R. WISBAUER University of D¨usseldorf, Germany e-mail address: wisbauer@math.uni-duesseldorf.de
Abstract. For a long period the theory of modules over rings on the one hand and comodules and Hopf modules for coalgebras and bialgebras on the other side developed quite independently. In this talk we want to outline how ideas from module theory can be applied to enrich the theory of comodules and vice versa. For this we considerA-coringsC with grouplike elements over a ring A, in particular Galois corings. IfAis right self-injective it turns out thatC is a Galois coring if and only if for any injective comoduleN the canonical map HomC(A, N)⊗BA→N is an isomorphism, whereB = EndC(A), the ring of coinvariants ofA. Together with flatness ofBAthis characterisesAas generator in the category of rightC-comodules. This is a special case of the fact that over any ringA, anA-moduleMis a generator in the categoryσ[M] (objects areA-modules subgenerated byM) if and only ifM is flat as module over its endomorphism ringSand the evaluation mapM⊗SHom(M, N)→N is an isomorphism for injective modulesN inσ[M].
1. Introduction
Not being born as a member of the Hopf family I lived for many years with modules and rings without paying attention to the developments in the theory of Hopf alge- bras. Somehow I had the impression that in the coalgebra world additive categories are not of central importance and that the inversion of arrows in the definition of comodules also turned the interest of researchers to different directions. It was only in recent years that - by comments of colleagues - I became aware of the fact that the central notion of my own work, the subgenerator of a module category, could also be of interest to comodule theory. In fact it was known from Sweedler’s book that for coalgebras over fields, every comodule is contained in a direct sum of copies ofC showing thatC is a cogenerator as well as a subgenerator for the comodules.
While for coalgebrasC over rings in general the cogenerator property ofCis lost, it is easy to see thatC is still a subgenerator. This was the motivation for me to have a closer look at this theory and to investigate how my experience from module theory could contribute to a better understanding of the coalgebraic world.
Seeing things from a different angle, it was not surprising that I sometimes came up with interesting answers to questions which native Hopf people had not previously considered. General (co-)module theory cannot make new contributions to the classification of finite dimensional (co-) algebras since in this special case the general notions coincide with more familiar ones. Probably because of this, quite a
2000Mathematics Subject Classification. 16W30.
Key words and phrases. Coring; semisimple coring; module categories; coinvariants functor;
structure theorem; Hopf algebras.
1
few traditionalists doubted if it makes any sense to study Hopf algebras over rings instead of fields. The situation is reminiscent of Jacobson’s definition of a radical for any ring, extending the nilpotent radical for finite dimensional algebras. While his radical did not contribute to the classification of simple algebras, it certainly deepened and widened the understanding of ring and module theory.
Familiarity with coalgebras over commutative rings needs only a small step to non-commutative base rings, leading to the notion of corings. The formalism and results from module theory readily apply to this more general situation and in what follows I’ll try to give some idea of how they can be used. Many of the observations to be reported result from cooperation and discussions with Tomasz Brzezi´nski and other colleagues.
2. Modules and comodules
LetAbe any associative ring and denote byMA andAM the categories of unital right and leftA-modules, respectively.
LetCbe anA-coring, i.e., an (A, A)-bimodule with coassociative comultiplication
∆ :C → C ⊗ACand counitε:C →A.
RightC-comodulesare rightA-modulesM with aright coaction
%M :M →M⊗AC,
which is coassociative and counital. The categories of left and right C-comodules are denoted byCMandMC, respectively.
The investigation of a ring A is strongly influenced by the fact that A is a projective generator for the left and for the right A-modules. AnA-coringC need not be a generator or cogenerator for theC-comodules nor is it projective or injective in general. However, every comodule is a subcomodule of a comodule which is generated byC and hence structural properties ofCmay transfer to comodules.
2.1. C is a subgenerator in MC. ForX ∈MA,X⊗AC is a rightC-comodule by IX⊗∆ :X⊗AC −→X⊗AC ⊗AC,
and for any M ∈ MC, the structure map %M : M → M ⊗AC is a comodule morphism.
