JOURNAL OF ALGEBRA 185, 108᎐124Ž. 1996 ARTICLE NO. 0315
Colocalization on Grothendieck Categories with Applications to Coalgebras
C. Nastasescu*˘˘
Facultatea de Matematica,˘ Str. Academiei 14, R. 70109 Bucharest 1, Romania
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Departamento de Algebra y Analisis,´´´ Uni¨ersidad de Almerıa, 04071 Almerıa, Spain
Communicated by Richard G. Swan
Received June 28, 1995
INTRODUCTION
Rings and modules of quotients with respect to an additive topology F or a localizing subcategory of R y Mod were introduced by Gabriel in his thesiswx 3 , and have been an important tool in ring theory for more than 20 years. If CF is the localizing subcategory of R y Mod associated to F, then we can consider the quotient category R y ModrCF and the canoni- Ž . cal functor TF : R y Mod ª R y ModrCF seewx 3 and wx 7 . It is well Ž known that TF has a right adjoint SF : R y ModrCF ª R y Mod in fact the existence of such a right adjoint functor is equivalent to the fact that . the subcategory we factor by is localizing; seewx 3 . The quotient ring RF associated to F is an approach to the quotient category in the sense that
under certain conditions RF y Mod is equivalent to the quotient category R y ModrCF . This situation represents the perfect localization of rings Žor the flat epimorphism of rings; seewx 7, p. 225. . Starting from the localization for rings, we develop a theory of ‘‘localization’’ for coalgebras,
* Part of this work was done while this author was visiting the University of Almerıa´ supported by DGICYT. † Supported by the grant PB91-0706 from the DGICYT.
108
0021-8693r96 $18.00 Copyright ᮊ 1996 by Academic Press, Inc. All rights of reproduction in any form reserved. GROTHENDIECK CATEGORIES 109 giving reasonable answers in this paper. More exactly, if C is a coalgebra over the field k, and T is a dense subcategoryŽ. or a Serre class of the category M C of right C-comodules, we can consider the quotient category C C C M rT and the canonical functor T: M ª M rT. The key step comes now: instead of considering that T is a localizing subcategoryŽ i.e., T is closed under arbitrary direct sums or, equivalently, T has a right adjoint. , we will ask T to be a colocalizing subcategoryŽ i.e., the functor T has a left adjoint H .. We will see later that a colocalizing subcategory is also localizing. The colocalizing subcategory T is called perfect if H is an exact functor. This situation is dual to the perfect localization of ringsŽ or the flat epimorphism of rings. . This paper is divided into five sections. In Section 1 we give some properties of left coflat monomorphisms of coalgebras. A study of colocal- ization in an abelian category is made in Section 2. We apply these results, in Section 3, to the category M C of right C-comodules, introducing the quotient coalgebra with respect to a colocalization as an analogue of the quotient ring. We also give the main properties of the quotient coalgebra. In Section 4 we study perfect colocalization on M C, proving that a perfect colocalization is given by aŽ. left coflat monomorphism. It follows from Section 1 that we can associate a perfect colocalization to any coflat monomorphism of coalgebras. The last section is concerned with applica- tions. A relevant example is the Goldie torsion theory, to which case we apply our theory.
NOTATION AND PRELIMINARIES
Let k be a field. By k-space Ž.k-map we mean a k-vector space Žk-linear map. . All unadorned tensor products, Hom, etc., will be over k. The reader is referred to the bookswx 1 and wx 8 for notions and notations concerning coalgebras and comodules. The category of k-coalgebras is denoted by
Cog k .IfCis a coalgebra, the categories of rightŽ. resp. left C-comodules is denoted by MC Žresp. C M.. The fact that a k-space M is an object of Ž. C such a category is denoted by MCCresp. M .If M,NgM , the k-space of C-comodules maps between M and N is denoted by ComCŽ.M, N .If C,Dare coalgebras, the category of Ž.C, D -bicomodules Ž i.e., left C- comodules, right D-comodules with compatible structures. is denoted by C D M ; an object in this category is represented by CDM . Ž. We will freely use ‘‘sigma notation’’: ⌬ c s Ý c12m c for the comulti- Ž. plication of a coalgebra C and m s Ý m01m m for the structure map of a right C-comodule M. For any abelian category A we denote ZŽ.A its centre, i.e., the commu- tative ring of all natural morphisms of the identity functor 1A : A ª A. 110 NASTASESCU˘˘ AND TORRECILLAS
