Robert Wisbauer
Abstract. The relations between various categoriesAandBare described by functors and important examples for these are the equivalences between cate- gories. They are often of the form Hom(P,−) :A→BwhereP is a generator (with additional properties) inA. In this talk we analyse generators in module and comodule categories. While in full module categories any generator is a flat module over its endomorphism ring, this need not be the case for genera- tors in comodule categories, even if they induce equivalences. As special cases Azumaya and Hopf algebras are considered.
1. Preliminaries
We recall some general notions from category theory. LetAandBdenote two categories. For any functorF :A→Band objectsA, A0 from Athere is a map
FA,A0 : MorA(A, A0)→MorB(F(A), F(A0)).
F is calledfaithful (full)if allFA,A0 are injective (surjective),A, A0∈A. An object P inAis agenerator in Aif MorA(P,−) :A→Setis a faithful functor.
1.1. Adjoint functors. A pair of functors F :A→B, G:B→Ais said to beadjoint if there is a functorial isomorphism
MorB(F(A), B)−→MorA(A, F(B)), B∈B, A∈A. Associated with such an adjoint pair there are natural transformations
unitη:IB→GF and counitε:F G→IA. (1) The following are equivalent:
(a) Gis full and faithful;
(b) ε:F G→IB is an isomorphism.
(2) The following are equivalent:
(a) F is full and faithful;
(b) η:IA→GF is an isomorphism.
(3) The following are equivalent:
(a) F andG are full and faithful;
(b) ε:F G→IB andη:IA→GF are isomorphisms;
(c) F andG are (inverse) equivalences betweenAandB.
1
InGrothendieck categorieswe have the following properties of generators (see [Nas, III, Teoremˇa 9.1]).
1.2. Gabriel Popescu Theorem. Let A be a Grothendieck category, P a generator inA, and S= EndA(P). ThenHomA(P,−) :A→SMhas a left adjoint T :SM→A and:
(1) ε:THomA(P,−)→IAis a functorial isomorphism;
(2) HomA(P,−)is fully faithful;
(3) T is left exact.
2. Generators in the category of A-modules
In this sectionAwill denote an associative ring with unit andAMdenotes the category of left A-modules. Morphisms of left modules should act from the right while those of right modules are acting from the left.
Throughout P will be a left A-module and S = EndA(P). Thus P is a right S-module and there is a canonical ring morphism
φ:A→B= EndS(M), a7→[m7→am].
P is calledbalancedprovidedφis an isomorphism.
2.1. Generators in AM - 1. For a module P inAM andS= EndA(P), the following are equivalent:
(a) P is a generator in AM;
(b) everyA-moduleN isP-generated, that is, there is an epimorphismP(Λ)→N, Λ some index set;
(c) (i) φ:A→B is an isomorphism and (ii) PS is finitely generated and projective.
The characterisation given in (c) is due to Morita [Mor] (e.g. [Fai, Proposition 3.26], [WiFo, 18.8]). The interesting aspect of it is that it describes theuniversal property ofP being a generator inAM byinternal properties ofP as an R- and an S-module. In particular, a generating property on the left side is related to a projectivity property on the right.
2.2. Adjoint pair P⊗S−,HomA(P,−). The functors P⊗S−:SM→AM, HomA(P,−) :AM→SM, form an adjoint pair with functorial isomorphism forX ∈SM,Y ∈AM,
HomA(P⊗SX, Y)→HomS(X,HomA(P, Y)), unit ηX :X →HomA(P, P⊗SX), x7→[p7→p⊗x];
counit εY :P⊗SHomA(P, Y)→Y, p⊗f 7→(p)f.
A special property of the category ofA-modules is the fact that for a generator P, the functor HomA(P,−) : AM → SM is not only faithful (required by the definition) but also full. To show this we recall the following (e.g. [WiFo, 15.9]).
2.3. Modules flat over the endomorphism ring. For anA-moduleP and S= EndA(P), the following are equivalent:
(a) P is a flat right S-module;
(b) for any f :Pn→P,n∈N,Kef isP-generated.
