed in 7.18, projective objects in M^{C} (if they exist) are also projective in M_{R}.

Over a Noetherian ring R, C is left and right locally Noetherian as a C^{∗}- module (by 8.15), and therefore we can apply 3.16 to obtain:

10.3. C as injective cogenerator in M^{C}. If R is Noetherian, then the following are equivalent:

(a) C is an injective cogenerator in M^{C};

(b) C is an injective cogenerator in ^{C}M;

(c) C is a cogenerator both in M^{C} and ^{C}M.

10.4. C as injective cogenerator in M_{C^{∗}}. If R is Artinian, then the following are equivalent:

(a) C is an injective cogenerator in M_{C^{∗}};

(b) _{C^{∗}}C is Artinian and an injective cogenerator in M^{C};

(c) C is an injective cogenerator in M^{C} and C^{∗} is right Noetherian.

If these conditions hold, then C^{∗} is a semiperfect ring and every right C^{∗}-module that is finitely generated as an R-module belongs to ^{C}M.

Proof. Since R is Artinian, C has locally finite length as a C^{∗}-module.

(a) ⇒ (b) Assume C to be an injective cogenerator in M_{C^{∗}}. Then, by 10.3, C is an injective cogenerator in M^{C}. Now 3.17 implies that _{C^{∗}}C is Artinian.

(b) ⇒ (a) and (b) ⇔ (c) follow again from 3.17.

Assume the conditions hold. C^{∗} is f-semiperfect, being the endomorphism ring of a self-injective module (see 3.15). So C^{∗}/Jac(C^{∗}) is von Neumann regular and right Noetherian, and hence right (and left) semisimple. This implies that C^{∗} is semiperfect.

Let L ∈ M_{C^{∗}} be finitely generated as an R-module. Then L is finitely cogenerated as a C^{∗}-module, and hence it is finitely cogenerated by C. This implies L ∈ ^{C}M. ⊔ ⊓

The decomposition of left semisimple coalgebras as a direct sum of (indecomposable) subcoalgebras (see 8.13) can be extended to more general situations. Recall that a relation on any family of (co)modules {M_{λ}}_{Λ} is defined by setting (cf. [3, 44.11])

M_{λ} ∼ M_{μ} if there exist nonzero morphisms M_{λ} → M_{μ} or M_{μ} → M_{λ},

and the smallest equivalence relation determined by ∼ is given by

M_{λ} ≈ M_{μ} if there exist λ_{1}, …, λ_{k} ∈ Λ,
such that M_{λ} = M_{λ_{1}} ∼ ⋯ ∼ M_{λ_{k}} = M_{μ} .

10.5. σ-decomposition of coalgebras. Let R be a Noetherian ring.

(1) There exist a σ-decomposition C = ⨁_{Λ} C_{λ} and a family of orthogonal central idempotents {e_{λ}}_{Λ} in C^{∗} with C_{λ} = C↼e_{λ}, for each λ ∈ Λ.

(2) Each C_{λ} is a subcoalgebra of C, C_{λ}^{∗} ≃ C^{∗} ∗ e_{λ}, σ[_{C^{∗}}C_{λ}] = σ[_{C_{λ}^{∗}}C_{λ}], and

M^{C} = ⨁_{Λ}σ[_{C^{∗}}C_{λ}] = ⨁_{Λ}M^{C_{λ}}.

(3) M^{C} is indecomposable if and only if, for any two injective uniform L, N ∈ M^{C}, L ≈ N holds.

(4) Assume that R is Artinian. Then M^{C} is indecomposable if and only if, for any two simple E_{1}, E_{2} ∈ σ_{C^{∗}}[C], Ê_{1} ≈ Ê_{2} holds.

Proof. (1),(2) By the Finiteness Theorem 8.11, C is a locally Noetherian C^{∗}-module. Now the decomposition of M^{C} (=σ[_{C^{∗}}C]) follows from module theory (see 44.14. in [3]).

Clearly the resulting σ-decomposition of C is a fully invariant decomposition, and hence it can be described by central idempotents in the endomorphism ring (= C^{∗}; see [3, 44.1]). Fully invariant submodules C_{λ} ⊂ C are in particular R-direct summands in C and hence are subcoalgebras (by 8.6). It is straightforward to verify that Hom_{R}(C_{λ}, R) = C_{λ}^{∗} ≃ C^{∗} ∗ e_{λ} is an algebra isomorphism. This implies σ[_{C^{∗}}C_{λ}] = σ[_{C_{λ}^{∗}}C_{λ}] = M^{C_{λ}}.

(3) is a special case of [3, 44.14].

(4) follows from [3, 44.14](3). Notice that Ê_{1} ≈ Ê_{2} can be described by extensions of simple modules (see [3, 44.11]). (The assertion means that the Ext quiver of simple modules in M^{C} is connected.) ⊔ ⊓

Transferring [3, 44.7] we obtain:

10.6. Corollary. Let C be a coalgebra with σ-decomposition C = ⨁_{Λ} C_{λ}. Then the left rational functor Rat^{C} is exact if and only if the left rational functors Rat^{C_{λ}} are exact, for each C_{λ}.

Even for coalgebras C over fields there need not be any projective comodules in M^{C}. We discuss the existence of (enough) projectives in M^{C} and the projectivity of C in M^{C} or in _{C^{∗}}M. Definition. A coalgebra C is called right semiperfect if every simple right comodule has a projective cover in M^{C}. If _{R}C is locally projective, this is obviously equivalent to the condition that every simple module in σ[_{C^{∗}}C] has a projective cover in σ[_{C^{∗}}C] (by 8.3), that is, M^{C} = σ[_{C^{∗}}C] is a semiperfect category.

Notice that a right semiperfect coalgebra C need not be a semiperfect left C^{∗}-module as defined in 3.10. The following characterisations can be shown.

10.7. Right semiperfect coalgebras. The following are equivalent:

(a) C is a right semiperfect coalgebra;

(b) M^{C} has a generating set of local projective modules;

(c) every finitely generated module in M^{C} has a projective cover. If R is a perfect ring, then (a)-(c) are equivalent to:

(d) M^{C} has a generating set of finitely generated C-projective comodules.

Proof. See 41.14., 41.16 and 41.22 in [3]. ⊔ ⊓

As an obvious application of 10.5 we obtain:

10.8. σ-decomposition of semiperfect coalgebras. Let R be Noetherian and C with σ-decomposition C = ⨁_{Λ} C_{λ}. Then C is a right semiperfect coalgebra if and only if the C_{λ} are right semiperfect coalgebras, for all λ ∈ Λ.

We finally turn to the question of when C itself is projective in M^{C} or C^{∗}M. Since C is a balanced (C^{∗}, C^{∗})-bimodule, we can use standard module theory to obtain some properties of C as a locally projective C^{∗}-module.

10.9. C locally projective as C^{∗}-module.

(1) If C is locally projective as a left C^{∗}-module, then C is a generator in CM.

(2) If C is locally projective as a left and right C^{∗}-module, then both Rat^{C} and ^{C}Rat are exact.

Proof. (1) If _{C^{∗}}C is locally projective, then, by [3, 42.10], C_{C^{∗}} is a generator in σ[C_{C^{∗}}] = ^{C}M.

(2) Assume that both _{C^{∗}}C and C_{C^{∗}} are locally projective. Then, by (1), C^{∗}C is a locally projective generator in σ[_{C^{∗}}C], and, by 9.11, Rat^{C} is an exact functor. Similar arguments show that ^{C}Rat is exact. ⊔ ⊓

10.10. C projective in M^{C}. Assume that C is projective in M^{C}.

(1) If C^{∗} is an f-semiperfect ring, or C is C-injective, then C is a direct sum of finitely generated left C^{∗}-modules.

(2) If C^{∗} is a right self-injective ring, then C is a generator in ^{C}M.

(3) If C^{∗} is a semiperfect ring, then _{R}C is finitely generated.

Proof. (1) This is a decomposition property of projective modules with f-semiperfect endomorphism rings (see [3, 41.19]). If C is self-injective, then C^{∗} is f-semiperfect.

(2) As a self-injective ring, C^{∗} is f-semiperfect. By 9.12, (1) implies that C is s-unital over the right trace ideal T′, and so T′ is a generator (by 9.8). Moreover, right injectivity of C^{∗} implies that T′ = Tr(^{C}M, C^{∗}) = Tr(C_{C^{∗}}, C^{∗}), and so C generates T′ (see [3, 42.7] for the definition of a trace).

(3) This follows from (1) and [3, 41.19]. ⊔ ⊓

11 Coalgebras over QF rings

Recall that a QF ring R is an Artinian injective cogenerator in M_{R}. We consider R-coalgebras C with _{R}C locally projective. If R is a QF ring, then this is equivalent to C being projective as an R-module.

11.1. Coalgebras over QF rings. If R is a QF ring, then:

(1) C is a (big) injective cogenerator in M^{C}.

(2) Every comodule in M^{C} is a subcomodule of some direct sum C^{(Λ)}.

(3) C^{∗} is an f-semiperfect ring.

(4) K := Soc_{C^{∗}}C ⊴ C and Jac(C^{∗}) = Hom_{R}(C/K, R).

(5) C^{∗} is right self-injective if and only if C is flat as left C^{∗}-module.

Proof. (1),(2) By 7.17, C is injective in M^{C}. Over a QF ring R, every R-module M is contained in a free R-module R^{(Λ)}. This yields, for any right C-comodule, an injection ϱ^{M} : M → M ⊗_{R} C ⊂ R^{(Λ)} ⊗_{R} C ≃ C^{(Λ)}.

(3) The endomorphism ring of any self-injective module is f-semiperfect (see 3.15).

(4) By 8.15, _{C^{∗}}C is locally of finite length and hence has an essential socle. By the Hom-tensor relations (see 7.9),

Jac(C^{∗}) = Hom^{C}(C/K, C) ≃ Hom_{R}(C/K, R).

(5) For any N ∈ M_{C^{∗}}, there is an isomorphism Hom_{R}(N ⊗_{C^{∗}} C, R) ≃ Hom_{C^{∗}}(N, Hom_{R}(C, R)) = Hom_{C^{∗}}(N, C^{∗}) (cf. [3, 40.18]). So, if C^{∗} is right self-injective, the functor Hom_{R}(− ⊗_{C^{∗}} C, R) : M_{C^{∗}} → M_{R} is exact. Since R is a cogenerator in M_{R}, this implies that − ⊗_{C^{∗}} C is exact, that is, _{C^{∗}}C is flat. Similar arguments yield the converse conclusion. ⊔ ⊓

By 4.17, For any injective cogenerator, fully invariant decompositions (coalgebra decompositions) are σ-decompositions (see [3, 44.8]). Consequently, 10.5 yields:

11.2. σ-decomposition of C. If R is a QF ring, then:

(1) C has fully invariant decompositions with σ-indecomposable summands.

(2) Each fully invariant decomposition (= decomposition into coalgebras) is a σ-decomposition.

(3) C is σ-indecomposable if and only if C has no nontrivial fully invariant decomposition, that is, C^{∗} has no nontrivial central idempotents.

(4) If C is cocommutative, then C = ⨁_{Λ}Ê_{λ} is a fully invariant decomposition, where {E_{λ}}_{Λ} is a minimal representing set of simple comodules in M^{C}, and Ê_{λ} denotes the injective hull of E_{λ}.

Proof. By 11.1, C is an injective cogenerator in σ[_{C^{∗}}C], and so (1), (2) and (3) follow from 4.17 and 10.5. In (4), C^{∗} is a commutative algebra by assumption, and so the assertion follows from [3, 43.7]. ⊔ ⊓

Over QF rings there is a bijective correspondence between closed subcategories of M^{C} and (C^{∗}, C^{∗})-sub-bimodules in C. However, the latter need not be pure R-submodules of C, and hence they may not be subcoalgebras. Recall that injectivity of C in M^{C} implies Tr(σ[N], C) = Tr(N, C), for any N ∈ M^{C}.

11.3. Correspondence relations. Let R be a QF ring and N ∈ M^{C}. Then:

(1) σ[N] = σ[Tr(N, C)].

(2) The map σ[N] ↦ Tr(N, C) yields a bijective correspondence between closed subcategories of M^{C} and (C^{∗}, C^{∗})-sub-bimodules of C.

(3) σ[N] is closed under essential extensions (injective hulls) in M^{C} if and only if Tr(N, C) is a C^{∗}-direct summand of _{C^{∗}}C. In this case Tr(N, C) is a subcoalgebra of C.

(4) N is a semisimple comodule if and only if Tr(N, C) ⊂ Soc(_{C^{∗}}C).

(5) If R is a semisimple ring, then Tr(N, C) is a subcoalgebra of C.

Proof. Since R is a QF ring, _{C^{∗}}C has locally finite length and is an injective cogenerator in M^{C}. Hence (1)–(4) follow from 4.13. Furthermore, if R is semisimple, the (C^{∗}, C^{∗})-sub-bimodule Tr(N, C) is an R-direct summand in C and so is a subcoalgebra by 8.6. This proves assertion (5). ⊔ ⊓

Since over a QF ring R any R-coalgebra C is an injective cogenerator in M^{C} and ^{C}M (by 11.1), the results from 10.4 simplify to the following.

11.4. C injective in M_{C^{∗}}. If R is QF, the following are equivalent:

(a) C is injective in M_{C^{∗}};

(b) C is an injective cogenerator in M_{C^{∗}};

(c) _{C^{∗}}C is Artinian;

(d) C^{∗} is a right Noetherian ring.

Proof. In view of the preceding remark the equivalence of (b), (c) and (d) follows from 10.4. The implication (b) ⇒ (a) is trivial, and (a) ⇒ (c) needs an argument from module theory (see [3, 8.3]). ⊔ ⊓

For finitely generated comodules, injectivity and projectivity in M^{C} may extend to injectivity, resp. projectivity, in _{C^{∗}}M.

11.5. Finitely presented modules over QF rings. Let R be a QF ring and M ∈ M^{C}.

(1) If M is projective in M^{C}, then M^{∗} is C-injective as a right C^{∗}-module and Rat^{C}(M^{∗}) is injective in ^{C}M.