Moreover any epimorphism A(Λ) → M of A-modules yields a diagram in MC with exact bottom row
M
%M
A(Λ)⊗AC //M⊗AC //0,
showing thatM is a subcomodule of aC-generated comodule, i.e.,Cis a subgenerator inMC.
Let us mention that over a quasi-Frobenius (QF) ringA, anyA-coringC is an injective cogenerator in MC and in CM. In fact any comodule is contained in a direct sum of copies ofC.
Both the duals ofC as leftA-module and as rightA-module can be defined and are of importance for comodule theory. We concentrate on one side.
2.2. The dual rings. Let C be anA-coring. ∗C=AHom(C, A)is a ring with unit εwith respect to the product (for f, g∈∗C,c∈ C)
f∗lg : C−→ C ⊗∆ ACI−→ CC⊗g −→f A, f∗lg(c) =X
f(c1g(c2)), and there is a ring anti-homomorphism ι:A→∗C, a7→ε(−a).
The bridge from comodules to modules is provided by the following observation.
2.3. C-comodules and ∗C-modules. Any M ∈MC is a (unital) left ∗C-module by
*:∗C ⊗AM →M, f⊗m7→(IM ⊗f)◦%M(m).
Any morphismh:M →N inMC is a left∗C-module morphism, so HomC(M, N)⊂∗CHom (M, N),
and there is a faithful functor fromMC toσ[∗CC], the full subcategory of∗CMwhose objects are submodules ofC-generated ∗C-modules.
Given the basic constructions we pause to think about what we can learn from module theory for comodules.
(1) In caseMC =σ[∗CC] we can transfer all theorems from module categories of typeσ[M] to comodules without extra proofs.
(2) More generally we can focus on the situation whenCis flat as leftA-module, in which case MC is a Grothendieck category. Many results and proofs in σ[M] can then easily be transferred in this case.
(3) We may study MC without any conditions on theA-module structure ofC and ask which notions still make sense and which problems can be handled in this general situation. Here the tranfer of results from σ[M] needs more caution since monomorphisms inMC need no longer be injective maps.
We will take a brief look at the first two situations and then concentrate on certain aspects of the third one in the last section.
To describe the coincidence ofMC andσ[∗CC] recall that anA-moduleM is said to belocally projectiveif, for any diagram of leftA-modules with exact rows
0 // F i // M
g
L f // N //0,
whereF is finitely generated, there existsh:M →Lsuch that g◦i=f ◦h◦i.
2.4. MC as full subcategory of ∗CM. The following are equivalent:
(a) MC=σ[∗CC];
(b) for allM, N ∈MC,HomC(M, N) =∗CHom(M, N);
(c) C is locally projective as leftA-module;
(d) every left∗C-submodule of Cn,n∈N, is a subcomodule of Cn; (e) the inclusion functor i:MC →∗CM has a right adjoint.
Proof. We refer to [10, 3.5], [2], or [3].
In the situation considered in 2.4 all theorems known for module categories of type σ[M] can be formulated for comodules. In particular the decomposition theorems for module categories yield decompositions of comodule categories and coalgebras (e.g., [8]).
In the following caseMC is a Grothendieck category.
2.5. C as a flatA-module. The following are equivalent:
(a) C is flat as a left A-module;
(b) every monomorphism in MC is injective;
(c) every monomorphismU → C in MC is injective;
(d) the forgetful functor(−)A:MC →MA respects monomorphisms.
If these conditions hold thenMC is a Grothendieck category.
Proof. See [10, 3.4] or [2].
We note that if the categoryMC is Grothendieck thenCneed not be flat as left A-module (e.g., [3]).
3. Generators in module categories
In any (additive) category a generatorP is characterised by the faithfulness of the functor Hom(P,−). In full module categories the following characterization (due to C. Faith) is well known (e.g., [6, 18.8]).
3.1. Generator in AM. For an A-module M with S = End(AM), the following are equivalent:
(a) M is a generator inAM;
(b) (i) MS is finitely generated and S-projective, and (ii) A'End(MS).
The characterisation of generators inσ[M] is more involved.
