JOURNAL OF ALGEBRA 190, 68᎐87Ž. 1997 ARTICLE NO. JA966882
Modules with Chain Conditions for Finite Matrix Subgroups
Wolfgang Zimmermann
Mathematisches Institut der Uni¨ersitat,¨ Theresienstrasse 39, View metadata, citation and similar papers D-80333at core.ac.uk Munich, Germany brought to you by CORE
Communicated by Kent R. Fuller provided by Elsevier - Publisher Connector
Received June 3, 1995
In this article we shall present new ways of describing modules with maximum resp. minimum condition for finite matrix subgroups. For vari- ous reasons these modules have attracted much attention. To give exam- ples, a module M satisfies the minimum condition for finite matrix subgroups iff it is Ý-pure-injective, i.e., every direct sum M Ž I . is pure- injectivewx 6, 14 . On the other hand such a module may be characterized by remarkable decomposition properties of its direct productswx 6, 15 . Mod- ules satisfying both of these chain conditions are of importance in repre- sentation theorywx 1, 4, 16 . Here we show that these chain conditions can be characterized by two well-known natural transformations, that have been repeatedly studied in Ž. Ž . literature. The first one is : Hom SRP, M m A ª Hom SRP, M m A Ž.Ž.Ž. Ž. given by maxsxmafor g Hom S P, M , a g A, and x g Ž I . P. Here SRM is a bimodule and SRP, A are modules. In case P sSS this I Ž.IŽŽ . . Ž . is the map ˜˜: M mRRiiA ª M m A , x m a s x m a , which has been studied, for instance, by Lenzingwx 7 , Raynaud and Grusonw 10 x , and Goodearlwx 5 regarding injectivity and surjectivity. Our starting point is the observation that for given RSRA and M the map is injective for all projective modules S P Žequivalently ˜ is injective for all sets I. if and only if M satisfies a weak maximum condition for certain finite matrix subgroups defined by A: we shall say for short that M has A-acc. Now the modules M having A-acc for all A are precisely those with maximum condition for finite matrix subgroups. This description reveals a curious parallelism between the class of modules with maximum condition for finite matrix subgroups and the class of Mittag᎐Leffler
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0021-8693r97 $25.00 Copyright ᮊ 1997 by Academic Press All rights of reproduction in any form reserved. CONDITIONS FOR MATRIX SUBGROUPS 69
modules: Recall that R A is a Mittag᎐Leffler module iff ˜ is injective for all MR and I wx10, Proposition 2.1.5 , in other words, iff every module MR has A-acc. Dual results are proven by studying the homomorphism : Ž. ŽŽ.. Ž.Ž. CmRSHom M, U ª HomSR Hom C, M , U with c m g s ŽŽ.. Ž. Ž. gc for c g C, g Hom SRM, U , and g g Hom C, M , where SMR is a bimodule and CRS, U are modules. The fact that is an isomorphism if C is finitely presented and U is injective is well known and useful in many respects. This time injectivity of for a fixed pair CRR, M and arbitrary injective modules U means that M satisfies a weak minimum condition for certain finite matrix subgroups given by C which we call
C-dcc. There is a connection between the two conditions: If SV is an injective cogenerator, then M satisfies C-dcc Ž.A-acc iffRS Hom Ž.M, V satisfies C-acc Ž.A-dcc . Again we have a global result: M has C-dcc for all modules C if and only if it satisfies the minimum condition for finite matrix subgroups. In this first section we have compiled the necessary technical informa- tion on matrix subgroups. Throughout R is a ring, the modules occurring in this article mostly have R as their ground ring, and instead of Hom RŽ M, N.Žwe shall use the shorter notation M, N .. The cardinality of a set X will be denoted by < 1. POINTED MODULES AND MATRIX SUBGROUPS We begin with collecting the facts about matrix subgroups needed in subsequent text. We shall describe these groups by what we call ‘‘pointed modules’’ because, compared to their representation by matrices, we feel this method to be more perspicuous and better suited to our purpose. It should be stressed that certain special cases of this material are scattered throughout literature. Throughout we fix a nonempty set I and call a pair Ž.A, a consisting of Ž. a left R-module A and a family a s aiigI of elements of A an I-pointed module. Given a right R-module M we define the map a: Ž I . Ž. Ý Ž. M ªMmRaAby m s m m a s igIim m a ifor m s miigIg ŽI. Ž. Ž. M and put TMA, aasKer . For every homomorphism f g M, MЈ ŽŽ..