Ais called a k-abelian category whenever there exists aŽ. preserving unit ring morphism : k ª ZŽ.A . Giving such a is equivalent to defining on Ž. any Hom A M, N a k-space structure such that the composition maps Ž. Ž. Ž. CC Hom A M, N = Hom A N, P ª Hom A M, P are k-bilinear. M , M, and C M D are instances of k-abelian categories. An abelian category A is called locally finite if it has a family of generators of finite lengthŽ seewx 7. . Following Takeuchiwx 9 , a k-abelian Ž. category A is of finite type if A is locally finite and Hom A M, N is finite dimensionalŽ. over k for any objects M, N of finite length.Ž It is easily seen Ž. that this last property is equivalent to End A M is finite dimensional for any simple object M of A.. We recall fromwx 9, Theorem 5.1 that a k-abelian category is of finite type if and only if it is k-equivalent to a C category M for some coalgebra C.If Ais of finite type, an object M g A Ž. is quasifinite if Hom A S, M is finite dimensional for all simple objects S ŽŽ. of A or equivalently Hom A X, M is finite dimensional for all objects X g A of finite length. . Let C be an arbitrary coalgebra, M be a right C-comodule, and N be a left C-comodule. The cotensor product M IC N is the kernel of the k-map MNm1y1m : MmNªMmCmN. Followingwx 2 , the cotensor C C product is a left exact functor and preserves inductive limits M = M ª Ž. MMkkis the category of k-spaces . Moreover the mappings m m c ª Ž. Ž. ⑀cm and c m n ª ⑀ cn yield natural isomorphisms M IC C , M C and CIC N , N. The cotensor product is associative. If N g M has finite dimension, then N* s HomŽ.N, k has a natural structure of left Ž. C-comodule and M ICCN* , Com N, M . A left C-comodule M is C called left coflat if the functor yIC M: M ª k y Mod is exact. It is proved by Takeuchi that M is left coflat if and only if M is an injective C object of M Žcf.wx 2. . C D Let now M g M . Then MD is quasifinite if and only if the functor MCDM Ž.Žᎏ . yIC M: ª has a left adjoint denoted by hMyD , wx9 . The Ž.ᎏ functor hMyD , is called the co-hom functor. The following descrip- D tion is given inwx 9 : if Y g M , then
U hMyDDiDiŽ.,Yslim Com ŽY , M .* s lim ŽM I Y .*, ªiªi where Ž.Yiiis the family of finite dimensional subcomodules of YD.We Ž. M D have in particular that hDyD ,Y,Yfor any Y g . The functor Ž.ᎏ hMyD , is right exact and commutes with inductive limits; it is an exact Ž. functor if and only if MDDis injective. If M is quasifinite, then eMyDs Ž. hMyD ,Mhas a natural structure of coalgebra, called the co-endomor- Ž .ŽŽ.. phism coalgebra of M seewx 9 ; M becomes then an eMyD ,D-bico- module. For anyŽ. bi comodules MCD, N , and DCX , with quasifinite MC, ␦ Ž. Ž. there exists a canonical map : hMyCD,NIXªNI DhMyC,X, GROTHENDIECK CATEGORIES 111
which is an isomorphism if either NDCis injective or M is quasifinite and injectiveŽ seewx 9, 1.13. .
1. COFLAT MONOMORPHISMS
C Let : C ª D be a coalgebra morphism. Then any M g M Žwith the .Ž structure map M : M ª M m C becomes a right D-comodule by 1 m .Ž.CD M:MªMmD. This defines an exact functor ᎏ : M ª M .In particular, C itself may be regarded as a left and right D-comodule. Since D C C D C g M , we can consider N s N IDC g M for any N g M , which Ž. Ž. is a right adjoint of ᎏ bywx 2, Proposition 6 . We observe that ᎏ s Ž. Ž. yIC C, where C is considered a C, D -bicomodule. C is a C, C -bico- C module, thus it is also a Ž.D, D -bicomodule via . Let now M g M . Since Ž.Ž.Ž.Ž. 1MCMCMMCMMmm1m1s1mm11m⌬, we have that Ž. M M:M IDMC. Denote by : M ª MIDC the co-restriction of Ä4C MM. Clearly the maps , M g M define a natural morphism : Ž.Ž. 1MCªᎏ(ᎏ. For M s C we have CCs ⌬ ; this yields a canonical morphism ⌬: Ž. CªCIDC, which is a C, C -bicomodule morphism. Let
p1 CIDC i C p2 be the restriction of the canonical maps C m C ª C defined by c12m c ª Ž. Ž Ž.. Ž. ⑀cc12resp. c 1m c 2ª c 1⑀ c 2. Clearly p1is a D, C -bicomodule mor- Ž. phism and p21is a C, D -bicomodule morphism. Also p ⌬ s p2⌬ s 1.C A characterization of monomorphism in the category of Cog k was given inwx 5, Theorem 3.5 . Assume now that is left coflat morphism, i.e., the Ä D 4 functor yIDC is exact. In this case Ts M g M < M IDC s 0isa D D D localizing subcategory of M . Let T: M ª M rT be the canonical functor. Bywx 5, Theorem 4.2 , there exists a subcoalgebra A of D such that Ä D Ž. 4 T sMgM< M M:MmA. Moreover, A is a co-idempotent coal- gebra, i.e., A s A n A. PROPOSITION 1.1. If : C ª D is a left coflat monomorphism, then
