This article was downloaded by: [University of California Santa Cruz]On: 07 November 2014, At: 23:39Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: MortimerHouse, 37-41 Mortimer Street, London W1T 3JH, UK

Communications in AlgebraPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/lagb20

Cousin complexes and flat ring extensionsHenrike Petzl aa Mathematical Institute , State University at Utrecht , Budapestlaan 6 PO Box 80.010,Utrecht, TA, 3508, The NetherlandsPublished online: 27 Jun 2007.

To cite this article: Henrike Petzl (1997) Cousin complexes and flat ring extensions, Communications in Algebra, 25:1,311-339, DOI: 10.1080/00927879708825856

To link to this article: http://dx.doi.org/10.1080/00927879708825856

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) containedin the publications on our platform. However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose ofthe Content. Any opinions and views expressed in this publication are the opinions and views of the authors,and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be reliedupon and should be independently verified with primary sources of information. Taylor and Francis shall not beliable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilitieswhatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising outof the use of the Content.

This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

COMMUNICATIONS IN ALGEBRA, 25(1), 31 1-339 (1997)

COUSIN COMPLEXES AND FLAT RING

EXTENSIONS

Henrike Petzl

Mathematical Instit,ute, St.ate Universit,~ a t Utrecht, Budapestlaan 6, PO Box 80.010, 3508 TA Utrecht, The Netherlands

0 Introduction

In this paper we shall deal wit,l~ Cousin complexes, which are algebraic analogues of the Cousin complex dealt with by Hart,shorne in [3, Chapt,er IV]; they were introduced by Sharp in [6 , $21 and in t,he Z-graded cont,ext. by Goto and Watan- abe in [2, Chapter 1, $31. The aut,hor wishes to express her profound gratitude t o R.Y. Sharp, for suggestsing t,his invest,igat,ion and many I~elpful comments.

Let R be a commut,at.ive, Noet.11erian ring and let M be an R-module. The Cousin complex of M , del~oted by CR(M)' is a complex of R-modules and R-homomorphisms of t?he form

such that,, for each n E No,

1991 M a i h e m a i i c ~ Subject C: luss i f icai~un: 13C14, 13D25, 13D45, 13E05

Address for corresponrfe~lce Pure Mathematics Sectlon, The Un~vers~ ty of Shefield, Shrffield S3 7RH, England

311

Copyright Q 1997 by Marcel Dekker, Inc.

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a Sa

nta

Cru

z] a

t 23:

39 0

7 N

ovem

ber

2014

PETZL

If hf is a rron-zero finitely gmrratetl R-motlrrlr then C E ( M ) ' is exacf if and only if A4 is Cohen-Macalllay. (See [7, Tlrrorem (2 .4 ) ] ) It has also been shown t,lrat - wif.11 tllr not,at.iorr from al)ovr - (Cokcn-9, E H:R'RV(Ad,). (See 111, Corollary].) 111 t.his palwr, we cor~sider t . 1 1 ~ s i t~at ior l where A antl B are two cornrntltative, Noet,herian rings and p : A -+ B is a flat ring homomorphism. Given an A-module M , we can t,l~ink of M @a B as an A- or as a B-module, and alt,ernat,ively consider the Cousin complexes C A ( M R A B)' or CB(M@a B)'. We investigat,e these complexes and find that t,lie first complex can be considered as a subcomplex of the latter, if all the relevant fibres of sat,isfy Serre's first condit,ion. In that case we obtkin an induced quot,ient complex, which turns out t.o be excact. if and only if those part.icular fibres are Cohen-Macaulay. Because we can interpret t . 1 ~ t,errns in t,he Cousin con~plex a s direct sums of certain local cohomology modules these resu1t.s car] he applied to local col~omology.

I Preliminaries

1.1 NOTATION. Throughout t,liis paper we sliall assume t,hat all rings are corn- mr~t,at.ive, Noet,l~erian and have m~~lt,iplicat,ive itlent,it,ies. We sliall also assume that ring homornorpl~is~ns preserve the n~ult,ipIicat,ive ident,ities. By R we shall always denot,e a ring; let N he a.n R-module and let p E Sup~>,q(N). We denote the N-l~eight of p (= dimR, N p ) by lltN p. For any q E Spec(R) we will denote the resitll~e field of Rq by ~ ( q ) . For any proper ideal I in R we define (see [13, Remarks '3.331)

assR(I) := AssR(R/I).

If A is a local ring with maximal ideal m , we shall denot,e t,his by writing ( A , m).

1.2 Rcn t t i l dc r ubo?lt thr cni~d7.11c.27o~t of 6 ' 0 7 1 ~ 9 1 ~ r o n l p l e z t s . Let L he a module over a ring S. The Cousin complex C(L)' (or Cs(L) ' ) for L has the form

with, f o ~ each it 6 No,

The Romon~orpltisms in t,his co~nplex have t . l x followi~~g propert,ies: for y E L antl a n ~ i l ~ i n ~ a l primr p of Supl)(L), the component of 6-1(y) in L, is ? / / I ; for n > 0, T E Ln-' antl q E S ~ ~ > I , ( L ) wit l~ htL q = 7 1 , the c o m p o ~ ~ e ~ i t . of e n - ' ( r ) in (Cok c ' ' - ~ ) ~ is ? / I , wf~ere 'ovrl.ltr~rs' are used t.o c h o t , e nat,rlral images of

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a Sa

nta

Cru

z] a

t 23:

39 0

7 N

ovem

ber

2014

COUSIN COMPLEXES AND FLAT RING EXTENSIONS 313

e l r m r n t . ~ of L"-I i l l Cok tn-'. Tlie fact that sl lcl~ a complex can l)e col~structed follows fro111 [(j. !2]. \Vr sllall nretl tile followi~lg propert.ies of C(L) '

(11) By [6, (2 7)(v1)], for each 11 E No and each p f Supp(L) wlth h t ~ p = n , every element of (Cok 1s a n n ~ h ~ l a t e t l by some power of p

(iii) It follows from (i) and t,he fact that 1ocalizat.ion commut,es wit,kl direct limit,s that,

Supp(Ln) C ( p E S~lpp(L) . IltL p 2 71) for all n E No

(iv) By [6, (2 .7 ) (~1 i ) ] .

S I I ~ ~ ( H " - ' ( C ( L ) * ) ) C ( p E Supp(L) htL p 2 n + 1 ) for all n E 140

1.3 C o u s i n corripltzcs u n d fi.oclioir forrriulion. Let t,he sit,uat.ion and notMion he as in 1.2. I t was proved in [6 , Tileorem (3.5)] tha t the formation of Cousin complexes 'com~nut,es ' wit,ll fraction for~nat, ion:

Let. T be a ~n~llt,iplicat~ively closed subset of S. There is an isomorphism of complexes of T- 'S-modnles and T - ' S - h o r n o m o r p h i s ~ ~ ~ s

T h e explicit construct,ion of t,his i s o m o r ~ ~ h i s ~ n is given in 16, pp . 346 - 3491.

The follow~ng lemma can easily be proved by tntluction

1.4 LEMMA. L f i R b c o rtiry uird 1ti L urid Ii bc R-7r~odu le s . W E shal l mt ' O ~ l € T ' ~ ~ 7 ~ f ~ ' ~ 0 d f710 f t c0~et .S I l l U ~ ~ ? F ' ~ ~ T ' I O ~ ~ cOktT'll t /s. I f f : L - I< I S U71 R- isortrorplri5rr1 i11rr.c an U I L incluccd tso~~ror .phis irr of c o r ~ r p l t z c s of R-rrrotlirlrs ond R - h o r r t o ~ 1 1 0 ~ 1 1 1 i s i t r s

( I ) f = @ ~ ( f ) - ' : L -+ Ii, urrd

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a Sa

nta

Cru

z] a

t 23:

39 0

7 N

ovem

ber

2014

314 PETZL

1.5 COIIS~IL C O I I I ~ ~ ~ T I S cl11c1 lo~ol (~o1101n01ogy. Let S be a r i ~ ~ g . For any ideal a in S, we denote by F', t,he a-torsiotr funct,or, wl~ich is nat t~rnlly isomorpl~ic to H i , t.he zerot,l~ local col~omology fuuctor wit,l~ respect, t,o a lfsing the hlat.lis- Gabriel st,ructure t,l~eory for ilfiect.ive rnodktles (see [4, Theorenls 18.4 and 18.51) it is not difficult to verify, t.l~at for any S-module M , such that h/l = r , ( M ) , all the t,errns in a n~i~ l in ia l injective resolution of M are a-t,orsion as well. Therefore we have in that case H b ( M ) = O for all integers i 3 1. M'e shall be using this occasionally without fltri.her comment,.

