An Auslander-type result for Gorenstein-projective modules

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1 Advances n Mathematcs 218 (2008) An Auslander-type result for Gorensten-projectve modules Xao-Wu Chen Department of Mathematcs, Unversty of Scence and Technology of Chna, Hefe , PR Chna Receved 6 November 2007; accepted 17 Aprl 2008 Avalable onlne 20 May 2008 Communcated by Mchael J. Hopkns Abstract An artn algebra A s sad to be CM-fnte f there are only fntely many, up to somorphsms, ndecomposable fntely generated Gorensten-projectve A-modules. We prove that for a Gorensten artn algebra, t s CM-fnte f and only f every ts Gorensten-projectve module s a drect sum of fntely generated Gorensten-projectve modules. Ths s an analogue of Auslander s theorem on algebras of fnte representaton type [M. Auslander, A functoral approach to representaton theory, n: Representatons of Algebras, Workshop Notes of the Thrd Internat. Conference, n: Lecture Notes n Math., vol. 944, Sprnger-Verlag, Berln, 1982, pp ; M. Auslander, Representaton theory of artn algebras II, Comm. Algebra (1974) ] Elsever Inc. All rghts reserved. Keywords: Gorensten-projectve modules; Trangulated categores; Dualzng varetes 1. Introducton Let A be an artn R-algebra, where R s a commutatve artnan rng. Denote by A-mod (resp. A-mod) the category of (resp. fntely generated) left A-modules. Denote by A-Proj (resp. A- proj) the category of (resp. fntely generated) projectve A-modules. Followng [21], a chan complex P of projectve A-modules s defned to be totally-acyclc, f for every projectve Ths project was supported by Chna Postdoctoral Scence Foundaton No , and was also partally supported by the Natonal Natural Scence Foundaton of Chna (Grant Nos , and ). The author also gratefully acknowledges the support of K.C. Wong Educaton Foundaton, Hong Kong. E-mal address: xwchen@mal.ustc.edu.cn /$ see front matter 2008 Elsever Inc. All rghts reserved. do: /j.am

2 2044 X.-W. Chen / Advances n Mathematcs 218 (2008) module Q A-Proj the Hom-complexes Hom A (Q, P ) and Hom A (P,Q) are exact. A module M s sad to be Gorensten-projectve f there exsts a totally-acyclc complex P such that the 0th cocycle Z 0 (P ) = M. Denote by A-GProj the full subcategory of Gorensten-projectve modules. Smlarly, we defne fntely generated Gorensten-projectve modules by replacng all modules above by fntely generated ones, and we also get the category A-Gproj of fntely generated Gorensten-projectve modules [17]. It s known that A-Gproj = A-GProj A-mod [14, Lemma 3.4]. Fntely generated Gorensten-projectve modules are also referred as maxmal Cohen Macaulay modules. These modules play a central role n the theory of sngularty [10 12, 14] and of relatve homologcal algebra [9,17]. An artn algebra A s sad to be CM-fnte f there are only fntely many, up to somorphsms, ndecomposable fntely generated Gorensten-projectve modules. Recall that an artn algebra A s sad to be of fnte representaton type f there are only fntely many somorphsm classes of ndecomposable fntely generated modules. Clearly, fnte representaton type mples CM-fnte. The converse s not true, n general. Let us recall the followng famous result of Auslander [3,4] (see also Rngel Tachkawa [27, Corollary 4.4]): Auslander s theorem. An artn algebra A s of fnte representaton type f and only f every A- module s a drect sum of fntely generated modules, that s, A s left pure semsmple, see [31]. Inspred by the theorem above, one may conjecture the followng Auslander-type result for Gorensten-projectve modules: an artn algebra A s CM-fnte f and only f every Gorenstenprojectve A-module s a drect sum of fntely generated ones. However we can only prove ths conjecture n a nce case. Recall that an artn algebra A s sad to be Gorensten [19] f the regular module A has fnte njectve dmenson both at the left and rght sdes. Our man result s Man theorem. Let A be a Gorensten artn algebra. Then A s CM-fnte f and only f every Gorensten-projectve A-module s a drect sum of fntely generated Gorensten-projectve modules. Note that our man result has a smlar character to a result by Belganns [9, Proposton 11.23], and also note that smlar concepts were ntroduced and then smlar results and deas were developed by Rump n a seres of papers [28 30]. 