ON FINITELY GENERATED MODULES OVER NOETHERIAN RINGS

Transcription

1 ON FINITELY GENERATED MODULES OVER NOETHERIAN RINGS BY J. P. JANS(i). Inroducion. If A is a module over a ring R, he module Homj;(/l,R) = A* is usually called he dual of A. The elemens of A can be considered as homomorphisms from A* o R so ha here is a naural map A-*A** of A ino is second dual. Following he erminology inroduced by Bass [2] we shall say ha A is orsionless if a is a monomorphism, reflexive if o is an isomorphism. We shall also refer o Ima as he orsionless facor of A. I is easy o see ha his is also A/A where A is he inersecion of kernels of he elemens of A*. In his paper we plan o coninue he sudy iniiaed in [5] relaing some of he conceps menioned above wih he funcor Ex^(A,R). Since i will only appear in his form in he presen paper we shall henceforh adop he noaion E"(A) = Ex"R(A,R). Also, in his paper we shall make he sanding assumpions ha he ring is boh lef and righ Noeherian and all modules under consideraion are finiely generaed. The reason behind he laer assumpion is o insure ha projecive modules are reflexive and ha he duals of projecive modules are projecive; see [2]. In 1 we relae double dual embeddings wih he orsionless facors of modules A such ha EX(A) =. The proof of his resul arose ou of Theorem 1.4 of [5] in which i is shown ha he firs dual A* is a direc summand of he "hird" dual A***. In 2 we inroduce he concep of D-class n and show ha he dual of a module of D-class n appears as he nh kernel in a projecive resoluion. I is hen clear ha properies of he ring which are phased in erms of he sor of kernels appearing in projecive resoluions can also be described in erms of modules of D-class n (and heir duals). In 3, we show ha, modulo a special condiion, a module T is of D-class n if and only if E\Tn) = for 1 g i ^ n - 1. Under Applicaions, 4, we relae properies of modules of D-class n o he global dimension, lef finiisic dimension and lef injecive dimension of he ring. Received by he ediors April 16, 1961 and, in revised form, January 8, (!) Research suppored in par by he Unied Saes Air Force Office of Scienific Research and in par by Naional Science Foundaion conrac NSF-G License or copyrigh resricions may apply o redisribuion; see hp://

2 ON FINITELY GENERATED MODULES 331 Mos of hese applicaions ake he form of generalizaions of heorems of [2; 5]. 5, Odds and ends, consiss of some resuls which are easily proved by he mehods of he paper. 1. W-moduIes and double dual embeddings. The following definiions will faciliae our exposiion: Definiion. A will be called a W-module if E1(A) =. Definiion. A monomorphism X** -* F* will be called a double dual embedding (D.D.E.) if i is he dual of an epimorphism F -» X*, X* a dual. The following heorem gives he he relaion beween double dual embeddings and he orsion facors of W-modules. p* Theorem 1.1. If Q** -* F* is a D.D.E. wih F projecive hen F*/lmp* is he orsionless facor of a W-module. Conversely, if T is he orsionless facor of a W-module hen here exiss a projecive module F such ha ->y-»f-> T -» is exac and X ->F is a D.D.E. Proof. Consider he exac sequence where ->T*->F->o*-> p* ^.Q**->F*_>r_> is also exac. Fis defined o be F*/Imp* and i urns ou ha Kerp = F*; his is really he argumen in Bass' paper [2]. Choose in Q** a submodule Q such ha q z+ Q** is he naural embedding. This gives rise o he following commuaive diagram wih exac rows and columns: p* ' -> Q** -* F* -» T -* ff * II ö i p*ff m i ->g >F* ->F'-> where T' is F/Im(p*cr) and 9 is he naural map of F/Im(p*a) ono F/Imp* = T. Now dualize he enire diagram o obain he following diagram: i p** o -> r*-> f -» g*** -+T'*-» F-> Q* -y E\T') -*. License or copyrigh resricions may apply o redisribuion; see hp://