From 1.2 we obtain
2.4. Generators inAM - 2. For anA-moduleP the following are equivalent:
(a) HomA(P,−) :AM→SM is faithful;
(b) HomA(P,−) :AM→SM is fully faithful;
(c) ε:P⊗SHomA(P, N)→N is an isomorphism for anyN ∈AM;
(d) ε:P⊗SHomA(P, N)→N is surjective for anyN ∈AM.
Proof. (a)⇒(b) follows from the Gabriel Popescu Theorem 1.2.
The other implications are obvious.
The properties of generators strongly depend on the categories they generate.
For example, consider Gen(P), the full subcategory of AM whose objects are the P-generated modules. It is the largest subcategory for whichP is a generator.
2.5. Generator in Gen(P).For anA-moduleP the following are trivial.
(1) P is a generator in Gen(P);
(2) HomA(P,−) : Gen(P)→SM is faithful;
(3) εN :P⊗SHomA(P, N)→N is surjective for anyN ∈Gen(P).
Here P need not be flat over its endomorphism ring unless Gen(P) has spe- cial properties, e.g., is a Grothendieck category. To study this consider the full subcategory σ[P] of AM whose objects are subgenerated by P, that is, they are submodules of P-generated modules. Since σ[P] is a Grothendieck category, the proof for 2.4 applies to show:
2.6. Generator inσ[P] - 1. For anA-moduleP the following are equivalent:
(a) HomA(P,−) :σ[P]→SM is faithful;
(b) HomA(P,−) :σ[P]→SM is fully faithful;
(c) εN :P⊗SHomA(P, N)→N is surjective,N ∈σ[P];
(d) εN :P⊗SHomA(P, N)→N is an isomorphism,N∈σ[P];
(e) PS is flat and εV is an isomorphism for all injectivesV ∈σ[P].
Proof. (e)⇒(d) LetN be any module inσ[P] and consider an exact sequence 0→N →I1→I2inσ[P] whereI1, I2 are (P-)injective modules inσ[P]. SincePS
is flat we obtain a commutative diagram with exact rows 0 //P⊗SHomA(P, N)
εN
//P⊗SHomA(P, I1) //
εI1
P⊗SHomA(P, I2)
εI2
0 //N //I1 //I2
By assumption, εI1 and εI2 are isomorphisms and hence so is εN. Thus P is a generator inσ[P] by 2.6.
The remaining assertions are obvious.
The question arises if one can - similar to Morita’s result for AM (see 2.1) - characterise generators inσ[P] by internal properties of P as anR- orS-module.
This can be derived from condition (e) in 2.6 by applying ideas from Zimmermann [Zim]. For this we recall from [Zim] and [WiSta]:
2.7. L-dcc and P-dcc. ForA-modulesL, V, P andS = EndA(P), consider the canonical map
νL,P,V :L⊗SHomR(P, V)→HomR(HomS(L, P), V), l⊗f 7→[g7→(g(l))f], which is an isomorphism providedLSis finitely presented andV isP-injective (e.g., [WiFo, 25.5]). Following Zimmermann [Zim, 3.2], we say thatPS hasL-dccifαL,P is a monomorphism for any injectiveV ∈σ[P]. The terminology indicates that the condition is related to descending chain conditions on certain matrix subgroups of P. In connection with the density property for P this can be understood as a Mittag-Leffler condition onPS (e.g. [WiSta, 5.4]).
PuttingL=P we see thatP-dcconP corresponds to the injectivity of νP,V :P⊗SHomA(P, V)→HomA(HomS(P, P), V).
With the canonical mapφ:A→B:= HomS(P, P) we obtain the commutative diagram
P⊗SHomA(P, V) νP,V //
εV
))R
RR RR RR RR RR RR
RR HomA(B, V)
φ∗
V.
As an injective module inσ[P],V isP-generated and soεV is always surjective. If φ∗ is an isomorphism, e.g., ifφ:A→B is dense, thenP hasP-dcc if and only if εV is an isomorphism for all injectivesV ∈σ[P].
Thus Morita’s description of generators inAMcan be generalised to generators inσ[P] in the following way (see [WiSta, Corollary 5.8]):
2.8. Generators inσ[P]- 2. For anA-moduleP the following are equivalent:
(a) P is a generator in σ[P];
(b) PS is flat and hasP-dcc and (i) φ:A→B is dense, or
(ii) φ∗: HomA(B, V)'HomA(A, V) for all (injective)V ∈σ[P].