(2) If M is finitely generated as an R-module, then:

(i) if M is injective in M^{C}, then M^{∗} is projective in M_{C^{∗}}. (ii) M is injective in M^{C} if and only if M is injective in _{C^{∗}}M. (iii) M is projective in M^{C} if and only if M is projective in _{C^{∗}}M.

Proof. (1) Consider any diagram with exact row in ^{C}M,

\xymatrix{0 \ar[r] & K \ar[r] \ar[d]^{f} & N \\  & M^{∗} ,}

where N is finitely generated as an R-module. Applying (−)^{∗} = Hom_{R}(−, R) we obtain – with the canonical map Φ_{M} : M → M^{∗∗} – the diagram

\xymatrix{M \ar[r] & M^{∗∗} \ar[d]^{f^{∗}} \\ N^{∗} \ar[r] & K^{∗} \ar[r] & 0 ,}

where the lower row is in M^{C} and hence can be extended commutatively by some right comodule morphism g : M → N^{∗}. Again applying (−)^{∗} - and

recalling that the composition M^{∗} ⟶^{Φ_{M^{∗}}} M^{∗∗∗} ⟶^{(Φ_{M})^{∗}} M^{∗} yields the identity (by [3, 40.23]) - we see that g^{∗} extends f to N. This proves that M^{∗} is N-injective for all modules N ∈ ^{C}M that are finitely presented as R-modules.

In particular, by the Finiteness Theorem 8.11, every finitely generated C^{∗}-submodule of C is finitely generated - hence finitely presented – as an R-module. So M^{∗} is N-injective for all these modules, and hence it is Cinjective as a right C^{∗}-module (see 10.1). Notice that M^{∗} need not be in ^{C}M (not rational). It is straightforward to show that Rat^{C}(M^{∗}) is an injective object in ^{C}M.

(2)(i) We know that M ⊂ R^{k}, for some k ∈ ℕ, and so there is a monomor-

phism in M^{C}, \xymatrix{M \ar[r]^{ϱ^{M}} & M ⊗_{R} C \ar[r] & R^{k} ⊗_{R} C ≃ C^{k}} , that splits in M^{C} and hence in _{C^{∗}}M (by 8.1). So the dual sequence (C^{∗})^{k} → M^{∗} → 0 splits in M_{C^{∗}}, and hence M^{∗} is projective in M_{C^{∗}}.

(ii) Let M be injective in M^{C}. Then M^{∗} is projective in M_{C^{∗}} (by (i)). Consider any monomorphism in M → X in _{C^{∗}}M. Then X^{∗} → M^{∗} → 0 is exact and splits in M_{C^{∗}}, and hence, in the diagram

\xymatrix{0 \ar[r] & M \ar[r] \ar[d]^{≃} & X \ar[d] \\ 0 \ar[r] & M^{∗∗} \ar[r] & X^{∗∗} ,}

the bottom row splits in _{C^{∗}}M and as a consequence so does the upper row, proving that M is injective in _{C^{∗}}M.

(iii) Let M be projective in M^{C}. Since M^{∗} is in ^{C}M (by 7.10), we know from (1) that it is injective in ^{C}M. Now we conclude, by the right-hand version of (i), that M ≃ M^{∗∗} is projective in _{C^{∗}}M. ⊔ ⊓

As shown in 11.5, for coalgebras over QF rings, finitely generated projective modules in M^{C} are in fact projective in _{C^{∗}}M. This is the key to the fact that in this case right semiperfect coalgebras are characterised by the exactness of the left trace functor (so also by all the equivalent properties of the trace functor given in 9.8).

11.6. Right semiperfect coalgebras over QF rings. Let R be QF and T = Rat^{C}(C^{∗}). Then the following are equivalent:

(a) C is a right semiperfect coalgebra;

(b) M^{C} has a generating set of finitely generated modules that are projective in _{C^{∗}}M;

(c) injective hulls of simple left C-comodules are finitely generated as Rmodules;

(d) the functor Rat^{C} : _{C^{∗}}M → M^{C} is exact;

(e) T is left C-dense in C^{∗};

(f) Ke T = {x ∈ C | T (x) = 0} = 0.

Proof. (a) ⇔ (b) If C is right semiperfect, there exists a generating set of finitely generated projective modules in M^{C} (see 10.7). By 11.5, all these are projective in _{C^{∗}}M. The converse conclusion is immediate.

(a) ⇒ (c) Let U be a simple left C-comodule with injective hull U → Û in ^{C}M. Applying Hom_{R}(−, R) we obtain a small epimorphism in _{C^{∗}}M,

Û^{∗} → U^{∗} → 0,

where U^{∗} is a simple left C^{∗}-module. Moreover, since R is QF, we know that Û is a direct summand of C_{C^{∗}}, and so Û^{∗} is a direct summand of C^{∗}, and hence is projective in _{C^{∗}}M. By assumption there exists a projective cover P → U^{∗} in M^{C}. Since P is finitely generated as anR-module and projective in M^{C}, it is also projective in _{C^{∗}}M (by 11.5), and hence Û^{∗} ≃ P. So Û^{∗} is finitely generated as an R-module and so is Û.

(c) ⇒ (a) Let V ⊂ C be a simple left C^{∗}-submodule. Then V^{∗} is a simple right C^{∗}-module in ^{C}M. Let V^{∗} → K be its injective hull in ^{C}M. By assumption, K is a finitely generated R-module, and so K^{∗} is a projective C^{∗}-module (by 11.5) and K^{∗} → V^{∗∗} ≃ V is a projective cover in M^{C}.

(b) ⇒ (d) The assumption implies that M^{C} has a generator that is projective in _{C^{∗}}M, and the assertion follows from 9.11.

(d) ⇔ (e) ⇔ (f) These equivalences follow from 9.8 and 9.6.

(d) ⇒ (c) Let V ⊂ C be a simple left C^{∗}-submodule. Then U = V^{∗} is a simple left C-comodule and there is a projective cover Û^{∗} → V in _{C^{∗}}M (see proof (a) ⇒ (c)). By 9.9, (d) implies that M^{C} is closed under small epimorphisms and hence Û^{∗} ∈ M^{C}. ⊔ ⊓

The conditions on left C^{∗}-modules (right C-comodules) posed in the preceding theorem imply remarkable properties of the left C-comodules.

11.7. Left side of right semiperfect coalgebras. Let C be right semiperfect, R a QF ring and T = Rat^{C}(C^{∗}). Then:

(1) the injective hull of any X ∈ ^{C}M is finitely R-generated, provided X is finitely R-generated.

(2) For every X ∈ ^{C}M that is finitely R-generated, Hom_{C^{∗}}(T, X) ≃ X.

(3) For every M ∈ ^{C}M, the trace of M^{C} in M^{∗} is nonzero.

(4) Any module in ^{C}M has a maximal submodule and has a small radical.

Proof. (1) Let X ∈ ^{C}M be finitely generated as an R-module. Then X has finite uniform dimension, and so its injective hull in ^{C}M is a finite direct sum of injective hulls of simple modules, which are finitely generated by 11.6(c).

(2) By (1), the C-injective hull X̂ of X is finitely R-generated and hence is C^{∗}-injective (see 11.5). So any f ∈ Hom_{C^{∗}}(T, X) can be uniquely extended to some h : C^{∗} → X̂ and h(ε) ∈ X̂, which is s-unital over T (see 9.10). Hence

h(ε) ∈ h(ε) · C^{∗} = h(ε) · T = h(T) = f(T) ⊂ X,

showing that h ∈ Hom_{C^{∗}}(C^{∗}, X) ≃ X.

(3) For every simple submodule S ⊂ M with injective hull Ŝ in ^{C}M, there are commutative diagrams

\xymatrix{0 \ar[r] & S \ar[r] \ar[d]_{i} & M \ar[dl]^{j} & M^{∗} \ar[r] & S^{∗} \\  & Ŝ & , & Ŝ^{∗} \ar[u]^{j^{∗}} \ar[ur]_{i^{∗}} & ,}

where i is injective and j is nonzero. By 7.10, Ŝ^{∗} belongs to M^{C} and so does its nonzero image under j^{∗}.

(4) Let M ∈ ^{C}M. By (3), there exists a simple submodule Q ⊂ M^{∗} with Q ∈ M^{C}. Then Ke Q = {m ∈ M | Q(m) = 0} is a maximal C^{∗}-submodule of M. This shows that all modules in ^{C}M have maximal submodules, and hence every proper submodule of M is contained in a maximal C^{∗}-submodule. This implies that Rad(M) is small in M. ⊔ ⊓

11.8. Finiteness properties. Let R be a QF ring.

(1) If C is right semiperfect and there are only finitely many nonisomorphic simple right C-comodules, then _{R}C is finitely generated.

(2) If C is right semiperfect and any two nonzero subalgebras have non-zero intersection (i.e., C is irreducible), then _{R}C is finitely generated.

(3) _{R}C is finitely generated if and only if M^{C} has a finitely generated projective generator.

(4) C^{∗} is an algebra of finite representation type if and only if there are only finitely many nonisomorphic finitely generated indecomposable modules in M^{C}.

Proof. (1) Since C_{C^{∗}} is self-injective, the socle of C_{C^{∗}} is a finitely generated R-module by [3, 41.23]. Hence Soc(C_{C^{∗}}) has finite uniform dimension, and since Soc(C_{C^{∗}}) ⊴ C, C is a finite direct sum of injective hulls of simple modules in ^{C}M that are finitely generated R-modules by 11.7.

(2) Under the given condition there exists only one simple right C-como-dule (up to isomorphisms), and the assertion follows from (1).

(3) If _{R}C is finitely generated, then M^{C} = _{C^{∗}}M. Conversely, assume there exists a finitely generated projective generator P in M^{C}. Then P is semiperfect and there are only finitely many simples in M^{C}. Now (1) applies.

(4) One implication is obvious. Assume there are only finitely many nonisomorphic finitely generated indecomposables in M^{C}. Since C is subgenerated by its finitely generated submodules, this implies that M^{C} has a finitely generated subgenerator. Now [10, 54.2] implies that there is a progenerator in M^{C}, and hence _{R}C is finitely generated by (3). ⊔ ⊓

Unlike in the case of associative algebras, right semiperfectness is a strictly one-sided property for coalgebras – it need not imply left semiperfectness. The next proposition describes coalgebras that are both right and left semiperfect.

11.9. Left and right semiperfect coalgebras. Let R be a QF ring, T = Rat^{C}(C^{∗}) and T′ = ^{C}Rat(C^{∗}). The following are equivalent:

(a) C is a left and right semiperfect coalgebra;

(b) all left C-comodules and all right C-comodules have projective covers;

(c) T = T′ and is dense in C^{∗};

(d) _{C^{∗}}C and C_{C^{∗}} are direct sums of finitely generated C^{∗}-modules.

Under these conditions, T is a ring with enough idempotents, and it is a generator in M^{C}.

Proof. (b) ⇒ (a) is obvious.

(a) ⇒ (b) By [3, 41.16], all finitely generated projective modules in M^{C} are semiperfect in M^{C}. According to 3.10 and 3.11, a direct sum of projective semiperfect modules in M^{C} is semiperfect provided it has a small radical. Since this is the case by 11.7, we conclude that every module in M^{C} has a projective cover. Similar arguments apply to the category ^{C}M.

(a) ⇔ (c) This is obvious by the characterisation of exactness of the rational functor in 9.8 and 11.6.

(c) ⇔ (d) follows from 9.12.

The final assertions follow from 9.12 and 9.8. ⊔ ⊓

For cocommutative coalgebras we can combine 11.9 with 11.2(4).

11.10. Cocommutative semiperfect coalgebras. Let R be QF and C cocommutative. The following are equivalent:

(a) C is semiperfect;

(b) C is a direct sum of finitely generated C^{∗}-modules;

(c) C is a direct sum of finitely R-generated subcoalgebras;

(d) every uniform subcomodule (C^{∗}-submodule) of C is finitely R-generated.

The trace functors combined with the dual functor (−)^{∗} define covariant functors ^{C}Rat ∘ (−)^{∗} : M^{C} → ^{C}M and Rat^{C} ∘ (−)^{∗} : ^{C}M → M^{C}. Over QF rings, these functors clearly are exact if and only if ^{C}Rat, respectively Rat^{C}, is exact, that is, C is left or right semiperfect. In this case they yield dualities between subcategories of M^{C} and ^{C}M.

Over a QF ring, projective comodules in M^{C} that are finitely generated as left C^{∗}-modules are also projective in _{C^{∗}}M (see 11.5). Moreover, any direct sum of copies of C is C-injective as a left and right C^{∗}-module.

11.11. Projective coalgebras over QF rings. If R is QF, the following are equivalent:

(a) C is a submodule of a free left C^{∗}-module;

(b) C (or every right C-comodule) is cogenerated by C^{∗} as a left C^{∗}-module;

(c) there exists a family of left nondegenerate C-balanced bilinear forms C × C → R;

(d) in M^{C} every (indecomposable) injective object is projective;

(e) C is projective in M^{C};

(f) C is projective in _{C^{∗}}M.

If these conditions are satisfied, then C is a left semiperfect coalgebra and C is a generator in ^{C}M.

Proof. (a) ⇔ (b) By 8.15, C is a direct sum of injective hulls of simple modules in M^{C}. If C is cogenerated by C^{∗}, then each of these modules is contained in a copy of C^{∗}, and hence C is contained in a free C^{∗}-module. Recall from 11.1 that C is a cogenerator in M^{C} and hence C^{∗} cogenerates any N ∈ M^{C} provided it cogenerates C.

(b) ⇔ (c) This is shown in [3, 6.6(2)].

(c) ⇒ (f) Let U be a simple left C^{∗}-submodule of C with injective hull Û ⊂ C in M^{C}. Then Û is a finitely generated R-module by [3, 6.6(3)]. Now we conclude from 11.5 that Û is injective in _{C^{∗}}M. Being cogenerated by C^{∗}, we observe in fact that Û is a direct summand of C^{∗}, and hence it is projective in _{C^{∗}}M. This implies that C is projective in _{C^{∗}}M.