In [11] it was show^^ t.llat t , l~r ter~ns in t,he Cousin complex can he int,erpret,ed as direct sums of rert,ain local cohomology modules. Let the sit,uation be as in 1.2 and assume t.l~at p E Sl~pp(L) wit,h l ~ t , ~ p = n . The11 i t follows frorn [ l l , Corollary] that (Cok e n - > ) , Z HFs,,(Lp). If S is local wit,l~ maximal ideal m, such that l > t , ~ rn = 11, t,llen 11 I , the or en^] stdates t,liat. for all int,egers j 3 O and for all int,egers O 6 i < 11, we l~nve H & ( L 1 ) = O.

1.6 The sirrdl ~vppoii of cr irrodrrlc Let M be a module over a rlng R For p E Spec(R), denote RP/pRP 11y i ( p ) , aud, for z E No, let pl(p, M) (or pk(p , M)) denote the 2-th B a r ~ l u n ~ l l t r of L ~ t ~ t l ~ tespect to p (see [4, T l ~ e o r e ~ n I8 71)

( M ) = {p E I ( R ) 3 ? E No such that E ~ t k , ~ ( i (p) , hi,) # 0)

= ( p E Sjwc(R) t h e ~ r ex~sts i E NO such that /r"p, M ) > 0)

The following propertlrs of t l ~ t , small support are not d~fficult to prove

Let O -+ L' M' - N' - O be an exact sequence of R-modules and R-Iiomomorpl~ist~ls Tllt111

(ii) Ass N' C Ass h i ' U supp(L1);

(iii) any one of supp(L1), sllpp(M') and supp(A") is col~t,ained in t,he union of t.he ot,her t,wo: and

(iv) if .S is a n i ~ ~ l t i ~ ~ l ~ c n t i \ ~ c ~ l y closed subset of R aud .V is at1 S-'R-module, then, w11e11 A' is rrgardrtl ;ts all R-n~odule in t , l~ r 11at11ral way,

Not,(% also that. for a11 t l ~ t l ( > s srt I and a family of R-niotlt~les ( M , ) , E I we have Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a Sa

nta

Cru

z] a

t 23:

39 0

7 N

ovem

ber

2014

COUSIN COMPLEXES AND FLAT RING EXTENSIONS 315

s t ~ p j ) ~ ( $ ; ~ , M , ) = UiEl supp(h-f,), since for any p E Spec(R) the funct.01 ExtkP(r;(p), . ) commut,es wit,ll direct. sums.

2 Cousin complexes and flat ring extensions

We begin this section by recalling a few fart,s about flat ring extensions.

2.1 PRELIMINAR~ES A N D NOTATION. Let A and B be rings and let cp : A -+ B be a flat ring l ~ o m o m o r p h i s ~ ~ ~ . Let p E Spec(A). We shall use the contraction and extension not,at,ion for ideals in A and B , for any ideal I in A, I' denotes y ( I ) B , and for any ideal J in B, t,he cont,raction p - ' ( J ) of J will be denoted by Je. The ring r ; ( p ) @ ~ B, will be referred t,o as the fibre (of y ) over p. Recall from [4, p.471 that t,he spectru~n of t , l~e fibre over p is in bijective correspondence with the set of all prime ideals in B which contract to p . Unless otherwise stat,ed, M will always tienot,e an A-module, such that M B := M @A B is non-zero

Proof . In the case where A4 is finitely generakd t,his is proved in [ I , Theorem A.111. However if M is not necessarily finitely generat,ed, M is the direct limit of it,s finit,ely generat,ed submodules, and if we denot,e the set of t,hese by C, then the set { L g A B : L E C ) is cofinal in t.he set of fiuit,ely generated submodules of M @A B. I t is not diffic.ult t,o deduce from that t,l~at. t,l~is more general stat,ement holds.

2.4 LEMMA. Lei tlrr S ~ ~ I ' U ~ I O ~ I hc ( I S 1 1 1 2.1 oird l f t p E Spec(A) be strch Ilrut pe # B . T l m r S := p(A \ p) i s u ~rr t r l l i p l~co t i r~r l y closcd slrb.sri of B und the irrdirctd rrrrg I r o r r ~ o i r ~ o r . ~ r l r ~ ~ ~ , r

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a Sa

nta

Cru

z] a

t 23:

39 0

7 N

ovem

ber

2014

316 PETZL

PT.oo~ . SIIICCJ p is flat i t 1s (stv [4, 1) 461) not difficl~lt to deduce that. pl' is fait.l~fully fiat. D P I I O ~ ~ . I,y 3 t lit. 11atural in~agr of S i l l Bjp? By [I), ( 2 2)(i)] t.llrre is a ring iso~~lorpl~ism

2.5 LEMMA Lrf f l t f r ~ f ~ r o l r o ~ r hc or 111 2 1 L t f p E Spec(A) hr snch t l i o t

Pe # B

( i ) Tlrc jibn of !p o f i r 7 . p I r l ~ o l l - ~ l ~ l i ~ l r n ~ ulrd p E SuppA(B). I f p 1.' d s o 111

S u ~ ) j > ~ ( h l ) , flrrn p I.\ 111 SuppA(AfB) us ~ricll .

Prnoj! ( I ) It f o l l o ~ s f r o r ~ ~ (9, (2.211 tliat h - ( p ) @ ~ B # 0. Consider t,he short exact seqllertce

O + pAp + Ap -+ ~ ( p ) + O

and t.ensor with B . Clearly t.he rrsnltirlg short exact secluellce implies that,

A p B Bp is 11011-zero m d so p E S U ~ ~ ) ~ ( B ) . Assume now t h a t p E S ~ ~ l > l > ~ ( B ) n S l t l > p ~ ( M ) . Define S := p ( A \ p ) . Recall from 2 . 4 , tliat, the induced ring I lomo~~~orphism : Ap - S - ' B is fait.hfrllly flat.. It, is ~f~raiglit~forwartl to see, that. t.lirre is an isomorphisn~ of ,Ap-moclules S - ' B r B p , autl since A t p # U, it, follows froln t . l~at that.

(ii) Since any ~r~illirnal prime of p' is in AssB(B/pB) it follows from 14, Tl~eorem 23 2(i)J t.l~at all the rnil~itrial prinles of pe are in the fibre over p.

Now assume that p E SuppA(hf) ant1 that q 1s a t n ~ r ~ ~ i n a l prllne of p' It follows from [9, Lemma 2 G , Lemma 2 51, that q E SrlppB(Ms} and that the tnduced

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a Sa

nta

Cru

z] a

t 23:

39 0

7 N

ovem

ber

2014

COUSIN COMPLEXES AND FLAT RING EXTENSIONS 3 17

ring homomorphism p' : A , - Bq is fait,l~ftllly flat and local. Therefore we know from 2 . 2 that,

dimB, M, @A,, Bq = dimap M, + dim B,/pB,.

Since t,here is a B,-isomorphism Mp @ a p Bq 2 MB @B Bq, it follows from the minirnalit,y of q that (ME)-ht q = M-ht p. The rest is now clear, as ace E a.