3. Bimodules and Azumaya algebras
LetAdenote an algebra over a commutative associative ringRwith unit, with multiplicationµ:A⊗RA→Aand unit mapη:R→A.
3.1. (A, A)-bimodules. Denote the category of (A, A)-bimodules by AMA. This can also be considered as category of left modules over the enveloping algebra A⊗RAo, that is we may identifyAMA=A⊗RAoM.
Thecenterof an (A, A)-bimoduleM is defined as
Z(M) ={m∈M|am=mafor alla∈A}
and it is easily veryfied that
AHomA(A, M)'Z(M), AEndA(A)'Z(A).
The multiplicationµfactors viaµ0:A⊗Z(A)A→A
ClearlyAitself is an (A, A)-bimodule but in general it is neither projective nor a generator in the categoryAMA. To describe this situation we apply the results from the previous section to the adjoint pair of functors
A⊗Z(A)−:Z(A)M→AMA, AHomA(A,−) :AMA→Z(A)M.
3.2. Azumaya algebras. For a ringA, the following are equivalent:
(a) A is a generator in AMA;
(b) A is a projective generator in AMA;
(c) AHomA(A,−) :AMA→Z(A)M is (fully) faithful;
(d) A⊗Z(A)−:Z(A)M→AMA is an equivalence;
(e) A⊗Z(A)Ao'EndZ(A)(A)andAZ(A) is finitely generated and projective;
(f) A is projective inAMA;
(g) µ:A⊗Z(A)A→Asplits inAMA.
Proof. The equivalence of (a), (c) and (e) is clear from 2.1 and 2.4.
(e)⇒(b) As a finitely generated, projective and faithful module over the com- mutative ringZ(A), Ais a generator in Z(A)M(e.g. [WiFo, 18.10]) and thus, by 2.4,Ais (finitely generated and) projective over EndZ(A)(A).
(b)⇒(f) and (f)⇔(g) are obvious.
(g)⇒(a) See, e.g., [WiBim, 28.7].
The theory sketched above can be extended in two different directions. Replace
AMA byσ[AAA], the full subcategory of bimodules subgenerated by A. The gen- erating property in this category does not imply finiteness over the endomorphism ring. Furthermore, replacing the ring A⊗RAo by the multiplication algebra, we need no longer require the algebraAto be associative.
3.3. Multiplication algebra. LetAbe a not necessarily associativeR-algebra with unit. Then left and right multiplications with anya∈AinduceR-linear maps
La :A→A, x7→ax; Ra :A→A, x7→xa.
Themultiplication algebraofA is the (associative) subalgebra M(A)⊂EndR(A) generated by{La, Ra|a∈A}.
We considerAas a left module overM(A) (finitely generated by 1A); EndM(A)(A) is isomorphic to the center ofA. Byσ[M(A)A], orσ[A] for short, we denote the full subcategory ofM(A)M subgenerated byA.
From 2.1 and 2.4 we derive immediately:
3.4. A as generator in M(A)M. For a (non-associative) algebra Awith unit the following are equivalent:
(a) A is a generator in M(A)M;
(b) A is a projective generator in M(A)M;
(c) HomM(A)(A,−) :M(A)M→Z(A)Mis (fully) faithful;
(d) εN :A⊗Z(A)HomM(A)(A, N)→N is sur(bi-)jective for any N∈M(A)M;
(e) φ:M(A)'EndZ(A)(A)andAZ(A) is finitely generated and projective.
The properties justify to call the algebras described in 3.4 (non-associative) Azumaya algebras(see [WiBim, 24.8]). Notice that - unlike inA⊗Z(A)AoM - pro- jectivity ofAin M(A)M need not imply the generating property.
Applying 2.6 and 2.8 we obtain:
3.5. Aas generator inσ[A]. For a (non-associative) algebraAwith unit, the following are equivalent:
(a) HomM(A)(A,−) :σ[A]→Z(A)M is (fully) faithful;
(b) εN :A⊗Z(A)HomM(A)(A, N)→N is sur(bi-)jective for any N∈σ[A];
(c) φ:M(A)→EndZ(A)(A)is dense andAZ(A)is flat and has A-dcc.
