MORITA DUALITY FOR GROTHENDIECK CATEGORIES

MORITA DUALITY FOR GROTHENDIECK CATEGORIES J . L. GÓMEZ PARDO AND P. A. GUIL ASENSIO Dedicated to the memory of Professor Pere Menal We survey some recent results on the theory of Morita duality for Grothendieck categories, comparing two different versions of this concept, and giving applications to QF-3 and Qf-3' rings. In recent years, two different extensions of the concept of Morita duality to the framework of Grothendieck categories have been proposed. These extensions have been motivated, on the one hand, by the desire of providing a unified background to new Morita-like dualities that have appeared in the literature (see e.g . [2], [15], [21]) and, on the other, by the realization that Morita duality for Grothendieck categories can, in fact, be applied to the study of rings and modules when one considera, say, a module category and looks at Morita dualities of some convenient quotient categories . This is the underlying idea in a series of papers by Colby and Fuller [4], [5], [6] in which they apply these techniques to the study of QF-3 and QF-3' rings . In this paper, we survey some recent results on this topic, that have been obtained in [9] and [10] . The first definition of Morita dualities for Grothendieck categories is due to Colby and Fuller [4], so we are going to speak of Colby-Fuller dualities . A right adjoint pair of contravariant functors D : A A' : D' between abelian categories A and A' defines a Colby-Fuller duality if both functors are exact and the full subcategories Ao and Aó of reflexive objects of A and A, are closed under subobjects, quotient objects and finite direct sums (Le ., they are finitely closed) and contain generating sets for A and A', respectively (Le ., they are generating) . The second extension has been recently proposed by Anh and Wiegandt in [3] . There Work partially supported by the DGICYT (PB90-0300) . 626 J . L. GÓMEZ PARDO, P . A. GUIL ASENSIO is an Anh-Wiegandt duality between two Grothendieck categories A and A' if there exist finitely closed generating full subcategories cp and cp' of A and A', respectively, and a contravariant equivalente F : cp , cp' : F' . In this case, the objects of co and cp' are also called reflexive . It is easy to check that any Colby-Fuller duality between Grothendieck categories defines an Anh-Wiegandt duality, but the converse is not so clear, oven when it is known to be true for module categories . In fact, several natural questions arise at this point : Question 1. Do Colby-Fuller and Anh-Wiegandt dualities coincide? . We will see below that the answer is no in general, but that they coincide in several important cases . This question is very Glose to a harder problem posed by Colby and Fuller in [4] and [6] : Question 2 . Is the exactness condition for the right adjoint functors of a Colby-Fuller duality a consequence of the remaining hypotheses? . We will also give a negative answer to this problem . Other natural questions are the following : Question 3 . Can the usual characterizations of Morita duality for module categories in terms of linear compactness be extended te these dualities? . Question 4. What happens when one of the Grothendieck categories is a module category? . We will show that Question 3 has a satisfactory answer and, regarding Question 4, that if a Grothendieck category has a Colby-Fuller duality with a module category, then it is a module category too ; whilst for Anh-Wiegandt dualities we exhibit a counterexample to this situation . Finally, we give some examples of the applications of this theory te the study of QF-3 and QF-3' rings . 1 . Preliminaries and notation Throughout, R denotes an associative unitary ring and R-Mod (ModR), the category of left (resp . right) R-modules . A ring R is said to be left QF-3 if it has a minimal faithful left R-module (Le ., a faithful module which is a direct summand of any other faithful module) and it is left QF-3' when its left injective hull, E(RR), is torsionless . In case that R is left and right QF-3 (QF 3'), we will simply say that it is QF-3 (resp . QF-3') and we will use a similar convention for other classes of rings . Recall that a pair of contravariant functors D : A : A' : D' between abelian categories A and A' is called right adjoint if there are natural isomorphisms 77A A' : Hom,A(A, D'A') -4 HomA, (A', DA), for each pair MORITA DUALITY FOR GROTHENDIECK CATEGORIES 6)27 of objects A E A and A' E A' . Associated te rl there are the arrows of right adjunction, T : 1,4 ---> D'D and T' : 1,4, --> D o D', defined as TA = r7A DA(1A) and Tá ' = r7D'A',A'(1D'A') (see [4]) . An object A of A (A' of .,4') is said to be reflexive if TA (resp . TA') is an isomorphism . We will denote by D(D') the Lambek localizing subcategory of R-Mod (Mod-R) associated to the filter of dense left (resp . right) ideals . We will follow, as in [4], the convention for the endomorphism ring of an object A i' an abelian category A of calling End,q(A) = HomA(A, A)° P . We refer te [1] and [20] for any undefined concept used in this paper.