2. Proof of Man theorem Before gvng the proof, we recall some notons and known results Let A be an artn R-algebra. By a subcategory X of A-mod, we mean a full addtve subcategory whch s closed under takng drect summands. Let M A-mod. We recall from [6,7] that a rght X -approxmaton of M s a morphsm f : X M such that X X and every morphsm from an object n X to M factors through f. The subcategory X s sad to be contravarantly-fnte n A-mod f each fntely generated modules has a rght X -approxmaton. Dually, one defnes the notons of left X -approxmatons and covarantly-fnte subcategores. The subcategory X s sad to be functorally-fnte n A-mod f t s contravarantly-fnte and

3 X.-W. Chen / Advances n Mathematcs 218 (2008) covarantly-fnte. Recall that a morphsm f : X M s sad to be rght mnmal, f for each endomorphsm h : X X such that f = f h, then h s an somorphsm. A rght X -approxmaton f : X M s sad to be a rght mnmal X -approxmaton f t s rght mnmal. Note that f a rght approxmaton exsts, so does rght mnmal one; a rght mnmal approxmaton, f n exstence, s unque up to somorphsms. For detals, see [6 8]. The followng fact s known. Lemma 2.1. Let A be an artn algebra. Then (1) The subcategory A-Gproj of A-mod s closed under takng drect summands, kernels of epmorphsms and extensons, and contans A-proj. (2) The category A-Gproj s a Frobenus exact category [22], whose relatve projectve-njectve objects are precsely contaned n A-proj. Thus the stable category A- Gproj modulo projectves s a trangulated category. (3) Let A be Gorensten. Then the subcategory A-Gproj of A-mod s functorally-fnte. (4) Let A be Gorensten. Denote by {S } n =1 a complete lst of parwse nonsomorphc smple A-modules. Denote by f : X S the rght mnmal A-Gproj-approxmatons. Then every fntely generated Gorensten-projectve module M s a drect summand of some module M, such that there exsts a fnte chan of submodules 0 = M 0 M 1 M m 1 M m = M wth each subquotent M j /M j 1 lyng n {X } n =1. Proof. Note that A-Gproj s nothng but X ω wth ω = A-proj n [6, Secton 5]. Thus (1) follows from [6, Proposton 5.1], and (3) follows from [6, Corollary 5.10(1)] (just note that n ths case, the regular module A A s a cotltng module). Snce A-Gproj s closed under extensons, thus t becomes an exact category n the sense of [22]. The property of beng Frobenus and the characterzaton of projectve-njectve objects follow drectly from the defnton, also see [14, Proposton 3.1(1)]. Thus by [18, Chapter 1, Secton 2], the stable category A- Gproj s trangulated. By (1) and (3), we see that (4) s a specal case of [6, Proposton 3.8]. Let R be a commutatve artnan rng as above. An addtve category C s sad to be R-lnear f all ts Hom-spaces are R-modules, and the composton maps are R-blnear. An R-lnear category s sad to be hom-fnte, f all ts Hom-spaces are fntely generated R-modules. Recall that an R-varety C means a hom-fnte R-lnear category whch s skeletally-small and dempotentsplt (that s, for each dempotent morphsm e : X X n C, there exst u : X Y and v : Y X such that e = v u and Id Y = u v). It s well known that a skeletally-small R-lnear category s an R-varety f and only f t s hom-fnte and Krull Schmdt (.e., every object s a fnte sum of ndecomposable