Note that the generating property ofA inσ[A] need not imply projectivity.
3.6. A as projective generator in σ[A]. For a (non-associative) algebra A with unit, the following are equivalent:
(a) A is a projective generator in σ[A];
(b) HomM(A)(A,−) :σ[A]→Z(A)M is an equivalence;
(c) A⊗Z(A)−:Z(A)M→σ[A]is an equivalence;
(d) εN :A⊗Z(A)HomM(A)(A, N)→N is surjective (bijective) for anyN ∈σ[A];
(e) φ:M(A)→EndZ(A)(A)is dense andAZ(A) is faithfully flat and hasA-dcc.
The rings considered in 3.6 are named Azumaya rings (e.g. [WiBim, 26.4]).
For example, any (non-associative) simple ring with unit is of this type.
Associative Azumaya rings are closely related to theideal algebrasstudied by M.L. Ranga Rao, a former student of Carl Faith (see [Rao]).
The theory sketched above can also be formulated for ringsAwithout unit. In this case the role of the center of A is taken over by the centroid of A, C(A) = EndM(A)(A) (see [WiBim]).
4. Corings and comodules
Consider any coring (C,∆, ε) over an associative ring A with unit, that is, an (A, A)-bimodule C with bimodule morphisms ∆ :C → C ⊗AC and ε: C → A satisfying the coassociativity and counitality conditions. We recall some elementary facts from [BrWi, Chapter 3].
4.1. Category of comodules. Right C-comodules ared defined as right A- modules M with an A-linear map (coaction) %M : M → M ⊗AC satisfying the coassociativity and counitality conditions.
Comodule morphisms f : M → N between C-comodules are those A-linear maps inducing commutative diagrams
M f //
%M
N
%N
M ⊗AC f⊗I //N⊗AC.
The category of all right C-comodules is denoted by MC. It is an additive category with direct sums and cokernels, and it is abelian providedC is flat as a leftA-module. There is a bijective natural map
ϕ: HomC(M, X⊗AC)→HomA(M, X), f 7→(IX⊗ε)◦f,
which shows that the functor − ⊗A C is right adjoint to the forgetful functor MC → MA. It also shows that the ring EndC(C) of comodule endomorphisms ofC is isomorphic to the dual ringC∗= HomA(C, A), the latter endowed with the convolution product forf, g∈ C∗,f ∗rg(c) =g((f⊗I)∆(c)).
Similarly, the left dual ∗C = AHom(C, A) is a ring with convolution product f∗lg(c) =f((I⊗g)∆(c)). The counitεis the unit for bothC∗ and∗C.
Every rightC-comodule (M, %M) has a left∗C-module structure given by
*:∗C ⊗RM →M, f ⊗m7→(IM ⊗f)◦%M(m).
Endowed with this structure, any C-comodule morphism f : M → N between comodules is∗C-linear, that is
HomC(M, N)⊆∗CHom(M, N), thus inducing a faithful functorMC →∗CM.
This functor is full if and only if C is locally projective as a left A-module (α-condition, [BrWi, 19.2]) and in this case MC is equivalent to σ[∗CC], the full subcategory of ∗CM subgenerated by C. The comodule category MC is equal to
∗CMif and only if AC is finitely generated and projective.
In general, MC is not an abelian category and hence the characterisations of generators considered in earlier sections need not apply. This kind of defect comes from the observation that monomorphisms in MC need not be injective maps. However, if AC is flat, then MC is abelian and results similar to those in section 2 can be obtained. In particular, ifAC is locally projective,MC is the same asσ[∗CC] and propositions like 2.6 or 2.8 do apply directly.
4.2. Generators in MC. Let C be an A-coring with AC flat, P ∈ MC and S= EndC(P). The following are equivalent:
(a) P is a generator in MC;
(b) HomC(P,−) :MC →MS is (fully) faithful;
(c) ε: HomC(P,−)⊗SP →IMC is an isomorphism;
(d) SP is flat andεV : HomC(P, V)⊗SP→V is an isomorphism for all injective comodules V ∈MC.