Ž. ŽI. we have fTA, aA M ;TM,aAЈ, i.e., T ,ais a subfunctor of V , where V Ž. denotes the forgetful functor Mod-R ª Mod-ޚ. In particular TMA, a is ŽI. an End RŽ.M -submodule of M . ⑀ Ž.I ⑀ Ž. Ž. ŽŽ .. Similarly, defining aai: A, N ª N by h s has ha igIand Ž. HNA, aasIm ⑀ for each left module N we obtain a subfunctor HA,a of V I. 70 WOLFGANG ZIMMERMANN The most important among these functors are those corresponding to the pairs Ž.A, a , where A is finitely presented and the tuple a is finite. It is well known and very useful in practice that, say, for a right module M the set of all HMC, cAŽ.coincides with the set of all TM,a Ž., where ŽC, c . resp. Ž.A, a runs through the class of all n-pointed finitely presented right resp. left R-modules. The following lemma in addition yields a correspon- dence between the n-pointed finitely presented right and left modules. wNote that the module A, which is constructed from the pair Ž.C, c , is the Auslander᎐Bridger transpose of CrCЈ.x LEMMA 1.1. Gi¨en a finitely presented n-pointed right moduleŽ. C, c there can be constructed a finitely presented n-pointed left moduleŽ. A, a such that Ž. Ž. Ž. Ž. HMC, cAsT,aC M for e¨ery right module M and T,cA N s H,a N for e¨ery left module N. Proof. We choose an exact sequence ␣ 6 6 6 GFC0, Ž. where F is free with a basis e1,...,ekii, kGn, such that e s c , 1FiFn, and G is free as well with a basis f1,..., fl. Let GЈ be a further ␥ Ј free module with a basis fy1 ,..., fyn and let : G ª F be defined by ␥Ž. fyiise,1FiFn. These data yield the following commutative dia- gram with exact rows and columns 6 6 6 6 0 GG=GЈ GЈ0 6 ␣6 sŽ.␣,␥ Fs F 6 6 6 CCrCЈ 6 6 00 where CЈ denotes the submodule of C generated by c and the unexplained maps are the obvious ones. If we dualize with respect to R we obtain the CONDITIONS FOR MATRIX SUBGROUPS 71 commutative diagram F* s F* 6  * 6 ␣ * 6 * 6 * 6 6 0GXX*G*=G*G*0 6 6 Ј X 6 6 G* AArAЈ 6 6 00 with exact rows and columns, and Ј denoting the cokernels of  * and ␣ Ј Ž. ŽU. *, respectively, and A s Im * . We put ai s fyi ,1FiFn, and Ž U .ŽUUUU. ajjsf,1FjFl, where fy1,..., fyn, f1,..., flis the dual basis of Ž. Ž᎐ . fy1 ,..., fyn, f1,..., fl . Applying the functor , M to the first diagram yields the diagram 00 6 6 6 X6 0Ž.CrC,MC Ž.,M 6 6 Ž.F,Ms Ž.F,M 6 Ž.,1 6 Ž.␣,1 6 Ž.,1 6 Ž.,1 6 6 0 Ž.GЈ,MG Ž=GЈ,MG . Ž.,M0 6 6 Ј 6 6 6 6 0Ž.GЈ,MMmRRAMmArAЈ0 6 6 00 Ž. Ýl Ž. with exact rows and columns, being given by h s js1hfjjmaq ÝnŽ. Ј Ž. is1 hfyi mayi and similarly. Putting a s ay1,...,ayn and identi- Ž. n Ž. fying GЈ, M with M it is easily seen that ,1 sa. Now the snake ( a ␦ n lemma yields the exact sequence 0 ª Ž.Ž.CrCЈ, M ª C, M ª M ª M mRRA ª M m ArAЈ ª 0 and an easy calculation shows that the connect- Ž. ing homomorphism ␦ coincides with ⑀cC. If follows that HM,cs TMA, aŽ.. Similarly it is proven that the sequence 0 ª Ž.ŽArAЈ, N ª A, 72 WOLFGANG ZIMMERMANN ⑀ a c .Žn . NªNªNmRRCªNmCrCЈª0 is exact, showing that HNA,␣ Ž. sTNC,c . A closer look at the proof also gives the original definition of the Ž. Ž. k,l subgroups HMC, cijby matrices. Let Q s r g R denote the repre- ␣ ␣Ž. Ýk senting matrix of , i.e., fjis s1eriij,1FjFl. Then obviously Ž. Ž. n HMC, c is the set of all x1,..., xn gM that can be complemented to a Ž. solution of the equation x1,..., xQk s0. Conversely each matrix Q g k, l R with n F k is the matrix of relations of a finitely presented n-pointed module Ž.C, c for which HC, c can be calculated in the former way. Given an arbitrary I-pointed module Ž.C, c the subgroup HMC, cŽ.may analo- gously be characterized by a column-finite R-matrix. On account of this connection HMC, cŽ.is called a matrix subgroup in case ŽC, c .is a 1-pointed module and called a finite matrix subgroupŽ or a subgroup of finite definition. if in addition C is finitely presented. We also want to indicate how this material is viewed by the model k,l theoristsŽ seewx 8, 9, 11 for detailed information. . Given a matrix Q g R Ž. with n F k as above let s Q ¨ 1,...,¨n denote the formula in n free variables, ᭚ ¨nq1,...,¨k, Ž.¨1,...,¨kQs0, in the language LRCof right R-modules. Then HM,cŽ.is the set of all Ž.n x1,..., xn gM satisfying . Formulae that are logically equivalent to formulae of this kind are called pp-formulae; therefore the subgroups HMC, cŽ.are called pp-definable and Ž.C, c is called a free realization of Ž.Ž . . The ‘‘dual’’ formula D ¨ 1,...,¨nRin the language L of left R- modules is given by En t ᭚w1,...,wl, Q Ž.