objects wth local endomorphsm rngs). See [26, p. 52] or [16, Appendx A]. Then t follows that any factor category [8, p. 101] of an R-varety s stll an R-varety. Let C be an R-varety. We wll abbrevate the Hom-space Hom C (X, Y ) as (X, Y ). Denote by (C op,r-mod) (resp. (C op,r-mod)) the category of contravarant R-lnear functors from C to R-Mod (resp. R-mod). Then (C op,r-mod) s an abelan category and (C op,r-mod) s ts abelan subcategory. Denote by (,C) the representable functor for each C C. A functor F s sad to be fntely generated f there exsts an epmorphsm (,C) F for some object C C; F s sad to be fntely presented (= coherent) [2,4], f there exsts an exact sequence of functors

4 2046 X.-W. Chen / Advances n Mathematcs 218 (2008) (,C 1 ) (,C 0 ) F 0. Denote by fp(c) the subcategory of (C op,r-mod) consstng of fntely presented functors. Clearly, fp(c) (C op,r-mod). Recall the dualty D = Hom R (,E): R-mod R-mod, where E s the njectve hull of R/rad(R) as an R-module. Therefore, t nduces dualty D : (C op,r-mod) (C,R-mod) and D : (C,R-mod) (C op,r-mod). TheR-varety C s called a dualzng R-varety [5], f ths dualty preserves fntely presented functors. The followng observaton s mportant. Lemma 2.2. Let A be a Gorensten artn R-algebra. Then the stable category A- Gproj s a dualzng R-varety. Proof. Snce A-Gproj A-mod s closed under takng drect summands, thus dempotent-splt. Therefore, we nfer that A-Gproj s an R-varety, and ts stable category A- Gproj s also an R- varety. By Lemma 2.1(3), the subcategory A-Gproj s functorally-fnte n A-mod, then by a result of Auslander and Smalø [7, Theorem 2.4(b)] A-Gproj has almost-splt sequences, and thus theses sequences nduce Auslander Reten trangles n A- Gproj (let us remark that t s Happel [19, 4.7] who realzed ths fact for the frst tme). Hence the trangulated category A- Gproj has Auslander Reten trangles, and by a theorem of Reten and Van den Bergh [25, Theorem I.2.4] we nfer that A- Gproj has Serre dualty. Now by [20, Proposton 2.11] (or [13, Corollary 2.6]), we deduce that A- Gproj s a dualzng R-varety. Let us remark that the last two cted results are gven n the case where R s a feld, however one just notes that the results can be extended to the case where R s a commutatve artnan rng wthout any dffculty. For the next result, we recall more notons on functors over varetes. Let C be an R-varety and let F (C op,r-mod) be a functor. Denote by nd(c) the complete set of parwse nonsomorphc ndecomposable objects n C. Thesupport of F s defned by supp(f ) ={C nd(c) F(C) 0}. The functor F s smple f t has no nonzero proper subfunctors, and F has fnte length f t s a fnte terated extenson of smple functors. Observe that F has fnte length f and only f F les n (C op,r-mod) and supp(f ) s a fnte set. The functor F s sad to be noetheran, f ts every subfunctor s fntely generated. It s a good exercse to show that a functor F s noetheran f and only f every ascendng chan of subfunctors n F becomes stable after fnte steps (one may use the fact: for a fntely generated functor F wth an epmorphsm (,C) F, then for any subfunctor F of F, F = F provded that F (C) = F(C)). Observe that a functor havng fnte length s necessarly noetheran by an argument on ts total length (.e., l(f) = C nd(c) l R(F (C)), where l R denotes the length functon on fntely generated R-modules). The followng result s essentally due to Auslander (compare [4, Proposton 3.10]). Lemma 2.3. Let C be a dualzng R-varety, F (C op,r-mod). Then F has fnte length f and only f F s fntely presented and noetheran. Proof. Recall from [5, Corollary 3.3] that for a dualzng R-varety, functors havng fnte length are fntely presented. So the only f follows. For the f part, assume that F s fntely presented and noetheran. Snce F s fntely presented, by [5, p. 324], we have the fltraton of subfunctors 0 = soc 0 (F ) soc 1 (F ) soc +1 (F )