The following notion generalises generators (see [WiGa3, 4.1]).
4.3. Galois comodules. LetC be anA-coring, P ∈MC and S = EndC(P).
P is said to be aGalois comoduleifεV : HomC(P, V)⊗SP →V is an isomorphism for all (C, A)-injective comodules V ∈ MC (i.e., V is injective with respect to A- splittingC-monomorphisms).
In casePA is finitely generated and projective,P being a Galois comodule can be characterised by the single isomorphism ([WiGa3, 5.3])
εC : HomC(P,C)⊗SP → C.
This implies that any coringC which is finitely generated and projective as a right A-module is a right Galois comodule.
Clearly, if the functor HomC(P,−) is fully faithful, thenPis a Galois comodule.
On the other hand, if P is a Galois comodule andSP is flat, then HomC(P,−) is fully faithful andAC is flat, i.e.,P is a generator in MC (see [WiGa3, 4.8]).
For a sketch of the way from Galois field extensions to Galois comodules we refer to [WiGa1].
4.4. A as C-comodule. Given an A-coring C one may ask whenA itself is a C-comodule. This is the case whenC has a grouplikeelement, that is, someg ∈ C with ∆(g) =g⊗g andε(g) = 1A. ThenA is a right (and left)C-comodule by the coactions
%A:A→ C, a7→ga, (A%:A→ C, a7→ag,)
and we denote A with this comodule structure by Ag (e.g. [WiGa2], [BrWi, Section 28]). ForM ∈MC, theg-coinvariantsofM are defined as
MgcoC ={m∈M |%M(m) =m⊗g}.
(1) ψM : HomC(Ag, M)→MgcoC, f 7→f(1A),is an isomorphism.
(2) EndC(Ag)'AcoCg ={a∈Ag|ga=ag}(= centraliser of gin A).
(3) For anyX ∈MA, (X⊗AC)coC'HomC(Ag, X⊗AC)'X, and CcoC 'HomC(Ag,C)'HomA(Ag, A)'A, which is a left A- and right EndC(Ag)-morphism.
Now one may ask whenAg is a Galois comodule. This leads to
4.5. Galois corings. For an A-coring C with a grouplike element g and B= AcoCg , the following are equivalent:
(a) εC : HomC(Ag,C)⊗BA→ C, f⊗a7→f(a),is an isomorphism;
(b) the (A, A)-bimodule map χ : A⊗B A → C, 1A⊗1A 7→ g, is a (coring) isomorphism;
(c) εN : HomC(Ag, N)⊗BA→N is an isomorphism, for every (C, A)-injective N ∈MC.
(C, g) is called aGalois coringif it satisfies the above conditions.
As seen before Galois comodules are close to generators. This is recalled in the 4.6. Galois Coring Structure Theorem. (e.g., [BrWi, 28.19]). Let C be anA-coring with grouplike elementg 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.
4.7. Ring extensions. To any ring extension B → A, anA-coring can be associated by puttingC=A⊗BAwith grouplike elementg= 1A⊗1A inC, called theSweedlerA-coring.
The category of right comodules over this coring is isomorphic to the category ofdescent datafor the given ring extension (e.g. [BrWi, 25.4]). In this context the results on Galois corings can be interpreted in the classicaldescent theory.
Recall that ifC satisfies the α-condition, then EndC(C) =∗CEnd(C)' C∗ and we get (e.g. [BrWi, 19.21]):
4.8. C as generator in MC. Let C be an A-coring withAC locally projective.
ThenMC =σ[∗CC]and the following are equivalent:
(a) C is a generator in MC;
(b) ∗CHom(C,−) :MC →MC∗ is (fully) faithful;
(c) C is flat as aC∗-module andεN :C ⊗C∗HomC(C, N)→N is an isomorphism forN ∈MC.
5. Bimodules and Hopf algebras
In this sectionRdenotes again a commutative associative ring with unit.