2.6 REMARK. Let t,he situat,ion be as in 2.1. Since we can consider MB as an A-motlde, t,he qitest,ion arises t,o what ext,ent we can compare the Cousin complexes Ca(MB)' and GB(MB)'. The next proposit,ion is a first step in the at,t,empt to answer that quest,ion. Recall that for any int,eger i 2 0 and a prime idea1 p E S ~ r p l > ~ ( M ~ ) of (ME)-height i the summand in t,he i-th term of CA(MB)., which corresponds to p is isomorphic t,o H ~ ( M E ) , 2 (HiC(MB)),. (See [a, Theorem 4.3, Theorem 5.11 and [ l l , Corotlary].) Clearly, this term vanishes if pe = B. If we denot,e CA(MB)* by

then it follows from 2 . 5 and 1.2 t , l~at for all int,egers i >, O

W e sliull cull a prtirie id fu1 p E SuppA(M), u~lrtch hos M - h t i g h t 2 relevant (for p and M), ~f p' # B clird HLA,,(MP) # O TIr f sc itlruuirl priiraes uirll be of perttc~tlar tn fercs t i n f A r r t - i ~ ~ ~ i i r d t r (f f h w pupcr

2.7 LEMMA Let t h f l l O ~ U ~ l 0 1 1 bt US t71 2.6. Tllc C o l l ~ l l l ~ O l l l p ~ ~ ~ CA(MB)* con be coitsrdercd us o cnrrrpltr of B-iriodales and B-liornoi~roryhzsms, such tltut MB, conszdercd us 'ntrrt7rs firslJ-tcrrrr in the Cous in c o r t ~ p l f x Ca(MB)' keeps rts oraytnul B - m o d d c structurt .

T h e new B-mod~rlr . s tr~rcf t lre oir t h f t e r i r ~ s iit Ca(MB)* trill1 also su f i s fy the fol lou~iny property. If i -1, (I E A u r ~ d r E Cok 8-', uir. m u y n t ~ d t i p l y z by a , litinkitty o f Cokdi-2 us ( I I I A-rrroddc, and on the o t l t f r hand we m a y consider the B-nt.odult: s t r v r 1 u r ~ on C ~ k d i - ~ , and rrrrtlttply T by p(u). I n tlrese circurnstoirccs IN^ huv t (LT = 17((1)1..

Proof. This is st,raight,forwartl

2.8 LEMMA Wc latc the rcri~cc irolution us ti^ 2 7 wttk lhc f o i h i z n g ntodt- fircattons L t t (A, m) bc n loco1 rairg clnd l t l p A - B be u fatllifully Pat ring llornoinorpf~tsiri Lr 1 M hr (111 A-rrrorlulc o l d let I, = dunA(M) Ustny 2 7 uir irray lhriik of na u B-ir~oclulc uiid ua such A ~ k ts B-asornorphlc to H:, (MB)

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a Sa

nta

Cru

z] a

t 23:

39 0

7 N

ovem

ber

2014

318 PETZL

The next lemma is a strargl~tforwartl motlificnttoi~ of T l ~ r o r e ~ n 1.7. in [12] an(! we shall omit tlte proof

Let t,he sit,uat,ion be as in 2.5. Tl~rougl~out t.he rest of tlie paper we sliall I)e using t,he followi~~g notat.ion. We writ,e t,he Cousir~ com1)lex of M e , considered as an A-module as described ill 2 . 6 , anti i f we think of as a B-n~otlule as

Proof. (When i t is clear from t,lie context,, wliicli r i ~ ~ g s A and B and which module M we are referring t,o, we will ab1)reviat.e @ : ( M ) by @.) Let 4- ' :=

idM,. Let 11s assulne that p E SuppA(M) has h!-llr>igl~t. O and t,liat pe # B Let ql, . . . , q,, be the ~niniinal prinle ideals of p e . Co~lsider the following well- defined B-homomorpliis~n t r p : (Me) , + @y=l(MB)q, , w l ~ i c l ~ maps an element, rn - t,o xy=l 3. (Here nl E M e a d s E A \ p.) Let 3 denote t,he set. of minimat primes in SuppA(M) which e x t e ~ ~ t l t,o proper ideals in B , and define 4" := CrES err. Since the cont,ract.ion of any prime ideal in S ~ l p p ~ ( M ~ ) having (ME)-height 0 also has M-height 0, it is clear t , l~at the diagram

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a Sa

nta

Cru

z] a

t 23:

39 0

7 N

ovem

ber

2014

COUSIN COMPLEXES AND FLAT RING EXTENSIONS 319

1 E N u . Assume t.l~at for all integers O < 1. < 1 a B-l~on~o~norphism d k : MA - (MB)I. has Lwen tlrfinetl suclr that the diagram

- commut,es. Let. 4' denot,e t.he induced map on t,he corresponding cokernels.

We shall now define @'f1. Let p E Supp(M) have M-height I+ 1, assume pr # B and (see 2.5(ii)) denot,e by ql, . . . , q,,, the minitnal prime ideals of pe (m E R). Let 1. E Cokril-' and 1 E A \ p and consider t,he - homomorphism c, :

(Cokdl-I), - & i ~ , ( ~ o k c ~ - ~ ) ~ , , wllicll maps $ t,o ELl $#. Define another

homoinorphisni qp : ( ~ o k d - ' ) ~ - M;+', which is such t,hat the component of q p ( f ) in (Coke1-'), is equal to t,he corresponding component of c p ( f ) if a is a minimal prime of pe , and 0 ot,lierwise. Let 3 denot,e t,he set of primes of M-height I + 1 in SuppA(M), which extend t,o proper ideals of B. Now define 4'+' := CpES ,jP. It suffices t,o verify t,hat t.he diagram

commutes In order to do t>hat tt w ~ l l be enough (see 2 ~ ( I I ) ) for us to show that for each a E S I I I > P ~ ( M ~ ) whch has hbB-height / + I , but for which htM aC < / + I , - the corresponding component in (Cok el-'), vanishes. (Here T is again an arbitrary element in Cokdl-I.) Let a ant1 3. be as mentioned and assume that -

# 0. This implies that a n n B ( T ( ~ ) ) a , and so annA(z) 2 annA(F(z)) E aC. Consequently aC E SuppA(Cok dl-'). This is a contxadiction, since all primes in the A-Support of Cok dl-' must have M-height a t least I + 1 by 1.2(i). This concludes the inductive step.

Since the chain map @ introduced in 2.10 is not. aIways injective, as easy exam- ples demonst,rate, our next t h k will be t,o ii~vest.igate, under which conditions it. is. We shall use t,he fact t,liat, t.11e srirnma~ttls in Cousin complexes can be described in t,errns of local cohomology mot l~~l r s . as explained in [ I l l

2.11 REMARK Let ( A , m) be a local rklig a ~ ~ d k t M he an A-module stlcll that litns m = 11 Let B hr a ling and p A - B a fa~thfully flat ring honio~norp l~~sm

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a Sa

nta

Cru

z] a

t 23:

39 0

7 N

ovem

ber

2014

320 PETZL

alttl C ~ S S I I ~ I I P tllat p i , , pl (1 E W ) arr t l ~ c t111nl111al prln~es of me Note that for each J E { 1 , , X } we hnvr r,. B,, = rp, B,,, Let @ be (lie c l ~ a ~ t l I I I ~ I ) descr~bed 111 2 10

Wesltall tlr cons~t l r r~~tg the ('ous~n complexes C'B~, ((Mu),,) ' for J E (1, , k } Recall t l ~ a t for any ~ n t e g e ~ 1 2 O thr 1-$11 terrii 111 tlint complex IS clr~rotetl by

( ( A f ~ ) ~ , 1' A

[I kf; - @(bf;),, ,=1

be tlrr ~ a n o n ~ r a l B - ~ I O ~ ~ I O I ~ ~ ~ ~ ~ ~ ~ I S I I ~ arttl consltlrr ( n s ~ n g 1 3) the ~ s o ~ n o r p l ~ i r m

Define T := W' o and 6 := T o 4"

L r t j E { I , . . . , k ) . For any B-mod111e A' we define

vjt : N - Np,

to he the canonirnl B-horno~norpl~is~n. It, is a rontinr check 1.0 verify t,Ilat,

with = H k , (Y;. )) ts a I ~ o r l ~ o ~ n o ~ p h ~ s m Letween ~ i e g a t ~ v e s t~ongly connected

seqwuces of fr~nctors lJs~ng [8, T l ~ e o ~ r m 4 31 we may also drdtlce the rxlstence of an ~somorpl t~sm

betwee11 negat~vesfrongly connected sequences of fnnctors, such that for any B- module N 3:',!(~) 1s the rdentlty on H:,(Np,) Define now for aH J E { I , , k)

and denote by 7 the following B-Iio~no~norpl~ism

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a Sa

nta

Cru

z] a

t 23:

39 0

7 N

ovem

ber

2014

COUSIN COMPLEXES AND FLAT RING EXTENSIONS 321

Proof. IJsilig [(i, L e ~ n ~ n a 3.21 i t is ~ o t . tl~Hicult, t,o verify t,llat. r, is surjec- t,ive. We wish t.o s l~ow t.l~at TI is ~~Vect,ive. Let i E { l , . . . , k } and recall t h t Ass((Coken-'),,) 5 {pi). Den0t.e Oy v, : - Mi t,lle canonical

i~lject,ion and denote by T , : @:=l(~.lg)p, + (Alg)p, t,he canonical project,ion.