(f) ⇒ (a) and (f) ⇒ (e) are obvious, and so is (d) ⇔ (e) (by 11.1).

(e) ⇒ (f) C is a direct sum of injective hulls Û ⊂ C of simple submodules U ⊂ C. By (e), Û is projective in M^{C}. Since it has a local endomorphism ring, we know from 3.9 that it is finitely generated as a C^{∗}-module and hence finitely generated as an R-module (by 8.11). Now we conclude from 11.5 that Û is projective in _{C^{∗}}M and so is C.

Finally, assume these conditions hold. By the proof of 11.11, the injective hulls of simple modules in M^{C} are finitely generated R-modules. By 10.7, this characterises left semiperfect coalgebras, implying that the right trace ideal T′ := ^{C}Rat(C^{∗}) is a generator in ^{C}M. Now, by 11.5(1), T′ is injective in ^{C}M, and hence it is generated by C and therefore C is a generator in ^{C}M. ⊔ ⊓

11.12. Corollary. Let R be QF and C projective in M^{C}. Then the following are equivalent:

(a) _{C^{∗}}C contains only finitely many nonisomorphic simple submodules;

(b) Soc(_{C^{∗}}C) is finitely generated as an R-module;

(c) C^{∗} is a semiperfect ring;

(d) _{R}C is finitely generated.

Proof. (a) ⇒ (b) By the Finiteness Theorem, the homogeneous components of the socle of C are finitely generated a R-modules.

(b) ⇒ (c) We know that C^{∗} is f-semiperfect. Clearly Soc(_{C^{∗}}C) ⊴ C, and hence C has a finite uniform dimension as a left C^{∗}-module. This implies that C^{∗} is semiperfect.

(c) ⇒ (a) For any semiperfect ring there are only finitely many simple left (or right) modules (up to isomorphisms).

(c) ⇒ (d) is shown in 10.10(3).

(d) ⇒ (b) follows from the fact that R is Noetherian. ⊔ ⊓

From 11.1 we know that, over a QF ring R, C is always an injective cogenerator in M^{C}. Which additional properties make C a projective generator?

11.13. C as a projective generator in M^{C}. Let R be QF and T = Rat^{C}(C^{∗}). The following are equivalent:

(a) C is projective as left and right C-comodule;

(b) C is a projective generator in M^{C};

(c) C is a projective generator in ^{C}M;

(d) C = T C and T has enough idempotents and is an injective cogenerator in M^{C}.

Proof. (a) ⇒ (b) This is obtained from 11.11 and 10.9.

(b) ⇒ (a) By 11.11, C is projective as a left C^{∗}-module and hence C^{∗} is C-injective as a right C^{∗}-module (by 11.5). To show that C is projective as a right C^{∗}-module we show that C^{∗} cogenerates C as a right C^{∗}-module. For this it is enough to prove that each simple submodule U ⊂ C_{C^{∗}} is embedded in C^{∗}. By 8.11, U is a finitely generated R-module. Clearly U^{∗} is a simple module in M^{C}, and hence there is a C^{∗}-epimorphism C → U^{∗}. From this we obtain an embedding U ≃ U^{∗∗} ⊂ C^{∗}, which proves our assertion.

(a) ⇔ (c) is clear by symmetry.

(a) ⇒ (d) From the above discussion we know that C is a left and right semiperfect coalgebra. Hence T is a ring with enough idempotents and M^{C} = σ[_{C^{∗}}T] by 11.9. Since C is projective, C ⊂ T^{(Λ)}, and hence T is a cogenerator in M^{C}. T is injective in M^{C} by 11.5.

(d) ⇒ (b) Since T is projective in M^{C}, injective hulls of simple modules in M^{C} are projective, and so C is projective in M^{C}. T is injective, and hence it is generated by C. By our assumptions T is a generator in M^{C} and so is C. ⊔ ⊓

In case C is finitely R-generated, M^{C} = _{C^{∗}}M and we obtain:

11.14. C as a projective generator in _{C^{∗}}M. If R is QF, the following are equivalent:

(a) C is a projective generator in _{C^{∗}}M;

(b) C is a generator in _{C^{∗}}M;

(c) C is a generator in M^{C} and _{R}C is finitely generated;

(d) C^{∗} is a QF algebra and _{R}C is finitely generated.

Proof. (a) ⇒ (b) is obvious.

(b) ⇒ (c) As a generator in _{C^{∗}}M, C is finitely generated as a module over its endomorphism ring C^{∗}, and hence _{R}C is finitely generated.

(c) ⇒ (d) Clearly C^{∗} is left (and right) Artinian. By assumption, C is an injective generator in _{C^{∗}}M. This implies that C^{∗} is self-injective and hence QF.

(d) ⇒ (a) As a QF ring, C^{∗} is an injective cogenerator in M^{C} = _{C^{∗}}M. From this it is easy to see that C is a projective generator in _{C^{∗}}M. ⊔ ⊓

12 Bialgebras

In this section we are concerned with the compatibility of algebra and coalgebra structures on a given R-module. In particular, we define bialgebras and study their most elementary properties.

12.1. Bialgebras. An R-module B that is an algebra (B, μ, ι) and a coalgebra (B, Δ, ε) is called a bialgebra if Δ and ε are algebra morphisms or, equivalently, μ and ι are coalgebra morphisms. For Δ to be an algebra morphism one needs commutativity of the diagrams

\xymatrix{B ⊗_{R} B \ar[r]^{μ} \ar[d]_{Δ⊗Δ} & B \ar[dd]^{Δ} & R \ar[r]^{ι} \ar[d]_{≃} & B \ar[d]^{Δ} \\ (B ⊗_{R} B) ⊗_{R} (B ⊗_{R} B) \ar[d]_{I_{B}⊗tw⊗I_{B}} &  & R ⊗_{R} R \ar[r]^{ι⊗ι} & B ⊗_{R} B, \\ (B ⊗_{R} B) ⊗_{R} (B ⊗_{R} B) \ar[r]^{μ⊗μ} & B ⊗_{R} B,}

where tw denotes the twist map. Similarly, ε is an algebra morphism if and only if the following two diagrams

\xymatrix{B ⊗_{R} B \ar[r]^{μ} \ar[d]_{ε⊗ε} & B \ar[d]^{ε} &  & B \ar[dr]^{ε} \\ R ⊗_{R} R \ar[r]^{≃} & R, & R \ar@{-->}[ur]^{ι} \ar[rr]^{=} &  & R}

are commutative. The same set of diagrams makes μ and ι coalgebra morphisms. For the units 1_{B} ∈ B, 1_{R} ∈ R and for all a, b ∈ B, the above diagrams explicitly mean that

\begin{matrix}Δ(1_{B}) = 1_{B} ⊗ 1_{B}, & ε(1_{B}) = 1_{R}, \\ Δ(ab) = Δ(a)Δ(b), & ε(ab) = ε(a)ε(b). \end{matrix}

Note that this implies that, in any R-bialgebra B, R is a direct summand of B as an R-module and hence B is a generator in M_{R}.

12.2. Bialgebra morphisms. An R-linear map f : B → B′ of bialgebras is called a bialgebra morphism if f is both an algebra and a coalgebra morphism.

An R-submodule I ⊂ B is a sub-bialgebra if it is a subalgebra as well as a subcoalgebra. I is a bi-ideal if it is both an ideal and a coideal.

Let f : B → B′ be a bialgebra morphism. Then:

(1) If f is surjective, then Ke f is a bi-ideal in B.

(2) Im f is a subcoalgebra of B′.

A remarkable feature of a bialgebra B is that the tensor product of Bmodules is again a B-module. First, recall that an R-module N is a B-module if there is an algebra morphism B → End_{R}(N).

12.3. Tensor product of B-modules. Let K, L be right modules over an R-bialgebra B.

(1) K ⊗_{R} L has a right B-module structure by the map

B ⟶^{Δ} B ⊗_{R} B → End_{R}(K) ⊗_{R} End_{R}(L) → End_{R}(K ⊗_{R} L) ;

we denote this module by K ⊗_{R}^{b} L. The right action of B is given by

! : K ⊗_{R} L ⊗_{R} B ⟶ K ⊗_{R} L, k ⊗ l ⊗ b ↦ (k ⊗ l)Δb,

where the product on the right side is taken componentwise, that is,

(k ⊗ l)!b := (k ⊗ l) Δb = ∑ kb\underline{1} ⊗ lb\underline{2} .

(2) For any morphisms f : K → K′, g : L → L′ in M_{B}, the tensor product map f ⊗ g : K ⊗_{R}^{b} L → K′ ⊗_{R}^{b} L′ is a morphism in M_{B}.

Proof. (1) follows easily from the definitions. Assertion (2) is equivalent to the commutativity of the following diagram:

\xymatrix{K ⊗_{R}^{b} L ⊗_{R} B \ar[r]^{f⊗g⊗I_{B}} & K′ ⊗_{R}^{b} L′ ⊗_{R} B \\ ! \ar[d] & ! \ar[d] \\ K ⊗_{R}^{b} L \ar[r]^{f⊗g} & K′ ⊗_{R}^{b} L′,}

which follows immediately from B-linearity of f and g. ⊔ ⊓

Of course similar constructions apply to left B-modules K, L, in which case the left B-multiplication is given by

! : B ⊗_{R} K ⊗_{R} L ⟶ K ⊗_{R} L, b ⊗ k ⊗ l ↦ Δb(k ⊗ l) .

Explicitly, the product comes out as b!(k ⊗ l) = ∑ b\underline{1}k ⊗ b\underline{2}l.

Dually, the tensor product of comodules has a special comodule structure.

12.4. Tensor product of B-comodules. Let K, L be right comodules over an R-bialgebra B.

(1) K ⊗_{R} L has a right B-comodule structure by the map (tensor over R)

\xymatrix{ϱ^{K⊗L} : K ⊗ L \ar[r]^{tw_{23}∘(ϱ^{K}⊗ϱ^{L})} & K ⊗ L ⊗ B ⊗ B \ar[r]^{I_{K}⊗I_{L}⊗μ} & K ⊗ L ⊗ B,}

where tw_{23} = I_{K} ⊗tw⊗I_{B}. This comodule is denoted by K ⊗_{R}^{c} L. Thus, explicitly, for all k ⊗ l ∈ K ⊗_{R}^{c} L,

ϱ^{K⊗_{R}L}(k ⊗ l) = ∑k\underline{0} ⊗ l\underline{0} ⊗ k\underline{1}l\underline{1}.

(2) For any morphisms f : K → K′, g : L → L′ in M^{B}, the tensor product map f ⊗ g : K ⊗_{R}^{c} L → K′ ⊗_{R}^{c} L′ is a morphism in M^{B}.

Proof. (1) This is proved by computing for all k ∈ K, l ∈ L,

(I_{K} ⊗ I_{L} ⊗ Δ) ∘ ϱ^{K⊗_{R}L}(k ⊗ l) = ∑ k\underline{0} ⊗ l\underline{0} ⊗ Δ(k\underline{1}l\underline{1})
= ∑ k\underline{0} ⊗ l\underline{0} ⊗ k\underline{1}\underline{1}l\underline{1}\underline{1} ⊗ k\underline{1}\underline{2}l\underline{1}\underline{2}
= ∑ k\underline{0}\underline{0} ⊗ l\underline{0}\underline{0} ⊗ k\underline{0}\underline{1}l\underline{0}\underline{1} ⊗ k\underline{1}\underline{1}l\underline{1}\underline{1}
= (ϱ^{K⊗_{R}L} ⊗ I_{B}) ∘ ϱ^{K⊗_{R}L}(k ⊗ l).

To prove (2), take any k ∈ K, l ∈ L and compute

ϱ^{K′⊗_{R}L′} ∘ (f ⊗ g)(k ⊗ l) = ∑ f(k)\underline{0} ⊗ g(l)\underline{0} ⊗ f(k)\underline{1}g(l)\underline{1}
= ∑ f(k\underline{0}) ⊗ g(l\underline{0}) ⊗ k\underline{1}l\underline{1}
= (f ⊗ g ⊗ I_{B}) ∘ ϱ^{K⊗_{R}L}(k ⊗ l).

This shows that f ⊗ g is a comodule morphism, as required. ⊔ ⊓

The coaction constructed in 12.4 is known as a diagonal coaction of a bialgebra B on the tensor product of its comodules.

In contrast to coalgebras, for a bialgebra B, any R-module K can be considered as B-comodule by K → K ⊗_{R} B, k ↦ k ⊗ 1_{B} (trivial coaction). In particular, the ring R is a right B-comodule, and this draws attention to those maps B → R that are comodule morphisms.

Definition. An element t ∈ B^{∗} is called a left integral on B if it is a left comodule morphism.

Recall that the rational part of B^{∗} is denoted by Rat^{B}(B^{∗}) = T and ϱ^{T} : T → T ⊗_{R} B denotes the corresponding coaction.

12.5. Left integrals on B. Let B be an R-bialgebra and t ∈ B^{∗}.

(1) The following are equivalent:

(a) t is a left integral on B;

(b) (I_{B} ⊗ t) ∘ Δ = ι ∘ t.

If B is cogenerated by R as an R-module, then (a) is equivalent to:

(c) For every f ∈ B^{∗}, f ∗ t = f(1_{B})t.

(2) Assume that _{R}B is locally projective.

(i) If t ∈ T, then t is a left integral on B if and only if ϱ^{T} (t) = t⊗1_{B}. (ii) If R is Noetherian or if t(B) = R, then any left integral t on B

is rational, that is, t ∈ T.

Proof. (1) (a) ⇔ (b) The map t is left colinear if and only if there is a commutative diagram

[figure: B; t; Δ; R; ι; B ⊗_{R} B; I_{B}⊗t; B; b; t(b); Δ(b); (I_{B} ⊗ t) ∘ Δ(b) = t(b)1_{B}.]