3.2. Generator inσ[M]. For an A-moduleM with S= End(AM), the following are equivalent:
(a) M is a generator inσ[M];
(b) for every N∈σ[M], the following evaluation map is an isomorphism:
M ⊗SHom(M, N)→N , m⊗f 7→f(m) ; (c) (i) MS is flat,
(ii) for every injective module V ∈σ[M], the canonical map M⊗SHom(M, V)→V , m⊗f 7→f(m), is injective (bijective).
If (any of ) these conditions are satisfied the canonical mapA→End(MS)is dense.
Proof. Most of the implications are well-known (see [6, 15.7,15.9], [2]). Because of its relevance for what follows, we show
(c)⇒(a) For anyK∈σ[M], there exists an exact sequence 0→K→Q1→Q2, whereQ1, Q2 are injectives inσ[M]. We construct an exact commutative diagram (tensoring overS)
0 //M⊗AHom(M, K) //
µK
M ⊗AHom(M, Q1) //
'
M⊗AHom(M, Q2)
'
0 // K // Q1 g // Q2 ,
showing thatµK is an isomorphism and soK isM-generated.
For a better understanding of condition (c)(ii) recall the following special case (e.g., [6, 25.5]).
3.3. Hom-tensor relation. Given M, V ∈ AM, S = End(M), and L ∈ MS, consider the map
L⊗SHom(M, V)→HomA(HomS(L, M), V), l⊗f 7→[g7→(g(l))f].
This is an isomorphism providedLS is finitely presented andV isM-injective.
IfA→End(MS) is dense, settingM =Lyields the map
M⊗SHom(M, V)→HomA(HomS(M, M), V)'V , m⊗f 7→mf . Since every M-injective module V ∈σ[M] isM-generated, this map is surjective for such modules. To make the map injective, it suffices, for example, to haveMS finitely presented or pure projective, and no flatness condition onMS is needed.
More generally (c)(ii) can be related to descending chain conditions on certain matrix subgroups ofM. For details we refer to [7] and [11].
Projectivity of a generator M is also reflected by properties of M as a module over its endomorphism ring.
3.4. Projective generator in σ[M]. For an A-module M with S = End(AM), the following are equivalent:
(a) M is a projective generator in σ[M];
(b) (i) MS is faithfully flat,
(ii) for every injective module V ∈σ[M], the canonical map M⊗SHom(M, V)→V , m⊗f 7→f(m), is injective (bijective).
Proof. (a)⇒(b) By the generator propertyM is a flat module overS (see 3.2).
Projectivity of M in σ[M] implies Hom(M, M I) = I, for every left ideal I ⊂ S, henceM I6=M ifI6=S. This shows thatM is faithfully flat (e.g., [6, 12.17]).
(b)⇒(a) In view of 3.2 it remains to show thatM is projective inσ[M]. For this consider any epimorphismL−→f N in σ[M]. We obtain the commutative diagram with exact rows
M ⊗SHom(M, L) //
'
M ⊗SHom(M, N) //
'
M ⊗SCoke Hom(M, f) //0
L f // N // 0 ,
where the vertical maps are the canonical isomorphisms (see 3.2). From this we con- cludeM⊗SCoke Hom(M, f) = 0 and faithfulness ofMS implies Coke Hom(M, f) = 0 which means that Hom(M, f) is surjective.
4. Galois corings
Given anA-coringC we may ask whenA is aC-comodule.
4.1. Grouplike elements. A non-zero element gof anA-coringCis said to be a grouplikeelement if ∆(g) =g⊗g andε(g) = 1A.
An A-coring C has a grouplike element g if and only if A is a right or left C- comodule, by the coactions
%A:A→ C, a7→ga, A%:A→ C, a7→ag.
For a proof we refer to [1] or [2]. We writeAg orgA, when we consider Awith the right or left comodule structure induced byg.
Example. LetB→Abe a ring extension, and letC=A⊗BAbe the Sweedler A-coring. Theng= 1A⊗1A is a grouplike element inC.
4.2. Coinvariants. Given anA-coringCwith a grouplike elementgandM ∈MC, theg-coinvariantsofM are defined as theR-module
MgcoC ={m∈M |%M(m) =m⊗g}= Ke(%M−(− ⊗g)), and there is an isomorphism
ψM : HomC(Ag, M)→MgcoC, f 7→f(1A).
The isomorphism is derived from the fact that anyA-linear map with sourceA is uniquely determined by the image of 1A.
