Algebr Represent Theor (2009) 12:343-356 DOI 10.1007/s10468-009-9153-6

Corings with Decomposition and Semiperfect Corings

L. El Kaoutit • J. Gómez-Torrecillas

Received: 17 March 2008 / Accepted: 16 September 2008 / Published online: 4 March 2009 © Springer Science + Business Media B.V. 2009

Abstract We give a characterization, in terms of Galois infinite comatrix corings, of the corings that decompose as a direct sum of left comodules which are finitely generated as left modules. Then we show that the associated rational functor is exact. This is the case of a right semiperfect coring which is locally projective and whose Galois comodule is a projective left unital module with superfluous radical.

Keywords Corings • Graded modules • Doi-Koppinen Hopf modules

Mathematics Subject Classifications (2000) 16W30 • 16W50 • 18E15

1 Introduction

When studying generalized module categories like Doi-Koppinen Hopf modules or, more generally, entwined modules (see [6] or [3] for detailed accounts of their fundamental properties), it is reasonable, after the observation that they are

Presented by Stefaan Caenepeel and Alain Verschoren.

Opgedragen aan Freddy Van Oystaeyen ter gelegenheid van zijn zestigste verjaardag.

This research is supported by the grants MTM2004-01406 and MTM2007-61673 from the Ministerio de Educación y Ciencia of Spain, and P06-FQM-01889 from the Consejería de Innovación, Ciencia y Empresa of Andalucía, with funds from FEDER (Unión Europea).

L. El Kaoutit (B)

Departamento de Álgebra, Facultad de Educación y Humanidades, Universidad de Granada, El Greco N° 10, 51002 Ceuta, Spain e-mail: kaoutit@ugr.es

J. Gómez-Torrecillas

Departamento de Álgebra, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain e-mail: gomezj@ugr.es

categories of comodules [4], to formulate our questions in terms of comodules over corings. However, it happens often that the answer is given in terms of abstract categorical concepts, which do not say much more on the concrete comodule algebra or entwining structure we are dealing with. In a more favorable situation, we could characterize categorical properties of the category of relative modules in terms of structural properties of the associated coring. But, sometimes, the structure of the corings is described in terms which have no direct relationship with the entwined algebra and coalgebra. One example of the sketched situation is the following: assume we want to study under which conditions the category M(^)A of entwined right modules over an entwining structure (A, C)y for an algebra A and a coalgebra C has a generating set of small projectives. Of course, when M(^)A is a Grothendieck category, a first answer is given by Freyd's theorem: it has to be equivalent to a category of modules over a ring with enough idempotents. Looking at M(^)A as the category of right comodules over a suitable coring [4], we have a characterization in terms of infinite comatrix corings in [8, Theorem 2.7]. However, none of these results give answers directly expressable in terms of A or C. In this paper, we describe a class of A-corings for which there exists a generating set of small projective objects for their category of right comodules. As a consequence, we will obtain that if C is a right semiperfect coalgebra over a field, then M(^)A has a generating set of small projective objects for every entwining structure (A, C)^ (Corollary 2.3). Of course, we think that the interest of our results go beyond the theory of entwined modules, as the categories of comodules over corings deserve to be investigated in their own right.

The idea comes from the theory of coalgebras. It is well known [17] that if C is a coalgebra over a field, then C is right semiperfect (or, equivalently, its category of right comodules ComodC has a generating set consisting of finite-dimensional projectives) if and only if C decomposes as a direct sum of finite-dimensional left subcomodules. We give a generalization of this characterization to the case of corings over an arbitrary ring A (Theorem 2.2). It is then natural to try to understand right semiperfect corings. This is done in Section 3. Our point of view here, in contrast with [9] and [5], deliberatively avoids the assumption of conditions on the ground ring A. Our approach rests upon the study of general semiperfect categories due to Harada [15], and on the study of corings having a generating set of small projective comodules developed in [8]. We also discuss the notion of a (right) local coring and the exactness of the rational functor.