5.1. Bialgebras and bimodules. AnR-moduleB is called abialgebra if it has an algebra structure (B, µ, η) and a coalgebra structure (B,∆, ε),
µ:B⊗RB →B, η:R→B, ∆ :B →B⊗RB, ε:B→R,
such that ∆ andεare algebra morphisms, or, equivalently, µand η are coalgebra morphisms where B ⊗RB is endowed with the canonical algebra and coalgebra structure (induced by the twist mapB⊗RB →B⊗RB).
A(mixed)B-bimoduleis anR-moduleM with structural maps ρM :M⊗RB→M, ρM :M →M ⊗RB.
satisfying the compatibility condition
ρM(mb) =ρM(m)·∆(b), forb∈B, m∈M.
Category of B-bimodules. The category of right B-bimodules is denoted byMBB. Its objects areB-bimodules and the morphisms are R-linear maps which are bothB-module andB-comodule morphisms.
MBB is an additive category with coproducts and cokernels. Similar to the cate- gory of comodules over a coalgebra (and for the same reasons),MBB has coproducts and cokernels but not necessarily kernels. It is abelian (and in fact Grothendieck) providedB is flat as anR-module.
Naturally, B is aB-bimodule and EndBB(B) =R. However, B need not be a subgenerator inMBB whileB⊗RB always is.
B⊗RBcan be considered as aB-coring and the categoryMBB is equivalent to the categoryMB⊗RB of rightB⊗RB-comodules. In particular, B is in MB⊗RB and 1B⊗1B is a group like element (see 4.5).
B induces the (free) functor
φBB :MR→MBB, M 7→(M⊗RB, IM ⊗µ, IM⊗∆), which is full and faithful by the natural isomorphisms forX, Y ∈MR,
HomBB(B⊗RX, B⊗RY)'HomBR(X, B⊗RY)'HomR(X, Y).
We have the adjoint pair of functors
φBB :MR→MBB, HomBB(B,−) :MBB→MR,
and sinceφBBis full and faithful we know (from 1.1) thatη :I→HomBB(B, B⊗R−) is an isomorphism.
Handling bialgebras there is a notion which does not show up in the settings considered so far.
5.2. Antipodes. Given a bialgebra B, besides composition, EndR(B) has another associative product, theconvolution product, given by
f ∗g(b) = (f⊗g)∆(b) forf, g∈EndR(B) andb∈B.
An elementS ∈EndR(B) is called anantipodeofB if it is inverse to IB with respect to the convolution product∗. So, by definition,
µ◦(S⊗IB)◦∆ =µ◦(IB⊗S)◦∆ =η◦ε .
Notice that as an endomorphism ofB,S need neither be injective nor surjective.
In the next theorem no assumptions are made on theR-module structure ofB (see [BrWi, 15.5 and 36.16]).
5.3. Hopf algebras - 1. For an R-bialgebra B the following are equivalent:
(a) HomBB(B,−) :MBB →MR is full and faithful;
(b) εN : HomBB(B, N)⊗RB →N is an isomorphism for anyN∈MBB; (c) HomBB(B,−) :MBB →MR is an equivalence (with inverse− ⊗RB);
(d) B⊗RB is a Galois B-coring;
(e) γB:B⊗RB→B⊗RB, a⊗b7→(a⊗1B)∆(b), is an isomorphism;
(f) B has an antipode.
BialgebrasB with these properties are calledHopf algebras.
The equivalence of (c) and (f) is known as theFundamental Theoremof Hopf algebras. Notice that here HomBB(B,−) is fully faithful but nevertheless B need not be flat over its endomorphism ringR. This comes from the fact that monomor- phisms inMBB need not be injective maps.
IfB is assumed to be flat as an R-module,B is a Hopf algebra if and only if B is a generator in the (Grothendieck) category MBB. AssumingB to be a locally projective R-module we can get more characterisations for such bialgebras. For this we consider the
5.4. Smash product. IfBR is locally projective, then the comodules inMB can be identified with theB∗-modules subgenerated by the comoduleB.
Thus, if B is a bialgebra, the objects inMBB can be considered asB-modules with a B∗-module structure satisfying some compatibility conditions. Thus the bimodules allow the structure of left modules over thesmash productB#B∗, which is defined as theR-moduleB⊗RB∗ with multiplication
(a⊗f)(b⊗g) := ((∆b)(a⊗f)) (1B⊗g).