Ž.1 Ker and Coker belong to T; D Ž.2 DrA is quasifinite in M ; D Ž.3 If E Ž DrA . is the injecti¨een¨elope of DrAinM and j: DrA ª Ž. EDrA is the inclusion map, then Coker j belongs to T; Ž. Ž . 4 T DD is a quasifinite injecti¨e cogenerator of the quotient category D M rT. 112 NASTASESCU˘˘ AND TORRECILLAS
Proof. Ž.1 We consider the functors
Ž.ᎏ 6 T 6 MMMCDD 6 6 T, Ž.ᎏ S r
Ž. Ž. where S is the right adjoint of T.If FsT(ᎏ and G s ᎏ (S, then bywx 5, Theorem 6.1 , F and G give an equivalence of categories between CD Mand M rT. Since is a monomorphism, we obtain byw 5, Theorem D 3.5x that Ker IDC s 0, hence Ker g M rT . Next consider the exact D sequence in M , C ª D ª Coker ª 0. Applying yIDC we get C C D C Coker C 0. Identifying D C with C, ID ª IDIDDª I ª I D IDCis replaced by p1: CIDC ª C. Since p1(⌬ s 1C and ⌬ is an isomorphism, we obtain that p1 is an isomorphism; therefore Coker IDC s 0 and Coker g T . Ž. Ž.Ž. 4 Since A g T , then TDD sTDrA. Since Ker , Coker g T , Ž. Ž . Ž. Ž. Ž . we obtain that TCDDsTD . Therefore FCCDsTC sTDrAs TDŽ.D . However, F is an equivalence; hence FCŽ.C is a quasifinite injective cogenerator. Thus TDŽ.D has the same properties. Ž. D Ž. 2 Let M be a simple object in M . Then Com D M, DrA s 0if MgT. Assume that M is T-torsion-free. Then the canonical morphism
Com Ž.M, D A Hom D Ž.TM Ž.Ž.,TDA , Dr ª MrT r
Ž. D fªTf,isinjective.SinceTM is simple in M rT , Hom D ŽŽTM .,TD ŽA ..is finite dimensional. It follows that D A is M rT r r quasifinite. Ž. Ž . 3 Applying T to the exact sequence 0 ª DrA ªj EDrAª Ž. Coker j ª 0, we obtain the exact sequence 0 ª TDrAªTŽj. ŽŽ .. Ž . Ž.Ž. TEDrA ªTCoker j ª 0. Since TDrA,TDD is injective, the essential monomorphism TjŽ.is an isomorphism; therefore TŽ.Coker j s 0, i.e., Coker j g T. PROPOSITION 1.2. Let C be a coalgebra and T a localizing subcategory of C C M . Then the quotient category M rT is a k-abelian category of finite type. C C Proof. Let T: M ª M rT be the canonical functor and S its right C Ž. adjoint. Followingwx 9 , it is sufficient to prove that End M rT X is finite C dimensional for any simple object X g M rT. The functor S is fully CŽ. ŽŽ.Ž.. Ž. faithful; thus Hom M rT X, X , ComC SX,SX . However, SX contains a nonzero simple object M. Since TSXŽŽ ..,X, we have that Ž. Ž .C Ž . SXrMgT and TM ,X. Finally Hom M r T X, X , C ŽŽ . . Ž Ž .. Ž . Hom M rT TM,X ,ComCCM, SX ,Com M, M , which is finite dimensional. GROTHENDIECK CATEGORIES 113
2. COLOCALIZATION IN ABELIAN CATEGORIES
Let A be an abelian category and C a dense subcategoryŽ or a Serre class. of A, i.e., for any exact sequence 0 ª MЈ ª M ª MЉ ª 0in A, MgCif and only if MЈ, MЉ g C Žseewx 3, p. 365. . We can construct the quotient category ArC and the canonical exact functor T: A ª ArC.We recall fromwx 3 that the objects of ArC are just the objects of A, and if M, N g A, then
Ј Ј Ј Ј Hom ArCŽ.M, N s limÄ4 Hom AŽ.M , NrN MrM g C , N g C . ªMЈ,NЈ
The quotient category ArC is abelian. The dense subcategory C is called localizing if T has a right adjoint. In the case where A is a Grothendieck category, the dense subcategory C is localizing if and only if C is closed under arbitrary direct sums. We are interested in the situation where C is a colocalizing subcategory, i.e., the canonical functor T has a left adjoint H: ArC ª A. The follow- ing result is not new; it is an exercise inwx 3, p. 369 .
PROPOSITION 2.1. Let C be a colocalizing subcategory of the abelian category A and let X be an object of ArC. The following assertions hold: Ž.a If Y g C is a quotient object of HŽ. X , then Y s 0. Ž. Ž. b If 0 ª M ªf P ª HX ª0is an exact sequence in A and M g C, then f splits. Ž. Ž Ž . . Ž Ž . c The canonical morphism Hom A HX,MªHom Ar C TH X , TŽ.. M sending f to TŽ. f is an isomorphism for any M g A. Ž. ⌿ ⌽ d If :1Ar C ªT(H and : H(T ª 1 A are the natural mor- phisms defined by the adjunction, then ⌿ is an isomorphism and, for any M g A, there exists an exact sequence
⌽Ž.M6 0ªKer ⌽Ž.M ª HT Ž. M M ª Coker ⌽Ž.M ª 0 with Ker ⌽Ž.M , Coker ⌽ Ž.M g C. Ž.e The functor H is fully faithful.
Proof. Everything follows byw 3, Lemmas 1 and 2 and Proposition 3, pp. 370, 371x using the dual category A o. In this case C remains a Serre class in A o, but H becomes a right adjoint of the canonical functor T: o o A ª A rC. 114 NASTASESCU˘˘ AND TORRECILLAS
PROPOSITION 2.2. Let C be a colocalizing subcategory of A. Then the following statements hold: Ž. ŽŽ.. 1 MgCif and only if Hom A HX,Ms0for any X g ArC. Ž.2 If A satisfies AB3* Ži.e., A has arbitrary direct products., then C is closed to direct products. Ž.3 If moreo¨er A is a Grothendieck category, then C is also a localizing subcategory.
