
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,È 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 m: Hom SRP, M m A ª Hom SRP, M m A Ž.Ž.Ž. Ž. given by mwmaxswxmafor w 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 mÄÄ: M mRRiiA ª M m A , m 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 m is injective for all projective modules S P Žequivalently mÄ 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 68 0021-8693r97 $25.00 Copyright Q 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 mÄ 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 n : Ž. ŽŽ.. Ž.Ž. CmRSHom M, U ª HomSR Hom C, M , U with n c m w g s ŽŽ.. Ž. Ž. wgc for c g C, w g Hom SRM, U , and g g Hom C, M , where SMR is a bimodule and CRS, U are modules. The fact that n is an isomorphism if C is finitely presented and U is injective is well known and useful in many respects. This time injectivity of n 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 <<X . 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 ta: Ž I . t 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 t . For every homomorphism f g M, M9 ŽŽ..Ž. ŽI. we have fTA, aA M ;TM,aA9, i.e., T ,ais a subfunctor of V , where V Ž. denotes the forgetful functor Mod-R ª Mod-Z. In particular TMA, a is ŽI. an End RŽ.M -submodule of M . e Ž.I e Ž. Ž. ŽŽ .. Similarly, defining aai: A, N ª N by h s has ha igIand Ž. HNA, aasIm e 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 CrC9.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 a 6 p 6 6 GFC0, Ž. where F is free with a basis e1,...,ekii, kGn, such that p e s c , 1FiFn, and G is free as well with a basis f1,..., fl. Let G9 be a further g 9 free module with a basis fy1 ,..., fyn and let : G ª F be defined by gŽ. fyiise,1FiFn. These data yield the following commutative dia- gram with exact rows and columns 6 h 6 r 6 6 0 GG=G9 G90 6 a6 bsŽ.a,g Fs F 6 p6 6 CCrC9 6 6 00 where C9 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 b * 6 a * 6 r* 6 h* 6 6 0GXX*G*=G*G*0 6 j6 j9 X 6 6 G* AArA9 6 6 00 with exact rows and columns, j and j 9 denoting the cokernels of b * and a 9 Ž.jr j ŽU. *, respectively, and A s Im * . We put ai s fyi ,1FiFn, and jŽ 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.b,1 6 Ž.a,1 6 Ž.r,1 6 Ž.h,1 6 6 0 Ž.G9,MG Ž=G9,MG . Ž.,M0 6 l6 l9 6 6 6 6 0Ž.G9,MMmRRAMmArA90 6 6 00 l lŽ. Ýl Ž. with exact rows and columns, being given by h s js1hfjjmaq ÝnŽ. l9 Ž. is1 hfyi mayi and similarly. Putting a s ay1,...,ayn and identi- Ž. n Ž. fying G9, M with M it is easily seen that l r,1 sta. Now the snake ( t a d n lemma yields the exact sequence 0 ª Ž.Ž.CrC9, M ª C, M ª M ª M mRRA ª M m ArA9 ª 0 and an easy calculation shows that the connect- Ž. ing homomorphism d coincides with ecC. If follows that HM,cs TMA, aŽ.. Similarly it is proven that the sequence 0 ª Ž.ŽArA9, N ª A, 72 WOLFGANG ZIMMERMANN e a t c .Žn . NªNªNmRRCªNmCrC9ª0 is exact, showing that HNA,a 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- a aŽ. 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.
Details
-
File Typepdf
-
Upload Time-
-
Content LanguagesEnglish
-
Upload UserAnonymous/Not logged-in
-
File Pages20 Page
-
File Size-