In recent years, two different extensions of the concept of Morita duality to the framework of Grothendieck categories have been proposed.These extensions have been motivated, on the one hand, by the desire of providing a unified background to new Morita-like dualities that have appeared in the literature (see e.g.[2], [15], [21]) and, on the other, by the realization that Morita duality for Grothendieck categories can, in fact, be applied to the study of rings and modules when one considera, say, a module category and looks at Morita dualities of some convenient quotient categories .This is the underlying idea in a series of papers by Colby and Fuller [4], [5], [6] in which they apply these techniques to the study of QF-3 and QF-3' rings .In this paper, we survey some recent results on this topic, that have been obtained in [9] and [10].
The first definition of Morita dualities for Grothendieck categories is due to Colby and Fuller [4], so we are going to speak of Colby-Fuller dualities.A right adjoint pair of contravariant functors D : A -A' : D' between abelian categories A and A' defines a Colby-Fuller duality if both functors are exact and the full subcategories Ao and Aó of reflexive objects of A and A, are closed under subobjects, quotient objects and finite direct sums (Le ., they are finitely closed) and contain generating sets for A and A', respectively (Le ., they are generating) .The second extension has been recently proposed by Anh and Wiegandt in [3] .There Work partially supported by the DGICYT (PB90-0300) .
is an Anh-Wiegandt duality between two Grothendieck categories A and A' if there exist finitely closed generating full subcategories cp and cp' of A and A', respectively, and a contravariant equivalente F : cp , cp' : F' .In this case, the objects of co and cp' are also called reflexive .
It is easy to check that any Colby-Fuller duality between Grothendieck categories defines an Anh-Wiegandt duality, but the converse is not so clear, oven when it is known to be true for module categories .In fact, several natural questions arise at this point : Question 1. Do Colby-Fuller and Anh-Wiegandt dualities coincide? .We will see below that the answer is no in general, but that they coincide in several important cases .This question is very Glose to a harder problem posed by Colby and Fuller in [4] and [6] : Question 2. Is the exactness condition for the right adjoint functors of a Colby-Fuller duality a consequence of the remaining hypotheses? .
We will also give a negative answer to this problem.Other natural questions are the following : Question 3 .Can the usual characterizations of Morita duality for module categories in terms of linear compactness be extended te these dualities? .
Question 4. What happens when one of the Grothendieck categories is a module category? .
We will show that Question 3 has a satisfactory answer and, regarding Question 4, that if a Grothendieck category has a Colby-Fuller duality with a module category, then it is a module category too ; whilst for Anh-Wiegandt dualities we exhibit a counterexample to this situation .Finally, we give some examples of the applications of this theory te the study of QF-3 and QF-3' rings.

. Preliminaries and notation
Throughout, R denotes an associative unitary ring and R-Mod (Mod-R), the category of left (resp .right) R-modules .A ring R is said to be left QF-3 if it has a minimal faithful left R-module (Le ., a faithful module which is a direct summand of any other faithful module) and it is left QF-3' when its left injective hull, E(RR), is torsionless.In case that R is left and right QF-3 (QF -3'), we will simply say that it is QF-3 (resp.QF-3') and we will use a similar convention for other classes of rings.
Recall that a pair of contravariant functors D : A : A' : D' between abelian categories A and A' is called right adjoint if there are natural isomorphisms 77A A' : Hom,A(A, D'A') -4 HomA, (A', DA), for each pair of objects A E A and A' E A' .Associated te rl there are the arrows of right adjunction, T : 1,4 ---> D'D and T' : 1,4, --> D o D', defined as TA = r7A DA(1A) and Tá ' = r7D'A',A'(1D'A') (see [4]) .An object A of A (A' of .,4') is said to be reflexive if TA (resp.TA') is an isomorphism .
We will denote by D(D') the Lambek localizing subcategory of R-Mod (Mod-R) associated to the filter of dense left (resp .right) ideals.We will follow, as in [4], the convention for the endomorphism ring of an object A i' an abelian category A of calling End,q(A) = HomA(A, A)°P .We refer te [1] and [20] for any undefined concept used in this paper.

Examples
We begin by giving a result that highlights the close relationship that exists between Question 1 and Question 2 .