Notations and basic notions We work over a commutative ground base ring with 1 denoted by K. The letters A, B are reserved to denote associative K-algebras with unit, which will referred to as rings. A module over a ring with unit means an unital module, and all bimodules are assumed to be central K-bimodules. The category of all right A-modules is denoted by ModA. A linear morphism acts on the left, so some conventions should be established. That is, if AN is a left A-module then its endomorphism ring End(AN) is considered as ring with the opposite multiplication of the usual composition law. In this way N is an (A, End(AN))-bimodule. While, if Na is a right A-module, then its endomorphism ring End(NA) has a multiplication the usual composition, and N becomes obviously an (End(NA), A)-bimodule. For any (B, A)-bimodule M, we consider in a canonical way its right and left dual modules M* = Hom (MA , Aa), *M = Hom (BM, BB) as (A, B)-bimodules.

We will also consider some rings without unit. When this is the case will be clear from the context.

For any category Q, the notation X e Q means that X is an object of Q, and the identity morphism of any object will be represented by the object itself.

Recall from [20] that an A-coring is a three-tuple (C, Ac,sc) consisting of an A-bimodule C and two homomorphisms of A-bimodules

C -^ C ® A c , C -A

such that (AC <<A C) o AC = (C <>A AC) o AC and (sc <A C) o AC = (C <A sc) o AC = C. A homomorphism of A-corings is an A-bilinear map 0 : C ^ C which satisfies sc o 0 = sc and AC o 0 = (0 <A 0) o AC.

A right C-comodule is a pair (M, pM) consisting of a right A-module M and a right A-linear map pM : M ^ M < A C, called right C-coaction, such that (M <A AC) o pM = (pM ®A C) o pM and (M ®a sc) o pM = M. A morphism of right C-comodules is a right A-linear map f : M ^ M satisfying pM o f = (f ® A C) o pM. The ^-module of all homomorphisms of right C-comodules from a comodule MC to a comodule M'C is denoted by HomC(M, M'). Right C-comodules and their morphisms form a ^-linear category ComodC which is a Grothendieck category provided AC is a flat module, see [11, Section 1]. Left C-comodules and their morphisms are symmetrically defined. If P is a right C-comodule such that PA is finitely generated and projective (profinite, for short), then its right dual P* admits, in a natural way, a structure of left C-comodule [3, 19.19]. The same arguments are pertinent for left C-comodules. The natural isomorphism P = *(P*) of right A-modules is in fact an isomorphism of C-comodules. Furthermore, if P and Q are two right C-comodules profinite as right A-modules, then the right dual functor induces a ^-module isomorphism HomC(P, Q) = HomC(Q*, P*).

2 Comatrix Corings and Corings with Decompositions

Let P be a set of profinite modules over a ring with unit A. Consider X = ®pep P and, for each P e P, let i p : P ^ X, np : X ^ P be, respectively, the canonical inclusion and the canonical projection. Consider the set {uP = iP o nP : P e P} of orthogonal idempotents of End(XA), and let T be a unital subring of End(XA) that contains the idempotents uP. Write

R = uqTUp,

a ring with enough idempotents. Its category of right unital R-modules is denoted by ModR. Unital here means MR = M, for M a right R-module. Let us recall one of the three constructions given in [8] for the (infinite) comatrix A-coring associated to P an T (or R). We have now the (R, A)-bimodule X and the (A, R)-bimodule Xf = ®Pe^P*. Both Xf and X are unital R-modules. In the A-bimodule Xf ®R X we have that 0 ®R x = 0 whenever 0 e P*, x e Q for P = Q. In this way, the formula

A(0 ® RX) = ^2,0 ®R ePi ® a e*P,i <8>r x (0 e P*, x e P),

where {ePj, e*Pi} is a dual basis for the profinite right A-module P, determines a ¿»multiplication

A : S* (R S -^ St (r S (a S* (R S

which, according to [8, Proposition 5.2], endows S* (R S with a structure of an A-coring referred to as the infinite comatrix coring associated to the set P and the ring R. Its counit is given by evaluation of the forms 0 e P* at the elements x e P, when P runs through P.