The commutativity of this diagram is expressed by condition (b). (b) ⇔ (c) For any f ∈ B^{∗} and b ∈ B,

\begin{aligned}f ∗ t(b) & = & (f ⊗ t) ∘ Δ(b) & = & f((I_{B} ⊗ t) ∘ Δ(b)), \\ f(1_{B})t(b) & = & f(t(b)1_{B}) & = & f(ι ∘ t(b)). \end{aligned}

From this (b) ⇒ (c) is obvious. If B is cogenerated by R, then (c) ⇒ (b).

(2)(i) If t ∈ T, that is, t is rational, then f ∗ t = (I_{T} ⊗ f) ∘ ϱ^{T} (t), for any f ∈ B^{∗}, and (1)(c) implies

(I_{T} ⊗ f)(ϱ^{T} (t)) = (I_{T} ⊗ f)(t ⊗ 1_{B}).

By local projectivity (α-condition; see 8.2) this means ϱ^{T} (t) = t ⊗ 1_{B}. The converse conclusion is obvious.

(ii) By (1)(c), B^{∗} ∗ t ⊂ R 1_{B}. If R is Noetherian, this implies that B^{∗} ∗ t is finitely presented as an R-module, and by 9.5(2) this implies that the element t ∈ Rat^{B}(B^{∗}) = T. If t(B) = R, then t⇀B = (I_{B} ⊗ t)Δ(B) = ι ∘ t(B) = R 1_{B} is finitely presented as an R-module and t ∈ T by 9.5(1). ⊔ ⊓

Throughout B will denotes an R-bialgebra with product μ, coproduct Δ, unit map ι and counit ε.

12.6. B-Hopf modules. An R-module M is called a right B-Hopf module if M is

(i) a right B-module with an action ϱ_{M} : M ⊗_{R} B → M, (ii) a right B-comodule with a coaction ϱ^{M} : M → M ⊗_{R} B, (iii) for all m ∈ M, b ∈ B, ϱ^{M}(mb) = ϱ^{M}(m)Δ(b).

Condition (iii) means that ϱ^{M} : M → M ⊗_{R}^{b}B is B-linear and is equivalent to the requirement that the multiplication ϱ_{M} : M ⊗_{R}^{c} B → M is B-colinear, or to the commutativity of either of the diagrams

\xymatrix{M ⊗_{R} B \ar[r]^{ϱ^{M} ⊗I_{B}} \ar[dd]_{ϱ_{M}} & M ⊗_{R}^{b} B ⊗_{R} B & M ⊗_{R}^{c} B \ar[r]^{ϱ^{M⊗^{c}B}} \ar[dd]_{ϱ_{M}} & M ⊗_{R}^{c} B ⊗_{R} B \ar[dd]^{ϱ_{M} ⊗I_{B}} \\  & ! \ar[d] \\ M \ar[r]^{ϱ^{M}} & M ⊗_{R} B , & M \ar[r]^{ϱ^{M}} & M ⊗_{R} B .}

An R-linear map f : M → N between right B-Hopf modules is a Hopf module morphism if it is both a right B-module and a right B-comodule morphism. Denoting these maps by

Hom_{B}^{B}(M, N) = Hom_{B}(M, N) ∩ Hom^{B}(M, N),

there are characterising exact sequences in M_{R},

0 → Hom_{B}^{B}(M, N) → Hom_{B}(M, N) ⟶^{γ} Hom_{B}(M, N ⊗_{R}^{b} B),

where γ(f) = ϱ^{N} ∘ f − (f ⊗ I_{B}) ∘ ϱ^{M} or, equivalently,

0 → Hom_{B}^{B}(M, N) → Hom^{B}(M, N) ⟶^{δ} Hom^{B}(M ⊗_{R}^{c} B, N),

where δ(g) = ϱ_{N} ∘ (g ⊗ I_{B}) − g ∘ ϱ_{M}.

Left B-Hopf modules and the corresponding morphisms are defined similarly, and it is obvious that B is both a right and a left B-Hopf module.

12.7. Trivial B-Hopf modules. Let K be any R-module.

(1) K ⊗_{R} B is a right B-Hopf module with the canonical structures

I_{K}⊗Δ : K⊗_{R}B → (K⊗_{R}B)⊗_{R}B, I_{K}⊗μ : (K⊗_{R}B)⊗_{R}B → K⊗_{R}B.

(2) For any R-linear map f : K → K′, the map f ⊗I_{B} : K⊗_{R}B → K′⊗_{R}B is a B-Hopf module morphism.

Proof. We know that K ⊗_{R} B is both a right B-module, and a comodule and the compatibility conditions are obvious from the properties of a bialgebra. It is clear that f ⊗ I_{B} is B-linear as well as B-colinear. ⊔ ⊓

12.8. B-modules and B-Hopf modules. Let N be any right B-module.

(1) The right B-module N ⊗_{R}^{b}B is a right B-Hopf module with the canonical comodule structure

I_{N} ⊗ Δ : N ⊗_{R}^{b} B → (N ⊗_{R}^{b} B) ⊗_{R} B, n ⊗ b ↦ n ⊗ Δb.

(2) For any B-linear map f : N → N′, the map f ⊗I_{B} : N ⊗_{R}^{b}B → N′⊗_{R}^{b}B is a B-Hopf module morphism.

(3) The map

γ_{N} : N ⊗_{R} B → N ⊗_{R}^{b} B, n ⊗ b ↦ (n ⊗ 1_{B})Δ(b) = (n ⊗ 1_{B})!b

is a B-Hopf module morphism.

Proof. (1) To show that I_{N} ⊗ Δ is B-linear one needs to check the commutativity of the following diagram:

\xymatrix{N ⊗_{R}^{b} B ⊗_{R} B \ar[r]^{I_{N} ⊗Δ⊗I_{B}} & (N ⊗_{R}^{b} B) ⊗_{R}^{b} B ⊗_{R} B \\ ! \ar[d] & ! \ar[d] \\ N ⊗_{R}^{b} B \ar[r]^{I_{N} ⊗Δ} & (N ⊗_{R}^{b} B) ⊗_{R}^{b} B .}

Evaluating this diagram at any a, b ∈ B and n ∈ N yields

(I_{N} ⊗ Δ) ((n ⊗ b)Δ(a)) = ∑ na\underline{1} ⊗ (ba\underline{2})\underline{1} ⊗ (ba\underline{2})\underline{2}
= ∑ na\underline{1} ⊗ b\underline{1}a\underline{2} ⊗ b\underline{2}a\underline{3}
= ((I_{N} ⊗ Δ)(n ⊗ b)) Δ(a),

by the multplicativity of Δ and the definition of the diagonal B-action on (N ⊗_{R}^{b} B) ⊗_{R}^{b} B (cf. 12.3).

(2) It was shown in 7.8 that f ⊗I_{B} is a comodule morphism, and we know from 12.3 that it is a B-module morphism.

(3) Clearly γ_{N} is B-colinear, and for any c ∈ B,

γ_{N}(n ⊗ bc) = (n ⊗ 1_{B})Δ(bc) = (n ⊗ 1_{B})(Δb)(Δc) = γ_{N}(n ⊗ b)Δ(c),

showing that γ_{N} is right B-linear. ⊔ ⊓

12.9. B-comodules and B-Hopf modules. Let L be a right B-comodule.

(1) The right B-comodule L⊗_{R}^{c} B is a right B-Hopf module with the canonical module structure

I_{L} ⊗ μ : L ⊗_{R}^{c} B ⊗_{R} B → L ⊗_{R}^{c} B, n ⊗ b ⊗ a ↦ n ⊗ ba.

(2) For any B-colinear map f : L → L′, the map f ⊗I_{B} : L⊗_{R}^{c}B → L′⊗_{R}^{c}B is a B-Hopf module morphism.

(3) There is a B-Hopf module morphism

γ^{L} : L ⊗_{R}^{c} B → L ⊗_{R} B, l ⊗ b ↦ ϱ^{L}(l)(1_{B} ⊗ b).

Proof. (1) To prove the colinearity of I_{L} ⊗ μ one needs to show the commutativity of the diagram

\xymatrix{L ⊗_{R}^{c} B ⊗_{R}^{c} B \ar[r]^{I_{L}⊗μ} \ar[d]_{ϱ^{L⊗^{c}B⊗^{c}B}} & L ⊗_{R}^{c} B \ar[d]^{ϱ^{L⊗^{c}B}} \\ L ⊗_{R}^{c} B ⊗_{R}^{c} B ⊗_{R} B \ar[r]^{I_{L}⊗μ⊗I_{B}} & L ⊗_{R}^{c} B ⊗_{R} B ,}

which follows from the multiplicativity of Δ.

(2) Clearly f ⊗ I_{B} is B-linear, and, as shown in 12.4, it is also B-colinear.

(3) Obviously γ^{L} is right B-linear, and colinearity follows from the commutativity of the diagram (which is easily checked)

[figure: L ⊗_{R}^{c} B; γ^{L}; ϱ^{L⊗B}; L ⊗_{R} B; I_{L}⊗Δ; L ⊗_{R}^{c} B ⊗_{R} B; γ^{L}⊗I; L ⊗_{R} B ⊗_{R} B; l ⊗ b; ∑ l\underline{0} ⊗ l\underline{1}b; ∑ l\underline{0} ⊗ b\underline{1} ⊗ l\underline{1}b\underline{2}; ∑ l\underline{0} ⊗ l\underline{1}b\underline{1} ⊗ l\underline{2}b\underline{2}.]

This completes the proof. ⊔ ⊓

Right B-Hopf modules together with B-Hopf module morphisms form a category that is denoted by M_{B}^{B}.

12.10. The category M_{B}^{B}. Let B be an R-bialgebra.

(1) The right B-Hopf module B ⊗_{R}^{b} B is a subgenerator in M_{B}^{B}.

(2) The right B-Hopf module B ⊗_{R}^{c} B is a subgenerator in M_{B}^{B}.

(3) For any M ∈ M_{B}^{B}, N ∈ M_{B},

Hom_{B}^{B}(M, N ⊗_{R}^{b} B) → Hom_{B}(M, N), f ↦ (I_{N} ⊗ ε) ∘ f,

is an R-module isomorphism with inverse map h ↦ (h ⊗ I_{B}) ∘ ϱ^{M}.

(4) For any M ∈ M_{B}^{B}, N ∈ M^{B},

Hom_{B}^{B}(N ⊗_{R}^{c} B, M) → Hom^{B}(N, M), f ↦ f(− ⊗ 1_{B}),

is an R-module isomorphism with inverse map h ↦ ϱ_{M} ∘ (h ⊗ I_{B}).

(5) For any K, L ∈ M_{R},

Hom_{B}^{B}(K ⊗_{R} B, L ⊗_{R} B) → Hom_{R}(K, L), f ↦ (I_{L} ⊗ ε) ∘ f(− ⊗ 1_{B}),

is an R-module isomorphism with inverse map h ↦ h ⊗ I_{B}.

Proof. (1) Let M ∈ M_{B}^{B}. For a B-module epimorphism f : B^{(Λ)} → M,

f ⊗ I_{B} : B^{(Λ)} ⊗_{R}^{b} B → M ⊗_{R}^{b} B

is an epimorphism in M_{B}^{B} (by 12.8), and so M ⊗_{R}^{b} B is generated by

B^{(Λ)} ⊗_{R}^{b} B ≃ (B ⊗_{R}^{b} B)^{(Λ)} .

Moreover, ϱ^{M} : M → M ⊗_{R}^{b}B is a (B-splitting) Hopf module monomorphism, and so M is subgenerated by B ⊗_{R}^{b} B.

(2) For any M ∈ M_{B}^{B}, there is a comodule epimorphism B^{(Λ)} → M ⊗_{R} B, and from this we obtain a Hopf module epimorphism

(B ⊗_{R}^{c} B)^{(Λ)} ≃ B^{(Λ)} ⊗_{R}^{c} B → (M ⊗_{R} B) ⊗_{R}^{c} B.

Moreover, there is a Hopf module monomorphism ϱ^{M} ⊗ I_{B} : M ⊗_{R}^{c} B → (M ⊗_{R} B) ⊗_{R}^{c} B and a Hopf module epimorphism M ⊗_{R}^{c} B → M, and hence M is subgenerated by B ⊗_{R}^{c} B.

(3) There is a commutative diagram with exact rows (⊗ means ⊗_{R}^{b}),

\xymatrix{0 \ar[r] & Hom_{B}^{B}(M, N ⊗B) \ar[r] \ar@{-->}[d] & Hom^{B}(M, N ⊗B) \ar[d]^{(I_{N} ⊗ε)∘−} \ar[r]^{β_{1}} & Hom^{B}(M ⊗B, N ⊗B) \ar[d]^{(I_{N} ⊗ε)∘−} \\ 0 \ar[r] & Hom_{B}(M, N) \ar[r] & Hom_{R}(M, N) \ar[r]^{β_{2}} & Hom_{R}(M ⊗B, N),}

where β_{1}(f) = f ∘ ϱ_{M} − ϱ_{N⊗B} ∘ (f ⊗ I_{B}) and β_{2}(g) = g ∘ ϱ_{M} − ϱ_{N} ∘ (g ⊗ I_{B}). As shown in 7.9, the second and third vertical maps are isomorphisms and hence the first one is also an isomorphism.

(4) Consider the commutative diagram with exact rows (⊗ for ⊗_{R}^{c}),

\xymatrix{0 \ar[r] & Hom_{B}^{B}(N ⊗B, M) \ar[r] \ar@{-->}[d] & Hom_{B}(N ⊗B, M) \ar[r]^{γ_{1}} \ar[d]^{−∘(−⊗1_{B})} & Hom_{B}(N ⊗B, M ⊗B) \ar[d]^{−∘(−⊗1_{B})} \\ 0 \ar[r] & Hom^{B}(N, M) \ar[r] & Hom_{R}(N, M) \ar[r]^{γ_{2}} & Hom_{R}(N, M ⊗B),}

where γ_{1}(f) = ϱ^{M} ∘ f − (f ⊗ I_{B}) ∘ ϱ^{N⊗B} and γ_{2}(g) = ϱ^{M} ∘ g − (g ⊗ I_{B}) ∘ ϱ^{N}. The second and third vertical maps are isomorphisms and hence the first one is an isomorphism, too.