Proof. Ž.1IfMgC, the desired relation follows by Proposition 2.1 Ž. a . Ž. ŽŽ.Ž.. Conversely, if M f C, then TM/0. Hence Hom ArC TM,TM / ŽŽ . Ž .. Ž Ž . . 0. However, Hom Ar C TM,TM ,Hom A HT M , M s 0, providing a contradiction. Ž.2 Directly fromŽ. 1 . Ž. Ž . Ž. 3 Let MiigI be a family of objects of C. We have by 2 that Ł M C, and since M is a subobject of Ł M , we obtain ig Iig [igIi igIi that M C, i.e., C is a localizing subcategory. [ig I i g From now on, we assume in this section that A is an abelian category with AB3*, and C is a colocalization subcategory of A.If MgAwe can Ž. define sM sFiiM, where Miare the subobjects of M with the property that MrMi g C. Since C is closed to direct product, we obtain Ž. Ž. from the exact sequence 0 ª MrsM ªŁiiMrM that MrsM gC. The correspondence M ª sMŽ.defines a subfunctor of the identity func- tor. Indeed, if u: M ª N is a morphism in A, we obtain from sMŽ.:M Ž. Ž . Ž. Ž . ªu Nª NrsN that Im (u : NrsN, hence Im (u g C. Therefore KerŽ.Ž. (u = sM. Thus Ž.ŽŽ.. (usM s0, showing that usMŽŽ ..:sN Ž .. PROPOSITION 2.3. Let M g A. Then there exists X g ArC such that M,HŽ. X if and only if s Ž. M s M and M is C-projecti¨e, i.e., for any diagram
M
f 6 6 6 PPuЈ0 with Ker u g C, there exists g: M ª P such that f s ug. Proof. If M , HXŽ., Proposition 2.1 Ž. a shows that sM Ž.sM. That M is C-projective follows from the fact that H is a left adjoint of T. GROTHENDIECK CATEGORIES 115
Conversely, Coker Ž.M g C and sM Ž.sMimply that Coker Ž.M s 0. The diagram M
1M 6 6 HTŽ. M6 M 0, Ž.M where Ker Ž.M g C shows that there is g: M ª HTŽ. M such that Ž. Ž. Ž. Ž. Mgs1M . Hence HT M , M [ Ker M ; that is, Ker M is a quotient object of HTŽ. M . Therefore Ker Ž.M s 0 and HT Ž. M , M. PROPOSITION 2.4. Let C be a colocalizing subcategory of A and M g A. The following properties hold:
Ž.1Im Ž.Mss Ž. M for any M g A. Ž.2 If C is closed under taking injecti¨een¨elopes, then ŽMisa . monomorphism for any M.
Proof. Ž.1 Let u: M ª Y be a morphism, where Y g C. Since ImŽŽ..u( M : Y, we obtain that Im ŽŽ..u( M s 0. Hence Im Ž.M : Ker u. Thus we have that Im Ž.M : sM Ž.. On the other hand sM Ž.: Im Ž.M , since Coker Ž.M g C. Ž.2 We consider the diagram
6 i 6 Ž.M 6 0 Ker Ž.MHTMM Ž. ,
j 6 EŽ.Ker Ž.M where EŽŽ..Ker M g C is the injective envelope of Ker Ž.M . There is some u: HTŽ. M ª E ŽKer Ž..M such that ui s j. Proposition 2.1 Ž. a shows that u s 0. Hence j s 0 and Ker Ž.M s 0.
3. COLOCALIZATION IN THE CATEGORY MC
Let C be a k-coalgebra and T a localizing subcategory of MC. We know Ä wx5 that there exists a subcoalgebra A of C such that TA s M g C Ž. 4Ä CH 4 H Ä M< M M:MmAsMgM We denote by TA the localizing subcategory associated to a co-idempo- tent subcoalgebra A. It is also proved inwx 5 that TA is a T.T.F. class, i.e., TA is closed under direct productsŽ cf.wx 7, VI.8. . Let us consider the C C C canonical functor TAA: M ª M rT , which has a right adjoint SAA: M rT C ªM, since TA is localizing. PROPOSITION 3.1. With the abo¨e notation, the following assertions are equi¨alent: Ž.1 TAA has left adjoint H Ž i.e., TAis a colocalizing subcategory .. Ž.2 CrA is a quasifinite right C-comodule. C Proof. By Proposition 1.2. Hence M rTA is a k-abelian category of finite C type, thenwx 9, Theorem 5.1 there exists a coalgebra D such that M rTA is equivalent via the pair of functors F 6 MCDTM. rA6 G Since TA is exact and commutes with direct sums, the functor F(TA: CD MªMalso has these properties. Bywx 9, Proposition 2.1 we obtain that ŽŽ .. F(TAD,yI P, where P DACs FT C . Now bywx 9, Proposition 1.9 D yIDDPhas a left adjoint if and only if P is quasifinite in M . Since F is an equivalence, PDAis quasifinite if and only if TCŽ.Cis quasifinite in CŽ. Ž . C MrTAA. Clearly TCCAsTCrA.If X is a simple object in M rTA, C Ž. then there exists a simple object M in M such that X s TMA . Then Hom CCŽX, TC ŽA .. Hom ŽTM Ž .,TC ŽA .. Com Ž M, M rTAAA r s M rT AAr , C Ž..Ž . STAA CrA ,Com AM, CrA , since CrA is an essential subobject of Ž. Ž. C STAA CrA. We conclude that TCAC is quasifinite in M rTAif and only if CrA is so. Remark 3.2. With the above notations TCACŽ.is a cogenerator of the CCŽ. category M rTAA. Indeed, if X g M rT , then X s TMAfor some C Ž.