Let 3. E (Cok c " - ~ ) ~ , lw a non-zero elr~i ie~i t . Cirarly, ~ ~ ( ~ ( v ~ ( z ) ) ) = 9 # 0 and for any j E {l , . . . , l :} wit,ll j # i we have *,(p(vi(r))) = 0, as JannB(vi (~) ) = Jallng(l.) = p , . From this t l ~ e claim follows easily.

2.13 T H E O R E M . t c l thc sili~c~tioit uird ttotntioic be u5 t i 1 2.11

F~tr t l~ermorc , rf HE,(MB) # 0 , y as ziyrrtivc zf and only zf {pl, ,pk) = asss(me) I n other 1nord5, y 25 ~rndcr thtsc rirrirmstu~~ces rnjectrvc zf and only if l l ~ c fibre rrny r;(rn)@A B sufiafics Scrie's fir$t tondzlzon S1 (Sec [4 , p 1831 )

(ii) Ass~iirlc thut Hg,(MB) # O Tltc irtrrp &n rs lhcn znjeclrve zf and only tf y is. ( I f ffG.(MBf = 0, Ihcit 4" as the zero hotrromorplttsm by [ l l , Corollary] und [8, Theorem 4 31.)

Proof. (1) Let J E 11, , k } lJsmg [(j, Theorem 3 51 we o b t a ~ n a cltain map

E = ( I CA(MB). 2 C B ( M ~ ) * + CB,, ((MB)P,)*, denote the 'differ-

entrat~on' maps in Cs, , hq ( f f ) ,>- l (For rlar~ty's sake we make for all s E (0, , n - 1) tllr following a l )brev~a t~ol~s

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a Sa

nta

Cru

z] a

t 23:

39 0

7 N

ovem

ber

2014

322 PETZL

I<;- 1 >-* * '1,fi - * Cok tl" -'

which we sl~all tlr11ot.e I>y ( I ) . Sinlilarly we have for all s E { O , . . . , n - 1) tlie following cornmut,at,ivr diagram wit.11 exact rows

wh~ch we w ~ l l refer to as (2) Let I E NU and s E (0, , n - 1) The preced~ng two d~agrams ~nduce d~agranls, wlth long exact sequences of local cohomology modules as then rows, from wltlch we obtam (after app l~ca t~on of the ~ndepen- dence theorem [8, Theorem 4 31) two commutat~ve dlagrarns

and

which will be referred to as (3) airtl ( 4 )

Note that (due to [ I 1 T h e o ~ r n ~ ] ) the f i ~ s t t e ~ i n \ 111 ttir lows of d ~ a g r a m (3) van~sfi, ~f s < 1.1 IVe tlwrrfore o1)ta111 f o ~ all s E { O , , I , - 1) and all i E No a c o m m ~ ~ t atlve dtagra~n

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a Sa

nta

Cru

z] a

t 23:

39 0

7 N

ovem

ber

2014

COUSIN COMPLEXES AND FLAT RING EXTENSIONS

Let s E (0 , . . . , i t - 11. Since tlie dimension of t,he BPI-module H,"-' is less than or equal t,o n - s - 1 (see 1.2(iv)), it. follows from [8, Theorem 6.11 t,hat,

( ~ f - 1 ) = 0 = j y n - " + l ( ~ j - l ) . Hp";B:?, I PJB,]

The HA-'-Iieiglit of m is by 1.2(iv) less t,han or equal t,o 11 - s - 1, and it follows therefore from [S, Theorem 6.11 anti t,lie intiepentlence theorem [8, Theorem 4.31 t,liat,

H;;"H;-~) H;;J++'(H;-~) = O .

Using t,hese vanishing propert,ies we obt,ain t,herrfore from diagram (4) t,he fol- lowing tliagratn

which commut.es. Comhiniilg t,hese last diagrams, while letting s take any value in (0 , . . . , 11 - 1) yields the conimutative diagram

Not,e that A A!; = H g . ( AM;) and that ( ( M B ) ~ , ) ~ = HpUJB,j.(((M~)p,)n).

Note that t,he vert.iral homomorpl~ism on the far left is in fact y', and denot,e the one on t.he far right by 6'. Considering t,he last diagram for all j E (1, . . . , k ) , it is not difficult t,o check t.llat 6 = zfZ1 6' : A(MB)" + $ F = l ( ( ~ ~ ) p , ) n . (See 2.11.) We t.lierefore have oht,ained a commut.at,ive diagram as described in the st,at,ement of 2.13(i)

Assume n o w , that H a , ( A f B ) # 0. It. remains to show, t,hat y is inject,ive, if and only if { p i . . . . . p i , } = assA(mt). Not,? t,l~at,

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a Sa

nta

Cru

z] a

t 23:

39 0

7 N

ovem

ber

2014

324 PETZL

Not.e, tallat. i t follows from (1) and 2.1 1 , that

As drl ( C $ ( c o ~ c n - 2 ) F , 1t follows f ~ o m the above in particular that we i = l

have a commutative diagram

It, uow suffices t,o show that X is i~ijectivr i f aucl olily il' { p i , . . ,p i ; } = assB(me). (We shall refer t,o t.liis eclr~ivnle~~ct~ as (**I . )

Note that Assa(HZ(Af)) = {m) a~ltl tliat there is 1)y [14 , Prupos~t.io~l 4.151 an isomor~)hism of B-motlu1t.s D

ownl

oade

d by

[U

nive

rsity

of

Cal

ifor

nia

Sant

a C

ruz]

at 2

3:39

07

Nov

embe

r 20

14

COUSIN COMPLEXES AND FLAT RING EXTENSIONS 325

It follows from t.his a ~ d [4, T l ~ r o r e ~ n 23.31 t.liat assB(me) = AssB(B/rne) = AssB(Hg(Ad)@ B) = A s s ~ ( H ~ , ( M ~ ) ) . Clearly, i f O # r is an elenie~it in Ker A , this implies that anns(r) p j for all j E { I , . . . , k). This implies (by [13, 9.441) that t,here is a t least one a.ssociat,etl prime of Hk,(MB) wl~ich does not belong to {pl, . . . , p k } . The remain i~~g clirect.ion of t , l~e equivalence st.at,ed in (**) can be dealt witsli in a similar way. This co~iclutles t,he proof of the first part of 2.13.

(ii) The st,at,ement.s in (ii) follow clirectly from (i) all(! t,he diagram (*), keep- ing in mind that, t,lie vert,ical hornomorphis~~ts in t,he lower Ilalf of t,he diagram are isomorphisms.