(5) View K as a trivial B-comodule. Then K ⊗_{R}^{c} B ≃ K ⊗_{R} B, and, by (4) and 7.9, Hom_{B}^{B}(K ⊗_{R} B, L ⊗_{R} B) ≃ Hom^{B}(K, L ⊗_{R} B) ≃ Hom_{R}(K, L), as required. ⊔ ⊓

12.11. M_{B}^{B} for B_{R} flat. Let B be flat as an R-module and M, N ∈ M_{B}^{B}. Then:

(1) M_{B}^{B} is a Grothendieck category.

(2) The functor Hom_{B}^{B}(M, −) : M_{B}^{B} → M_{R} is left exact.

(3) The functor Hom_{B}^{B}(−, N) : M_{B}^{B} → M_{R} is left exact.

Proof. (1) For any morphism f : M → N in M_{B}^{B}, Ke f is a B-submodule as well as a B-subcomodule (since B_{R} flat) and hence Ke f ∈ M_{B}^{B}.

(2) Any exact sequence 0 → X → Y → Z in M_{B}^{B} induces the commutative diagram with exact columns

\xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & Hom_{B}^{B}(M, X) \ar[r] \ar[d] & Hom_{B}^{B}(M, Y) \ar[r] \ar[d] & Hom_{B}^{B}(M, Z) \ar[d] \\ 0 \ar[r] & Hom_{B}(M, X) \ar[r] \ar[d] & Hom_{B}(M, Y) \ar[r] \ar[d] & Hom_{B}(M, Z) \ar[d] \\ 0 \ar[r] & Hom_{B}(M, X ⊗_{R}^{b}B) \ar[r] & Hom_{B}(M, Y ⊗_{R}^{b}B) \ar[r] & Hom_{B}(M, Z⊗_{R}^{b}B).}

The columns are simply the defining sequences of Hom_{B}^{B}(M, −) in 12.6. The second and third rows are exact because of the left exactness of Hom_{B}(M, −) and − ⊗_{R} B. Now the diagram lemmata imply that the first row is exact.

(3) This is shown with a similar diagram that uses − ⊗_{R}^{c} B instead of − ⊗_{R}^{b} B. ⊔ ⊓

12.12. Coinvariants of comodules. For M ∈ M^{B}, the coinvariants of B in M are defined as

M^{coB} := {m ∈ M | ϱ^{M}(m) = m ⊗ 1_{B}} = Ke (ϱ^{M} − (− ⊗ 1_{B})).

This is clearly an R-submodule of M and there is an isomorphism

Hom^{B}(R, M) → M^{coB}, f ↦ f(1),

where R is considered as a B-comodule. In particular, this implies that B^{coB} = R1_{B}. Furthermore, for any R-module K,

Hom^{B}(K, M) ≃ Hom_{R}(K, M^{coB}),

where K is considered as a trivial B-comodule.

The last isomorphism follows by the fact that f ∈ Hom^{B}(K, M) is equivalent to the commutativity of the diagram

[figure: K; f; −⊗1_{B}; M; ϱ^{M}; k; f(k); K ⊗_{R} B; f⊗I_{B}; M ⊗_{R} B; k ⊗ 1_{B}; f(k) ⊗ 1_{B} = ϱ^{M}(f(k)) .]

12.13. Coinvariants of Hopf modules. For any M ∈ M_{B}^{B}, the map

ν_{M} : Hom_{B}^{B}(B, M) → M^{coB}, f ↦ f(1_{B}),

is an R-module isomorphism with the inverse ω_{M} : m ↦ [b ↦ mb]. Furthermore, the diagram

[figure: Hom_{B}^{B}(B, M) ⊗_{R} B; ν_{M} ⊗I_{B}; M; f ⊗ b; f(b); M^{coB} ⊗_{R} B; M; f(1_{B}) ⊗ b; f(1_{B})b]

is commutative. In particular, Hom_{B}^{B}(B, B) ⟶^{≃} B^{coB} = R 1_{B} is a ring isomorphism.

Proof. The isomorphism ν_{M} is obtained from the following commutative diagram of R-module maps with exact rows:

\xymatrix{0 \ar[r] & Hom_{B}^{B}(B, M) \ar[r] \ar@{-->}[d] & Hom_{B}(B, M) \ar[r]^{γ_{1}} \ar[d]^{≃} & Hom_{B}(B, M ⊗_{R} B) \ar[d]^{≃} \\ 0 \ar[r] & M^{coB} \ar[r] & M \ar[r]^{γ_{2}} & M ⊗_{R} B,}

where γ_{1}(f) = ϱ^{M} ∘ f − (f ⊗ I_{B}) ∘ Δ, that is, the top row is the defining sequence of Hom_{B}^{B}(B, M), and γ_{2}(m) = ϱ^{M}(m) − m ⊗ 1_{B}. ⊔ ⊓

12.14. Coinvariants of trivial Hopf modules.

(1) For any K ∈ M_{R}, Hom_{B}^{B}(B, K ⊗_{R} B) ≃ K as R-modules.

(2) For all L ∈ M_{R} and M ∈ M_{B}^{B}, there are R-module isomorphisms

Hom_{B}^{B}(L⊗_{R}B, M) ≃ Hom_{R}(L, M^{coB}) and End_{B}^{B}(B⊗_{R}B) ≃ End_{R}(B).

(3) There is an adjoint pair of functors

− ⊗_{R} B : M_{R} → M_{B}^{B}, Hom_{B}^{B}(B, −) : M_{B}^{B} → M_{R},

and Hom_{B}^{B}(B, − ⊗_{R} B) ≃ I_{M_{R}}.

Proof. (1) Consider R as a B-comodule as in 12.10(4). Then the Homtensor relation 7.9(1) implies

Hom_{B}^{B}(B, K ⊗_{R} B) ≃ Hom_{B}^{B}(R ⊗_{R}^{c} B, K ⊗_{R} B) ≃ Hom^{B}(R, K ⊗_{R} B) ≃ K.

(2) Combining 12.10(4) and 12.12, one obtains the chain of isomorphisms

Hom_{B}^{B}(L ⊗_{R} B, M) ≃ Hom^{B}(L, M) ≃ Hom_{R}(L, M^{coB}).

(3) By 12.13, the adjointness is just an interpretation of the isomorphism in (2), and, by (1), the composition of the two functors is isomorphic to the identity functor on M_{R}. ⊔ ⊓ 12.15. Coinvariants and B-modules. For any N ∈ M_{B}, the map

ν′_{N⊗B} : Hom_{B}^{B}(B, N ⊗_{R}^{b} B) → N, f ↦ (I_{N} ⊗ ε) ∘ f(1_{B}) ,

is an R-isomorphism with the inverse n ↦ [b ↦ ∑ nb\underline{1} ⊗ b\underline{2}]. Furthermore, the diagram

[figure: Hom_{B}^{B}(B, N ⊗_{R}^{b} B) ⊗_{R} B; ν′_{N⊗B}⊗I_{B}; N ⊗_{R}^{b} B; N ⊗_{R} B; γ_{N}; N ⊗_{R}^{b} B; g ⊗ b; g(b); (I_{N} ⊗ε)g(1_{B})⊗b; g(1_{B})Δb ,]

where γ_{N} is described in 12.8(3), is commutative. This yields in particular

(B ⊗_{R}^{b} B)^{coB} ≃ Hom_{B}^{B}(B, B ⊗_{R}^{b} B) ≃ B ,

and the commutative diagram

[figure: Hom_{B}^{B}(B, B ⊗_{R}^{b} B) ⊗_{R} B; B ⊗_{R}^{b} B; B ⊗_{R} B; γ_{B}; B ⊗_{R}^{b} B; h ⊗ a; h(a); (I_{B} ⊗ ε)h(1_{B}) ⊗ a; h(1_{B})Δa.]

Proof. By 12.10, Hom_{B}^{B}(B, N ⊗_{R}^{b} B) ≃ Hom_{B}(B, N) ≃ N and commutativity of the diagrams is shown by a straightforward computation. ⊔ ⊓

12.16. Invariants. Let A be an R-algebra A and φ : A → R a ring morphism. Considering R as a left A-module, one may ask for the A-morphisms from R → M, where M ∈ _{A}M. Define the invariants of M by

AM = {m ∈ M | am = φ(a)m for all a ∈ A}.

Then the map _{A}Hom(R, M) → ^{A}M, f ↦ f(1), is an R-module isomorphism.

12.17. Invariants for bialgebras. For any bialgebra B, the counit ε is a ring morphism and hence induces a left and right B-module structure on R. Therefore, for any left B-module M, the invariants of M corresponding to ε come out as

BM = {m ∈ M | bm = ε(b)m for all b ∈ B}.

Furthermore, the map _{B}Hom(R, M) → ^{B}M, f ↦ f(1), is an R-module isomorphism. The left invariants ^{B}B of B are called left integrals in B,

BHom (R, B) ≃ ^{B}B = {c ∈ B | bc = ε(b)c for all b ∈ B}.

Right invariants and right integrals in B are defined symmetrically .

On the other hand, for the dual algebra B^{∗}, the map φ : B^{∗} → R, f ↦ f(1_{B}), is a ring morphism. Coinvariants of right B-comodules are closely related to invariants of left B^{∗}-modules corresponding to φ.

12.18. Invariants and coinvariants. Let B be a bialgebra that is locally projective as an R-module (cf. 8.2).

(1) For any M ∈ M^{B}, ^{B^{∗}}M = M^{coB}.

(2) For the trace ideal T = Rat^{B}(B^{∗}), ^{B^{∗}}T = T^{coB}.

(3) If B_{R} is finitely generated, then ^{B^{∗}}B^{∗} = (B^{∗})^{coB}.

Proof. (1) Let m ∈ ^{B^{∗}}M and f ∈ B^{∗}. From f ⇀m = ∑ m\underline{0}f(m\underline{1}) we conclude

(I_{M} ⊗ f)ϱ^{M}(m) = (I_{M} ⊗ f)(m ⊗ 1_{B}).

Now local projectivity of B implies that ϱ_{M}(m) = m⊗1_{B}, that is, m ∈ M^{coB}, as required. Conversely, take any m ∈ M^{coB} and compute

f ⇀m = (I_{M} ⊗ f)ϱ^{M}(m) = mf(1_{B}) = mφ(f).

This shows that m ∈ ^{B^{∗}}M, and therefore ^{B^{∗}}M = M^{coB}.

(2) From the definition of the trace ideal we know that T ∈ M^{B}; hence the assertion follows from (1).

(3) If B_{R} is finitely generated and projective, then T = B^{∗} and the assertion follows from (2). ⊔ ⊓

Every Hopf module M ∈ M_{B}^{B} is a right B-comodule, and hence it is a left B^{∗}-module (in the canonical way). This yields an action of B^{op} ⊗_{R} B^{∗} on M,

B^{op} ⊗_{R} B^{∗} ⊗_{R} M → M, (a ⊗ f) ⊗ m ↦ (a ⊗ f)ϱ^{M}(m) = ∑ m\underline{0}af(m\underline{1}).

This action is obviously an R-linear map, but it does not make M a module with respect to the canonical algebra product in B^{op} ⊗_{R} B^{∗}. On the other hand, there exists a different multiplication on B^{op} ⊗_{R} B^{∗} that makes M a module over the new algebra. Denote this product by “?”. For all a ∈ B, f, g ∈ B^{∗}, and m ∈ M, a product ? has to satisfy the associative law

[(a ⊗ f)?(b ⊗ g)](m) = (a ⊗ f)((b ⊗ g)m) = ∑(a ⊗ f)(m\underline{0}bg(m\underline{1}))
= ∑ m\underline{0}b\underline{1}a f(m\underline{1}b\underline{2})g(m\underline{2})
= ∑ m\underline{0}b\underline{1}a (b\underline{2}⇁f) ∗ g(m\underline{1})
= [∑ b\underline{1}a ⊗ (b\underline{2}⇁f) ∗ g](m).

From this we can see how the multiplication ? on B^{op} ⊗_{R} B^{∗} should be constructed in order to possess the desired properties.

12.1. Smash product B^{op}#B^{∗}. Consider an algebra B^{op}#B^{∗}, which is isomorphic to the tensor product B^{op} ⊗_{R} B^{∗} as an R-module and has the product

(a#f)(b#g) := ((Δb)(a#f)) (1_{B}#g) = ∑ b\underline{1}a#(b\underline{2}⇁f) ∗ g,

where a#f = a⊗f is the notation. Then B^{op}#B^{∗} is an associative R-algebra with unit 1_{B}#ε, and the maps

\begin{aligned}B^{op} & → & B^{op}#B^{∗}, & a ↦ a#ε, \\ B^{∗} & → & B^{op}#B^{∗}, & f ↦ 1_{B}#f, \end{aligned}

are injective ring morphisms, making every left B#B^{∗}-module a right Bmodule and a left B^{∗}-module. The algebra B^{op}#B^{∗} is called a smash product.

Every M ∈ M_{B}^{B} is a left B^{op}#B^{∗}-module, and therefore M_{B}^{B} is embedded in σ_{B^{op}#B^{∗}}[B ⊗_{R}^{b} B] ⊂ _{B^{op}#B^{∗}}M. If B_{R} is locally projective, then

M_{B}^{B} = σ_{B#B^{∗}}[B ⊗_{R}^{b} B] = σ_{B#B^{∗}}[B ⊗_{R}^{c} B].

In particular, M_{B}^{B} = _{B^{op}#B^{∗}}M provided that B_{R} is finitely generated and projective.

Proof. The first assertions are immediate consequences of the action considered above and the definition of the product #. The local projectivity implies that the right B-comodule structures correspond to left B^{∗}-module structures.