¨1,...,¨n,w1,...,wls0, ž/0 n, n where En denotes the unit matrix in R .If Nis some left module the set Ž.n Ž. of all y1,..., ynCgN satisfying D is just TN,c.w This follows from  Ž.En our proof of Lemma 1.1, the representing matrix of * being 0 Q .x k, l Ž. Similarly, given a matrix U g R with n F l let ¨ 1,...,¨n denote the formula ᭚ t ¨nq1,...,¨l, UŽ.¨1,...,¨ls0 Ž.Ž . in R L and let D ¨ 1,...,¨n denote the formula En 0 ᭚w1,...,wl, Ž.¨1,...,¨n,w1,...,wls0 ž/U in LR. It is easily seen that DD and are equivalent, just as DD and are. This means that D describes a duality, the so-called elementary duality. Lemma 1.1 may be regarded as a consequence of this duality. CONDITIONS FOR MATRIX SUBGROUPS 73 Next we derive some rules of calculating for the functors TA, aAand H ,a. First we fix a morphism ␣: Ž.Ž.A, a ª B, b of I-pointed modules to be a Ž. Ž. Ž. homomorphism ␣ g A, B such that ␣ a s b, i.e., ␣ aiis b for all igI. A morphism of this kind implies inclusions of functors TA, aB; T ,b and HA, aB> H ,b. In order to describe sums and intersections we define the direct sum Ž.S, s and the direct product Ž.P, p of the pointed modules Ž.A,aand Ž.B, b . We put S s A = BrU, where U is the submodule of Ž. Ž. A=Bgenerated by the pairs aii, yb , i g I, and let s s siigIwith Ž. Ž. Ž .Ž. siisa,0qUs0, biq U. Now the maps A, a ª S, s , x ¬ Ž.x,0qUand ŽB, b .Ž.ª S, s , y ¬ Ž.0, y q U are morphisms making Ž.S,sa direct sum of Ž.Ž.A, a and B, b . The direct product is given by Ž. Ž. PsA=B,pspiigIiiiwith p s a , b , and the usual projections. The proof of the next lemma is evident. LEMMA 1.2. Ž. 1 TS,sAsT,aBqT,bP,T,pAsT,aBlT,b. Ž. 2 HS,sAsH,aBlH,bP,H,pAsH,aBqH,b. We shall frequently use the following conclusion, which is to be illus- Ž. Ž. trated by means of the first formula. We assume that TMA,aB;TM,b Ž. Ž. holds for some module M. Then TA, aS; T ,sSand TM,sBsTM,b, i.e., if TB, bSis substituted by T ,seven the inclusion of functors can be achieved without changing TA, aBand the value of T ,bon M. In a similar way the other formulae can be used. LEMMA 1.3. Let ␣: Ž.Ž.A, a ª B, b be a morphism of I-pointed modules Ј Ž. Љ and J be a nonempty proper subset of I. Moreo¨er, let a s aiigJ, a s Ž.Ј Ž. Љ Ž. Ј Ј aiigI_Jii,bsbgJi,and b s b igI_J, and let A ; A and B ; Bbe the submodules generated by aЈ and bЈ, respecti¨ely. ŽJ.ŽI. Ž.1a Ž. For e¨ery left module M the canonical maps j: M ª M Ž I . Ž I J . and q: M ª M _ induce the exact sequence jq 0ªTMA,aЈŽ.ªTMA,aA Ž.ªTMrAЈ,aЉ Ž.ª0. Ž. Ž. Ž. Ž. b The equality TA, aB M s T,bA M is equi¨alent to T,aЈ M s Ž. Ž. Ž. TB,bЈ M and TArAЈ,aЉ M s TMBrBЈ,bЉ. I JI Ž.2a Ž. For e¨ery right module N the canonical maps N _ ª N and I J N ª N induce the exact sequence 0 ª HNArAЈ,aЉŽ.ªHNA,aA Ž.ªHN,aЈ Ž.ª0. Ž. Ž. Ž. Ž. b The equality HA, aB N s H,bA N is equi¨alent to H,aЈ N s Ž. Ž. Ž. HB,bЈ N and HArAЈ,aЉ N s HNBrBЈ,bЉ. ŽFor simplicity of notation we write aЉ and bЉ for the corresponding tuples of residue classes mod AЈ and mod BЈ, respecti¨ely.. 74 WOLFGANG ZIMMERMANN Proof. We only showŽ. 1 . Assertion Ž. a is left as an easy exercise. The morphism ␣ induces morphisms Ž.Ž.ŽA, aЈ ª B, bЈ and ArAЈ, aЉ .ª Ž.BrBЈ,bЉwhich leads to the commutative diagram 0 ªTMA,aЈŽ.ªTMA,aA Ž.ªTMrAЈ,aЉ Ž.ª0 ll l 0ªTMB,bЈŽ.ªTMB,bB Ž.ªTMrBЈ,bЉ Ž.ª0 from whichŽ. b is readily inferred. In the following direct limits of pointed modules will play an important Ž. Ž . Ž role. Let L, F be an upward directed partially ordered set and ml: .Ž . AlmlªAFmlllbe a direct system with limit : A ª A gL. If each IŽ.Ž. module Allis pointed by a g Alsuch that the ml: Al, alª Am, am are morphisms we speak of a direct system of I-pointed modules. Putting Ž. Ž . Ž . asppa for some p g L, the l: A l, a lª A, a are morphisms Ž. ŽŽ.. as well. We shall write A, a s lim l g LllA , a , lor more simply Ž. Ž . ª A,aslim All, a to denote this situation. The existence of such limits ª is an easy consequence of the ‘‘unpointed’’ case. Ž.n LEMMA 1.4. Let A be a left module and a s a1,...,an gA . Ž.1 There exists an upward directed ordered setŽ. L, F and a direct ŽŽ .Ž .. system ml: A l, a l ª Am, aofn m lFm -pointed finitely presented mod- Ž. ŽŽ.. ules and an isomorphism A, a ª lim l g LlllA , a , . ª Ž.2 For each morphism ␣: Ž.Ž.Ž.B, b ª A, a , where B, bisan n-pointed finitely presented module andŽ. A, a is represented as a direct limit Ž. Ž.