2.14 LEMMA Let the s ~ t u a t ~ o n antl notation be as In 2 1 and 2.10 Let p E SuppA(M) be such that p' # B antl let S = cp(A \ p) Denote by

the faithfully flat rntg hornomorpl~~st~t tlescr~betl In 2 4 Let m E M, 6 E B, n E A and s E A \ p It 1s not d~fficult to ver~fy that

mapping an element rn @ 6 @ to F @ is an Ap-isomorphism. Similarly, ' 4 s )

it is a rout,ine check t.o verify that.

mapping an element nr @ b @ to F @ (for b' E B) IS an S-'B- 4.) 44.) ~somorphtsm These last two ~somorphisn~s mtluce by 1 4 ~ s o m o r p h ~ s m s

Recall 1.3. Denote by R the ~somorph~srn between complexes of Ap-modules

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a Sa

nta

Cru

z] a

t 23:

39 0

7 N

ovem

ber

2014

PETZL

Proof Let 11 2 0 be an llltegrr Let p E SuppA(hf) of M-height 11 not he relevant, I e (by [8, Theor r~n 5 1 1 ) ( I $ F ( M ) ) p = O Since, by [ 1 1 , Corollary]

and by [8, Theore111 4 3, T h e o ~ r ~ n 5 1) and [ I 4 P ~ o p o s ~ t ~ o n 4 151 there are A- isomorphisms

Clearly, 4" is t,herefore inject,ive if and only if for all relevant p E Supp,(M) wit.h htaM p = 1 1 t,lie rest,rict,io~l

4: : (chk ( P - ~ ), - nf;;

is it~.ject,ive.

Let p E SuppA(hf) be a rekva~lt prime (for hl a ~ ~ t l 9) of M-heigl~t n. Denote by ql, . . . , qk the m i ~ ~ i m a l primes of p'. Lrt S := !p(.4 \ p) ant1 denot.e by y" the induced fait.lifully flat ring I ~ o ~ n o n ~ o r p l ~ i s ~ ~ i

which was discussed in 2.4. Let M denot,e the tlirrct sum of all t,he summands of M g , which correspo~~d t,o prime ideals ill S ~ t p p ( M ~ ) , W I I ~ C I I have a coutraction st,rict,ly cont,ainetl in p. l ising [(j, L e ~ ~ l r u a 3.2, (3 .6) j autl 2.14 i t is not difficdt to check t,hat t.he tliagrxm (wit.h otwious l ~ o n ~ o ~ n o r l ~ l ~ i s ~ ~ i s )

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a Sa

nta

Cru

z] a

t 23:

39 0

7 N

ovem

ber

2014

COUSIN COMPLEXES AND FLAT RING EXTENSIONS 327

Note that by 2 4 tlre maxullal ltleal pAp is relrval~t for Mp a~it l p", and that (by 114, Propovt~on 4 151) HFA,, (Alp) # O I I ~ I ~ ~ I P S that

since pi' is faithfi~lly flat,.

I t follows now from 2.13 and 2.4 t,hat ( c $ ~ ~ ' ~ ) ~ is injective if and only if ~ ( p ) @ a B sat,isfies S1 Theorem 2.15 follows from this.

3 Conditions for the exactness of the quotient complex

3.1 REMARK. Let t.lie sit.uation be as i l l 2.10. If is inject,ive we can consider the induced quot,ie~t, complex CB(hfBf'/@(Ca(MB)'), which we shaH tlenot,e by Q* (or Q:(M)' t,o rule out any ambiguity). For complet,eness' sake define Q-I := Q-\:= 0, and denot,e t,Ile maps of Q' by ( f i ) j 2 - 2 . Let a E Supp,(M) wit.h a' # B have htM a = k and let p l , . . . ,pi he the minimal primes of ae. Define

I

Q(a) = ( @ ( ~ o k f ' - ' ) , , ) /4k( (~ok d k - ' ) , ) t=l

The final sect1011 of 1111s paper IS dethcated to t h ~ s cpo t~ent complex and we shall see that 11 has qiilte a few ~ n t e ~ e s t ~ n g propert~es Our main ~esu l t In this sectloll states that the quot~ent complex Q' 1s exact ~f and only ~f all the fibres of p, whlcli co~respond t o p m e s relevaut for M and p are Collen-hlacaulay

We begin wit.11 a t,echnical lemma which will be useful t,hror~gliout t,he remainder of t,ltis paper.

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a Sa

nta

Cru

z] a

t 23:

39 0

7 N

ovem

ber

2014

PETZL

I t fo l lows t 7 1 purlzr~i lnr ilroi for n l i i ir t tycrs 7 2 O

Proof It follows from 2 11 a11t1 [( I , ' llitwre~n 3 51 that there are ~ s o m o r p l ~ ~ s r ~ s of rornplexrs of B v - ~ ~ ~ o t l u I e s

the C O I ~ ~ O S I ~ I O I I of ul11e11 w r clrnotr. by r i By r i WP d r ~ ~ o t e the catiou~cal ~sonrorphrsnl (see 1 3)

Also thr followrng tecll~l~cal I r~nma I < ~rrrclrtl for t l ~ e proof of t f ~ e maln theorrm

3.3 LEMMA Lf t ( A , m) rrird ( B 11) hf tocot aird uar71711f thu i p A - B Z P u f ~ i f h f i ~ l l ~ P(11 riirg /toirroi11071)A~~ir1 L t l A4 bf (111 .4-1110d1ilt und u5sit7ue tho1 m h a s h f -Ac ty l t l b , r~ird Iltol l~t ,+ , , n >, J + 2 , u l l~ t r r 1 3 1. zs ail i n t c y c r Assiurtc t h e l @ 2 ( ~ 4 ) (irc 2 10) t r 1i t~c t t i7 , r , uitd t h a t tlir fibre ~ ( m ) @ A B 1s

Cohe i r -Mnru i rhy i f m i i r t / r t o i 1 1 1 0 7 M (I1 follows 111 thnl cur f froria 2 2 t h u t Orptll ~ ( m ) @ A B 3 J + 2 - 1 /

P w o [ Lef, i E { 0 , . . . , j) a ~ ~ t l co~ls~tlrr the short exact. sequeuce

Corl\~der now the c a w wlir~it~ j - I # k I t follows fro111 I&, Ttieo~rm 3 21 that

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a Sa

nta

Cru

z] a

t 23:

39 0

7 N

ovem

ber

2014

COUSIN COMPLEXES AND FLAT RING EXTENSIONS

. . If j - i > k we may use 2.6 t,o see that = 0. Thus we may assume that j - i < k . Let p E SuppA(M) have M-l~eight j - i ; clearly p must be strictly contained in m = nC, and so there exist,s an element (1 E m \ p. Since multipli- cation by a is an automorphism on the A-module (Cok &-'-'),., mult,iplication by p (u ) constitut.es a B-isomorphism on t,he B-module ( C o k c l ~ - ~ - ~ ) , . (Recall from 2.7 that Cok rlj-i-2, ant! t , l~us (Cok rlj-i-2)p have cano~iical B-module st,ruct,ures.) Consequent.ly, mult.iplicat.ion by p(u) E n is a B-isomorphism on ~ ; + l ( ( ~ ~ k (lj-i-2 ),). But as every element in t2his local col~onlology module is

annihilated by a power of n , it. follows t,hat H;+'((Cok dl-'-' )p) = 0.

To fir~ish the proof assume no? tliaf, j - i = k . In that caw t.here is by 2.8 a B-module isomorphisln A ML-' E H ~ . ( M ~ ) antl it suffices t,o show that Hi+'(Hi .(MB)) = O. This is clear, if m is not relevant,. Assume therefore t,hat m is relevant,. Recall that Hk,(MB) E H ~ ( M ) @ A B antl t,hat every A- (or B-) module is the direct limit of its finit,ely generated A- (or B-) submodules. By C we shall denote t.he set of finit,ely generat,ed A-submodules of H,(,(M) and by 6 t,he set of finitely gelmated B-submotlules of H k ( M ) @ A B . It is not difficult to verify t#l~at t,he set {N @ A B : N E C) is a cofinal subset of @. Now let T # 0 be in 2. IJsing the Ilypot,l~esis ant1 the adtlit,ional assumption that j - i = 1: if, follows that,

T h ~ s iniphes (see [I, Theorem 3 5 71) 111 part~cular that H;+'(T @ A B ) = 0 Therefore

Proof. Let a E SuppA(M) have litM a = k , assume t.l~at at # B and let p l , . . . , p i he tslte minimal pri~ltes of' a t . Let S := p(A \ a) altd ronsitler the induced faithfdly flat ring hornomorpl~is~n cp" : A , - S- 'B . As there are canonical ring l~omomorpl~isms S-' B - Bp, for all j E 11, . . . , I ) and because of 2.7 it is not difficult t.o show t.l~at Q(a) (see 3.1) has st,ructurr as S-'B-module. It follows in particular from l.e(vi) t.11at ally q E suppB(Q(a)) ha cont.raction D

ownl

oade

d by

[U

nive

rsity

of

Cal

ifor

nia

Sant

a C

ruz]

at 2

3:39

07

Nov

embe

r 20

14

330 PETZL

cont.ainetl in a . For any b E SuppB(MB) of I l t n r , b = 1. wit11 l ~ t , ~ * bC < k , 11 IS

clear t,hat. has T-'B-st.rl~cti~re, wl~ere T := p(A \ bC) . Thr lemma follows from t,l~ese co~lsitlerat,io~~s, [14, Lr~nnla 3.3.71 atid t.he fact that.