4.3. Coinvariants ofA and C. Let C be an A-coring with a grouplike element g.
Then:
(1) EndC(Ag)'AcoCg ={a∈Ag|ga=ag},
i.e., subalgebra ofA given by the centraliser ofg inA.
(2) For any X∈MA,(X⊗AC)coC 'HomC(Ag, X⊗AC)'X, and forX =A,
CcoC 'HomC(Ag,C)'HomA(Ag, A)'A, which is a left A- and rightEndC(Ag)-morphism.
4.4. The induction functor. For an A-coring C with grouplike element g, let B =AcoCg . Given any rightB-module M, M ⊗BA is a right C-comodule via the coaction
%M⊗BA:M⊗BA→M ⊗BA⊗AC ∼=M⊗BC, m⊗a7→m⊗ga.
For any morphismf :M →N inMB,
f⊗IA:M⊗AC →N⊗AC, m⊗a7→f(m)⊗a,
is a morphism in MC and hence the assignment M →M ⊗BA andf →f⊗BIA
defines a functor − ⊗BA:MB→MC known as aninduction functor.
Theg-coinvariants provide a functor in the opposite direction.
4.5. The g-coinvariants functor. LetC be an A-coring with a grouplike element g andB=AcoC. The functor
HomC(Ag,−) :MC →MB,
is the right adjoint of the induction functor− ⊗BA:MB →MC. Notice that for M ∈MC, the rightB-module structure ofHomC(Ag, M)is given byf·b(a) =f(ba).
This functor is isomorphic to the coinvariant functor Gg:= (−)coCg :MC →MB, M 7→McoC, which acts on morphisms by restriction of the domain, i.e.,
for any f :M →N in MC,Gg(f) =f |MgcoC. ForN ∈MB the unit of the adjunction is given by
ηN :N →(N⊗BA)coC, n7→n⊗1A, and forM ∈MC, the counit reads
σM :McoC⊗BA→M, m⊗a7→ma.
Notice that for any rightB-module N, there is a left A-module isomorphism HomC(N⊗BA,C)∼= HomA(N⊗BA, A)∼= HomB(N, A).
The structure of anA-coringCwith a grouplike elementginvolves two rings, the algebra Aitself and itsg-coinvariants algebraB, and a ring mapB →A. On the other hand, to any ring extension B→A one can associate its canonical Sweedler A-coring A⊗BA which also has a grouplike element. Thus we have two corings with grouplike elements: the originalA-coringC we started with and the canonical coring associated to the related algebra extensionB→A. It is natural to study the relationship between these corings, and, in particular, to analyse corings for which this relationship is given by an isomorphism. This leads to the notion of a Galois coringintroduced in [1].
Recall thatM ∈MC is said to be (C, A)-injectiveif for every C-comodule map i:N→L which is a coretraction inMA, every diagram
N i //
fBBBBBB
BB L
M
in MC can be completed commutatively by some g : L → M in MC. This is equivalent to%M :M →M ⊗AC being a coretraction inMC.
4.6. Galois corings. For anA-coringC with a grouplike elementg andB=AcoCg , the following are equivalent:
(a) The following evaluation map is an isomorphism:
ϕC : HomC(Ag,C)⊗BA→ C, f⊗a7→f(a);
(b) the (A, A)-bimodule map defined by
χ:A⊗BA→ C, 1A⊗1A7→g, is a (coring) isomorphism;
(c) for every (C, A)-injective comoduleN ∈MC, the evaluation ϕN : HomC(Ag, N)⊗BA→N, f⊗a7→f(a), is an isomorphism.
(C, g) is called aGalois coringif it satisfies the above conditions.
Proof. (a)⇔(b) Observe that the canonical isomorphism h: HomC(Ag,C)→A, f 7→ε◦f(1A), is rightB-linear, and we get the commutative diagram
HomC(Ag,C)⊗BA ϕ //
h⊗I
C,
=
f ⊗a
_ f(a)
=
A⊗BA χ //C, ε◦f(1A)⊗a ε◦f(1A)ga , where the last equality is obtained by colinearity off, which implies
ε◦f(1A)ga= (ε⊗I)(f(1A)⊗ga) = (ε⊗I)◦∆f(a) =f(a).