If B_{R} is finitely generated and projective, then there is a right coaction (see 7.10) B^{∗} → End_{R}(B) ≃ B^{∗} ⊗_{R} B, g ↦ (I_{B^{∗}} ⊗ g) ∘ Δ. The map

B^{∗} ⊗_{R}^{c} B → B^{op}#B^{∗}, f ⊗ b ↦ b#f,

is an isomorphism of left B^{op}#B^{∗}-modules. Indeed, note that, for any b, x ∈ B and f, g ∈ B^{∗},

∑(b⇁f)(g\underline{0}(x)g\underline{1}) = (b⇁f)(I_{B} ⊗ g)Δ(x) = (b⇁f) ∗ g(x).

Using these identities we compute

(a#f)(g ⊗ b) = (a ⊗ f)ϱ^{B^{∗}⊗B}(g ⊗ b) = ∑g\underline{0} ⊗ b\underline{1}af(g\underline{1}b\underline{2})
↦ ∑b\underline{1}a#(b\underline{2}⇁f)(g\underline{1})g\underline{0} = ∑b\underline{1}a#(b\underline{2}⇁f) ∗ g
= (a#f)(b#g),

that is, the map defined above is a morphism of left B^{op}#B^{∗}-modules. Clearly it is an isomorphism. Therefore, B^{op}#B^{∗} ∈ M_{B}^{B} and hence M_{B}^{B} = _{B^{op}#B^{∗}}M.

⊔ ⊓

13 Antipodes and Hopf algebras

13.1. The ring (End_{R}(B), ∗). For any R-bialgebra B, (End_{R}(B), ∗) is an associative R-algebra with product, for f, g ∈ End_{R}(B),

f ∗ g = μ ∘ (f ⊗ g) ∘ Δ,

and unit ι ∘ ε, that is, ι ∘ ε(b) = ε(b)1_{B}, for any b ∈ B (cf. 5.3). If B is commutative and cocommutative, then (End_{R}(B), ∗) is a commutative algebra.

Definitions. An element S ∈ End_{R}(B) is called a left (right) antipode if it is left (right) inverse to I_{B} with respect to the convolution product on End_{R}(B), that is, S ∗ I_{B} = ι ∘ ε (resp. I_{B} ∗ S = ι ∘ ε). In case S is a left and right antipode, it is called an antipode. The corresponding conditions are

μ ∘ (S ⊗ I_{B}) ∘ Δ = ι ∘ ε, μ ∘ (I_{B} ⊗ S) ∘ Δ = ι ∘ ε.

Explicitly, for all b ∈ B, an antipode S satisfies the following equalities:

∑ S (b\underline{1}) b\underline{2} = ε(b)1_{B} = ∑ b\underline{1}S (b\underline{2}) .

Left and right antipodes need not be unique, whereas an antipode is unique whenever it exists. A bialgebra with an antipode is called a Hopf algebra.

Antipodes are related to the right Hopf module morphism

γ_{B} : B ⊗_{R} B → B ⊗_{R}^{b} B, a ⊗ b ↦ (a ⊗ 1_{B})Δb = ∑ ab\underline{1} ⊗ b\underline{2}.

Notice that γ_{B} is also a left B-module morphism in an obvious way.

13.2. Existence of antipodes. Let B be an R-bialgebra.

(1) B has a right antipode if and only if γ_{B} has a left inverse in _{B}M.

(2) If B has a left antipode, then γ_{B} has a right inverse in _{B}M.

(3) γ_{B} is an isomorphism if and only if B has an antipode.

Proof. (1) If β is a left inverse of γ_{B}, for all b ∈ B, 1_{B} ⊗ b = β ∘ γ_{B}(1_{B} ⊗ b) = β(Δb) holds. This implies that ι ∘ ε(b) = (I_{B} ⊗ ε) ∘ β(Δb). Then S = (I_{B} ⊗ ε) ∘ β(1_{B} ⊗ −) : B → B is a right antipode since

μ ∘ (I_{B} ⊗ S) ∘ Δ(b) = ∑ b\underline{1}((I_{B} ⊗ ε)β(1_{B} ⊗ b\underline{2})) = (I_{B} ⊗ ε) ∘ β(Δb) = ι ∘ ε(b),

where we used that β is left B-linear.

Now suppose that S : B → B is a right antipode. Then

β : B ⊗_{R}^{b} B → B ⊗_{R} B, a ⊗ b ↦ (a ⊗ 1_{B})(S ⊗ I_{B})(Δb) = ∑ aS(b\underline{1}) ⊗ b\underline{2}, is a left inverse of γ_{B}, since for any b ∈ B,

β ∘ γ_{B}(1_{B} ⊗ b) = β(Δb) = (μ ⊗ I_{B}) ∘ (I_{B} ⊗ S ⊗ I_{B}) ∘ (I_{B} ⊗ Δ)(Δb)
= (μ ⊗ I_{B}) ∘ (I_{B} ⊗ S ⊗ I_{B}) ∘ (Δ ⊗ I_{B})(Δb)
= ∑ μ ∘ (S ⊗ I_{B})(Δb\underline{1}) ⊗ b\underline{2}
= ∑ ε(b\underline{1})1_{B} ⊗ b\underline{2} = 1_{B} ⊗ b .

(2) Let S be a left antipode, that is, μ ∘ (S ⊗ I_{B})(Δb) = ι ∘ ε(b), for b ∈ B. Then

β : B ⊗_{R}^{b} B → B ⊗_{R} B, 1_{B} ⊗ b ↦ (S ⊗ I_{B})(Δb) = ∑ S(b\underline{1}) ⊗ b\underline{2},

is a right inverse of γ_{B}, since

\begin{aligned}γ_{B} ∘ β(1_{B} ⊗ b) = γ_{B}((S ⊗ I_{B})(Δb)) & = & ∑ S(b\underline{1})b\underline{2} ⊗ b\underline{3} \\ & = & ∑ ε(b\underline{1})1_{B} ⊗ b\underline{2} = 1_{B} ⊗ b . \end{aligned}

(3) Suppose that γ_{B} is bijective. Take any f ∈ End_{R}(B) and observe that if μ ∘ (I_{B} ⊗ f)(Δb) = 0, for all b ∈ B, then f = 0. Indeed, any element in B ⊗_{R} B can be written as a sum of elements of the form (a ⊗ 1_{B})(Δb) and

μ ∘ (I_{B} ⊗ f)((a ⊗ 1_{B})(Δb)) = a(μ((I_{B} ⊗ f)(Δb))) = 0,

implying μ(I_{B} ⊗ f)(B ⊗_{R} B) = Bf(B) = 0, and so f = 0, as claimed. By (1), there exists a right antipode S, and for this we compute

μ ∘ (I_{B} ⊗ μ ∘ (S ⊗ I_{B}) ∘ Δ)(Δb)
= μ ∘ (I_{B} ⊗ μ) ∘ (I_{B} ⊗ S ⊗ I_{B}) ∘ (I_{B} ⊗ Δ)(Δb)
= μ ∘ (μ ⊗ I_{B}) ∘ (I_{B} ⊗ S ⊗ I_{B}) ∘ (Δ ⊗ I_{B})(Δb)
= ∑ ε(b\underline{1})b\underline{2} = b = μ ∘ (I_{B} ⊗ ι ∘ ε)(Δb).

By the preceding observation this implies μ∘(S ⊗I_{B})∘Δ = ι∘ε, thus showing that S is also a left antipode. ⊔ ⊓

13.3. Properties of antipodes. Let H be a Hopf algebra with antipode S. Then:

(1) S is an algebra anti-morphism, that is, for all a, b ∈ H, S(ab) = S(b)S(a), and S ∘ ι = ι.

(2) S is a coalgebra anti-morphism, that is, tw ∘ (S ⊗ S) ∘ Δ = Δ ∘ S and ε ∘ S = ε.

(3) If S is invertible as a map, then, for any b ∈ H,

∑ S^{−1}(b\underline{2})b\underline{1} = ε(b)1_{H} = ∑ b\underline{2}S^{−1}(b\underline{1}) .

Proof. (1) Consider the convolution algebra H̃ := (Hom_{R}(H⊗_{R}H, H), ˜∗) corresponding to the canonical coalgebra structure Δ_{H⊗_{R}H} on H ⊗_{R} H with the counit ε̃ = ε ⊗ ε. In particular, the unit in H̃ comes out as

ι̃: H ⊗_{R} H ⟶^{ε⊗ε} R ⟶^{ι} H .

In addition to the product μ : H ⊗_{R} H → H, consider the R-linear maps

ν : H ⊗_{R} H → H, a ⊗ b ↦ S(b)S(a), ρ : H ⊗_{R} H → H, a ⊗ b ↦ S(ab) .

To prove that S is an anti-multiplicative map, it is sufficient to show that ρ˜∗μ = μ˜∗ν = ι̃∘ ε̃ (the identity in H̃). By the uniqueness of inverse elements we are then able to conclude that ν = ρ. Consider the R-linear maps

\xymatrix{H ⊗_{R} H \ar[r]^{Δ_{H⊗H}} & H ⊗_{R} H ⊗_{R} H ⊗_{R} H \ar[r]^{ρ⊗μ} \ar[r]_{μ⊗ν} & H ⊗_{R} H \ar[r]^{μ} & H .}

Take any a, b ∈ H and compute

a ⊗ b ↦ ∑ a\underline{1} ⊗ b\underline{1} ⊗ a\underline{2} ⊗ b\underline{2}

⟼^{ρ⊗μ} ∑ S(a\underline{1}b\underline{1})a\underline{2}b\underline{2} = S ∗ I_{H}(ab) = ε(ab)1_{H} ,

⟼^{μ⊗ν} ∑ a\underline{1}b\underline{1}S (b\underline{2}) S (a\underline{2})

= ∑ a\underline{1}S (a\underline{2}) ε(b) = ε(a)ε(b)1_{H} .

Thus ν = ρ, and S is an anti-multiplicative map, that is, S(ab) = S(a)S(b). Furthermore, 1_{H} = ι ∘ ε(1_{H}) = (I_{H} ∗ S)(1_{H}) = S(1_{H}), so that S is a unital map and hence an algebra anti-morphism.

(2) This is a dual statement to (1), and we use a similar technique as for the proof of (1). In this case consider the convolution algebra corresponding to H as a coalgebra and H ⊗_{R} H as an algebra, (Hom_{R}(H, H ⊗_{R} H), \underline{∗}). Let ν := tw ∘ (S ⊗ S) ∘ Δ and ρ := Δ ∘ S. Direct computation verifies that ρ \underline{∗} Δ = ι_{H} ∘ ε_{H⊗H} = Δ \underline{∗} ν . From this we conclude that ρ = ν, so that S is an anti-comultiplicative map. Furthermore, for all a ∈ H, we know that ε(ι ∘ ε(a)) = ε(a), and ι ∘ ε(a) = ∑ S(a\underline{1})a\underline{2} . This implies

ε(a) = ε(ι ∘ ε(a)) = ∑ ε (S(a\underline{1})) ε(a\underline{2}) = ε ∘ S(a) ,

hence S is a coalgebra anti-morphism, as stated.

(3) Apply S^{−1} to the defining properties of S. ⊔ ⊓

We now prove that Hopf algebras are precisely those R-bialgebras for which the category M_{B}^{B} is equivalent to M_{R}. It is interesting to notice that this can be seen from a single isomorphism.

13.4. Fundamental Theorem of Hopf algebras. For any R-bialgebra B the following are equivalent:

(a) B is a Hopf algebra (that is, B has an antipode);

(b) γ_{B} : B ⊗_{R} B → B ⊗_{R}^{b} B, a ⊗ b ↦ (a ⊗ 1_{B})Δb, is an isomorphism in M_{B}^{B};

(c) γ^{B} : B ⊗_{R}^{c} B → B ⊗_{R} B, a ⊗ b ↦ Δa(1_{B} ⊗ b), is an isomorphism in M_{B}^{B};

(d) for every M ∈ M_{B}^{B}, M^{coB} ⊗_{R} B → M, m ⊗ b ↦ mb, is an isomorphism in M_{B}^{B};

(e) for every M ∈ M_{B}^{B}, there is an isomorphism (in M_{B}^{B})

φ_{M} : Hom_{B}^{B}(B, M) ⊗_{R} B → M, f ⊗ b ↦ f(b);

(f) φ_{B⊗B} : Hom_{B}^{B}(B, B ⊗_{R}^{b} B) ⊗_{R} B → B ⊗_{R}^{b} B is an isomorphism in M_{B}^{B};

(g) Hom_{B}^{B}(B, −) : M_{B}^{B} → M_{R} is an equivalence (with inverse − ⊗_{R} B). If B is flat as an R-module, then (a)-(g) are equivalent to:

(h) B is a (projective) generator in M_{B}^{B};

(i) B is a subgenerator in M_{B}^{B}, and φ_{M} is injective for every M ∈ M_{B}^{B}. If B_{R} is locally projective, then (a) − (i) are equivalent to:

(j) B is a subgenerator in M_{B}^{B} and the image of B^{op}#B^{∗} → End_{R}(B) is dense (for the finite topology).

For any Hopf module M over a Hopf algebra B, the coinvariants M^{coB} are an R-direct summand of M.

Proof. (a) ⇔ (b) was shown in 13.2, and by symmetry (see 12.9) the same proof implies (a)⇔(c). (b) ⇔ (f) is clear by 12.15.

(d) ⇔ (e) This follows from the commutative diagram in 12.13.

(a) ⇒ (d) For any M ∈ M_{B}^{B}, consider ϕ : M → M^{coB}, m ↦ ∑ m\underline{0}S(m\underline{1}). The following equalities show that the image of ϕ is in M^{coB}:

\begin{aligned}ϱ^{M}(ϕ(m)) = ϱ^{M}(∑ m\underline{0}S(m\underline{1})) & = & ∑ m\underline{0}S(m\underline{3}) ⊗ m\underline{1}S(m\underline{2}) \\ & = & ∑ m\underline{0}S(m\underline{1}) ⊗ 1_{B} = ϕ(m) ⊗ 1_{B}. \end{aligned}

Now we show that the map

(ϕ ⊗ I_{B}) ∘ ϱ^{M} : M → M^{coB} ⊗_{R} B

is the inverse of the multiplication map ϱ_{M} : M^{coB} ⊗_{R} B → M. For m ∈ M,

ϱ_{M} ∘(ϕ⊗I_{B})(ϱ^{M}(m)) = ∑ ϕ(m\underline{0})m\underline{1} = ∑ m\underline{0}S(m\underline{1})m\underline{2} = ∑ m\underline{0}ε(m\underline{1}) = m.