Praof. Asswnr t , l~at Q;(hrf)' is not exact. Let. i E No, such t.llat~

Note also that for any p E S ~ p p ~ ( h ! ~ ) wit11 q = p' we may conclude t.l~at, p' : Aq -+ Rp is faithfully flat antl t.lmt all t,I~e fibres correspontling t>o primes in A q whir11 are relrvant. for Mq ant1 y' arr Col~en-hlacaulay. (See [10. Lemlna 4.21.)

It. is t,lterefore st~ffirient,, i f we rest,rict ourselves ill t.lie proof of 3 . 5 t.o t,he case of a fait,lifulIy flat I~omomorpltis~n of local ri~lgs. This means, t,llat we asstune A , B and p : A - B to be local, a ~ t d show by i ~ ~ t l ~ ~ r t i o n o ~ t t . l~e tlimrllsion of M t,liat t,he cluot.irnt. complex in t.liose special cases is exact..

I) Bcyinnitrg of ttrc iird~~~clioii. Let. us assumr t, l~at A is local (with maximal ideal m) such t, l~at h4-l~eigl~t m = 0. Let. also B be local wit.11 maximal ideal n antl assume p is local antl fait.11fully flat,, a11t1 t,liat all t,lte fibres corresponding t80 primes relevant, for h4 in SuppA(M) are Colien-Macaulay.

We shall abbreviate Q;(hl)' by Q' antl a):(A4) by a).

Suppose that AssB(Hn(Q')) # (4 a~ltl look for a co~~tratlict.iorl. Since

any p E AssB(H0(Q')) will have Mo-lreigl~t, at. least. 2 (see 1.2(iv)). By our 1ocalizat.ion 1r1n11ln 3.2 we may thxefore asslilnc: t l ~ a t Ms-llt n 3 2 altd t l ~ a t n E AssB(H1'jQ')). Clrarly t l~ i s inlplirs t,l~nt. H:(Qu) # 0, wliich is a contradict,ior~ 1.0 3.3.

We insert uow - still assuming that. dim IZI = 0 - a s e c o ~ ~ d induction to show, t,hat all t , l~r homology modules of Q' wit11 i ~ ~ t l r x greakr t l m ~ zero must vanlsl~

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a Sa

nta

Cru

z] a

t 23:

39 0

7 N

ovem

ber

2014

COUSIN COMPLEXES AND FLAT RING EXTENSIONS 33 1

as well. We have already perfornletl t l ~ r beg~nning of t111s ~nduction, by showing that HO(Q*) = 0

Assume therefore that k 2 O is all int,egrr and we have shown Hi(Q') = 0 for all 0 < i < k . We shall prove under t,l~at asst~rnpt.ion that also H ~ + ' ( Q * ) = 0. Again using 1.2(iv), we may assume wit.hout. loss of general it,^ t,hat the dimension of MB is great,er t h n or equal t,o 1: + 3 and that n is an associated prime of HI'+'(Q'). Consecl~~ent,ly, n E Ass(Cok f') and H:(Cok f k ) # 0. Because of the short exact seqllence

and the fact tallat, H~(Q' ;+ ' ) = O (si~tce QE+' = ( M ~ ) ~ + I and because of [ l l , Theorem] it follows that, ~ k ( C o k f k - ' ) # 0. Successively dealing with short exact sequences of t,he t,ype S, for 0 6 i < L - 1 and applying similar arguments we may eventually conclude t, l~at HL+'(Cok f - I ) = H;+'(Q') # 0. This is again a ~ontradict~ion to 3 . 3 (wit,l~ k = 0 , i = j = P + 1 in the not,at,ion used there).

This proves, that in the local cmr, wl~ere M is of dimension O t,he quot,ient complex Q* is always exact. C;onseclr~ent,ly, after considering Lemma 3.2 no prime ideal in S 1 1 p p ~ ( h 4 ~ ) wliich c~nt,rart,s t,o a prime ideal of M-height 0 can support Hi (Q*) (for any int.eger i ) in t.he general case.

II) Irtduclioe Aypolkcsis. Let k E No. Let us assume we have shown for all (local) cases, where dim M < k t,l~at. Q' is exact. So - in t,he general case - only t,hose prime ideals in S ~ l p p ~ ( M ~ ) which have c~nt~ract~ions wit,h M-height a t least k + 1 coultl possibly s u ~ ~ p o r t coliornology motl~des of Q' by 3.2.

III) Inthrl i~ic s fxp . Ass~lme again t.llat (A, rn) and (5, n) are local and that

p : A B is faithfully flat.. Assume also t h t M has tlimension k. + 1. We have Mi = Q j for j > k + 1 ant1 so we deal wit,h the first 1: + 1 cohomology modules of Q' in a fi~lit~e induct~iot~ and t,hen discuss t , l~e vanishing of Hi (Q*) for i 2 k + 1 in all induct,ion, whiclr is s~rnilar t,o the one in part I) of t,his proof.

By finit,e indnct,ion on i we show that. H'(Q0) = 0 for all int,egers 0 < i 6 k

Let. i = 0. Since t,he small support suppB(QO) - if uot empt,y - consists of primes of Suppn(MB), whose cont.ract,ions have M-height 0 in Spec(A) (see 3.4) and because H0(Q') 5 Q", it is clear from II) that Ho(Q') = 0. Since by 3.4 supp(Cok f o ) C supp(Q1) ~ s u p p ( Q " ) , we may conclude t.hat any prime ideal in the small support of Cok f" lies in SuppB(Me) and contracts to a prime, whose M-height does not exceed 1.

In our induct,ive assumptioil we z~wlrnr now t , l~at ! E (0,. . . , k - 1) (if = 0 t,l~ere is nothing to show at t,llis point) and t , l~at we have shown H ~ ( Q ' ) = 0 for all j E ( 0 , . . . , 1 ) . Furthermore we assume t,llat it has been shown that, for all such j, all the primes in the small s t~pport of Cok fj- ' lie in SuppB(MB) and cont,ract t.o prime ideals of h4-lleigllt not exceeding j.

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a Sa

nta

Cru

z] a

t 23:

39 0

7 N

ovem

ber

2014

332 PETZL

We I I O W 11ave 1.0 show II'+'(Q') = O a ~ ~ t l that any prinle ideal in t,l~e small support of Cok f ' lies in SuppB(AIB) ai~tl coi~t.ract.s t.o a priine ideal in Spec(A) whose Ali-heigl~t. is at ~iiost. 1 + 1 Co~~sitler t.l~erefore t l i v sliort exact sequelice

By t.l~e 'overall' inductive Iiypotl~esis I I ) any prime ideal i l l ,4ss(Ht+'(Q')) must, lie in S I I ~ I ) ~ ( M ~ ) and c011t.ract to a prime of M - l ~ t 3 L + 1. By l.B(iv), 3.4 and the ir~tluct.ive hypotliesis it. follows I~owrver t.llat every prime in the small support, of Cok f ' lies in SttppB(MB) and co~rt,ract,s t,o a prime ideal i r i SuppA(hf ) , which has h4-lieight not, excretling 1 + 1 < r(.. This is a co~~t.radict,ion and clearly completes t l ~ e i~~duc t ive stel) i l l our fii~it,e intluctioil.

We now comj)let~e t,he 'overall' iutluction by showing (by a~iot.lier ~ ~ ~ d u c t i o i ~ on j 2 1) t , l~at Hh+J(Q ' ) = O.