When P consists of right comodules over an A-coring C, we have that S = ®Pep P is a right C-comodule. We consider then the infinite comatrix coring by putting T = End(SC). We have that

R = 0 uQEnd(SC)uP = 0 HomC (P, Q) (2.1)

P,QeP P,QeP

It follows from [8, Lemma 4.7] and [8, diagram (5.12)] that there is a canonical homomorphism of A-corings

can : Sf (R S ^ C, can(y (Rx) = y (A C)pS(x), (2.2)

where y e S* acts on y e S by evaluation in the obvious way.

Definition 2.1 The comodule S is said to be R — C-Galois if the canonical map can is bijective.

Recall from [11, Section 2] that a three-tuple T = (C, B, {—, —)) consisting of an A-coring C, an A-ring B (i.e., B is an algebra extension of A) and a balanced A-bilinear form {—, —) : C x B ^ A, is said to be a right rational pairing over A provided

(1) pa : B ^ *C is a ring anti-homomorphism, where *C is the left dual convolution ring of C defined in [20, Proposition 3.2], and

(2) aM is an injective map, for each right A-module M,

where a— and p— are the following natural transformations

Pn : B (aN -3- Hom (AC, AN), aM : M (A C -s- Hom (Ba , Ma)

b (A n -[c ^ {c, b )n] m (a c -[b ^ m{c, b)].

Given a right rational pairing T = (C, B, {—, —)) over A, we can define a functor called the right rational functor as follows. An element m of a right B-module M is called rational if there exists a set of right rational parameters {(c, mi)} C C x M such that mb = J] ¿mi {ci, b), for all b e B. The set of all rational elements in M is denoted by RatT(M). As it was explained in [11, Section 2], the proofs detailed in [12, Section 2] can be adapted in a straightforward way in order to get that RatT(M) is a B-submodule of M and the assignment M ^ RatT (M) is a well defined functor

RatT : ModB ^ ModB,

which is in fact a left exact preradical [19, Ch. VI]. Therefore, the full subcategory RatT(ModB) of ModB whose objects are those B-modules M such that RatT(M) = M is a closed subcategory. Furthermore, RatT(ModB) is a Grothendieck category which is shown to be isomorphic to the category of right comodules ComodC as [11, Theorem 2.6'] asserts (see also [1, Proposition 2.8]). We say that a set S of objects of a Grothendieck category A is a generating set of A if the coproduct 0XeS X is a generator of A.

We are ready to state and prove our main theorem.

Theorem 2.2 The following statements are equivalent for an A-coring C:

(i) C = 0 Ee£ E, for a family ofsubcomodules E of c C such that a E is profinite for every E e E;

(ii) a C is projective and there exists a generating set P of small projective objects in Comodc such that the right comodule S = 0 p^ P, considered as a left module over R = 0 p Qep Homc(P, Q), admits a decomposition as direct sum of finitely generated R-submodules;

(iii) a C is projective and there exists a generating set P of Comodc, whose members are profinite as right A-modules, such that S = 0^^p P admits, as a left module over R = 0 p Qep Homc(P, Q), a decomposition as direct sum of finitely generated R-submodules;

(iv) a C is projective and there exists a set P of right C-comodules profinite as right A-modules such that C is R — S-Galois for S = 0^^p P and R = ©P QeP Homc(P , Q), and S admits, as a left R-module, a decomposition as direct sum of finitely generated R-submodules.