On the other hand, for n ⊗ b ∈ M^{coB} ⊗_{R} B,

\begin{aligned}(ϕ ⊗ I_{B}) ∘ ϱ^{M}(nb) & = & (ϕ ⊗ I_{B})(∑ nb\underline{1} ⊗ b\underline{2}) = ∑ ϕ(nb\underline{1}) ⊗ b\underline{2} \\ & = & ∑ nb\underline{1} S(b\underline{2}) ⊗ b\underline{3} = ∑ nε(b\underline{1}) ⊗ b\underline{2} = n ⊗ b. \end{aligned}

(e) ⇒ (f) is trivial (take M = B ⊗_{R}^{b} B).

(e) ⇔ (g) From 12.14 we know Hom_{B}^{B}(B, − ⊗_{R} B) ≃ I_{_{R}M}. Condition (f) induces Hom_{B}^{B}(B, −) ⊗_{R} B ≃ I_{M_{B}^{B}}, and the two isomorphisms characterise an equivalence between M_{R} and M_{B}^{B}.

(g) ⇒ (h) Obviously (g) always implies that B is a generator in M_{B}^{B} and that B is projective in M_{B}^{B} (that is, Hom_{B}^{B}(B, −) : M_{B}^{B} → M_{R} preserves epimorphisms).

Now suppose that _{R}B is flat. Then M_{B}^{B} has kernels and Hom_{B}^{B}(B, −) is a left exact functor.

(h) ⇒ (i) Suppose that B is a generator in M_{B}^{B}. Of course any generator is in particular a subgenerator. For any M ∈ M_{B}^{B}, the set Λ = Hom_{B}(B, M) yields a canonical epimorphism

p : B^{(Λ)} → M, b_{f} ↦ f(b).

Choosing Λ′ = Hom_{B}(B, Ke p) we form – with a similar map p′ – the exact sequence in M_{B}^{B},

\xymatrix{B^{(Λ′)} \ar[r]^{p′} & B^{(Λ)} \ar[r]^{p} & M \ar[r] & 0 .}

Now apply Hom_{B}^{B}(B, −) to obtain the exact sequence

\xymatrix{Hom_{B}^{B}(B, B^{(Λ′)}) \ar[r] & Hom_{B}^{B}(B, B^{(Λ)}) \ar[r] & Hom_{B}^{B}(B, M) \ar[r] & 0 .}

By the choice of Λ and Λ′, this sequence is exact. Now tensor with − ⊗_{R} B to obtain the commutative diagram with exact rows (⊗ for ⊗_{R}),

\xymatrix{Hom_{B}^{B}(B, B^{(Λ′)})⊗B \ar[r] \ar[d]_{≃} & Hom_{B}^{B}(B, B^{(Λ)})⊗B \ar[r] \ar[d]_{≃} & Hom_{B}^{B}(B,M)⊗B \ar[d]^{φ_{M}} \ar[r] & 0 \\ B^{(Λ′)} \ar[r] & B^{(Λ)} \ar[r] & M \ar[r] & 0 .}

The first two vertical maps are bijective since Hom_{B}^{B}(B, −) commutes with direct sums. By the diagram properties this implies the bijectivity of φ_{M}.

(i) ⇒ (f) Assume that B is a subgenerator in M_{B}^{B} and that φ_{M} is injective for all M ∈ M_{B}^{B}. Then clearly φ_{N} is bijective for all B-generated objects N in M_{B}^{B} and M is a subobject of such an N. Choose an exact sequence 0 → M → N → L in M_{B}^{B} where N and L are B-generated. Then clearly φ_{N} and φ_{L} are bijective and there is a commutative diagram with exact rows,

\xymatrix{0 \ar[r] & Hom_{B}^{B}(B, M)⊗_{R}B \ar[r] \ar[d]^{φ_{M}} & Hom_{B}^{B}(B, N)⊗_{R}B \ar[r] \ar[d]^{φ_{N}} & Hom_{B}^{B}(B, L)⊗_{R}B \ar[d]^{φ_{L}} \\ 0 \ar[r] & M \ar[r] & N \ar[r] & L .}

From this we conclude that φ_{M} is also bijective.

(h) ⇔ (j) If B_{R} is locally projective, then, by 4.7(g), B is a generator in σ[_{End_{R}(B)}B]. Moreover, M_{B}^{B} = σ[_{B^{op}#B^{∗}}B ⊗_{R}^{b} B]. Now assume (h). Then M_{B}^{B} = σ[_{B^{op}#B^{∗}}B] and the density property follows by [3, 43.12]. On the other hand, given the density property and the subgenerating property of B, one has σ[_{End_{R}(B)}B] = σ[_{B^{op}#B^{∗}}B] and B is a generator in M_{B}^{B}.

The R-linear map ϕ : M → M^{coB} considered in the proof (a)⇒(d) splits the inclusion M^{coB} → M, thus proving the final statement. ⊔ ⊓

Notice that parts of the characterisations in 13.4 apply to Hopf algebras that are not necessarily flat as R-modules (see 13.7 for such examples).

13.5. Finitely generated Hopf algebras. For an R-bialgebra B with B_{R} finitely generated and projective, the following are equivalent:

(a) B is a Hopf algebra;

(b) γ_{B} : B ⊗_{R} B → B ⊗_{R}^{b} B is surjective;

(c) B has a left antipode;

(d) B^{op}#B^{∗} ≃ End_{R}(B);

(e) B is a generator in _{B^{op}#B^{∗}}M.

Proof. (a)⇔(b)⇔(c) follow from 13.2 and the fact that, for finitely generated projective R-modules, any surjective endomorphism is bijective.

(a) ⇒ (d) As a generator in M_{B}^{B} = _{B^{op}#B^{∗}}M, B is a faithful B^{op}#B^{∗}- module and the density property of B^{op}#B^{∗} (see 13.4) implies B^{op}#B^{∗} ≃ End_{R}(B).

(e) ⇔ (d) Since B is a subgenerator in σ[_{End_{R}(B)}B], the assertion follows from 13.4(j).

(e) ⇒ (a) Under the given conditions M_{B}^{B} = _{B^{op}#B^{∗}}M (see 12.1) and the assertion again follows from the Fundamental Theorem 13.4. ⊔ ⊓

Clearly, if B is a finitely generated projective R-module, then M^{B} = _{B^{∗}}M has (enough) projectives and 12.5 implies the following corollary.

13.6. Semigroup bialgebra. Let G be a semigroup with identity e. The semigroup algebra R[G] is the R-module R^{(G)} together with the maps (defined on the basis G and linearly extended)

μ : R[G] × R[G] ⟶ R[G], (g, h) ↦ gh and ι : R → R[G], r ↦ re .

Since R[G] is a free R-module, there are also linear maps (see 5.6)

Δ : R[G] ⟶ R[G] ⊗_{R} R[G], g ↦ g ⊗ g, and ε : R[G] ⟶ R, 1 ↦ 1, g ↦ 0.

It is easily seen from the definitions that Δ and ε are algebra morphisms.

If G is a group, then S : R[G] → R[G], g ↦ g^{−1}, is an antipode, that is, in this case R[G] is a Hopf algebra.

13.7. Polynomial Hopf algebra. As noticed in 5.8, for any commutative ring R, the polynomial algebra R[X] is a coalgebra by

\begin{matrix}Δ_{2} : R[X] ⊗_{R} R[X] → R[X], & 1 ↦ 1, X^{i} ↦ (X ⊗ 1 + 1 ⊗ X)^{i}, \\ ε_{2} : R[X] → R, & 1 ↦ 1, X^{i} ↦ 0, i = 1, 2, …. \end{matrix}

Together with the polynomial multiplication this yields a (commutative and cocommutative) bialgebra that is a Hopf algebra with antipode

S : R[X] → R[X], 1 ↦ 1, X → −X.

For any a ∈ R, denote by J the ideal in R[X] generated by aX. Since

Δ_{2}(aX) = 1 ⊗ aX + aX ⊗ 1, ε_{2}(aX) = 0 and S(aX) = −aX ,

it is easily seen that J is a Hopf ideal. Therefore H = R[X]/J is a Hopf algebra over R. Notice that H need no longer be projective or flat as an R-module. In particular, if R is an integral domain and 0 ≠ a ∈ R, then Hom_{R}(R/aR, R) = 0 and H^{∗} = Hom(H, R) ≃ R, and H-subcomodules of H do not correspond to H^{∗}-submodules.

Exercises

13.8. Exercises for Modules and Tensor Products.

(1) Let M, M′, M′′ be right and N, N′, N′′ left R-modules. Prove:

(i) I_{M} ⊗ I_{N} = I_{M⊗_{R}N}.

(ii) For any morphism f : M → M′, f ⊗ 0 = 0.

(iii) For morphisms f : M → M′, f′ : M′ → M′′ and g : N → N′, g′ : N′ →

N′′,

(f′ ⊗ g′) ∘ (f ⊗ g) = f′ ∘ f ⊗ g′ ∘ g.

(iv) If f and g are isomorphisms, then f ⊗ g is an isomorphism, and

(f ⊗ g)^{−1} = f^{−1} ⊗ g^{−1}.

(v) For f_{1}, f_{2} : M → M′ and g_{1}, g_{2} : N → N′,

(f_{1} + f_{2}) ⊗ g = f_{1} ⊗ g + f_{2} ⊗ g and f ⊗ (g_{1} + g_{2}) = f ⊗ g_{1} + f ⊗ g_{2}.

(2) Let K ⊂ R_{R} and L ⊂ _{R}R be right and left ideals of R. Prove

R/K ⊗_{R} R/L ≃ R/(K + L) as ℤ-modules.

Conclude that ℤ/mℤ⊗_{ℤ}/nℤ ≃ ℤ/g(m, n)ℤ, where g(m, n) denotes the greatest common divisor of m, n ∈ ℤ.

(3) Let _{R}M_{S} be a bimodule and _{S}N an S-module. Prove: If _{R}M and _{S}N are flat modules, then _{R}M ⊗_{S} N is a flat R-module.

(4) Let μ_{ℚ} : ℚ ⊗_{ℤ} ℚ → ℚ and μ_{ℂ} : ℂ ⊗_{ℝ} ℂ → ℂ be the multiplication maps.

Prove:

(i) μ_{ℚ} and μ_{ℂ} are ring homomorphisms; (ii) μ_{ℚ} is an isomorphism, μ_{ℂ} is not monic.

(5) Prove:

(i) ℚ is flat as a ℤ-module.

(ii) For abelian torsion groups M (every element has finite order),

M ⊗_{ℤ} ℚ = 0. (iii) ℚ/ℤ ⊗_{ℤ} ℚ/ℤ = 0. (iv) For finite ℤ-modules K, L, K ⊗_{ℤ} L ≃ Hom_{ℤ}(K, L).

116 Chapter 3. Exercises

13.9. Exercises for Modules and Algebras.

(1) Let I be an ideal of the ring R. Put An_{M}(I) = {m ∈ M | Im = 0} for a left

R-module M. Prove:

(i) An_{M}(I) is an R/I-module. (ii) The map ψ : Hom_{R}(R/I, M) ⟶ An_{M}(I), f ⟼ f(1 + I),

is an isomorphism of R/I-modules. (iii) The assignment M ⟼ An_{M}(I) defines a functor

Hom_{R}(R/I, −) : R−Mod ⟶ R/I−Mod.

(iv) The functor Hom_{R}(R/I, −) respects essential monomorphisms and in-

jective modules.

(2) Let M be a left R-module and M^{∗} := Hom_{R}(M, R). Prove:

(i) M^{∗} is a right R-module. (ii) If M is finitely generated and projective, then M^{∗} is also finitely gen-

erated and projective. (iii) If M is a generator in R-Mod, then M^{∗} is a generator in Mod-R.

(3) Prove that for a right ideal J ⊂ R the following are equivalent:

(a) R/J is flat as right R-module;

(b) the exact sequence 0 → J → R → R/J → 0 is pure in Mod-R;

(c) for every left ideal I ⊂ R, we have JI = J ∩ I.

(4) Let T be any associative ring (without unit). A left T -module N is called s-unital if u ∈ T u for every u ∈ N. T itself is called left s-unital if it is s-unital as a left T -module. Prove that for a left T -module N, the following are equivalent:

(a) N is an s-unital T -module;

(b) for any n_{1}, …, n_{k} ∈ N, there exists t ∈ T with n_{i} = tn_{i} for all i ≤ k;

(c) for any set Λ, N^{(Λ)} is an s-unital T -module. Hint for (a) ⇒ (b): Assume the assertion holds for k − 1 elements. Choose t_{k} ∈ T such that t_{k}n_{k} = n_{k} and put m_{i} = n_{i}−t_{k}n_{i}, for all i ≤ k. Choose t′ ∈ T satisfying m_{i} = t′m_{i}, for all i ≤ k − 1. Then consider t := t′ + t_{k} − t′t_{k} ∈ T.

(5) Prove that for an ideal T in an algebra A, the following are equivalent: (a) T is left s-unital;

(b) for every left ideal I of A, T I = T ∩ I;

(c) A/T is a flat right A-module.

13. Antipodes and Hopf algebras 117

13.10. Exercises for Coalgebras.

(1) Let g : A → A′ be an R-algebra morphism. Prove that, for any R-coalgebra

C,

Hom(C, g) : Hom_{R}(C, A) → Hom_{R}(C, A′)

is an R-algebra morphism.

(2) Let f : C → C′ be an R-coalgebra morphism. Prove: If f is bijective then f^{−1} is also a coalgebra morphism.

(3) Let C be a free R-module with basis {g_{i}, d_{i} | i ∈ ℕ} and define

Δ : C → C ⊗_{R} C, g_{i} ↦ g_{i} ⊗ g_{i},
d_{i} ↦ g_{i} ⊗ d_{i} + d_{i} ⊗ g_{i+1};
ε : C → R g_{i} ↦ 1,
d_{i} ↦ 0 .

Prove that (C, Δ, ε) is a coalgebra.

(4) Let C be a free R-module with basis {s, c} and define

Δ : C → C ⊗_{R} C, s ↦ s ⊗ c + c ⊗ s,
c ↦ c ⊗ c − s ⊗ s;
ε : C → R s ↦ 0,
c ↦ 1 .

Prove that (C, Δ, ε) is a coalgebra.

(5) Let C be an R-module and C^{∗} = Hom_{R}(C, R). For an R-submodule D ⊂ C

define

D^{⊥} := {f ∈ C^{∗} | f(D) = 0} = Hom_{R}(C/D, R) ⊂ C^{∗} ,

and for any subset J ⊂ C^{∗}, put

J^{⊥} := ⋂ {Ke f | f ∈ J} ⊂ C .

Prove that D ⊂ D^{⊥⊥}, and D = D^{⊥⊥}, provided that C/D is cogenerated by R.

(6) Now let C be an R-coalgebra and D ⊂ C be an R-submodule. Prove: (i) If D is a left C^{∗}-submodule of C, then D^{⊥} is a right C^{∗}-submodule. (ii) If D is a (C^{∗}, C^{∗})-sub-bimodule of C, then D^{⊥} is an ideal in C^{∗}. (iii) If D is a coideal in C, then D^{⊥} is a subalgebra of C^{∗}. (Hint: Recall that for f, g ∈ C^{∗} and c ∈ C, f ∗ g(c) = f(g⇀c).)

(7) With the notation above prove:

(i) If J ⊂ C^{∗} is a right (left) ideal, then J^{⊥} is a left (right) C^{∗}-submodule of C.

(ii) If J ⊂ C^{∗} is an ideal, then J^{⊥} is a (C^{∗}, C^{∗})-sub-bimodule of C.

(8) Let P, N be R-modules where P is projective. Prove:

(i) For any index set Λ, the canonical map R^{Λ} ⊗_{R} P → P^{Λ} is injective. (ii) For any family f_{λ} ∈ Hom_{R}(N, R), λ ∈ Λ,

⋂_{Λ}(Ke f_{λ} ⊗_{R} P) = (⋂_{Λ}Ke f_{λ}) ⊗_{R} P.

(9) Let C = (C, Δ, ε) be an R-coalgebra. Prove that the category of right Ccomodules is equivalent to the category of left comodules over the opposite coalgebra C^{cop} = (C, tw ∘ Δ, ε). (Hint: If ϱ^{M} : M → M ⊗_{R} C defines a right C-comodule then tw ∘ ϱ^{M} : M → C ⊗_{R} M yields a left C^{cop}-comodule.)

(10) Let R be a von Neumann regular ring and f : M_{R} → M′_{R} and g : _{R}N → _{R}N′ two R-module homomorphisms. Prove that

Ke f ⊗ g = Ke f ⊗_{R} N + M ⊗_{R} Ke g,

(11) Let R be von Neumann regular, C ∈ M_{R}, C^{∗} = Hom_{R}(C, R), D ⊂ C, and J ⊂ C^{∗}. Recall the definitions

D^{⊥} := {f ∈ C^{∗} | f(D) = 0} ⊂ C^{∗} , J^{⊥} := ⋂ {Ke f | f ∈ J} ⊂ C .

Prove:

(i) (J_{1} ⊗_{R} J_{2})^{⊥} = J_{1}^{⊥} ⊗_{R} C + C ⊗_{R} J_{2}^{⊥}. (ii) If C is a coalgebra and J ⊂ C^{∗} is a subalgebra, then J^{⊥} is a coideal. (iii) If C is a coalgebra and R is semisimple, then D ⊂ C is a coideal if and

only if D^{⊥} is a subalgebra.

(12) Let (A, μ, ι) be an R-algebra with_{R}A is finitely generated and projective with dual basis a_{1}, …, a_{n} ∈ A and π_{1}, …, π_{n} ∈ A^{∗}, where Hom_{R}(−, R) = (−)^{∗}. Recall the isomorphism

ψ : A^{∗} ⊗_{R} A^{∗} → (A ⊗_{R} A)^{∗}, f ⊗ g ↦ [a ⊗ b ↦ f(a)g(b)].

Prove:

(i) Describe ψ^{−1} in terms of the dual basis. (ii) A^{∗} is a coalgebra with coproduct

\xymatrix{A^{∗} \ar[r]^{μ^{∗}} & (A ⊗_{R} A)^{∗} \ar[r]^{ψ^{−1}} & A^{∗} ⊗_{R} A^{∗}}

and counit

ε := ι^{∗} : A^{∗} → R, f ↦ f(1_{A}).

(iii) The dual of the coalgebra A^{∗} is isomorphic (as an algebra) to A.

13. Antipodes and Hopf algebras 119

(13) For C ∈ M_{R}, C^{∗} = Hom_{R}(C, R), and subsets D ⊂ C, S ⊂ C^{∗} recall the definitions

D^{⊥} := {f ∈ C^{∗} | f(D) = 0} ⊂ C^{∗} , S^{⊥} := ⋂ {Ke f | f ∈ S} ⊂ C .

Prove:

(i) S^{⊥} =< S >^{⊥} and D^{⊥} =< D >^{⊥}, where < X > is the R-linear closure of X.

(ii) S^{⊥} = ((S^{⊥})^{⊥})^{⊥} and D^{⊥} = ((D^{⊥})^{⊥})^{⊥}.

(14) Let K be a field and V, W vector spaces over K. Prove:

For any u ∈ V ⊗_{K} W there exist n ∈ ℕ, linearly independent x_{1}, …, x_{n} ∈ V, and linearly independent y_{1}, …, y_{n} ∈ W, such that

u = ∑_{i=1}^{n} x_{i} ⊗ y_{i}.

(15) Let K be a field and C a K-coalgebra. An non-zero element g ∈ C is called grouplike if Δ(g) = g ⊗ g. Prove:

(i) For any grouplike element g ∈ C, ε(g) = 1. (ii) The set of grouplike elements is linearly independent.

(16) Let A be a finite dimensional algebra over the field K and A^{∗} the dual coalgeba of A. Prove that the grouplike elements of A^{∗} correspond to the algebra morphisms A → K.

(17) Let A be a ring and M a faithful left A-module. Prove that for an ideal T ⊂ A the following are equivalent:

(a) T is M-dense in A, i.e., for any a ∈ A and m_{1}, …, m_{n} ∈ M there exists t ∈ T such that

tm_{i} = am_{i}, for all i = 1, …, n.

(b) M is s-unital as T -module.

13.11. Exercises for Hopf Algebras.

(1) Let H be a Hopf algebra with antipode S that is finitely generated and projective as an R-module. Show that H^{∗} with the canonical structure maps is again a Hopf algebra.

(2) Prove that for a Hopf algebra H with antipode S, the following are equivalent:

(a) for any h ∈ H, ∑ S(h\underline{2})h\underline{1} = ε(h)1_{H};

(b) for any h ∈ H, ∑ h\underline{2}S(h\underline{1}) = ε(h)1_{H};

(c) S ∘ S = I_{H}.

(3) Let H, K be Hopf algebras with antipodes S_{H}, S_{K}, respectively. Prove that, for any bialgebra morphism f : H → K, S_{K} ∘ f = f ∘ S_{H}.

(4) Let H be a Hopf R-algebra that is finitely generated and projective as an R-module. Prove:

(i) The antipode S of H is bijective.

(ii) The right coinvariants (H^{∗})^{coH} of H^{∗} form a finitely generated projec-

tive R-module of rank 1. (iii) If (H^{∗})^{coH} ≃ R, then H ≃ H^{∗} as left H-modules (that is, H is a

Frobenius algebra).

Bibliography

[1] Abe, E., Hopf Algebras, Cambridge University Press, Cambridge (1977)

[2] Bourbaki, N., Algebra I, Addison-Wesley, Reading, MA (1974)

[3] Brzeziński, T., Wisbauer, R., Corings and Comodules, Cambridge University Press (2003)

[4] Dǎscǎlescu, S., Nǎstǎsescu, C., Raianu, S., Hopf Algebras. An Introduction, Marcel Dekker, New York-Basel (2001)

[5] Kassel, C., Quantum Groups, Springer, Berlin (1995)

[6] Lusztig, G., Introduction to Quantum Groups, Birkhäuser, Basel (1993)

[7] Majid, S., A Quantum Group Primer, Cambridge University Press, Cambridge (2002)

[8] Montgomery, S., Hopf Algebras and Their Actions on Rings, Reg. Conf. Series in Math., CBMS 82, AMS, Providence RI (1993)

[9] Sweedler, M.E., Hopf Algebras, Benjamin, New York (1969)

[10] Wisbauer, R., Foundations of Module and Ring Theory, Gordon and Breach, Reading-Paris (1991) (Grundlagen der Modul- und Ringtheorie, Verlag Reinhard Fischer, München (1988))

[11] Wisbauer, R., Modules and Algebras: Bimodule Structure and Group Actions on Algebras, Pitman Mono. PAM 81, Addison Wesley, Longman, Essex (1996)

[12] Wisbauer, R., Decompositions of modules and comodules, in Algebra and Its Applications, Huynh et al. (Eds.), AMS Contemp. Math. 259, 547– 561 (2000)

Index

(A, B)-exact sequences, 28 (A, B)-finite modules, 28 (A, B)-injective modules, 28 (A, B)-projective modules, 28 (A, B)-semisimple modules, 29 (C, R)-injective comodule, 58 (M, B)-injective modules, 28 (M, B)-projective modules, 28 C locally projective as C^{∗}-module.,

84 C projective in M^{C}, 84 C-comodules and C^{∗}-modules, 66 C^{∗}-modules, 63 L-pure morphism, 12 M-flat modules, 9 N-projective, 81 R as coalgebra., 41 Σ-notation, 38 α-condition, 64 σ-decomposition of modules, 36 σ-indecomposable, 36 Homrelations, 53

trivial coaction, 97

algebra morphism, 16 algebras, 15 antipode, 108 associated algebras, 39

balanced map, 1 bi-ideal, 95 bialgebra, 95 bialgebra morphism, 95 big cogenerator, 35 bimodules, 98 category M^{C}, 56 category σ[M], 22 category M_{B}^{B}, 101, 102 coaction and C^{∗}-modules, 66 coaction, right, 49 coactions and tensor products, 52 coalgebra, 38 coalgebra morphism, 44 coalgebra, opposite, 43 coalgebra. cocommutative, 43 coalgebras over QF rings, 87 coassociative, 38 coassociative coaction, 49 cocommutative, 38 coideal, 45 coinvariants of a comodule, 103, 104 comodule morphism, 50 comodule, N-injective, 81 comodule, relative injective, 58 comodule, right, 49 comodule, semisimple, 71 comodules, 49 comodules finitely presented as R-

modules, 54 convolution algebra, 41 coproduct of coalgebras, 47 coproduct of comodules, 52 correspondence relations, 35 counital coaction, 49

decomposition of subcategories, 36 decompositions over Noetherian rings,

79 diagonal coaction, 97 direct sum of coalgebras, 47 dual algebra, 41 duals of algebras, 43

endomorphism ring of C, 55 existence of tensor products, 1

factor coalgebra, 69 factorisation theorem, 46 Finiteness Theorem, 71 flat factor rings, 30 fully invariant submodule, 35 fundamental theorem of Hopf alge-

bras, 111

generator, 111, 113 graded modules, 49

Hom-tensor relation, 8 Hom-tensor relations over algebras,

18 Hopf algebra, 108 Hopf module morphism, 99

idempotents and comodules, 70 injective cogenerator in M^{C}, 81 injective cogenerator in _{C^{∗}}M, 82 injective comodules, 60 injective in M^{C}, 60 injective modules, 22 integral, left, 97, 105 intersection property, 13 invariants, 105 invariants of a module over a bial-

gebra, 105 irreducible coalgebra, 90

kernels and cokernels, 51

left C-comodules and right C^{∗}-modules,

66 left antipode, 108 left integral, 97, 105 left integrals in B, 105 local module, 25 locally Artinian, 25 locally Noetherian, 24 locally of finite length, 25 locally projective, 77

M-injective, 22 M-projective, 23 matrix coalgebra, 43 modules over algebras, 17 Morita dualities, 27

opposite coalgebra, 43

perfect category, 26 polynomial coalgebra, 42 progenerator, 23 projective coalgebras over QF rings,

91 projective comodules, 61 projective cover in σ[M], 25 projective in M^{C}, 61 projective modules, 23 properties of the antipode, 109 pure morphism, 12 pure morphisms, 12 pure submodule, 12

Quasi-Frobenius, QF module, 27

rational element, 74 rational functor, 74 rational functor exact, 78 rational submodule, 74 rational submodule of C^{∗∗}, 75 relative injective comodule, 58 right antipode, 108 right rational C^{∗}-module, 76 right semiperfect coalgebra, 83 right semiperfect coalgebras, 83, 88 right trace ideal, 76

s-unital, 30 s-unital ideals, 30 scalar extension, 41 self-generator, 22 semigroup bialgebra, 113 semigroup coalgebra, 42 semiperfect category, 26 semiperfect module, 25 semisimple comodule, 71 semisimple modules, 24 simple coalgebras, 73 small submodule, 25 smash product, 106, 107 sub-bialgebra, 95 subcoalgebra, 46 subcomodule, 52 subgenerator, 23 superfluous in M, 25 superfluous submodule, 25 supplement in M, 25

tensor functors, 5 tensor product, 1 tensor product and direct products,

5 tensor product and direct sums, 3 tensor product of B-comodules, 96 tensor product of B-modules, 96 tensor product of algebras, 16 tensor product of coalgebras, 48 tensor product of comodules, 96 tensor product of homomorphisms,

2 tensor product over commutative rings,

12 tensor product with cyclic modules,

8 trace ideal, 76 trace ideal and decompositions, 79 twist map, 13

universal property of the tensor product, 17

weak QF modules, 27

zero in the tensor product, 7