Let. j = 1. Since Hk+'(Q') Z f l k t 1 ( C ( h l i ~ ) * ) we may rt.js~~me (see 1.2(iv)) w i t , l ~ o ~ ~ t loss of generality t,lrat dim MB 2 k +3. Again we may also assume t,hat, n E A s s ( ~ ' + l ( Q ' ) ) and so H:(Cok f k ) # O. Following t.he same ideas as earlier we may use t.lw short. exact sequence

t,o show t,ltat eit.ller H:(Qktl) # 0 or Hk(Cok f k - ' ) # 0. However, using Lenima 3.3 we know t,hat for all i ~ ~ t e g r r s O 6 i 6 1. + 1 t,he local cohoniology module HA(Q"+'-') vanishes. Tlilis, we may go on t,o t,he s l~ort exact sequence

and use the fact that HA(C'ok f k l ) # O antl H;(QL) = O to deduce that H;;'(Cok fL- -" ) # O

Not,e t h t wit>l~ t.he 11t'Ip of the result of o w precrtling fi1iit.e induct,ion we are now in a position to oht,ai~i for all v = -1 , . . . , I. sliort exact sequences of t,he form

S , : O ---+ ~ o k f"-' - Q'+' i Cok fi' i 0,

C o n h u i n g this process eventually allows 11s 1.0 conclude that Hk+'(Qo) # 0, which is a c.ont,radict,ion t.o Lemn~a 3.3 . We may t,lierefore assume that H'+'(Q.) = 0, and t,llis comptet.es the begin~ling of otlr i~iductive proof wit,hin t,lre 'overall' irttluctive st.rp.

So assume that a 1.+1 and that we have shown for all 1 < I that H J (Q') = O It rerllalns toshow that HX+'(Q') = 0 A g a ~ n , we Inay assunle that dmAdE, 3 7+3, antl that n is in A ~ S ( H ' ~ ' ( Q ' ) ) 1Js1ng the ~ n c l t ~ c t ~ v r I~ypothesls w~ In~xy also conclude that we have short exact seqlwnces of type .5(, for all 3 E 1-1, , 7 }

Thus Hc(Cok f ' ) # 0 a id ritlwr I1: (QL+') # 0 or HA((hk fl-') # 0 We can rule out, the first. of those caw>, sillre H:(Qttl) = 0. 1,y 3.3 As in t,lie previous

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a Sa

nta

Cru

z] a

t 23:

39 0

7 N

ovem

ber

2014

COUSIN COMPLEXES AND FLAT RING EXTENSIONS 333

streps we can now go back t,llrol~gh t Ire short. exact. sequences of t,ype S t , - 1 6 t ,< i , successively applying 3 . 3 1111ti l we can C O I I C I I I ~ ~ t.l~at H,!,+'(Cok f - ' ) = H~+' (Q") # O , which is a contratlirtio~r t.o 3 .3 . This cornplrt.es t,he proof.

In our next. theorem we sllall prove t.hr converse of 3 .5 , so t,llat we can -roughly speaking- say, t,hat, exact~~ess of the iluot,ient complex is equivalent to Col~en- Macaulayness of all t,lte rrlevant, fibres.

Praof. Let p E SuppA(M) be relevant for Af and p. The fibre ring ~ ( p ) @A B is Cohen-Macaulay, if and only if for all q E Spec(r;(p) @A B) t,he 1ocalizat.ion ( ~ ( p ) @A B), is Colien-h4acaulay. By (9, (2.2)(iv)] it suffices t,herefore t.o show that for each q E Spec(B) wit.11 qC = p t.he fibre r;(pAp) @A, Bq of t,he fait,l~fttlly flat ring homomorphism y' : Ap -+ Bq is Cohen-Macanlay. From 3.2 and tzhe

B hypothesis it follows that QA"(MP) is exact., anti so it is clear t,hat. we can rest,rict ourselves t,o the local case i n t.he proof of 3.6.

Assume t,herefore t,hat (A, m) autl (B, n ) axe local, antl that p is fait,hfdly flat. We also a s u m e t,liat m is reIevant anti t.l~at C := tc(m)gA B sat,isfies S1. Wit,hout loss of gerreralit,y we may assume t.l~at dim A4 := k 2 0 and t,llat dim M B := k + i , wit,h i 3 2 . (If i = 1, it follows t.lrat d i m C = 1 = tleptah C , as C sat,isfies S1. In t.hat case C is already COII~II-Rlacaulny.) Not,e t,hat wit,h these hypotheses dim C = i 3 2 antl depf 1iC 2 1.

We shall show that the exact,ness of Q' implies that depth C = i. \Ve begin with a few preliminaries.

(1) Since Q' is exact. we obtain short exact sequences

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a Sa

nta

Cru

z] a

t 23:

39 0

7 N

ovem

ber

2014

334 PETZL

(2 ) Since wr have (for all j E ( 0 , . . . , A . ) ) sl~ort exact, seq1trllcrs of' t l ~ e for111

11 follows fiorn [ I 1, T l~ro l rm] ' I I I ~ I IS, Tl~rorem 3 21 t11~1f for all 111tegrrs 3 E { U , , A . - 1) autl for all 111trgrls f 2 U we hn\r H L ( Q J ) = U

(3) Since M:" H ; + ' ( I \ ~ ~ ) ~t follows that for all positive int,egers 1 L + i - HA(M, ) - 0.

(4) Let. T t1euot.e a h i t r l y grl~rrat,ed non-zero sul~module of H ~ ( M ) . Since r,(T) = T we may concl~~de t l ~ i ~ t . drptllAT = 0, and so (hy 2.3)

Similarly, if T' is anotlrrr noli-zero finitely generai,etl proprr sul~modr~le of T, t l ~ r n tlept.llA(T/T') = 0, and we have a short exact secltlri~cr

O + T ' % ~ U - T @ ~ B ~ T / T ' @ ~ B - ~ .

After a l)~>l~cat lon of focal colromology we the~rfore /lave ffoi all r~~tegers 1 2 O ) an exact sequence

H,!,(T/T' B) - H;+'(T' B) - H,!,+'(T MA B)

(5) Recall (see also [ I l , Tl~t 'oir~~ll 3 21 aud 114, Propos~ttol~ 4 151) that for all ~ntegers 1 3 0

H,!,(llrA,(~i,)) = lrn1 - (H , ! , (L @ A B ) ) , LEL

where 2 denotes t,he set of fiuitely generated A-submodules of H k ( ~ 4 )

Let u := depth C , and recall t , l~at 1 < u. Assume now that u < i . We shall prove that this leads t,o a co~lt,ratfict,ion. For all non-zero T E 2, we have by 2.3 depth,T @J B = v Consrc1ue~~t.ly u is minimal with the property t-hat H,"(T @ B) # 0. Since the first, term of tile short exact sequence in (4) with I = v - 1 vanishes, and all t,l~t, const,it.uent homomorphisnls in the direct limit const,ruction of ('5) (with 1 = v) are inject,ive, it follows from (4) and (5) that u is act,ually minimal wit,h the proptxrt,y t,llat H:(N:.(Aifs)) # 0 (See [ I , Theorem 3.5.71.)

To ol,t,ain a c~nt~ratI ic t . io~~ it will he sufficient, t,o sllow t l ~ a t H;(H;,(M,)) = 0. IJsir~g t,he preliminaries we s l~al l sl~ow t,l~at for all integers 11 E 11, . . . , i - 1) t,he local co110n101ogy 1nod111r H:(N$, (MB)) va~lis l~rs . By [ l l : Tlirorelnl and 1.4 we have an exact sequrncr

ant1 i t suffices Ily [ l l , l ' l ~ r o r t . ~ ~ ~ ] a11t1 2.8 t,o sl~ow t,l~;it for all infegers t E {O, . . . , i - 2) we have ,V;(Qk) = 0. lisillg ( E " ' ~ ' ) 1,) ( 1 ) ~t will be enongh to s l ~ o n t.llai, for all il~f~gt'l.:: 1 E 10, . . , i - 2) D

ownl

oade

d by

[U

nive

rsity

of

Cal

ifor

nia

Sant

a C

ruz]

at 2

3:39

07

Nov

embe

r 20

14

COUSIN COMPLEXES AND FLAT RING EXTENSIONS

(G) Note that one can conclr~tlr ~ntluct~vely from (1) and (2) that when I . 2 1, we have

v l 2 o , v ~ E {-2, , L - 2) H ; ( C O ~ ~ J ) = o (For k = O t,here is not,liirtg t,o show as Cok f-' = 0.) It is t,herefore sufficient t,o show tlhat H:(C:ok cbl) = O for every 1 E {O,. . . , i - 2 ) .