(i) ^ (ii) Associated to the given decomposition of left comodules C = ®Ee£ E, there is a family of orthogonal idempotents {eE : E e E} in End(CC), where eE = iE o nE, for iE : E ^ C the canonical injection and nE : C ^ E the canonical projection for each E e E. The ring End(CC) is endowed with the multiplication opposite to the composition. Since AC is projective we have the canonical rational pairing T = (C, End(CC), {—, —)), where {c, f) = s(f (c)), for c e C, f e End(CC). Thus each right C-comodule admits a right End (CC)-action, and so is in particular for the right C-comodules * E, with E e E. Consider the set of right C-comodules P ={* E : E e E} and the right C-comodule S = ®Ee£ * E. Each of the maps

eEEnd(cC) -Homc(E, C) = *E (2.3)

eEf I-£ o f o eE o ie

is an isomorphism of right End(C C) -modules, which means that each eEEnd(CC) is actually a rational right End(CC)-module, and so a right C-comodule. In this way, we get an isomorphism of right C-comodules

0eEEnd(cC) ^ 0 *E = S. (2.4)

EeE EeE

Since eEEnd(CC) is a small object in the category ComodC for every E e E, we deduce that

eEEnd(cC) = 0 eE End(€C)eF. (2.5)

Consider in the ring End(SC) (with multiplication the usual composition), the set of idempotents u*E = i* E o n* E, for i* E :* E ^ S the canonical injection and n*E : S ^ * E the canonical projection for each E e E. We have the ring with enough orthogonal idempotents

R = u*EEnd(SC)u*F.

E, F e £

Clearly we already have, for every pair of comodules E, F e E, ^-linear isomorphisms

eEEnd(CC)eF = HomC(E, F), (2.6)

u*EEnd(SC)u*F = HomC(*F, *E). (2.7)

Using Eqs. 2.6 and 2.7 we define, taking into account the canonical ^-linear isomorphism HomC(E, F) = HomC (* F, * E), an isomorphism

eE End(CC)eF = u* E End(SC)u. F (2.8)

for each pair E, F e E.

In view of equality (Eq. 2.5) and the family of isomorphisms (Eq. (2.8)), we deduce a ^-linear isomorphism

0 eEEnd(CC) =0 u*EEnd(SC)u.F = R (2.9)

Ee£ Ee£

Now, if we compose the isomorphisms given in Eqs. 2.4 and 2.9 we get an isomorphism f : S —> R. In order to give an explicit expression for the isomorphism f, we should take into account that the inverse map of Eq. 2.4 is defined as follows: to each element 0E e* E it corresponds (C ®A QE) ◦ A o eE e eEEnd(CC). From this, given Q = J2e e £ Qe e ®E e £ * E = S, we get a map

f (Q) : 0 * F 0 * F

Fe £ Fe £

defined by f (0)

( E vA = E VE ◦ nE o (C <g> A 0e) ◦ A ◦ IE, \F e E ' E, F e E

F e£ Fe£

E VF e F (2.10)

Using the expression 2.10 we can easily show that f : S ^ R is in fact a left R-module isomorphism. In particular, we deduce from the

decomposition R = ®Fe£Ru*F that RS admits a decomposition as direct sum of finitely generated R-submodules.

Consider now the infinite comatrix coring St ®R S, for St = ®E e £ (*E)*. We have in fact an isomorphism St = C of left comodules. A routine computation show that the following diagram

St ®R S -C

St ®Rf =

St ®R R -St

is commutative, and so can is an isomorphism. Since RS is a faithfully flat module because RS = rR, we can apply [8, Theorem 5.7((iii) ^ (i))] to deduce that {*E : E e E} is a generating set of a small projec-tives for ComodC.

(ii) ^ (iii) This is clear, since each P e P is, as right A-module, finitely generated

and projective (see [8, Theorem 5.7]).

(iii) ^ (iv) This is deduced from [8, Theorem 4.8] and [8, diagram (5.12)].

(iv) ^ (i) We can consider R as the (no unital) as a subring (without unit) of

End (SC) given by

R = uPEnd(SC)uQ,

P, Q e £

where uP = iP o nP for nP : S ^ P (resp. iP : P ^ S) is the canonical projection (resp. injection). Consider the decomposition RS = eISi as direct sum of finitely generated R-submodules RSi. We have the following decomposition of the infinite comatrix coring as a left comodule

St ®r S = 0 St ®r Si (2.11)

For each i e i there exists a presentation of the left R-module Si

Fi Si 0, (2.12)

where Fi = ®P e piRup, with a finite subset Pi c P. Applying the functor St ®R — to the sequence 2.12, we obtain an exact sequence of left A-modules

St ®r Fi St ®r Si 0. (2.13)

Since there is an A-module isomorphism

St ®RFi Z ®p e PiP*

for each i e I, we deduce from the sequence 2.13 that St ®R Si is a finitely generated A-module for each i e i.