ŽI. C MgM. We have an exact sequence 0 ª M ª CC in M for some set I. Since TA is exact and commutes with direct sums, we obtain an exact Ž . ŽŽ ..Ž I . C Ž. sequence 0 ª TMAACAACªTC in M rT . Thus TC is a cogener- C ator in M rTA. C For the rest of the section we assume that CrA is quasifinite in M . With the above notations we have two pairs of adjoint functors: TA 6 F 6 MMCCM D 6 rTA6 HGA Ž.Ž. and the isomorphisms HAA(T , H A(G ( F(TAAAand T ( H , Ž.Ž.FTHG1C . Moreover H G is a left adjoint of the (AA((,MrTA A( Ž.Ž. functor F(TAA. Therefore the object H (TCACdoes not depend on the choice of the coalgebra D and the functors F, G. Let us denote PD s GROTHENDIECK CATEGORIES 117 Ž.Ž. F(TCAC. Since CrA is quasifinite, PDis a quasifinite D-comodule; D also PD is a cogenerator of M . We obtain bywx 9, Proposition 1.10 that M D Ž. HAD(Gis the co-hom functor sending Y g to hPyDDD,Y. In partic- Ž.Ž.Ž.Ž. ular HAAC(TC,hPyDD,P DsePyDD, which has a natural coalge- bra structure. This coalgebra structure does not depend on the choice of D. Indeed, DЈ is another coalgebra such that there exists an equivalence DDЈ XŽ.Ž. X U:MªMsuch that FЈ s U( F.If PDЈ s FЈ(TCAC, then P DЈs Ž. Ž. ŽX. UPD. Since U is an equivalence we clearly have ePyDD,ePyDЈDЈ. C D As a consequence we can assume that M rTAAs M . Since T is exact and commutes with inductive limits, we have that TACDsyI P , where Ž. Ž.ᎏ MD PDACsTC and H As hPyDD,.If X Dg the natural morphism Ž. Ž . Ž . X:XD ªhPyDD,X DI CDPis an isomorphism from Section 2 . In C fact Ž.X is just the natural morphism defined inwx 9, 1.7 . Let M g M . Ž. Ž . From Section 2 we have the canonical map M : hPydD,MI CDPª Ž. Ž. M, with Ker M , Coker M g T. We can write M s D M, where Ms are the finite dimensional subcomodules of M. Then M ICDP s lim Ž.M P . Since the functor hPŽ.,ᎏcommutes with inductive ª ICD yDd limits, we obtain that hPŽ.,MP lim hP Ž,M P ..On yDDI CDsªyDD ICD C the other hand the functor TA sends any simple object S of M either to 0 C Ž. or a simple object of M rTAA. Therefore TMhas finite dimension for C any finite dimensional M g M . However, TACDsyI P ; thus MICDP Ž. is a finite dimensional D-comodule for any . Hence hPyDD,MI CDP lim Com Ž.M P , P *. sª D ICD D On the other hand M hCŽ.,M lim Com ŽM , C .*. Since P s yC sª C D Ž. sCICDP, then we have for each a natural morphism ComCM, C ª Com Ž.M P , P , u u 1 . By dualizing we obtain a natural D I CD D ª I CPD Ž.Ž. morphism Com D M ICDP , P D* ª ComCM, C *. Taking inductive lim- Ž. Ž . its we just obtain the morphism M : hPyDD,MI CDPªM. In par- Ž. Ž . ticular, for M s CC we obtain the morphism C : ePyDDªC. Since Ž. TACDsyI P , it follows from the above facts that C is a coalgebra ␦ Ž. morphism. Bywx 9, 1.13 we have a natural morphism : hPyDD,MI CDP Ž. Ž. MC ªMIC hPyDD,P DsMI CePyDD. Then for any M g , the mor- Ž. Ž. phism M can be obtained as follows: hPyDD,MI CDPª␦ MhPŽ.,P MeP Ž. MCM.Hence IC yDD DsI CyDDª1ICŽC. IC, Ž. Ž Ž..␦ Ž. Ž . Ms1IC C(. Moreover HT M s hPyDD,MI CDPis a right Ž. Ž. ePyDD-comodule, and the right C-comodule structure that HT M has via Ž.C is just the original C-comodule structure. Summarizing we have the following result: C PROPOSITION 3.3. Assume that CrA is quasifinite in M . With the preceding notation we ha¨e: Ž. Ž .Ž. Ž. 1 EsHAA(T C has a natural coalgebra structure and C : Ž. Ž. EªC is a coalgebra morphism. Moreo¨er Ker C , Coker C g TA. 118 NASTASESCU˘˘ AND TORRECILLAS Ž.2Im Ž.C sC Ž. Ker f ACH ,where A is re- s sFf g Com CŽC, A. s garded as a right C-comodule. Ž. Ž .Ž . 3 HAA(T M has a natural right E-comodule structure for any C MgM. Proof. It remains to show only the second assertion. Clearly sCŽ.: Ker f. Ff g Com CŽC, A. Conversely, let us take some M g TA and g: C ª M a comodule Ž I . I morphism. We have a monomorphism u: M ª A : A for some set Ž. I. Then Ker g s Fig IiKer ug , where iis the canonical projec- tion. Hence Ker f Ker g sCŽ.. FfgCom CCŽC, A. : FggCom ŽC, M ., M g TAs HŽH.HŽ. H Next ACrACs0 implies that CrACgTA. Thus sC :AC. On the other hand if f: C ª A is a comodule morphism then fACŽ H . :AfCHŽ.:AAHHs0. Hence AC:Ker f. This shows that ACH:sCŽ., proving the desired equality. The coalgebra E is called the quotient coalgebra with respect to the localizing subcategory TA. We recall that a coalgebra C is called right semiperfect if the category MC has enough projective objectsŽ seewx 4. . The next result characterizes the localizing subcategories for these coalgebras. PROPOSITION 3.4. If C is a right semiperfect coalgebra, then CrAisa right quasifinite comodule, i.e., TA is a colocalizing subcategory. Proof. Let M be a simple object of MC. Since C is right semiperfect, M has a projective cover P ª M ª 0wx 4 . Moreover, since M is finite dimensional, P is also finite dimensional. The fact that C is quasifinite Ž. yields that ComC P, C is finite dimensional. The exact sequence C ª Ž. Ž . CrAª0 produces the exact sequence ComCCP, C ª Com P, CrA ª Ž.