(b)⇒(c) First observe that for anyX∈MA,χyields the isomorphisms HomC(Ag, X⊗AC)⊗BA'X⊗BA'X⊗A(A⊗BA)'X⊗AC.
Now assumeN ∈MC to be (C, A)-injective and consider the commutative dia- gram
0 //HomC(Ag, N)⊗BA //
ϕN
N⊗BA //
'
(N⊗AC)⊗BA
'
0 //N //N⊗AC //N⊗AC ⊗AC,
where the top row is exact by the purity (splitting) property shown in 4.7 below, and bijectivity of the two vertical maps follows from the preceding remark. From this, bijectivity ofϕN follows.
(c)⇒(a) This is obvious sinceCis always (C, A)-injective.
Let us mention that weak Galois corings are considered in [9, 2.4]. For such corings the action ofA onC is not required to be unital.
The purity condition needed above arises from the following splitting property (forL=A).
4.7. Splitting induced by (C, A)-injectivity. For anA-coring C, let M ∈MC be(C, A)-injective. Then for anyL∈MC, the canonical sequence
0 //HomC(L, M) i //HomA(L, M) γ //HomA(L, M ⊗AC), splits inMB, whereB= EndC(L)andγ(f) =%M ◦f −(f ⊗IC)◦%L.
A similar result holds for relative injective left comodules.
Proof. Denote byh:M ⊗AC →M the splitting map of%M in MC. Then the map
HomA(L, M)'HomC(L, M⊗AC)→HomC(L, M), f 7→h◦(f⊗IC)◦%L, splits the first inclusion inMB, and the map
HomA(L, M⊗AC)→HomA(L, M), g7→h◦g,
yields a splitting map HomA(L, M⊗AC)→HomA(L, M)/HomC(L, M), since for anyf ∈HomA(L, M),
h◦γ(f) =f−h◦(f⊗IC)◦%L∈f+ HomC(L, M).
The next theorem shows which additional condition on A is sufficient to make Aga comodule generator for a GaloisA-coring (C, g). The second part is essentially [1, Theorem 5.6].
4.8. The Galois Coring Structure Theorem. Let C be an A-coring with grou- plike element g andB=AcoCg .
(1) The following are equivalent:
(a) (C, g)is a Galois coring andBAis flat;
(b) AC is flat and Ag is a generator inMC. (2) The following are equivalent:
(a) (C, g)is a Galois coring andBAis faithfully flat;
(b) AC is flat and Ag is a projective generator inMC;
(c) AC is flat and HomC(Ag,−) :MC →MB is an equivalence with inverse − ⊗BA:MB →MC (cf. 4.5).
Proof. (1) (a)⇒(b) Assume (C, g) to be a Galois coring. Then in the diagram of the proof of 4.6, (c)⇒(b), the top row is exact by flatness of BA without any condition onN∈MC. So HomC(Ag, N)⊗BA→N is surjective (bijective) showing thatAgis a generator. Moreover the isomorphism− ⊗AC ' − ⊗A(A⊗BA) implies thatAC is flat.
(b)⇒(a) If AC is flat then monomorphisms in MC are injective. As for module categories one can show that the generator Ag in the categoryMC is flat over its endomorphism ringB, and HomC(Ag, M)⊗BA'M, for allM ∈MC.
(2) The proof for 3.4 also works for comodules.
If the ringAis right self-injective, thenCis injective inMC and the reformulation of the characterization of Galois corings and the Structure Theorem is just the description of generators in module categories (compare 3.2, 3.4).
4.9. Corollary. Assume Ato be a right self-injective ring and letC be anA-coring with grouplike elementg.
(1) The following are equivalent:
(a) (C, g)is a Galois coring;
(b) for every injective comodule N∈MC, the evaluation ϕN : HomC(Ag, N)⊗BA→N, f⊗a7→f(a), is an isomorphism.
(2) The following are equivalent:
(a) (C, g)is a Galois coring andBAis (faithfully) flat;
(b) BA is (faithfully) flat and for every injective comodule N ∈ MC, the following evaluation map is an isomorphism:
ϕN : HomC(Ag, N)⊗BA→N, f⊗a7→f(a).