Since C is R — S-Galois, the map we have can : St ®R S ^ C is an isomorphism of an A-corings. Therefore, the decomposition in Eq. 2.11 can be transferred via can to a decomposition of CC = e IEi as a

direct sum of subcomodules which are finitely generated as left Amodules. Each one of the E¿'s is of course a projective A-module as AC is projective, and this finishes the proof. □

According to [17, Theorem 10] a coalgebra C over a field admits a decomposition as a direct sum of finite-dimensional left subcomodules if and only if ComodC has a generating set of finite-dimensional projective right comodules (C is already right semiperfect). Thus, in the coalgebra case, the condition on X in statement (ii) of Theorem 2.2, namely that RX is a direct sum of finitely generated R-submodules, may be deleted. We do not know if this is also the case for corings over a general ring (for Quasi-Frobenius ground rings, the answer is positive [9, Theorem 3.5]). Note that the additional condition on the left R-module structure of X cannot be avoided in statement (iii) even in the coalgebra case, as there is a generating set for ComodC of finite-dimensional comodules for any coalgebra C over a field. This is also the case for statement (iv), since any coalgebra is R — X-Galois (by [8, Theorem 4.8] and [8, diagram (5.12)]).

As an application of Theorem 2.2, we obtain the following remarkable fact concerning the existence of enough projectives for categories of entwined modules. For the definition of an entwining structure and a discussion of their properties and their relationships with corings, we refer to [3].

Corollary 2.3 Let ^ : A ® C ^ C ® Abe an entwining structure between an algebra A and a coalgebra C over a commutative ring K. If C admits a decomposition as a direct sum of left subcomodules profinite as K-modules (e.g., if K is a field and C is right semiperfect), then the category of right entwined modules M(^)A has a generating set of small projective objects.

Proof By [4, Proposition 2.2], A ® C is endowed with the structure of an A-coring such that ComodC is isomorphic to the category of right entwined A — C-modules. The comultiplication on A ® C is given by

A ® C -s- A ® C ® C = A ® C ® a A ® C

so that every decomposition of C as direct sum of left C-comodules leads to such a decomposition of the A-coring A ® C. Obviously, if the direct summands in C are profinite K-modules, then the corresponding direct summands of A ® C are profinite as left modules over A. Now, the Corollary is a consequence of Theorem 2.2. □

Right semiperfect coalgebras over a field are characterized by the fact that the rational functor Rat: *CMod ^ .CMod is exact [13, Theorem 3.3]. The exactness of the rational functor associated to a general rational pairing in the context of corings has been considered recently in [10]. The rational functor canonically associated to a coring becomes exact for the corings characterized in Theorem 2.2, as the following Proposition shows. The exactness of the rational functor was given in [3] (see Remark 2.6 below).

Proposition 2.4 Let C be an A-coring admitting a direct sum decomposition C = ®e6£ E, for a family of subcomodules E of cC such that aE is profinite for every

E e E. Consider the canonical right rational pairing T = (C, End(cC), {—, —)), and denote by a := RatT (End(cC)gnd(Cc^ the rational ideal. Then

(a) R is a right rational End (cC) -module injected in a.

(b) a is generated as a bimodule by {eE}Ee£, that is a = EeS End(cC)eEEnd(cC), and a is a pure left submodule of End(cC).

(c) The functor RatT : ModEnd(CC) ^ ModEnd(CC) is exact.

(a) Follows directly from the isomorphisms 2.3 and 2.9.