Ž. 0 since P is projective . Hence ComC P, CrA is finite dimensional. From Ž. Ž. the exact sequence 0 ª ComCCM, CrA ª Com P, CrA , we conclude Ž. that ComC M, CrA is finite dimensional. Thus CrA is quasifinite. 4. PERFECTŽ. EXACT COLOCALIZATION Through this section we keep the notations of Section 3. PROPOSITION 4.1. The following assertions are equi¨alent: Ž. CC 1The functor TAA: M ª M rT has an exact left adjoint HA: CC MrTAªM. Ž. Ž. Ž. 2iCrA is quasifinite right C-comodule and ii Coker j g TA, where j: CrA ª ECŽ.rA is the inclusion morphism of CrA into its injecti¨e en¨elope. GROTHENDIECK CATEGORIES 119 Ž. Ž. Ž. Proof. 1 « 2 . Clearly i follows from Proposition 3.1. Since HA is exact, then TA carries injective objects to injective objects. In particular Ž. Ž. Ž . Ž . TCACis injective. However, TCACsTC ArA; therefore TAA rCis injective. The morphism j is essential, so TjAŽ. is essential. Since ŽŽ .. Ž. TECAArA is injective, we obtain that Tjis an isomorphism. Thus Coker j g TA. Ž. Ž. C 2«1 . By Proposition 3.1, TAAhas a left adjoint H . Let Q g M ŽI. be an injective object. Then Q is a direct summand of Ž.CC for some nonempty set Ž.I . Since TA commutes with direct sums, we obtain that Ž. Ž Ž ..ŽI. Ž. Ž . TQAAis a direct summand of TCC. Now TCACsTCArAand by Ž. Ž Ž .. Ž . ii they are equal to TECA rA. Since ECrA is T-torsion-free, we Ž Ž .. Ž . obtain that TECAArA is injective in T seewx 3, Chapter III . We have seen that TAAcarries injectives to injectives; therefore H is an exact functor. The colocalizing subcategory T of MC will be called perfect Ž.or exact if any of the equivalent conditions in the last proposition holds. PROPOSITION 4.2. The condition Coker j g T is satisfied in the following cases: Ž.1 C is a hereditary coalgebra Ž i.e., gl.dim C F 1,wx 6. . Ž. 2 TA is closed under injecti¨een¨elopes. Ž. C 3 The quotient category M rTA is a semisimple category. Proof. Ž.1 CrA is a rightŽ. and left injective C-comodule since C is a hereditary coalgebra. Ž.2 Since T is closed under injective envelopes, we obtain that A is an injective right C-comodule and CC s A [ CrA. Therefore CrA is also an injective right C-comodule. Ž.3 TAAsemisimple implies that TjŽ.is an isomorphism; therefore Coker j g T. We can state now the main result of this section. THEOREM 4.3. Let C be a coalgebra and A be a co-idempotent subcoalge- Ž bra such that TA is a colocalizing subcategory i.e., CrA is a quasifinite right C-comodule.. Then the following assertions are equi¨alent: Ž.1 TA is a perfect colocalizing subcategory. Ž.2 If E is the quotient coalgebra associated to TA, then the natural coalgebra morphism : E ª C is a left coflat monomorphism. Moreo¨er, if Ž.1 and Ž.2 hold, the localizing subcategory T associated to is TA. 120 NASTASESCU˘˘ AND TORRECILLAS Proof. Ž.2 « Ž.1 . Follows from Propositions 1.1 and 4.1. Ž. Ž. 1«2 . With the notations from Section 3, we have TACDsyI P . Since Ker , Coker g TAC, then Ker I PDDs 0. Since P is injective and quasifinite,wx 9, Proposition 1.14 shows that the natural morphismŽ. * ␦ : hPyDDŽ.,MI CDPªMI CePyDD Ž. C is an isomorphism for any MCDg M . Then P is an injective cogenerator. Hence M ICDP s 0 if and only if M ICE s 0. Indeed, M ICDP s 0 Ž. implies M ICE s 0. The converse follows from the fact that hPyDD,X Ž. s0 if and only if X s 0. This is clear since hPyDD,Xs lim Com Ž.X , P *, where X X and Ž.X is the family of ª ig ICiD s D igIi iigI all finite dimensional subcomodules of X Ž.ordered by inclusion . If X / 0, since PDcis an injective cogenerator we have Com Ž.Xi, PD* / 0 for some Ž. igI. However, this implies that hPyDD,X/0. Since Ker I CDP s 0, then Ker ICE s 0. Bywx 5, Proposition 3.5 we obtain that is a monomorphism in the category Cog k . Let us consider the diagram Ž. hPDD,ᎏ6 MMEDy 66 Ž. yICE ᎏ 6 TA MC Ž. Žᎏ . The isomorphism * shows that yIC E , hPyDD,(T A. Since Ž.