5 X.-W. Chen / Advances n Mathematcs 218 (2008) where soc 1 (F ) s the socle of F, and n general soc +1 s the premage of the socle of F/soc (F ) under the canoncal epmorphsm F F/soc (F ). Snce F s noetheran, we get soc 0 F = soc 0 +1(F ) for some 0, and that s, the socle of F/soc 0 (F ) s zero. However, by the dual of [5, Proposton 3.5], we know that for each nonzero fntely presented functor F, the socle soc(f ) s necessarly nonzero and fntely generated semsmple. In partcular, soc(f ) has fnte length, and thus t s fntely presented. Note that fp(c) (C op,r-mod) s an abelan subcategory, and thus F/soc 1 (F ) s fntely presented. Applyng the above argument to F/soc 1 (F ), we obtan that soc 2 (F ), as an extenson between the socles of two fntely presented functors, has fnte length. In general, one proves that F/soc (F ) s fntely presented and soc +1 (F ) has fnte length for all. Hence soc(f/ soc 0 (F )) = 0 wll mply that F/soc 0 (F ) = 0,.e., F = soc 0 (F ), whch has fnte length. Let us consder the category A-GProj. Smlar to Lemma 2.1(1), (2), we recall that A- GProj A-Mod s closed under takng drect summands, kernels of epmorphsms and extensons, and t s a Frobenus exact category wth (relatve) projectve-njectve objects precsely contaned n A-Proj. Consder the stable category A- GProj, whch s also trangulated and has arbtrary coproducts. Recall that n an addtve category T wth arbtrary coproducts, an object T s sad to be compact, f the functor Hom T (T, ) commutes wth coproducts. Denote the full subcategory of compact objects by T c. If we assume further that T s trangulated, then T c s a thck trangulated subcategory. We say that the trangulated category T s compactly generated [23,24], f the subcategory T c s skeletally-small and for each object X, X 0 provded that Hom T (T, X) = 0 for every compact object T. Note that n our stuaton, we always have an ncluson A- Gproj A- GProj, and n fact, we vew t as A- Gproj (A- GProj) c. Next lemma, probably known to experts, states the converse n Gorensten case. It s a specal case of [14, Theorem 4.1] (compare [10, Theorem 6.6]). One may note that n the artn case, the category A- Gproj s dempotent-splt. Lemma 2.4. Let A be a Gorensten artn algebra. Then the trangulated category A- GProj s compactly generated and A- Gproj (A- GProj) c s dense (.e., surjectve up to somorphsms) Proof of Man theorem Assume that A s a Gorensten artn R-algebra. Set C = A- Gproj. By Lemma 2.2, C s a dualzng R-varety. For a fntely generated Gorensten-projectve module M, we wll denote by (,M)the functor Hom C (,M); for an arbtrary module X, we denote by (,X) C the restrcton of the functor Hom A (,X)to C. For the f part, we assume that each Gorensten-projectve module s a drect sum of fntely generated ones. It suffces to show that the set nd(c) s fnte. For ths end, assume that M s a fntely generated Gorensten-projectve module. We clam that the functor (,M)s noetheran. In fact, gven a subfunctor F (,M), frst of all, we may fnd an epmorphsm (,M ) F, I where each M C and I s an ndex set. Compose ths epmorphsm wth the ncluson of F nto (,M), we get a morphsm from I (,M ) to (,M). By the unversal property of coproducts