(b) From the isomorphisms 2.3 we get that ^Ee£eEEnd(CC) is a rational right ideal, which implies that ^Ee£ End(CC)eEEnd(CC) c a. Thus, we only need to check the reciprocal inclusion. So let b e a = RatT (End(CC)gnd(£C)) an arbitrary element with a right rational system of parameters {(bi, ci)}i=1,...n c End(CC) x C. Let E' c E be a finite subset such that ci e ®Ee&E for every i = 1,...,n, and take e = ^ Ee £, eE. One easily checks that c^e = ci, for all i = 1, ■■■ , n. Therefore,

be bi c, e) = bisc(cie) = J^ b^O = b, (2.14) i i i

which gives the needed inclusion. Equation 2.14 also implies that a is a pure left End(CC)-submodule of End(CC).

(c) In view of (b) and the equality Ca = C, we can apply [10, Theorem 1.2] to get the exactness of the rational functor. □

Example 2.5 (compare with [5, Example 5.2]) Let C be a cosemisimple A-coring. By [11, Theorem 3.1], AC and CA are projective modules. So we can consider its right canonical rational pairing T = (C, End(CC), {—, —)). The structure Theorem of cosemisimple corings [7, Theorem 4.4] implies that C is a direct sum of left C-comodules where each of them is a finitely generated and projective left A-module. Thus for a cosemisimple coring RatT has to be exact.

Remark 2.6 Under the anti-isomorphism of rings End(CC) = *C the rational pairing considered in Proposition 2.4 goes to a rational pairing ((*C)op, C) that gives rise to the "more usual" rational functor Rat: *CMod ^ *CMod considered for instance in [3, Section 20], where its exactness was studied. In particular, one deduces from [3, 20.8, 20.12] that Rat: *CMod ^ *CMod is exact whenever CC admits a direct decomposition as assumed in Proposition 2.4. Their arguments run on a different road than ours.

3 Local Corings and Semiperfect Corings

Let C be an A-coring such that its category ComodC of all right C-comodules is a Grothendieck category. The coring C is said to be a right semiperfect coring if every finitely generated right C-comodule has a projective cover. That is, C is right semiperfect if and only if (by definition) ComodC is a Grothendieck semiperfect

category in the sense of M. Harada [15, Section 3, page 334]. Following [18, page 347] and [15, Section 1, page 330], a right C-comodule P is said to be a semiperfect right comodule if PC is a projective comodule and every factor comodule of P has a projective cover. A right C-comodule P is said to be a completely indecomposable comodule if its endomorphisms ring End(PC) is a local ring (i.e., its quotient by the Jacobson radical is a division ring).

Recall from [15, Corollary 1] that if A is a locally finitely generated Grothendieck category, then A is semiperfect if and only if A has a generating set of completely indecomposable projective objects. Of course, this result can be applied in particular to ComodC, whenever it is a locally finitely generated Grothendieck category.

Remark 3.1 Assume that ComodC is a Grothendieck category. It seems to be an open question if it is locally finitely generated. In the case when AC is locally projective in the sense of [21], ComodC is isomorphic to the category of all rational left *C-modules (by [1, Lemma 1.29] and [11, Theorem 2.6']) and, therefore, the set of all cyclic rational left *C-comodules generates the category of right C-comodules. In fact, in this case, a right C-comodule is finitely generated if and only if it is finitely generated as a right A-module (see, e.g., [9, Lemma 2.2]). Thus, ComodC has a generating set of comodules that are finitely generated as right A-modules, whenever AC is locally projective.

The following theorem is a consequence of [15, Corollary 2], [15, Theorem 3] and [8, Theorem 5.7] (see also [14, Theorem 6.2]).

Theorem 3.2 Let C be an A-coring and P a set of right C-comodules. The following statements are equivalent.

(i) a C is aflat module and P is a generating set of small completely indecomposable projective comodules for Comode;

(ii) a C is a flat module, Comod e has a generating set consisting of finitely generated objects, C is a right semiperfect A-coring and P contains a set of representatives of all semiperfect completely indecomposable right C-comodules;

(iii) each comodule in P is finitely generated and projective as a right A-module, C is (R, X)-Galois, where X = 0P and R = 0F Qep Home (P, Q), rX is faithfully flat, and End( Pe) is a local ring for every P e P.