Ž. as in 1 there is some l g L and a morphism : B, b ª All, a with ␣sl . Proof. Ž.1 It is well known that there exist an upward directed set N Ž. and a direct system ml: A lª A m lFm in N of finitely presented modules having direct limit A. Let : Al ª A, l g N, be the canonical maps. Now there is some g N such that a1,...,an belong to the image of .We Ž. Ä 4 choose ajg Awith a jjs a ,1FjFn. Putting L s l g Nl Next we record some results on the behavior of the subgroups TMA,aŽ. and HMA, aŽ.under dualization; further results will follow in Section 3. First some notation. Letting SRM be a bimodule and SU a module we put qŽ. ŽI. MsSSM,U.If X is a subset of M the set of the families s Ž. qI Ž.Ý Ž . iigIigM satisfying x s gIix is 0 for all x g X will be CONDITIONS FOR MATRIX SUBGROUPS 75 0 I Ž I . denoted by X , and for a subset Y ; Mq the set of the x g M with 0 Ž.xs0 for all g Y will be denoted by Y . The following lemma and its corollary are modifications ofwx 16, Proposition 3 ; certain special cases are also common in model theory, for instance, seewx 11, 1.5 . LEMMA 1.5. Ž. Ž q.Ž.0 1IfSA U is injecti¨e, then H,aA M s TM,a. Ž. Ž q.0 Ž. 2 IfSA U is a cogenerator, then H,aA M s TM,a. Ž. Ž. Proof. 1 Let SSU be injective. Applying the functor y, U to the exact sequence j Ž I . a 0 ª TMA,aRŽ.ªMªMmA Ž.Žq. and taking into account the isomorphism SRSM m A, U , RRA, M we ⑀a Ž.j,1 Ž.qqI ŽŽ.. obtain the exact sequence 0 ª A, M ª M ª TMA,aS,U.It Žq.Ž.Ž.0 follows that HMA,␣ sIm ⑀aAs Ker j,1 sTM,a. Ž. 2IfSUis a cogenerator the following equivalences hold for m g Ž I . Ž. Ž q. Ž .Ž . M : m g TMA,aRRmmmas0m᭙hgA,M:ha ms0m Ž.Ž.Ž.qq Ž.0 ᭙hgA,M:⑀aAhms0mmgHM,a. COROLLARY 1.6. We assume thatS U is an injecti¨e cogenerator and Ž I . denote by T the subset of all subgroups of M of the form TA, aŽ. M and H qI q the subset of all subgroups of M of the form HA, aŽ M .. Then the map 0 0 T ª H, T ¬ T , is a lattice anti-isomorphism with in¨erse H ª T, H ¬ H . Proof. By the preceding lemma both of the maps are well defined and because of the equations 00 q 0 TMA,aAŽ.sHM,aA Ž .sTM,a Ž.and qq00 0 HMA,aAŽ.sTM,aA Ž.sHM,a Ž. they are inverses of each other. The formulae of Lemma 1.2 show that T 0 and H are lattices and that the map T ¬ T is an anti-isomorphism. OROLLARY Ž. Ž . C 1.7. Let A, a s lim l g LlA , a l be a direct limit of ª I-pointed left modules. Ž.1 The familyŽ T Ž M .. is upward directed and TŽ. M A ll, algLA,as TMŽ.. DlgLAll,a Ž.2 In case U is injecti e the familyŽŽ H Mq.. is downward SA¨ ll,algL directed and HŽ Mq.ŽHMq.. A,als F gLAll,a Ž. Ž.Ž. Proof. 1 The canonical maps All, a ª A, a imply the inclusion TMŽ.TM Ž.. Conversely, if m TMŽ., i.e., m a 0, Dl g LAll,aA;,aAg ,am s 76 WOLFGANG ZIMMERMANN then there is some l g L such that m m alls 0in MmA; hence m g TMŽ.. All,a Ž. Ž q.Ž.0 2 Using the formula just proved we have HMA,aAsTM,as TMŽ.0 HMŽ.q. FlgLAll,alsFgLAll,a OROLLARY Ž .Ž. Ž. C 1.8 comparewx 16, p. 704 . Let A, a s lim l g LlA , a l. ª Then we ha eHŽ. N HŽ. N for each pure-injecti e left mod- ¨ A, alsFgLAll,a ¨ ule N. Proof. We only have to show the inclusion > . To this end we use the evaluation map c: N ª Nqqs ŽŽN, ޑrޚ ., ޑrޚ ., which is known to be a pure monomorphism. Because N is pure-injective there is a homomor- qq phism d: N ª N with dc s 1.N By Corollary 1.7 we have HNŽ qq .ŽHNqq .. Letting n A, alsFgLAll,a g HNŽ.we may infer cnŽ. HNŽqq .; hence there is F l g LAll,algFgLAll,a some ␥ g ŽA, Nqq .Ž.Ž.such that cns␥a and we arrive at n s dcŽ. n s Ž. Ž . d␥agHNA,a . The class of modules satisfying the intersection property in Corollary 1.8 properly contains the class of pure-injective modules: For instance it also contains arbitrary direct sums of pure-injective modules. We intend to take up the study of this module class in a separate paper. 2. INJECTIVITY OF Now we will show the announced ‘‘functorial’’ characterization of mod- ules with maximum condition for finite matrix subgroups. The following preparatory lemma is a ‘‘pointwise’’ version of the main result. It has been brought to our attention that the lemma overlaps with work by Rothmaler wx11 , a fact to be clarified after the lemma has been stated. Since Roth- maler’s proofs use the language of model theory, we present a complete algebraic proof. LEMMA 2.1. The following statements are equi¨alent for a bimoduleSR M and a moduleR A: Ž. Ž.Ž . 1 The map : SSP, M m RA ª SSP, M m RA is injecti¨e for all projecti¨e modulesS P. Ž. I Ž.I Ž 2The map ˜: M mRRA ª M m A is injecti¨e for all sets I for all sets I with<< I F maxÄ<<< Ž. ގ n Ž. Ž 4For all n g , a g A and all presentations A, a s lim l g LlA , .Ž . ª all with finitely presented modules A , respecti¨ely there is an l g L with TMŽ.TM Ž.. A,aAsll,a n Ž.5 For all n g ގ and a g A there exists an n-pointed finitely pre- Ž. Ž.