We call a right C-comodule N semisimple (in MC) if every C-monomorphism 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. Simple and semisimple left C-comodules and (C,C)- bicomodules are defined similarly.
The coringCis said to beleft (right) semisimpleif it is semisimple as a left (right) comodule. Cis called asimple coalgebraif it is simple as a (C,C)-bicomodule.
From [2, 3] we recall:
4.10. Semisimple corings. For anA-coring C the following are equivalent:
(a) C is right semisimple;
(b) AC is projective andC is a semisimple left∗C-module;
(c) CA is projective andC is a semisimple rightC∗-module;
(d) C is left semisimple.
Note that not every canonical coring associated to an algebra extensionB→Ais a Galois coring with respect to a grouplike 1A⊗1A. However, if the extensionB →A is faithfully flat than (A⊗BA,1A⊗B1A) is a Galois-coring. As a particular example of this one can consider a Galois coring provided by Sweedler’s Fundamental Lemma (cf. [5, 2.2 Fundamental Lemma]).
4.11. Fundamental Lemma. Let A be a division ring. Suppose that C is anA- coring generated by a grouplike element g as an (A, A)-bimodule. Then (C, g)is a Galois coring.
Proof. Under the given conditionA is simple as left C-comodule and it sub- generatesC and hence MC. This implies that C is a simple and right semisimple coring andAis a projective generator inM. So (C, g) is a Galois coring by 4.8.
More general simple corings with grouplike elements can be characterised (com- pare also [3]) in the following way.
4.12. Simple corings. Let C be an A-coring with grouplike element g. Then the following are equivalent:
(a) C is a simple and left (or right) semisimple coring;
(b) (C, g)is Galois and EndC(Ag) is simple and left semisimple;
(c) χ:A⊗BA→ C is an isomorphism andB is a simple left semisimple subring of A;
(d) CA is flat, gA is a projective generator in CM, and EndC(gA) is simple and left semisimple.
Proof. LetCbe simple and left semisimple. Then there exists only one simple comodule (up to isomorphism) and so every non-zero comodule is a projective
generator in MC. In particular Ag is a finite direct sum of isomorphic simple comodules and hence EndC(Ag) is simple and left (and right) semisimple. So the assertions follow by 4.8 and 4.6.
As a special case we will consider Hopf algebras. For this we recall the conditions for bialgebras.
4.13. Bialgebras. Let R be a commutative ring. An R-module B which is an algebra and a coalgebra is called abialgebraifB⊗RBis aB-coring with bimodule structure
a0(a⊗b)b0 =X
a0ab10⊗bb20, fora, a0, b, b0∈B, comultiplication
∆ :B⊗RB→(B⊗RB)⊗B(B⊗RB)'B⊗RB⊗RB, a⊗b7→X
a⊗b1⊗b2, and counitε:B⊗RB →B, a⊗b7→aε(b).
Clearly 1B⊗1B is a grouplike element and the ring of B⊗RB-covariants of B is isomorphic toR.
B⊗RB is a subgenerator in the categoryMB⊗RB which can be identified with the category ofMBB right Hopf modules, the subcategory ofMBconsisting of those comodulesM whose structure maps are rightB-module morphism, i.e.,
%M(mb) =%M(m)∆(b), form∈M, b∈B.
By 4.6 and [9, 5.10] we obtain:
4.14. Hopf algebras. For a bialgebraB the following are equivalent:
(a) B⊗RB is a Galois B-coring;
(b) the following canonical map is an isomorphism:
γB:B⊗RB→B⊗RB, a⊗b7→(a⊗1)∆(b);
(c) B is a Hopf algebra (has an antipode);
(d) HomBB(B,−) :MBB →MR is an equivalence (with inverse− ⊗RB).
If (any of ) these conditions hold, B is a projective generator in MBB.
Notice that the coinvariantsBB⊗RB=Rand we get the generator property ofB without requiring any flatness condition forBR. Characterization (d) is essentially the Fundamental Theorem for Hopf algebras (e.g., [2]). Of course there are examples of Hopf algebras which are not flat over the base ring (e.g., [4, Beispiel 1.2.7]).
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