Theorem 2 .1.Let F : cp : cp' : F' be an Ahn-Wiegandt duality between Grothendieck categories A and A .Then F and F' can be extended to a right adjoint pair of contravariant functors D : A ;--A' : D' in a unique way .Sketch of Proof: For the extension, observe that any object A of A can be written as A = lim A¡, where the A Z are reflexive subobjects of A .Define then DA = l f im FAZ and similarly D' .One may then check that this definition is independent of the representation of A as direct limit of objects of cp and that, in fact, D and D' are a right adjoint pair of contravariant functors .
It might appear, at first sight, that Theorem 2 .1 makes Question 1 and Question 2 equivalent but this is not the case .The reason is that, when one extends an Anh-Wiegandt duality te a contravariant right adjoint pair, the enlarged subcategories of reflexive objects .,40and Aó (which contain co and cp', respectively) are not necessarily finitely closed .This is shown by the following simple example that preves that Colby-Fuller dualities are stronger than Anh-Wiegandt dualities.
Example 2.2 .Let A be the Grothendieck category of all abelian p-groups and let D : A -A , be the (composition) functor D = T(Homíz(-,7Z(p'))), where T denotes the torsion functor of abelian groups and 7L(p') is the Prilfer group.Then, D forms a right adjoint pair with itself, and the full subcategory W of A consisting of the artinian p-groups is finitely closed and generating.Further, D establishes a contravariant equivalence of cp with itself and, thus, it defines an Anh-Wiegandt duality in A. However, the functor D is not exact and so, it does not define a Co1by-Fuller duality.In fact, one may check that, if i : ZL(pn) ---, 7L(p -) is the inclusion of the cyclic p-group ZL(pn) in 7L(p'), D(i) is not an epimorphism, since D(ZL(pn)) -7Z(pn) and D(7L(p')) = 0 .But this adjoint pair has also another pathology 00 that prevents it from being a Co1by-Fuller duality.Indeed, T( H 7L(pn)) n=1 is a reflexive object in A (Le ., it belongs te Ao) but has a subobject, 00 E) Z(pn), which is not reflexive .Thus, the full subcategories of reflexive objects are not finitely closed .
Example 2 .2provides a negative answer to Question 1 but, since .,40 is not finitely closed, it says nothing about Question 2, Le., it does not solee the problem posed by  and [6, p. 185] asking whether the exactness of the functors is a superfluous hypothesis in the definition of a Co1by-Fuller duality (recall that this is just the case for Morita dualities between module categories) .The following example completely solves this problem.Then, the functors D and D' are right adjoint and it is not difficult to see that they induce a contravariant equivalente between the full subcategories of A and Zp-Mod consisting in the artinian p-groups and the noetherian p-adic modules, respectively.Flrther, by an argument similar to Example 2 .2,D' is not exact.Now, we only need to show that any reflexive object of Zp-Mod is noetherian.But, by the structure theorem of Fzchs for character groups (see [7,Theorem 47.1]), if X is a reflexive object of 7L p -Mod, X -rj7L(pni) ® JIZp and hence it I suffices to prove that I and J must be finite.If this is not the case, let r, = max(III, ¡J¡) > k~o .Then Homzp (((B7L(pni)) ® (®ZL p ), ZL(pw)) contains a subgroup A -rjZL(p), so that ¡Al = 2r -and pA = 0.By the k injectivity of the p-adic module 7Z (p') one can then show that D'X contains a subgroup of cardinality > 2" and hence IDD' (X) 1 > 22w .Since J XJ = 2", this shows that X cannot be reflexive and hence I and J must be both finite .
Remark.Observe that 7L p -Mod has a self Morita duality.Since A cannot be equivalent to 7L p -Mod, Example 2.3 also shows that a Grothendieck category (or even a module category with Morita dual-ity) may have Anh-Wiegandt dualities with nonequivalent Grothendieck categories.