Ž. Ž. sented module B, b and a morphism ␣: B, b ª A, a with TA, a M s TMB,bŽ.. Ž.6 For all finitely generated submodules AЈ ; A, a g A and all presen- Ž.Ј Ž.Ž tations ArA , a s lim l g LlA , a l with finitely presented modules Al ª resp..Žthere is some l L with T M .TMŽ.. g ArAЈ,aAs ll,a Ž.7 For all finitely generated submodules AЈ ; A and a g A there is a pointed finitely presented moduleŽ. B, b and a morphism ␣: Ž.ŽB, b ª ArAЈ, .Ž.Ž. a with TArAЈ,aB M s TM,b. A module R A satisfying our unrestricted conditionŽ. 2 is called an M-Mittag᎐Leffler module inwx 11, p. 39 . Hence the equivalence of the unrestricted conditionŽ. 2 with the restricted one resembles the equiva- lenceŽ. v m Žvi . ofwx 11, Theorem 2.2 , while our equivalenceŽ. 2 m Ž.5 coincides withŽ. v m Žviii . ofwx 11, Theorem 2.2 . Furthermore, our implica- tionŽ. 2 « Ž.3 generalizesw 11, Corollary 2.4Ž. ex . Proof. It is obvious thatŽ. 1 and the unrestricted version of Ž. 2 are equivalent. Now we assume thatŽ. 2 holds for sets I with < n Ž.Ž.x n a Ž.xa 0, ÝÝkkxgMmkksž/m s ž/ n ks1 k xgM Ý ŽŽ.. n hence, kkxx gMkm a s 0 by assumption. It follows that there is I Ý ŽŽ.. n Ž. some l g L such that kkxx gMlkm a s 0in M mA l. Hence x TMŽ.for all x M n, i.e., TMŽ.TM Ž.. The other inclusion gA ll, aAg ,aA;ll,a being obvious, assertionŽ. 4 is proved. Ž.4« Ž.5 Because we can write ŽA, a .as a direct limit of pointed finitely presented modules by Lemma 1.4,Ž. 5 is an immediate consequence ofŽ. 4 . Ž. Ž. ŽÝn Ž. . Ýn 5«2 Let ks1 mki igIkm a s 0, i.e., ks1mkim a k s 0 for all Ž. Ž.Ž.Ž. igI. Putting a s a1,...,an and choosing ␣: B, b ª A, a as in 5 Ž . Ž. Ž. we obtain mki kg TM A, aBsTM,bkkik, hence Ý m m b s 0 for all IŽ.I igI. Because B is finitely presented, ˜: M mRRB ª M m B is an Ž. I Ž. isomorphism, hence Ýkkiikm m b s 0in M mBand Ýkkiikm m a s Ž.ŽŽ.. 1m␣Ýkkiikmmbs0. 78 WOLFGANG ZIMMERMANN Ž.2« Ž.3 Every finitely generated submodule AЈ ; A gives rise to the commutative diagram with exact rows: I I I M m AЈ ª M m A ª M m ArAЈ ª 0 Ј ˜ x IIx x I Ž.Ž.Ž.MmAЈªMmAªMmArAЈª0 As Ј is surjective and is injective, ˜ is injective Ž.3« Ž.6« Ž.7 are shown likeŽ. 2 « Ž.4 « Ž.5. Ž. Ž. Ž . Ž . 7«6 Let ArAЈ, a s lim All, a with finitely presented modules Ž.Žª . Ž. Al and ␣: B, b ª ArAЈ, a as in 7 . By Lemma 1.4 there is some Ž.Ž. lgLand a morphism : B, b ª All, a with ␣ s l, hence TMŽ.TM Ž.TM Ž.TM Ž.. ArAЈ,aBs,bA;ll,aA;rAЈ,a Ž.6« Ž.4 According to our proof ofŽ. 4 « Ž.5 we have only to show the Ž. n Ž. Ž . restricted version of 4 . Let a g A with n ) 1 and A, a s lim All, a , Ž. ŽX.Žª the Al being finitely presented. Then A, aЈ s lim all, a and ArAЈ, .ŽX. XŽ.XªŽ. anllslim A rA , aln, where a s a1,...,an1, alls a1,...,aln 1 , and X y X y AЈ;A and All; A are the submodules generated by aЈ and al, respec- tively. Proceeding by induction we can assume that there is some u g L with TMŽ.TMX Ž.; by Ž. 6 there is a L with TM Ž. A,aЈsAuu,aA¨ g rAЈ,an s TMX Ž.. Since L is directed we can achieve u hence Lemma 1.3 A¨¨¨r A , a n s ¨ yields TMŽ.TM Ž.. A,aAs¨¨,a The statements of Lemma 2.1 describe a sort of maximum condition for matrix subgroups. For instance, conditionŽ. 4 says that the directed family ŽŽ..TM has a maximal, hence a greatest element. Therefore we A ll, algL shall say that M has A-acc in case it satisfies these properties. Next we derive a number of corollaries of Lemma 2.1. First we recall the I Ž.I fact that ˜: M mRRA ª M m A is surjective for all I iff A contains a finitely generated submodule AЈ such that M mR ArAЈ s 0w 3, Lemma 1.1x . Together with Lemma 2.1 this yields a description of the modules M such that ˜ is an isomorphism for all I. COROLLARY 2.2. p Ž.1 If the sequence 0 ª MЈ ª M ª MЉ ª 0 of right modules is pure- exact then M has A-acc iff MЈ and MЉ ha¨e this property ŽI. I Ž.2 If I is some set then A-acc passes o¨er from M to M , M , and I Ž I . M rM . Proof. Ž.1 Given a set I the diagram X I I Y I 0 ª M m A ª M m A ª M m A ª 0 Јx ˜x Љx XYII I 0ªŽ.Ž.ŽMmAªMmAªMmArAЈ .ª0 CONDITIONS FOR MATRIX SUBGROUPS 79 in commutative with exact rows. Hence Ј is injective if ˜ is and, conversely, ˜ is injective provided that Ј and Љ are. To show that A-acc goes over from M to MЉ we use conditionŽ. 5 of Lemma 2.1. Letting n agAwe know that there is a morphism ␣: Ž.Ž.B, b ª A, a , where B is Ž. Ž. finitely presented, such that TMA, aBsTM,b. It is easy to see that Ž Ž .. Ž . Ž Ž .. Ž . Ž. pTA,aA M sTM,aBЉand pT,bB M sTM,bAЉ, hence TM,aЉs TMB,bŽ.