ᎏ hPyDD, is an equivalencewx 9, Theorem 3.5 , then the functor yICE is exact, i.e., E is a left coflat C-comodule. Ž. Ä C 4 The isomorphism * also shows that T s M g M COROLLARY 4.5. Let C be a hereditary coalgebra. If TA is a colocalizing C subcategory in M , then TA is a perfect localizing subcategory. COROLLARY 4.6. Let TA be a localizing subcategory closed under injecti¨e en¨elopes. Then TA is a perfect colocalizing subcategory. Moreo¨er the associ- ated quotient coalgebra is a subcoalgebra E of C such that E is a left coflat C-comodule and E l A s 0. GROTHENDIECK CATEGORIES 121 Ž. Proof. Since TA is closed under injective envelopes, then A s EA, C where A is regarded as a right C-comodule. Then C s A [ CrA in M . Hence CrA is an injective and quasifinite right C-comodule. This means that TA is a perfect colocalizing subcategory. The second part follows from Proposition 2.4 and Theorem 4.3. Also E l A s 0 since A and E are subcoalgebras with AIC E s 0. 5. GOLDIE TORSION THEORY Let A be a Grothendieck category and P be the class of all objects of A of the form MrL, where M, L g A and L is an essential subobject of M. Then P is closed under homomorphic images. Let G be the smallest localizing subcategory of A containing P. It is easily seen that G is the class of all M g A with the property that MrMЈ has a subobject belonging to P for any MЈ ; M. The localizing subcategory G is called the Goldie torsion theory of A.If MgG, since M is essential in EMŽ.i.e., EM Ž.rM gP, the exact sequence 0 ª M ª EMŽ.ªEM Ž.rMª0 yields that EMŽ.gG. Thus G is closed under injective envelopes. C We focus our attention on the case A s M , C a coalgebra. Let us denote by SpecŽ.C the class of all types of simple right C-comodules, C i.e., SpecŽ.C s ÄwxSS< gM,Sis a simple right comodule4 , where wxS s C ÄSЈgM< SЈ,S4. Clearly SpecŽ.C is a set. If X is a subset of SpecŽ.C , we denote by CX the smallest localizing subcategory containing X.Itis Ä C easily seen that CX s M g M C PROPOSITION 5.2. The Goldie torsion theory G of M is generated by the MC simple objects of , which are not projecti¨e, i.e., G s CSpecŽC .yP . In C particular if P s л, then G s M . C Proof. Let M g M be a simple projective object such that M g G. Then M , XrY, where Y is essential in X. The exact sequence 0 ª Y ª X ª XrY ª 0 shows that Y is a direct summand of X. Hence X s Y and 122 NASTASESCU˘˘ AND TORRECILLAS M s 0, which is a contradiction. It remains to show that any simple M C with M f G is projective. Let us consider the following diagram in M : M f 6 6 6 u 6 6 0 Ker uXXЉ 0. Consider the pullback diagram of M Ł X Љ X: 6 6 qM 6 6 0 Ker qMMXŁ 0 MXЉ f qX 6 6 6 6 6 u6 6 0 Ker u XXЉ0. This shows that Ker qMXis not essential in M Ł ЉX. However, M is simple. Then there exits a morphism ¨: M ª M Ł X Љ X such that qM ¨ s 1.IfMXgsq¨, then ug s uq XM¨ s fq ¨ s f. Thus M is projective. Since G is closed under injective envelopes, Corollary 4.6 shows that G Ž. is a perfect colocalizing subcategory. We denote E s sCG . Then G s Ä C 4 M g M < M IC E s 0 and E is a subcoalgebra with E a left coflat C- comodule. PROPOSITION 5.3. Let C be a coalgebra, G the Goldie torsion theory, and Ž. EssCG .The following properties hold: Ž.1 E is a cosemisimple subcoalgebra of C. CE Ž.2MrGis equi¨alent to M . Ž.3 E is the sum of all simple subcoalgebras B of C with the prop- erty that B is a sum of minimal left co-ideals isomorphic with simple left C- comodules of the form M* s HomŽ.M, k , where M is a simple projecti¨e right C-comodule. C C Proof. Ž.1 and Ž. 2 . Any object of M rG is injectiveŽ i.e., M rG is a C spectral category,wx 7. and contains a nonzero simple object. Thus M rG is a semisimple category. Since G is a perfect colocalizing subcategory, C E E M rG is equivalent to M . Hence M is also a semisimple category and E is a cosemisimple coalgebra. Ž.3 Let EЈ be the sum of all the coalgebras with the mentioned property inŽ. 3 . If M is a projective right C-comodule, then M*isan injective left C-comodule. Therefore EЈ is an injective left C-comodule Ä C 4 and it is left coflat. Let GЈ s M g M < M IC EЈ s 0 , which is clearly a Ž. Ž. perfect colocaling subcategory. Since sCG sEand sCG Ј sEЈ, we only GROTHENDIECK CATEGORIES 123 C have to prove that G s GЈ. Let M g M . Then M g G if and only if Ž. ComC S, M s 0 for any simple projective right comodule S. However, Ž. ComCCS, M s M I S*. Thus M g G if and only if M ICS* s 0 for any projective right comodule S, which coincides with M IC EЈ s 0, i.e., M g GЈ. The next result gives a sufficient condition in order to have that the coalgebra E is a direct summandŽ. as coalgebra of C. Let A be a Grothendieck category and let C be a localizing subcategory of A.We Ž. denote by tMC the sum of all subobjects N : M such that N g C. Since Ž. Cis closed under direct sums and quotients it is clear that tMC gC. ŽŽ.. Also tMC rtMC s0. Therefore the left exact functor tC is a radicalwx 7 . LEMMA 5.4. Let A be a locally finite Grothendieck category and let C1 and C2 be two localizing subcategories of A with the following properties: Ž. 1 C12lCs0. Ž. 2 If S is a simple object in A, then S g C12or S g C . Ž. 3 C12and C are closed under injecti¨een¨elopes. Then we ha e the decomposition M M M , where M tŽ. M and ¨ s 12[ 1s C1 M tMŽ.. 2sC2 Ž. Ž. Proof. By 1 and 2 clearly M12l M s 0. Assume that M12[ M / M. Then there exists a nonzero simple object S g C and MЈ : M such that M M MЈand MЈ M M S. Since M tMŽ.for i 1, 2, we 12[ : r 12[ , is Ci s can assume MЈ s M. Let N1 be a maximal subobject with the properties M11: N and Ž. N12lMs0N 1always exits by Zorn’s lemma . If X : N1is a nonzero simple object, then X l M211s 0. Hence X g C and X : M . Thus N1is Ž. an essential extension of M11and by 3 it follows that N g C1. We obtain that M11s N . Analogously it results that M2is maximal with the property that M21l M s 0. Assume now that S g C11. Since M is maximal with the property M12lMs0, then the canonical monomorphism 0 ª M21ª MrM is Ž. essential. By 3 it follows that MrM12g C . The exact sequence MrM1 ªMrM12[M,Sª0 yields S g C2. Hence S g C12l C s 0. This is a contradiction. If C is a coalgebra with the property that the Goldie torsion theory G of MC is exactly the whole category MC, then we say that C is a singular right Ž. coalgebra. Followingwx 1 or wx 8 , C* s Hom k C, k has a natural ring struc- ture. Indeed if f, g g C*, then fg s Ž.Žf m g ⌬ here k m k , k .. The co-unit ⑀ is the identity element of this ring. Inwx 1 or wx 8 it is proved that the category MC is isomorphic with a subcategory of left C*-modules, 124 NASTASESCU˘˘ AND TORRECILLAS denoted by RatŽ.C* y Mod . The objects of Rat Ž.C* y Mod are called left rational C*-modules. In the case when dim C - ϱ, then MC is isomorphic to C* y Mod. In this case C is a singular right coalgebra if and only if the ring C* is left singularwx 7 . We have the very satisfactory decomposition: PROPOSITION 5.5. If the ring C* is reducedŽ i.e., C* has no nonzero nilpotent elements., then the coalgebra C is a direct sum of two subcoalgebras C s D [ E, where D is right singular coalgebra and E is cosemisimple. Proof. We first show that if S is a simple projective right C-comodule, then S is also an injective object in MC. Bywx 2, Proposition 4 it follows that S is a left projective simple C*-module. Therefore C* s I [ J, where I and J are left ideals in C* and I , S Ž.as left C*-modules . Then there exit two orthogonal idempotents e, f g C* such that 1 s e q f, I s C*e, J s C*f, and ef s fe s 0. If b g J then b s f for some g C*. Since 2 Ž.efsefe f s 0, by hypothesis e f s 0; hence IJ s 0. Analogously we have JI s 0; hence I and J are two-sided ideals. Since JS s 0, then S is a left C*rJ-module. However, C*rJ , I , S is a division ring; therefore Sis a leftŽ. and right projective module over the ring C*rJ. On the other hand the canonical ring morphism : C* ª C*rJ is left and right flat. Hence S is also left injective as the C*-module. Thus S is injective in MC. We denote by C the localizing subcategory of MC generated by simple projective right C-comodules. By Proposition 5.2 it follows that G l C s 0. The preceding result implies that C is closed under injective envelopes. Ž. Now we can apply Lemma 5.4, C s D [ E where D s tCG C and E s Ž. tCC C . Bywx 5, Theorem 4.2 , D and E are subcoalgebras. 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