3 332 J. P. JANS [February Noe ha he map o* is an epimorphism and ha Imp = Imp** = (2* embedded naurally in Q*** [5]. I follows herefore ha o*p** is an epimorphism and consequenly T is a W-module. The "Five lemma" implies ha * is an isomorphism, and, in his way, T can be considered as he orsionless facor of T wih T' a lf-module. To prove he converse saemen, consider Tas he orsionless facor of T' wih F^T') =. Selec F, projecive mapping ono T and consruc he following commuaive diagram wih exac rows and columns: j -» X -> F P -+ T -* k II ->g -» F -» T'-> where p = p. In he following we shall esablish ha X U F is a D.D.E. Firs form duals o obain he diagram: ö p* -* T* -* F*-> F*/Imp* -* lö*. II -> T'* -> F* -» Ö* -*. Exacness of he boom row comes from he hypohesis E1^') =. Also * is an isomorphism by hypohesis. Since Imp* = Imp* we have he following isomorphisms, F*/Imp* = F*/Imp* = Q*. Thus we see ha F*/Imp* is a dual iq*) and i follows from [2] ha he dual of he epimorphism F* -» F*/Imp* is he monomorphism X -> F. The laer is herefore a D.D.E. The above proof yields several corollaries. c* Corollary 1.2. // Q** -* F* is a D.D.E. wih F projecive and Q** reflexive hen F*/lmp* is a orsionless W-module. Conversely, if T is a orsionless W- module, -> X -*F -» T-* exac wih F projecive hen X -* F is a D.D.E. Proof. Recall ha in he proof of he heorem we seleced a module Q in Q** so ha Q->(2** was he naural embedding of Q in is double dual. In he corollary we can le Q = Q** and he resuling diagram has only one row insead of wo. In his case he W-module T' coincides wih is orsionless facor Tand he firs par of he corollary follows. The converse par of he corollary follows from he proof of he corresponding par of he heorem using he fac ha T is is own orsionless facor. In [2], Bass showed how o injec any orsionless module ino a projecive module by using he following consrucion. Le T be orsionless and le T* be is dual. Find a projecive F mapping ono T*, F ~* T* and dualize o obain License or copyrigh resricions may apply o redisribuion; see hp://

4 1963] ON FINITELY GENERATED MODULES 333 he D.D.E. T** -» F*. Since T is orsionless he naural mapping of T ino T** is a monomorphism which ogeher wih p* gives an embedding of T ino F*. We shall call such an embedding of a orsionless module a sandard embedding. Noice ha he above consrucion makes up par of he diagrams used in he proof of Theorem 1.1 above. By arguing from hese diagrams we ge anoher corollary. Corollary 1.3. The exac sequence -» T * F is a sandard embedding of he orsionless module T in he projecive F if and only if F/lmj is a W-module. We remark a his poin ha no all embeddings of orsionless modules in projecives are sandard embeddings. For example, if P ^* F is an embedding of a projecive P in he projecive F he embedding will be sandard if and only if he embedding splis. If he embedding does no spli hen F/lmj has dimension one and modules of dimension one are never ly-modules [4, p. 123]. Of course if he embedding splis i will be sandard. 2. D-classes. In his paper Bass proved he following exremely useful heorem connecing arbirary finiely generaed modules and duals. Shifing heorem. Le A be finiely generaed, B orsionless C* a dual. If any one of he modules A, B or C* is given he oher wo exis and are conneced by he exac sequence where F and F' are projecive. ->C*->F->B-*->B-*F'->,4-> We would like o exend his heorem (and some of is numerous corollaries) by shifing hrough projecive modules even farher back. Tha is, we wish o examine he srucure of he modules D such ha ->-D*->F"->C*-* is exac wih F" projecive. While we are a i we migh jus as well shif back n seps. The invesigaion will be faciliaed by he following definiion. Definiion. We shall say ha he orsionless module Tn is of D-class n if i can be fied ino an exac diagram of he form -, T^y -+ Fn.y -* Tn - (1)... " f -2 - T.-l - -* Tf* -*... _> Tf* -+ Fy -> T2 -> where each F is projecive, he horizonal monomorphisms are D.D.E.'s and he License or copyrigh resricions may apply o redisribuion; see hp://