Љ. I ŽI. Ž.2 The statement is obvious for M ; as the sequence 0 ª M ª I I Ž I . Ž I . I Ž I . M ª M rM ª 0 is pure-exact, it holds for M and M rM as well. COROLLARY 2.3. LetSR M be a bimodule such that MR has A-acc. If NS is finitely presented or flat then N mS M has A-acc, too. Proof. Clearly the assertion is true if NS is finitely presented. Now let NS be flat and 0 ª K ª F ª N ª 0 be an exact sequence with a free module F. By purity the sequence 0 ª K mSSM ª F m M ª N m SM ª 0 is also pure-exact. Now the assertion follows from Corollary 2.2 because Ž I . F mS M , M for some set I. The next statement is proven like Corollary 2.2. COROLLARY 2.4Ž see alsow 11, Corollary 2.4Ž. bx. . Let 0 ª AЈ ª A ª AЉ ª 0 be a pure-exact sequence of left modules. Ž.1 If M has A-acc then it has AЈ-acc. Ž.2 If M has AЈ-acc and AЉ-acc then it has A-acc. In general M does not satisfy AЉ-acc if it satisfies A-acc: Let R be a ring over which there exists a module M which does not have maximum condition for finite matrix subgroups. By Theorem 2.5 there exists a module AЉ such that M does not have AЉ-acc. However, there is a pure-exact sequence 0 ª AЈ ª A ª AЉ ª 0 with pure-projective middle term A and obviously M has A-acc. Next, considering modules with A-acc for all A we arrive at modules with maximum condition for finite matrix subgroups. In order to keep the following theorem lucid we shall ‘‘globalize’’ only part of the statements of Lemma 2.1. THEOREM 2.5. The following assertions are equi¨alent for a module MR: Ž.1 M has maximum condition for finite matrix subgroups. Ž. I Ž.I 2The map : M mRRA ª M m A is injecti¨e for all modules A and sets I. Ž. ގ Ž.ގ 3The map : M mRRA ª M m A is injecti¨e for all denumer- ably generated modules A. 80 WOLFGANG ZIMMERMANN Ž.4 For e¨ery n g ގ and e¨ery n-pointed moduleŽ. A, a there exists a morphism ␣: Ž.Ž.B, b ª A, a , where Ž. B, bisann-pointed finitely presented Ž. Ž.Ž module such that TA, aB M s TM,b.It is sufficient to take 1-pointed modules only.. Proof. Ž.1 « Ž.4 If we can verify conditionŽ. 4 for n s 1 it holds for Ž. Ž . arbitrary n by Lemma 1.3. Letting a g A and A, a s lim All, a with ª finitely presented AlAthe group TM,aŽ.is the union of the directed family of finite matrix subgroups ŽŽ..TM. By assumption the latter has a A ll, a maximal element TMŽ.that has to coincide with TMŽ.. All,aA,a The equivalenceŽ. 2 m Ž.4 follows from Lemma 2.1. Ž.2« Ž.3 is trivial. Ž.3 « Ž.1 We assume that there is a proper ascending chain ŽŽ..TM with finitely presented modules A and a A . A ll, alG1 lllg Lemma 1.2 allows to assume that additionally there are morphisms ␣l: Ž.Ž . Ž. Ž. All,aªA lq1,alq1,lG1. Putting A, a s lim lG1All, a we ob- ª serve that A is denumerably generated. Now we choose m TMŽ. lAg ll,a _ TMŽ.for l 2 and put m Ž0, m , m ,.... Mގ. By choice of A ly1, aly1 G s 23 g the mllwe have m m als 0in MmA ll; hence m m a s 0in MmA.As Ž.Ž. this holds for all l G 1 we have m m a s mllm a G1s 0. Our as- sumption that is mono implies m m a s 0; hence there is some l G 1 with m m alls 0. In particular this gives m q1m alls 0, i.e., m q1g TMŽ., contradicting the choice of m . A ll, alq1 Remark 1. As each module is a pure submodule of a pure-injective one, it is sufficient to require conditionŽ. 2 of Theorem 2.5 for pure-injec- tive modules A and I s ގ only. Remark 2. Corollaries 2.2 and 2.3 can also be interpreted as statements concerning modules with maximum condition for finite matrix subgroups. Remark 3. We omit theŽ. routine proof of the following description of the flat modules satisfying the conditions of Theorem 2.5. Let MR be flat Ž. Ž . during this paragraph. This means that the equality TMA, aAsMT,aR R holds for all 1-pointed modules Ž.A, a ; hence the set T Ž.M of all TMA,a Ž. equals the set of subgroups ML, L running through the left ideals of R. Furthermore we recall that the set F of all left ideals F with MF s M is a Ž. Gabriel topology. We need the set SF R R of the F-saturated left ideals L; they can be characterized by the condition ᭙r g R: r g L m Mr ; ML Ž. Ž. in the present situation. It is easy to see that the map SF R R ª T M , L¬ML, is a lattice isomorphism.Ž For more details, seewx 13 .. The following statements are equivalent: Ž.a Mhas a maximum condition for finite matrix subgroups. CONDITIONS FOR MATRIX SUBGROUPS 81 Ž. b Given a chain L12; L ; иии of left ideals of R the chain ML12; ML ; иии is stationary. Ž.c Each left ideal L contains a finitely generated left ideal LЈ such that ML s MLЈ. II Ž.d For each left ideal L and each set I we have Ž.ML s ML. Ž. Ž . e The lattice SF R R is noetherian. To give an example, these conditions are valid if R is left noetherian. We further note that M is Ł-flat, i.e., that all products M I are flatwx 3 if Ž.