(i) ^ (iii) This is a consequence of [8, Theorem 5.7].

(iii) ^ (ii) By [8, Theorem 5.7], AC is a flat module and -<g>R X : ModR ^ ComodC is an equivalence of categories. From Eq. 2.1 we get the decomposition R = ® uPR, and, since End(uPRR) = uPRuP = End(PC) for every P e P, we deduce that {uPR : P e P} is a generating set of completely indecomposable finitely generated projective objects for ModR. Therefore, {uPR ®R X : P e P} becomes a generating set of completely indecomposable finitely generated projective objects for Comod C. Now, uPR ®R X = P as a right C-comodule, for every

P G P. This implies that P is a generating set of finitely generated completely indecomposable projective comodules. By [15, Corollary 1], C is right semiperfect. Now, if U is any completely indecomposable projective right C-comodule, then U is an epimorphic image of a direct sum of comodules in P. We get thus that U is isomorphic to a direct summand of a sum of comodules in P. By the Krull-Schmidt-Azumaya theorem, we obtain that U is isomorphic to one comodule in P. (ii) ^ (i) From the proof of [15, Theorem 3], it follows that there is a set of finitely generated projective completely indecomposable generators of ComodC. This obviously implies that P is a generating set of projective finitely generated indecomposable comodules. □

Let us record some information deduced from Theorem 2.2 as a kind of "Structure Theorem" of right semiperfect A-corings.

Corollary 3.3 Assume aC to be locally projective. Then C is right semiperfect if and only if there is a set P of right C-comodules such that

1. Every P G P isprofinite as a right A-module,

2. C is R — X-Galois, where X = 0pGp P and R = 0p QGp Home (P, Q),

3. rX is faithfully flat, and

4. End(Pe) is a local ring, for every P G P.

As mentioned before, a coalgebra over a field is right semiperfect if and only if its rational functor is exact. This characterization has been extended to the case of corings over QuasiFrobenius rings, see [9, Theorem 4.2], [5, Theorem 4.3], [3, 20.11]. For a general ground ring A, we obtain the following result.

Corollary 3.4 Let C be a right semiperfect A-coring such that aC is a locally projective module. Consider the canonical right rational pairing T = (C, End(cC), (—, —)) and let X and R be as in Corollary 3.3. Assume that rX is projective and its Jacobson radical Rad (rX) is a superfluous R-submodule of rX. Then the rational functor RatT : ModEnd(CC) ^ ModEnd(Ce) is exact.

Proof By Corollary 3.3, R = 0PePuPR, where {uP : P g P} is a set of orthogonal local idempotents. By [16, Theorem 2], {RuP : P g P} becomes a generating set of completely indecomposable (finitely generated) projective objects for RMod. Since RX is projective, it must be a direct summand of a direct sum of left R-modules of the form RuP. Now, the non-unital version of [18, Theorem 5.5] gives that RX has to be a direct sum modules of the form RuP. By Theorem 2.2, CC is a direct sum of

comodules, profinite as left A-modules. Proposition 2.4 gives the exactness of RatT.

Next, we deal with the case where P contains only one comodule. This naturally leads to the notion of a local coring.

Definition 3.5 A right semiperfect A-coring C whose category of right comodules has a unique (up to isomorphism) type of completely indecomposable semiperfect comodule, will be called a right local coring.

The following is the version of Theorem 3.2 for the case where P is a singleton.

Corollary 3.6 Let C be an A-coring and (Xe, pX) a right C-comodule with endomor-phism ring T = End(Xc). The following statements are equivalent

(i) aC is a flat module and Xe is a small completely indecomposable projective generator in the category of right comodules Comode;

(ii) a C is a flat module, the category Comode is locally finitely generated, C is a right local A-coring and Xe the unique semiperfect completely indecomposable right comodule (up to isomorphism);

(iii) Xa is a finitely generated and projective module, C is T — X-Galois, tX is a faithfully flat module, and T is a local ring.