5 334 J. P. JANS [February verical maps are all naural embeddings of he T in heir second duals T**. We shall say ha any orsionless module is of )-class 1. The following heorem esablishes a connecion beween (lef) modules of D-class n and he kernels in a projecive resoluion of a (righ) module. Theorem 2.1. If here exiss a collecion of exac sequences (1*) -> T,iy -» F -> Tf-*, lgign-1 wih each F projecive hen T can be seleced of D-class n. Conversely, if Tn is a module of D-class n hen T* can be embedded in a collecion of he form (1*). Proof. As a he beginning of he proof of Theorem 1.1, he sequence O^Tiy^Fi-^T^O induces he sequence -» T**-*F*-* T +1 - where he monomorphism is clearly a D.D.E. Now we can obain a diagram of he form (1) by sringing hese sequences ogeher wih he naural verical maps j: _ j;**. Thus T can be chosen o be of D-class n. Conversely, dualizing a sequence of he form -> T **-> F ^> Tl+1 -> gives rise o a sequence of he form -+ T*+ y -> Ff -* T***. Bu since we are assuming ha he injecion in he firs sequence is a D.D.E., he map from F* o j;*** in he second sequence is he second dual of an epimorphism of F* ono T*. Because projecives are reflexive ha second dual will coincide wih he original map, and we obain he sequence -> T * y ->F *-> T *->. Tha is, if T is a module of D-class n, T* can be embedded in a diagram of he form (1*). I should be noed ha every module involved in he definiion of D-classes and in he proof of Theorem 2.1 is orsionless. The whole hing can be rephrased in nonorsionless erms and a corresponding heorem can be proved. For he sake of compleeness, we indicae briefly how his can be done. Definiion. Tn' will be of D'-class n if i can be fied ino an exac diagram of he form -+ Tn_, -+ F., -+ T' ^ f _2 -+ r;., -> o d') - T2 - - Ty -* Fy - T'2 - where each T\ is a IT-module for 2 ^ i ^ n and all he verical maps are he mappings of he T' on he orsionless facors T. We shall allow any module o be of D'-class 1. The heorem ha would go wih he above definiion would read like Theorem 2.1 wih D' and T' replacing D and T. Tha one can jump back and forh License or copyrigh resricions may apply o redisribuion; see hp://

6 1963] ON FINITELY GENERATED MODULES 335 beween he diagrams (1) and (1') is he conen of Theorem 1.1. Moreover ha heorem also shows ha a module is of D-class n if and only if i is he orsionless facor of a module of D '-class n. Noe ha he modules of D '-class 2 are he W-modules. 3. Ex of modules of D-class n. If one can find a orsionless module T wih he propery ha '(TB) = for 1 ^ i ^ n - 1 hen i is easy o see ha he module is of D class n. For if ^r1^f1->f2->...fb_1^tb-> is par of a projecive resoluion for T wih F projecive hen a repeaed applicaion of Corollary 1.2 yields he diagram -* rn**1-^fn_1^t -> (2)... F _2^T **1-> o-^r1**->f1->r2**->o where he T/**are he appropriae kernels. In his secion we invesigae a condiion which will insure ha all modules T of D-class n have he propery ha '(T ) = for 1 g i ^ n - 1. Tha is we shall ry o collapse he diagram (1) wha is really an exac sequence (2). A his poin he exposiion is faciliaed by he concep of grade, defined by Rees [6] for commuaive Noeherian rings. Definiion. The module M has grade r if '(M) = for i ^ r - 1 and E'(M) /. The module S has reduced grade r if '(S) = for 1 ^ i ^ r 1 and Er(S) #. Noe ha orsionless modules always have grade (since (T)# ) bu he reduced grade of a orsionless module may be large. In fac our goal in his secion is o prove (under cerain condiions) ha if T is of D-class n hen he reduced grade of T is greaer han or equal o n 1. Theorem 3.1. If R has he propery ha for all r and for all finiely generaed righ modules M he grade of Er(M) is greaer han r 1 hen every lef module of D-class n has reduced grade greaer han n 1. Proof. Since for n = 1 here is nohing o prove, we begin an inducion a n = 2. In his case he hypohesis implies ha ( 2(M)) = for all finiely generaed righ modules M. Now by Corollary 1.5 of [5] we know ha he duals of all lef modules are reflexive. Bu hen T** is reflexive for each lef module T and we can conclude from Corollary 1.2 ha every lef module T of D-class n(n _ 2) is a W-module. In paricular if F2 is of D-class 2 hin EX(T2) = and he reduced grade of T2 is greaer han 1. License or copyrigh resricions may apply o redisribuion; see hp://