Linear compactness for Grothendieck categories
In Examples 2 .2 and 2 .3,we have shown that Anh-Wiegandt dualities are weaker than Colby-Fuller dualities.However, it is well known that they coincide for module categories .In this section we are going to obtain conditions under which an Anh-Wiegandt duality can be extended to a Colby-Fuller duality.We recall from [8] that an object A of a Grothendieck category A is called linearly compact when, for each inverse system of epimorphisms {pi : X , Xá}z in A, the induced morphism 1¡mm pi : X --> 1li Xi is also an epimorphism.We will say that a subcategory C of A is linearly compact if, for each epimorphism of inverse systems {pi : Xi -Yi}I in A, with the Xi in C, the induced morphism pim pi is an epimorphism .We remark that any object of a linearly compact subcategory of A is linearly compact but, as we will see below, the converse is not true.We begin with the following proposition that is easy to prove (see also [10, Lemma 2]) : Proposition 3.1 .Let D : A : A' : D' be a right adjoint pair of contravariant functors between Grothendieck categories A and A' and let cp and cp' be finitely closed full subcategories of A and A' consisting of reflexive objects.Then any object of cp or cp' is linearly compact.Now, we are in position to prove the following theorem: Theorem 3.2.Let D : A z± A' : D' be a right adjoint pair of con- travariant functors between Grothendieck categories .The following conditions are equivalent: i) There exist finitely closed, generating, linearly compact full subcategories cp and cp' of A and A' such that D and D' induce a contravariant equivalente between them .ii) D and D' define a Colby-Fuller duality between A and A' .
Further, when these equivalent conditions hold, the reflexive objects for the right adjunction coincide with the linearly compact objects.Sketch ofproof. .i) =:> ii) First, notice that since we already know that D and D' are left exact (by the adjunction), in order to show that they are exact we only need to prove that they transform monomorphisms in epimorphisms .But this is a consequence of the fact that cp and W' are linearly compact and generating and D and D' establish a contravariant equivalente between them.Now, using an argument similar to [4, Prop .2.3 i)], we can check that D and D' are faithful and, applying [6,Prop. 4], we see that D and D' define a Colby-Fuller duality.
ii) => i) We only have to show that the full subcategories Ao and Aó of A and A' consisting in the reflexive objects for the adjunction are linearly compact .Consider an epimorphism of inverse systems {pi : Xi -> Yi}I in A with each Xi in Ao .Then, as {Dpi} is a monomorphism of direct systems in A', we get that lim Dpi is a monomorphism .Now, since D' is exact, we see that limD'Dpi -D'(limDpi) is an epimorphism.But, as all the Xi are reflexive, it is easy to see that D'Dpi = pi canonically .Similarly, Aó is linearly compact .
Finally, observe that, since ..40 and Aó are finitely closed, any reflexive object is linearly compact by Proposition 3 .1.To prove the converse it is enough to write the dual of any linearly compact object as a direct limit of reflexive subobjects and then use the linear compactness and the exactness of D and D' .
Remark.In view of Theorem 3.2 it is clear that the category of artinian p-groups considered in Examples 2.3 and 2.4 is not a linearly compact subcategory of the category of abelian p-groups, despite the fact that each artinian p-group is linearly compact .
If we combine Theorem 2.1 and Theorem 3.2, we get a necessary and sufflcient condition for both kind of dualities to agree .Corollary 3.3.An Anh-Wiegandt duality F : cp z± ep' : F' between Grothendieck categories A and A' can be extended to a Colby-Fuller duality if, and only if, cp and cp' are linearly compact subcategories of A and A', respectively .
The reason for the fact that for module categories both kinds of dualities coincide is that a full subcategory of a category of modules is linearly compact if and only if all his objects are linearly compact (see e.g.[11,Theorem 7.1] and [18, Lemma 5.7]).The following corollary extends this fact .
Corollary 3.4.Let F : W ; W' : F' be an Anh-Wiegandt duality between Grothendieck categories A and .,4'which contain generating sets of projective objects.Then this duality can be extended to a (unique) Colby-Fuller duality between A and A' .
Sketch ofproof-By the Gabriel-Popescu Theorem [20, Theorem X.4.1] the categories A and A' can be represented as quotient categories of module categories and, since they contain generating sets of projective objects, the functors of the representation can be chosen to be exact .Now the linear compactness of cp and ep' is a consequence of Proposition 3.1 and the fact that, for module categories, any full subcategory consisting of linearly compact objects is linearly compact .
Remark.We note that this is just the situation considered in [2], [15] and [21] .