Our last proposition gives a connection between right local corings and simple cosemisimple corings. Every cosemisimple coring is a right and left semiperfect coring by [11, Theorem 3.1]. A simple cosemisimple coring is a cosemisimple coring with one type of simple right comodule, or equivalently, left comodule. The structure of these corings was given in [7, Theorem 4.3].

Proposition 3.7 Let X a be a non zero finitely generated and projective right A-module and T C S = End(XA) a local subring of its endomorphism ring such that tX is a faithfully flat module. Consider the comatrix A-coring C := X* ®t X, and denote by J the Jacobson radical of T, and by D := T/J the division factor ring. Then we have

(i) X/JX admits the structure of a simple right (X* ®t X)-comodule whose endomorphism ring is isomorphic to the division ring D, that is End((X/JX)c) = D.

(ii) If (X/JX)a is a projective module, then the canonical map

cari£/J£ : (X/JX)* ®d (X/JX) -^ X* ®t X

is a monomorphism of A-corings with domain a simple cosemisimple coring.

(iii) There is a short exact sequence of X* ®t X-bicomodules

0 -X* ®t (JX) -a- X* ®t X -a- X* ®t D ®t X -a- 0

whose cokernel is an A-coring without counit.

(iv) tS is a faithfully flat module and the canonical Sweedler S-coring S ®t S is a right local S-coring with S$®ts the unique (up to isomorphism) semiperfect completely indecomposable comodule.

Proof We first make some useful observations. Recall that X is a right C-comodule with coaction pX (u) = ^i u ®A u* ®T u, for every u e X, where {(ui, u*)}i C X x X* is any finite dual basis for XA, while X* is a left C-comodule with coaction XX* (u*) = u* ®T ui ®a u*, for every u* e X*. It is clear that pX is left T-linear while XX* is right T-linear. Since TX is a faithfully flat module, [7, Theorem 3.10] implies that T = End(XC) and — ®T X : ModT ^ ComodC is an equivalence of categories with inverse the cotensor functor —nCX* : ComodC ^ ModT .This means, by Corollary 3.6,

that C is a right local A-coring with XC the unique type of semiperfect completely indecomposable right comodule (up to isomorphisms).

(i) The right C-coaction of the right A-module X/JX is given by the functor — ®T X, since X/JX = D ®T X is an isomorphism of right A-modules. This comodule is the image under the equivalence —®T X of the simple right T-module DT, so it is necessarily a simple right C-comodule. Its endomorphism ring is computed using the following isomorphisms:

HomC (D ®T X, D ®T X) = HomT (D, HomC (X , D ®T X)^j

= HomT (D, HomT ((XBcX*), D)) = HomT (D, D),

where we have used the adjunction —DCX H — ®T X in the second isomorphism and the right T-linear isomorphism XDCX* = T in the last one.

(ii) Since (X/JX)A is a finitely generated and projective module, we can construct the canonical map canX/JX. This map is a monomorphism by (i) and [2, Theorem 3.1]. Finally, [7, Proposition 4.2] implies that (X/JX)* ®D (X/JX) is simple cosemisimple A-coring, since D is already is division ring.

(iii) The stated sequence is obtained by applying — ®T X to the following short exact sequence of right T-modules

0 -X*J -> X* -> X*®tD -s- 0.

This gives in fact an exact sequence of right C-comodules; the C-bicomodule structure is then completed by the compatible left C-coaction of CX*. The C-bicomodule X* ®T D ®T X admits a coassociative comultiplication given by

A : X* ®t D ®t X -(X* ®t D ®t X) <g>a (X* <s>t D ®t X)

u* <g>T d <g>T v I-u* ®t d ®t ut ®a u* ®t 1 ®t v.

(iv) This is an immediate consequence of [7, Theorem 3.10] and Corollary 3.6. □

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