6 2048 X.-W. Chen / Advances n Mathematcs 218 (2008) and then by Yoneda s lemma, we have, for each, a morphsm θ : M M, such that F s the mage of the morphsm (,θ ) : (,M ) (,M). I I Note that I (,M ) (, I M ) C, and the morphsm above s also nduced by the morphsm I θ : I M M. Form a trangle n A- GProj K[ 1] I M I θ M φ K. By assumpton, we have a decomposton K = j J K j where each K j s fntely generated Gorensten-projectve. Snce the module M s fntely generated, we nfer that φ factors through a fnte sum j J K j, where J J s a fnte subset. In other words, φ s a drect sum of M φ K j and 0 K j. j J j J \J By the addtvty of trangles, we deduce that there exsts a commutatve dagram I M I θ M M ( j J \J K j )[ 1] (θ,0) M where the left sde vertcal map s an somorphsm, and M and θ are gven by the trangle ( j J K j )[ 1] M θ M φ j J K j. Note that M C, and by the above dagram we nfer that F s the mage of the morphsm (,θ ) : (,M ) (,M), and thus F s fntely generated. Ths proves the clam. By the clam, and by Lemma 2.3, we deduce that for each M C, the functor (,M) has fnte length, n partcular, supp((,m)) s fnte. Assume that {S } n =1 s a complete lst of parwse nonsomorphc smple A-modules. Denote by f : X S the rght mnmal A-Gprojapproxmatons. By Lemma 2.1(4), the module M s a drect summand of M and we have a fnte chan of submodules of M wth factors beng among X s. Then t s not hard to see that supp((,m)) supp((,m )) n =1 supp((,x )) for every M C. Therefore we deduce that nd(c) = n =1 supp((,x )), whch s fnte. For the only f part, assume that the Gorensten artn algebra A s CM-fnte. Then the set nd(c) s fnte, say nd(c) ={G 1,G 2,...,G m }. Set B = End C ( m =1 G ) op. Then B s also an artn R-algebra. Note that for each C C, the Hom-space Hom C ( m =1 G,C) has a natural left B-module structure, moreover, t s a fntely generated projectve B-module. In fact, we get an equvalence of categores ( m ) Φ = Hom C G, - : C B-proj. =1

7 X.-W. Chen / Advances n Mathematcs 218 (2008) Then the equvalence above naturally nduces the followng equvalences, stll denoted by Φ, Φ : fp(c) B-mod, Φ: ( C op,r-mod ) B-Mod. In what follows, we wll use these equvalences. By [24, p. 169] (or [13, Proposton 2.4]), we know that the category fp(c) s a Frobenus category. Therefore, va Φ, we get that B s a self-njectve algebra. Therefore by [1, Theorem 31.9], we get that B-Mod s also a Frobenus category, and by [1, p. 319], every projectve-njectve B-module s of form m =1 Q (I ), where {Q 1,Q 2,...,Q m } s a complete set of parwse nonsomorphc ndecomposable projectve B- modules such that Q = Φ(G ), and each I s some ndex set, and Q (I ) s the correspondng coproduct. Take {P 1,P 2,...,P n } to be a complete set of parwse nonsomorphc ndecomposable projectve A-modules. Let G A-GProj. We wll show that G s a drect sum of some copes of the modules G and P j. Then we are done. Consder the functor (,G) C, whch s cohomologcal, and thus by [13, Lemma 2.3] (or [24, p. 258]), we get Ext 1 (F, (,G) C ) = 0 for each F fp(c), where the Ext group s taken n (C op,r-mod). VaΦ and applyng the Baer s crteron, we get that (,G) C s an njectve object, and thus by the above, we get an somorphsm of functors m (,G ) (I) (,G) C, =1 where I are some ndex sets. As n the frst part of the proof, we get a morphsm θ : m =1 G (I ) G such that t nduces the somorphsm above. Form a trangle n A- GProj m =1 G (I ) θ G X ( m =1 G (I ) ) [1]. For each C C, applyng the cohomologcal functor Hom A- GProj (C, ) and by the property of θ, we obtan that Hom A- GProj (C, X) = 0, C C. By Lemma 2.4, the category A- GProj s generated by C, and thus X 0, and hence θ s an somorphsm n the stable category A- GProj. Thus t s well known (say, by [15, Lemma 1.1]) that ths wll force an somorphsm n the module category m =1 G (I ) P G Q, where P and Q are projectve A-modules. Now by [1, p. 319], agan, P s a drect sum of copes of the modules P j. Hence the combnaton of Azumaya s theorem and Crawlay Jønsson Warfeld s theorem [1, Corollary 26.6] apples n our stuaton, and thus we nfer that G s somorphc to a drect sum of copes of the modules G and P j. Ths completes the proof.

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