7 336 J. P. JANS [February Assume now ha he heorem has been esablished for inegers less han n and ha n is greaer han 2. Since T _ y in he diagram (1) is of D-class n 1 we have E\Tn-y) = for 1 ^j z% n - 2. To finish he proof we shall show ha EjiTn-y) = EJiT**y) for 1 ^ j< ^ n 2. This will be sufficien for we see ha he equaion E\T**y ) = EJ+1iT ) holds for all j ^ 1. Also we know ha E\Tn) = from he firs par of he proof. If we examine he diagrams (1) and (1*) and apply he dualiy heorem of [5] we obain he shor exac sequence (3) ^T -y^t y-+e1itï-2)^. Also, from he sequence (1*) we ge he isomorphism E\T*^2) = En~2iT*). By an applicaion of Bass' Shifing Theorem we can raise he superscrip by wo o obain E"iM) = E"~2iT*y) for a suiable righ module M. If we pu his ino he sequence (3) we have he shor exac sequence (3') ->T _1^T*_*1^Fn(M)^. Now apply EJ o his sequence and use he hypohesis ha E\E\M)) = for Oz%jz%n 1. From he exac sequence of homology we obain he desired isomorphisms E\Tn-y) =E\T**y) holding for l^' «-2. This complees he proof of he heorem. We remark ha he proof of he heorem did no use he full force of he hypohesis "all r" bu uses insead he hypohesis "all r up o and including n." The raher srange hypohesis "grade E\M) greaer han r - 1" in he preceding heorem brings up he quesion of which rings have his condiion. H. Bass has consruced a proof of he fac ha for a commuaive Noeherian ring R his condiion is equivalen o he condiion ha Rp has finie injecive dimension over iself for every prime p where Rp is he localizaion of R a p. The proof of his can be based on he resuls of [3]. We know of no analogous heorem for noncommuaive rings. 4. Applicaions. In his secion we relae modules of D-class n o various invarians of he ring. Among hese invarians are he global dimension, lef (and righ) finiisic dimension and he lef (and righ) injecive dimension of he ring as a module over iself. Recall ha he global dimension of R, gl-dim-r, is defined o be he supremum of he projecive dimensions of all he R-modules Auslander showed [1], ha for he rings we consider his can be compued by aking he supremum of he dimensions of he cyclic lef R-modules. The lef finiisic dimension of R, IfPDiR), is he supremum of he projecive dimensions of he finiely generaed lef modules of finie projecive dimension. See Bass' paper [2] for a number of relaions beween hese and oher dimensions. The following heorem connecs a propery of modules of D-class n wih he global dimension of he ring. License or copyrigh resricions may apply o redisribuion; see hp://