Ž.a᎐e are satisfied. The converse of this implication fails: think of a von Neumann regular ring that is not semisimple. Remark 4. Let R be a noetherian algebraŽ i.e., an algebra over a commutative noetherian ground ring over which it is finitely generated as a module. . Then each finitely generated R-module has maximum condi- tion for finite matrix subgroups. Further modules with this chain condition can be constructed from these building blocks by use of Corollaries 2.2 and 2.3. 3. INJECTIVITY OF In order to obtain a ‘‘functorial’’ characterization of the minimum condition for finite matrix subgroups as well we dualize the results of Section 2. We begin with two general observations concerning injectivity of and then prove the analogue of Lemma 2.1. PROPOSITION 3.1. LetSR M be a bimodule, SRU, C be modules, and : q Ž.q Ž.q CmRRRMªC,M ,where ᎐ denotes dualization with respect toS U. Ž. Ž q.Ž.0 n 1is injecti¨e iff TC, cC M s H,c M for all n g ގ and c g C . Ž.The inclusion ; is ¨alid without assumption. Ž. n Ž. Ž . 2Let c g C and C, c s lim l g LlC , c l. If is injecti¨e then ª HMŽ.0 HMŽ.0.If in addition U is a cogenerator then C, cl;DgLCll,cS HMŽ. HMŽ.. C,clsl gLCll,c Proof. Ž.1 This follows from the fact that Ž.c m s 0 is equivalent Ž.0 n qn to g HMC, c for c g C and g M . Ž. Ž.0 Ž q. 2 Injectivity of implies HMC,cCsTM,clsDgL TMŽq.ŽHM.0; hence HMŽ.00 HMŽ.00.If Cll,cl;j gLCl,cl1 F gLCll,cC;,c Uis a cogenerator the latter means HMŽ.HM Ž.; the other F l g LCll,cC;,c inclusion being trivial the proof is complete. 82 WOLFGANG ZIMMERMANN LEMMA 3.2. LetSR M be a bimodule, SU be an injecti¨e cogenerator, and CR be a module. Then the following statements are equi¨alent: Ž. Ž.ŽŽ.. 1 The map : C mRSM, SV ª SC R, M R, SV is a monomor- phism for all injecti¨e modulesS V. Ž. Ž I.ŽŽ . I. 2The map : C mRSM, SU ª SC R, M R, SU is injecti¨e for all sets IŽ all sets I with<< I F maxÄ<<< Since the modules Cl inŽ. 4 can be chosen finitely presented the implicationsŽ. 4 « Ž.5 is clear. Ž. Ž. n Ž.n 5«1 Let SSV be injective, c g C and g M, SV such that Ž.cms0. Furthermore let ␥ : Ž.Ž.D, d ª C, c be a morphism as inŽ. 5 Ž. Ž. Ž.0 Ž.0 with HMC, cDsHM,dC.AsgHM,cDsHM,dthe element Ž.ŽŽ dmis contained in the kernel of the map Ј: D mRSM, SV ª SD R, .. Ž MRS,V, which is an isomorphism. Hence d m s 0 and c m s ␥ m 1.Žd m . s 0. CONDITIONS FOR MATRIX SUBGROUPS 83 We shall say that a module M has C-dcc if it satisfies the conditions of the preceding lemma. COROLLARY 3.3. LetSR M be a bimodule. Ž. Ž. Ž. 1aIf MRR has A-acc and S V is injecti¨e thenR M, V has A-dcc. Ž.b IfRS Ž M, V . has A-dcc and V is a cogenerator then M has A-acc. Ž. Ž. Ž . 2aIf M has CRS-dcc and V is injecti¨e thenR M, V has C-acc. Ž.b IfRS Ž M, V . has C-acc and V is a cogenerator then M has C-dcc. Ž. Ž. q Ž. n Proof. 1 a Let SV be injective, M s M, V , and a g A .By Lemma 3.2 there is an n-pointed finitely presented module Ž.B, b and a Ž.Ž. Ž. Ž. morphism ␣: B, b ª A, a with TMA, aBsTM,b. Now Lemma 1.5 Žq.Ž.0 Ž.0 Žq. q yields HMA, aAsTM,aBsTM,bBsHM,b, showing that M has A-dcc. Ž.b is shown similarly, using the second formula of Lemma 1.5. Ž.2 Ž. a Is proven like Ž.Ž. 1 a though by use of Proposition 3.1. Ž.b Let SSV be a cogenerator, T be some ring such that VTis a bimodule, and WTTbe an injective module with V ; WT. For an arbitrary set I we have the commutative diagram I 6 I Ž.Ž.M, V T, WTRm C Ž.Ž.RC,RTŽ.M,VT,W 6 X I cI1 Ž.Ž.SMmRSC,VT,W T m 6 c˜I 6 I I MmRC Ž.MmRC ŽŽ . . ŽŽ .. where c: M ª M, V TT, W and ˜c: M mRC ª M mRC, V TT, W denote the evaluation maps that are known to be pure monomorphisms. It I Ž. follows that c m1 is injective. If R M, V has C-dcc then is injective as well; hence is injective. The shows that M has C-acc. COROLLARY 3.4. Ž.1 Letting 0 ª MЈ ª M ª MЉ ª 0 be a pure-exact sequence, M has C-dcc if and only if MЈ and MЉ ha¨e. ŽI. IIŽI. Ž.2If M has C-dcc and I is some set then M , M and M rMha¨e C-dcc as well. Proof. Ž.1 The given sequence being pure, the exact sequence 0 ª Ž.Ž.Ž.MЉ,ޑrޚªM,ޑrޚªMЈ,ޑrޚª0 is split; hence our assertion follows from Corollaries 2.2 and 3.3. 