Similar to the case of comodules we may identify
MBB =σB#B∗[B⊗RB]⊂B#B∗M,
whereMBB=B#B∗Mif and only ifBRis finitely generated (and projective).
In the situation described above we can refer to 2.6 and 2.8 to describe Hopf algebras (see also [BrWi, 15.5]). The results are similar to the characterisations of Azumaya rings (see 3.6).
5.5. Hopf algebras - 2. For an R-bialgebra B with BR is locally projective, the following are equivalent:
(a) B is a generator inMBB;
(b) HomBB(B,−) :MBB →MR is (full and) faithful;
(c) B is a subgenerator inMBB andφ:B#B∗→EndR(B)is dense;
(d) εV : HomBB(B, V)⊗RB→V is an isomorphism for injectivesV ∈MBB. Special cases of the preceding results can be found in [WiMod] where further applications of these techniques to module algebras and group actions on algebras are outlined.
In [WiAlg] and [MeWi] the theory of bialgebras and Hopf algebras is gener- alised to endofunctors of arbitrary categories.
References
[BrWi] Brzezi´nski, T. and Wisbauer, R.,Corings and Comodules, London Math. Soc. LN Series 309, Cambridge University Press (2003)
[Fai] Faith, C., Algebra: Rings, modules and categories. I, Springer-Verlag, Berlin-Heidelberg- New York (1973)
[MeWi] Mesablishvili, B. and Wisbauer, R., Bimonads and Hopf monads on categories, arXiv:0710.1163
[Mor] Morita,K.,Duality for modules and its applications to the theory of rings with minimum condition, Sci. Rep. Tokyo Kyoiku Daigaku, Sect. A 6, 83-142 (1958)
[Nas] Nˇastˇasescu, C., Rings. Modules. Categories. (Inele. Module. Categorii.). (Romanian) Bu- curesti: Editura Academiei Republicii Socialiste Romania (1976)
[Rao] Rao, M.L. Ranga,Azumaya, semisimple and ideal algebras, Bull. Am. Math. Soc. 78, 588- 592 (1972)
[WiFo] Wisbauer, R.,Foundations of module and ring theory, Algebra, Logic and Applications.
3, Gordon and Breach, Philadelphia (1991)
[WiBim] Wisbauer, R.,Modules and algebras: bimodule structure and group actions on algebras, Pitman Mon. Pure Appl. Math. 81, Addison Wesley Longman, Harlow, Essex (1996) [WiSta] Wisbauer, R., Static modules and equivalences, Interactions between ring theory and
representations of algebras. Proc. conf. in Murcia, Van Oystaeyen, F. (ed.) et al., Marcel Dekker, New York, LN Pure Appl. Math. 210, 423-449 (2000)
[WiMod] Wisbauer, R., Module and comodule categories – a survey, Proc. 2. Palest. Intern.
Conference, West Bank, Palestine, 1998, Elaydi, S. (ed.) et al., World Scientific, Singapore, 277-304 (2000)
[WiGa1] Wisbauer, R.,From Galois field extensions to Galois comodules, Advances in ring theory, Proc. 4th China-Japan-Korea conference, Nanjing, China, Chen, J.L. (ed.) et al., World Scientific, New York, 263-281 (2005)
[WiGa2] Wisbauer, R., On Galois corings, Hopf algebras in noncommutative geometry and physics. Proc. Conf. Brussels 2002, Caenepeel, S. (ed.) et al., Marcel Dekker, New York, LN Pure Appl. Math. 239, 309-320 (2005)
[WiGa3] Wisbauer, R.,Galois comodules, Commun. Algebra 34(7), 2683-2711 (2006)
[WiAlg] Wisbauer, R.,Algebras versus coalgebras, Appl. Categ. Struct. 16(1-2), 255-295 (2008) [Zim] Zimmermann, W., Modules with chain conditions for finite matrix subgroups, J. Algebra
190, 68-87 (1997)
Department of Mathematics, Heinrich Heine University, 40225 D¨usseldorf, Ger- many
E-mail address:wisbauer@math.uni-duesseldorf.de