8 1963] ON FINITELY GENERATED MODULES 337 Theorem 4.1. For he rings R under consideraion he following are equivalen for inegers n ^ 1: (a) gl.dim. (R)=n4-1. (b) Duals of modules of D-class n are projecive. Proof. Combining Bass' Shifing Theorem wih Theorem 2.1 we see ha if T* is he dual of a righ module of D-class n hen T* appears as he kernel in he nh projecive module of a projecive resoluion of some lef R-module A. And, conversely, given a projecive resoluion of some lef R-module A he nh kernel is he dual of a righ module of D-class n. Thus (a) and (b) are equivalen. We remark ha we mean by he nh module in a projecive resoluion, he one wih subscrip n. The subscrips sar a zero. This is he usual noaion, bu i is a poor way o coun. In he above argumen, we did no need o disinguish beween lef and righ because of Auslander's resul menioned above. In he nex heorem, we do have o make such a disincion since he lef and righ finiisic dimensions need no be he same. The following heorem can be considered as a generalizaion of Theorem 5.3 of [2]. Theorem For he rings under consideraion he following are equivalen for n 2: 1 : (a) lfpd(r) Í n. (b) The only righ modules of D-class n wih projecive duals are he projecives. Proof. For n = 1 his is exacly Bass' Theorem. Assume condiion (a) and le T be of D-class n wih T* projecive. By he proof of Theorem 4.1, T* is he kernel in he nh projecive module of a projecive resoluion of some lef module A, and A is herefore of finie projecive dimension. By he assumpion (a) we see ha he sequence -» T*-*F -y -* Tnly -> is exac and splis. Bu hen he dual sequence -» T**y -* F*_ y -* T -> used in showing T o be of D-class n also splis, and T is projecive. Conversely, assume condiion (b) and le A be a lef module of projecive dimension less han or equal o n + 1 for n g 2. We will show ha is dimension is acually less han n + 1. If F is he nh projecive in a projecive resoluion of A, we have he sequence -» T *-> F -> T * y where T is of D-class and T*is projecive. By (b) we conclude ha T is projecive and he sequence -» T ** -> F* -*Tn -* used in exhibiing he D-class of T splis. Thus we see ha T ** is projecive. Bu T *_! is a direc summand of T*** so boh of hese are projecive. Tha is, he projecive dimensions of A is less han n +1. This concludes he proof of he heorem. In [5] we showed ha he difference beween a orsionless module and is second dual is a module of he form E1(B). We were able o use his o connec License or copyrigh resricions may apply o redisribuion; see hp://

9 338 J. P. JANS [February he vanishing of E1(B) and [El(B)~\* wih reflexiveness. The following heorem can be hough of as an exension of Corollary 1.3 of [5]. Theorem 4.3. For he rings R under consideraion he following are equivalen for n _ 1: (a) The lef modules of D-class n are reflexive. (b) E"(B) = Ofor all orsionless righ modules B. (c) En+1(C) = Ofor all righ modules C. (d) The righ injecive dimension of he ring is less han or equal o n. Proof. For n = 1 his is exacly Corollary 1.3 of [5]. Also he equivalence of (b), (c), and (d) for all n ^ 1 follows from Bass' Shifing Theorem. I is herefore sufficien o show he equivalence of (a) and (b) for n ^ 2. If T is of D-class n hen we are assured of he wo sequences -* T *_*, -> F* -» T ->, -T * ->F -*T Lx-4. Under hese circumsances he conclusion of Theorem 1.1 of [5] holds and we obain he addiional exac sequence -+ T -* r**-+ E\T*- x) -. Since T was of D-class n, T?- x can be hough of as a kernel in a projecive resoluion of a orsionless module B. By using he exac sequence of homology on '( ) we see ha EX(T*-X) "(B). Thus we arrive a he exac sequence (*) -* Tn - T *-> E"(B) -y. The above consrucion can be reversed in he sense ha we could have sared wih he orsionless righ module B and worked backwards o ge a lef module of D-class n. I is clear now ha he sequence (*) gives he equivalence of (a) and (b). We know from [5] ha we can hang sars on he sequence (*) o ge he following exac spli sequence, (**) -> [ "(B)]* -* T***-y T* -y. From his sequence we ge immediaely he following corollary. Corollary 4.4. For he rings under consideraion he following are equivalen for n ^ 1 : (a) [ (B)]* = Ofor all orsionless righ modules B. (b) T*is reflexive for all lef modules T of D-class n. 5. Odds and ends. We include he following because he mehods of proof appear o be relaed o he preceding resuls. License or copyrigh resricions may apply o redisribuion; see hp://