84 WOLFGANG ZIMMERMANN Ž.2 As the functors HC, c commute with direct sums and products Ž I . IIŽI. M and M inherit C-dcc from M;by1,Ž. M rM has C-dcc as well. COROLLARY 3.5. Let M ª N be a pure-injecti¨e hull of M. If M has A-accŽ. C-dcc then N has the respecti¨e property, too. Proof. Letting Mqqs ŽŽM, ޑrޚ ., ޑrޚ . it is well known that N is isomorphic to a direct summand of Mqq. By Corollary 3.3 A-acc Ž.C-dcc passes from M to Mqq, hence to N. COROLLARY 3.6. Let 0 ª CЈ ª C ª CЉ ª 0 be a pure-exact sequence of right modules. Ž.1 If M has C-dcc it has CЈ-dcc. Ž.2 If M has CЈ-dcc and CЉ-dcc , then it has C-dcc. Proof. This follows from Corollaries 2.4 and 3.3. Recall that a left module A is called Mittag᎐Leffler module in case : I Ž.I MmRRAªMmAis a monomorphism for all right modules M and sets I wx10, Proposition 2.1.5 . Obviously it is sufficient to require this property for pure-injective modules M only. COROLLARY 3.7. The following are equi¨alent for a left module A: Ž.1 A is a Mittag᎐Leffler module. Ž.2 E¨ery Ž pure-injecti¨e . right module has A-acc. Ž.3 E¨ery Ž pure-injecti¨e . left module has A-dcc. Proof. The equivalenceŽ. 1 m Ž.2 is a direct consequence of Lemma 2.1. Observing that every module M admits a pure embedding into MqqsŽŽM,ޑrޚ .,ޑrޚ .the equivalenceŽ. 2 m Ž.3 follows from Corollar- ies 3.3 and 3.4. THEOREM 3.8. LettingSR M be a bimodule andS U be an injecti¨e cogenerator the following statements are equi¨alent: Ž.1 M has minimum condition for finite matrix subgroups, i.e., Mis Ý-pure-injecti¨e. Ž. Ž . ŽŽ . . 2 The map : C mRSM, SV ª CRRS, M , V is a monomor- phism for all modules CRS and all injecti¨e modules V. Ž. Ž ގ.ŽŽ . ގ. 3The map : C mRSM, SU ª CRRS, M , U is a monomor- phism for all denumerably generated modules CR. Ž. n 4 For all modules CR and all n g ގ and c g C there is an n-pointed finitely presented moduleŽ. D, d and a morphism ␥ : Ž.Ž.D, d ª C, c with Ž. Ž.Ž . HMC, cDsHMn,ds1is sufficient . Proof. Ž.1 « Ž.4 By Lemma 3.2 we have to showŽ. 4 only for n s 1. Let Ž. Ž . cgCand C, c s lim Cll, c with finitely presented modules Cl.As M ª CONDITIONS FOR MATRIX SUBGROUPS 85 is pure-injective it follows from Corollary 1.8 that HMC, cŽ.is the intersec- tion of the downward directed family of finite matrix subgroups ŽŽ..HM . This family has a minimal element HM Ž.; hence Cll, clgLCll,c HMŽ.HM Ž.. C,cCsll,c Ž.4« Ž.2 follows from Lemma 3.2. Ž.2« Ž.3 is obvious. Ž. Ž. 3«1 Letting CR be a denumerably generated module we have the commutative diagram Ž.ŽŽ..ގގ 6 CmRSM, SUCRRS,M,U 6 6 ŽŽ ..ގ ގ CmRSM, SU By assumption is injective, hence is injective, and by Theorem 2.4 the dual Ž.M, U has maximum condition for finite matrix subgroups. Now Corollary 1.6 implies that M has minimum condition for finite matrix subgroups. We note that Corollaries 3.3, 3.4, and 3.5 can be interpreted ‘‘globally’’ thus expressing properties of modules with minimumŽ. maximum condition for finite matrix subgroups. In the same way Corollary 3.6 has a global version which is well known: COROLLARY 3.9Žwxwx 12, Theorem 6.3 and 2, Theorem 8. . The following are equi¨alent for R: Ž.1 E¨ery left R-module is a Mittag᎐Leffler module. Ž.2 E¨ery left R-module has minimum condition for finite matrix sub- groups. Ž.3 E¨ery right R-modules has maximum condition for finite matrix subgroups. To conclude we show that a module M with M-dcc induces a certain equivalence. The simplest description of such a module is as follows: For n all n g ގ and m g M there exists an n-pointed finitely presented Ž. Ž. Ž. Ž. module D, d such that End M и m s HMM, mDsHM,d. Examples are the Ý-pure-injective modules but also the Mittag᎐Leffler modules, in particular the pure-projective modules. Ž. q Ž. Let S s End RSM , U be an injective cogenerator, and M s SSM, U . We denote the full subcategory of S-Mod of the SSM-injective and M- generated modules by L and the full subcategory of R-Mod of those modules which are isomorphic to direct summands of powers MqI by D. 86 WOLFGANG ZIMMERMANN COROLLARY 3.10. The following statements are equi¨alent: Ž.1 M has M-dcc. Ž. Ž . 2 The functorsSR M, y and M m ᎐ induce in¨erse equi¨alences between L and D. Proof. If L is some left S-module we shall denote the image of : Ž.Ž. Ž. MmRSM, SLª SS, SL,Lby L . Ž.1« Ž.2 Let L g L and L ; LЈ be an S-injective hull. As Ž.LЈ s L Ž. and the map : M mRSM, SLЈ ª LЈ is injective by Lemma 3.2, the map Ž. :MmRSM, SLªLis an isomorphism. Next, given X g D we show Ž.Ž.Ž. that M mRSX g L and that : X ª M, SM mRX , xmsmmx,is I a bijection. Let X g D and j: X ª Mq be a split monomorphism. 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