We finalize this section with a theorem that extends Müller's characterization of Morita dualities [17, Theorem 1] to Colby-Fuller dualities with reflexive generators.Observe that this particular case of Colby-Fuller dualities is of special interest because of its application to the study of QF-3 and QF-3' rings (see [4], [5] and [6]) .First, we need the following result that is implicit in [6] (see also [4]) : Theorem 3.5 (Colby-Fuller) .A ring R is QF-3' iff the R-dual functors induce a Colby-Fuller duality with reflexive generators between R-Mod /D and Mod-R/D' .Further, R is also its own maximal quotient ring if only if it is closed for the localizing subcategories D and D' .ii) => i) Take R = EndA(U) .By [16,Theorem 8.3], R-Mod /D is equivalent to A, where D is the Lambek localizing subcategory of R-Mod and R is left QF-3' and its own left maximal quotient ring.Further, identifying R-Mod /D with A, we get that the smallest finitely closed subcategory of R-Mod /D containing R is linearly compact .We want to see that the R-double dual functor ( )** : Mod-R -> Mod-R preserves monomorphisms or, equivalently, that if j is a monomorphism in Mod-R, then j* is an epimorphism in R-Mod /D .We are going to prove it in five steps: Step 1.Consider the inclusion j : J -RR where J = a¡ R is -i a finitely generated right ideal of R. If j** is not a monomorphism, there is a non zero homomorphism co : J* -`RR such that cp o j* = 0. Let f E J* such that W(f) qÉ 0 and define 0 : J* -RR' the monomorphism given by O(f) = (f (ai))i=1 ,n .Then, since the injec- tive envelope of RR, E(RR), is torsionless, one can construct a homo-morphism a : RR' -`RR such that a o V) 7~0, but a o 0 o j* = 0. n Take h E J* such that a o O(h) :~0 and define h : ® ai R ---> Rn i=1 as h(ri, . . ., rn) = (h(ai)ri, . . . .h(an)rn) .Then it is easy to see that (a o 0)(h) = a(h(ai), . ..,h(an)) = a(h(ai, . ..,an)) = h(ca(ai, . ..,an) = h(a o 0 o j* (l» = 0, which is a contradiction .Thus, j** is a monomorphism.
Step 2 .If J is a right ideal of R and j : J --> R is the inclusion, then we can write J : lim Ji where the Ji are finitely generated idéals of J. Now, using Step 1 and the fact that R is linearly compact in R-Mod /D, it is easy to show that j* is also an epimorphism in R-Mod/D, i.e., j** is a monomorphism.
Step 3. If j : X -+ Rn is a monomorphism in Mod-R, we can prove using induction and Step 2 that j** is a monomorphism .
Step 4. If j : XR --+ RR) is a monomorphism, for some set I, then we can write j = liM jF : XF -> RF, where F ranges over the finite subsets of I, and use Step 3 and the linear compactness of the smallest finitely closed subcategory of R-Mod /D containing R to show that j** is a monomorphism.
Step 5 .Finally, if j : X -> Y is an arbitrary monomorphism in Mod-R, we can see that j** is a monomorphism writing Y as a quotient of R(I), considering the inverse image of X by this epimorphism, and then using Step 4.
Thus, we have shown that ()** : Mod-R -i Mod-R preserves monomorphisms .Using now Theorem 3 .5 and [6, Theorem 1], we see that there is a Colby-Fuller duality between R-Mod /D and Mod-R/D' with reflexive generators.Combining Theorems 3.5 and 3 .6,we get the following corollary that extends to QF-3' maximal quotient rings the characterization of QF-3 maximal quotient rings given in [19, Theorem 2.1] .
Corollary 3.7.-A ring R is QF-3' and its own maximal quotient ring if and only if it is the endomorphism ring of a generator-cogenerator U of a Grothendieck category such that the smallest finitely closed full subcategory containing U is linearly compact .
4 .Applications to QF-3 and QF-3' rings In this section, we exhibit some consequences of the previous results for the study of QF-3 and QF-3' rings .We begin by answering the fourth question posed in the introduction .First, we need the following result, due to Colby and Fuller (see [5,Theorem 2]) : the ring R = z is a left QF-3' ring with a left dominant module RP = o .However, it is not right QF-3' nor left QF-3.
Theorem 4.2 and 4.3 also can be used for obtaining new characterizations of particular classes of QF-3 rings .We finalize this paper with two examples of them (see also [9]).

Theorem 3. 6 .
Let A be a Grothendieck category .The following are equivalent : i) There exists a Grothendieck category A' and a Colby-Fuller duality D : A --A' : D' with reflexive generators .ii) A has a generator-cogenerator U such that the smallest finitely closed subcategory of A containing U is linearly compact.Skecth ofproof.i) =* ii) By Proposition 3 .1 and [4, Proposition 2 .3] .