10 1963] ON FINITELY GENERATED MODULES 339 Theorem 5.1. For he rings under consideraion he following condiions are equivalen: (a) All lef W-modules are orsionless. (b) All righ orsionless modules are W-modules. (c) All orsionless lef modules are reflexive. (d) The righ injecive dimension of he ring is less han or equal o one. Proof. We showed he equivalence of (b), (c) and (d) in Corollary 1.3 of [5]. In he following we shall esablish he equivalence of (a) and (b). Le A be a orsionless righ module. From Theorem 1.1 of [5] here is a orsionless lef module B such ha -> B -+ ß**-> E\A) -> is exac. By he proof of Theorem 1.1 we can embed his sequence in an exac diagram E\A) ->B**-> F -> M -y ->B ->F->M'-> E\A) where E1iM') =, M is he orsionless facor of M', and F is projecive. If condiion (a) holds, we have M = M' and B = B**so ha E\A) =. Then (a) implies (b). By Theorem 1.1 we could have sared wih he lef lf-module M' and formed he above diagram wih A orsionless righ module. Assuming condiion (b) E\A) = and i follows ha M' is is own orsionless facor. This complees he proof of he heorem. The following may be well known, bu we include i for laughs. I seems o be relaed o Theorem 4.1. Theorem 5.2. For he rings R under consideraion he following are equivalen: (a) All W modules are projecive. (b) gl.dim.(r) ál. Proof. If (b) holds hen R has only modules of dimensions zero and one. If A has dimension one hen by [4, p. 123] EliA) # and (a) follows: Now assume (a). Le B be a orsionless module, we shall show ha B is projecive. By Corollary 1.3, we can embed B in a projecive so ha he facor is a W- module. Bu hen he embedding splis and B is herefore projecive. License or copyrigh resricions may apply o redisribuion; see hp://

11 34 J. P. JANS In [5] we showed ha if B*= hen B = E1(A) for a suiable A of dimension one (or zero if B is projecive). Some of our above argumens show ha we can find modules of he form E[(A) in anoher way. Theorem 5.3. If M' is a W-module hen he kernel of he map of M' on is orsionless facor is E1 (A) for some orsionless A. Conversely, for every orsionless A here is a W-module M' such ha El(A) is he kernel of he map of M' on is orsionless facor. Proof. The proof consiss of examining he big diagram used in he proof of Theorem 5.1 and he fac ha he diagram can be consruced saring eiher wih he righ module A of wih he lef W-module M'. I should be noed ha in he above heorem M' and A are modules of he opposie hand. Bibliography 1. M. Auslander, On he dimension of modules and algebras. III. Global dimension, Nagoya Mah. J. 9 (1955), H. Bass, Finiisic dimension and a homological generalizaion of semi-primary rings, Trans. Amer. Mah. Soc. 95 (196), , Injecive dimension in Noeherian rings, Trans. Amer. Mah. Soc. 12 (1962), H. Caran and S. Eilenberg, Homological algebra, Princeon Univ. Press, Princeon, N.J., J. P. Jans, Dualiy in Noeherian rings, Proc. Amer. Mah. Soc. 12 (1961), D. Rees, The grade of an ideal or module, Proc. Cambridge Philos. Soc. 53 (1957), Universiy of Washingon, Seale, Washingon License or copyrigh resricions may apply o redisribuion; see hp://