To do so c o n s ~ d e ~ the short exact sequence (Ek+('-')) and apply the I",-functor We o b t a ~ n the f o l l o ~ lng long exart sequellce

of local col~on~ology ~notlules Since by [ll, Theorem] H;((MB)"+('-I)) = O for all 1 E No and H,'((MB)~+') = O for all ~ntegers r 2 1 by (3), i t follows that

for all s E No \ (1). In t2he next step we use this while repeating e~sent~ially the same procedure wit.11 the short exact sequence and show that for all non-negative integers j # 2 we have H ~ ( C O ~ c"+('-~)) = 0. We continue in this manner, until we have used t,he short exact sequence (EL-') and deduced t.hat for all non-negative int.egers j # i - 1 we have Hi(Cok ek-') = 0. In part,icular we have shown that

It follows now from (6) t.hat H',(QL) = O for all mtegers j E (0,. . . , i -21 , which concludes the proof

Corrsequc:nres f o r loccll colroirrology rrrodttlts. Let t,he uot,at,ion be as in 3.5. If t,he quot,irnt con~y,lr>; Q z ( M ) ' is exact,, we obt.ain - rougldy speaking - an exact resolut,ion of rert,ain loral col~on~ology motlules.

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a Sa

nta

Cru

z] a

t 23:

39 0

7 N

ovem

ber

2014

PETZL

Proof Let a E S I I ~ I > ~ ( M ) have Ad-ltr~glit k a11t1 br W C I L t l~dt ae # B Define for s E Nr, to be the set of q E Supp(AfB) of MB-hetght a and w ~ t h qC = a Recall from 3 1 f hat Q(a) = ($4EOI, (Cak 6 k-2)q}/c+4L((C~k d - ? ),) F~nally

define R", @ ... (Cok c'-'), C'o~ts~ctrr the short exact sequence 9 E b .

If a # rn apphcat~on of the r,.-fullctor to t111s s l to~ t exact sequeucr ~ntluces a l o ~ g rxacf sequence of local col~onlology motlrlles 111 wtl~cl~ all the terms must vanish, smce m ~ ~ l t i p l ~ c a t ~ o n by an element a E m \ a is an auton-rorpImm on each of the first two terms In the short exact sequence above

Also, ~f s > k and a # m, we can show w1f11 stmilar methods and [8, Theorem 4 31 that for all integers r 2 O

On the otlre~ Iland, ~f a = rn ~t can be seen w ~ t h the help of 1 2(11) that H:,(Q(m)) = Q(m) and that H:.(Rk) = R k for all s 3 t whle for any integer T 2 1 we have Hk,(Q(m)) = O and HL,(R&) = 0.

In concltrs~on t,hese co~isltleratmns sliow that rm. (Qk) = Rk ~f k > t , Tm.(Qt) = Q(m), anti thaf for aH 0 6 1. 6 f - 1 we have T,. (Qr) = O

We have therefore s l~owr~ that Q' 1s a I'm--acychc resolut~on of O It 15 now clear, that for ail Integers r we have

H;,(O) = 0 = Hr(r,. (Q')).

Since it follows from 2.13(ii) t,llat tltrrr is at] iso~iiorpl~ism

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a Sa

nta

Cru

z] a

t 23:

39 0

7 N

ovem

ber

2014

COUSIN COMPLEXES AND FLAT RING EXTENSIONS 337

3.8 EXAMPLE. Lc1 r l 2 2 h c uir zirt(ycr. uild l r 1 X I , . . .,A',, bc iitdetcr.irriitutcs. Denolc @[XI, . . . , .Yn- I ] b y ui~d @[XI1 , . . . , A',,] b y q X n ] . Let r be the ideal in C[X,,] yc~ccrcrt~cl b y X I , . . . , ?(,,-1 uiid g tile tdcul iic C[X,,-I] ycnerutcd by X I , . . . , X,-1. Ltt 3 hc tbc ~ d r d rn @[I,] w1rir.h zs gcitcrulrd b y XI1,. . . , S,. Tlrcre 1.5- un isoirtoi~p/ri.sirr of @[X,,]-irrodrr/ts

Proof. Consider t l ~ e fait.l~fully flat ring l ~ o ~ n o n ~ o r p l ~ i s ~ n

i. : @[En-I] -- c[fin].

All the fibres of ip are Cohen-Macaulay, and t,hat also holds for tAe induced fait,hfully flat ring homomorpl~ism

Let A := M := CIXn-l], antl B := C[EnIj, and let rn := gA. Note that. X,-l ipl(rn)B = (+, . . . , -i-) = yB.

We obtain tallerefore from 3.7 a short exact sequence

IJsing [8, Theorem 4.31 antl [14, Proposit.ion 4.151 we have the following qX,,]-isomorphisins

and consequentSy we have

Define a mult,iplicatively closed sr~bset. S := @[Xn] \ 3 of ClX,,]. Since H , ; ~ ~ ( B ) is a B-module, it is clear t.l~at

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a Sa

nta

Cru

z] a

t 23:

39 0

7 N

ovem

ber

2014

PETZL

II&(B) E H.y([C[X,L])j.

Let. L := H.:'(C[X,]). Since S I I ~ ~ ~ ~ ~ ~ , , ~ ( L ) = {3 ) , ~ t . follows t,hat

H.yB(B) S 11.;(C[X,,]).

The claim follo\+.s from this a~ttl (t).

REFERENCES

[I] W BRUNS, J HERZOO, ~ o h ~ i ~ - A f u ( o i ~ / ~ y i znys , (Cambridge Untverstty Press 1993)

[2] S GOTO, I i WATANAEIE, 'On gratled rltlgs, I ' , J . M o f h . Soc J o p n 30 (1978) 179-213

[3] R HARTSHORNE, R , Rfsrd1115 (11td d11(1111y, Lecture Notes In Mathemat~cs 20 (Spr~nger, Berl~n, 1966)

[5] D C; NORTHCOTT, A N iirl7orlrrtl1oi~ t o Iioi~~oloqrcol c~lycbru, (Canllmtlge l Jn~vr rs~ ty Press 19h0)

[6] R . Y . SHARP, 'The Cousit~ conrplrx for a lnotf~rle over a commutative Noetlterian ring', Aftrlh. Z( i t . 112 (l!)(j!i) 340-356

[8] R. Y . S H A R P , 'Local col~omology t.ltrory in conln~~~tat , ive algrl,ra,' Q u c ~ r t . .I. Molh . O ~ j f o i d (2) 21 (11170) 425-434

[lo] R . Y S t r a w , 'Accel~talrle R ~ I I ~ S i r t t t l f lon~ornorpl~ic 11nagt:s of' C;orelrsteirt Ri~tgs' , .I. of Alych ic~ ( I ) 44 (1!177) 24C -2(il

[ I l l I? Y S H A R P , 'Local col~o~r~ology < I I I ( I ~ I I P (Co~~si~t c0111p1ex for a c o n ~ n ~ r ~ t a - tive ~ o t ~ l l ~ t ~ r i a t ~ r l l~g ' , 1\10tlr. Z f ~ t 1:)3 (1977) 151-22

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a Sa

nta

Cru

z] a

t 23:

39 0

7 N

ovem

ber

2014

COUSIN COMPLEXES AND FLAT RING EXTENSIONS 339

[14] J . R.. STROOKER, Ho1rro1oyiccl.l Qucsiious i n Loctl.1 A l y e h r a , (Caml)ridge [Ji~iversit~y Press 10'30)

Received: March 1996

Revised: August 1996

Dow

nloa

ded

by [

Uni

vers

ity o

f C

alif

orni

a Sa

nta

Cru

z] a

t 23:

39 0

7 N

ovem

ber

2014