奈良教育大学学術リポジトリNEAR
An Application of Vasconcelos' Lemma
著者 HINOHARA Yukitoshi, KAWASAKI Ken‑ichiroh, TAKAHASHI Kazuyoshi
journal or
publication title
奈良教育大学紀要. 自然科学
volume 49
number 2
page range 1‑3
year 2000‑11‑10
URL http://hdl.handle.net/10105/1414
奈良教育大学紀要 第49巻 第2号(自然)平成12年
Bull. Nara Univ. Educ,Vol.49, No. 2 (Nat), 2000
An Application of Vasconcelos Lemma
Yukitoshi HINOHARA* , Ken‑ichiroh KAWASAKI and Kazuyoshi TAKAHASHI ' {Departmen t of Mathematics, Nara University of Education, Nara, 630‑8528, Japan)
(Received April 6, 2000)
Abstract
In this paper we prove that there holds Supp//o(A.)⊃‑⊃SuppHn(Kサ),where Hi{K,) is the homology of the
K(‑szul complex:A'. ‑K*(X. M ), tor a sequence ot elementsX xi,‑,Xn ofa commutative noethenan ring, and a finitely
generated it‑odule M. This is a generalization of Vasconcelos Lemma. Further＼ we give a proof of the non‑negativity of
partia一 Euler‑ characteristics of the Koszul complex as an application of Vasconcelos lemma.
Key Words: rigidity, alternating sum, multiplicity, KOszul complex.
We assume that all rings are commutative and noethenall
with identity throughout this paper.
Let M be a finitely generated module over a ring A.Then an endomorphism ot M onto itself is always an automorphism.
This is the well‑known fundamental resull due to Vasconcebs (cf.[ V. Proposition 1.2仕This lemma is written in the follow‑
ing generalized form:
Lemma 1. Let A be a ring, M afinitely generatedA‑module,
and 6 an A‑endomorphism ofM. Then wピ/me Supp(Coker
中日⊃ Supp(Ker(中)・
Proof. Suppose that p￠Supp(Cokcr(恒i, so we have Cokcr
‑ Cokcr{ yi )p‑ O.Thcn the mappii一g函:Mp→M¥> is s両ective Thus the n一之Ippin再p is injective by[V.Proposition 1.2]. Therefore, it
followゝthat Ker( <j> )p‑0. that is p瑳Supp(Ker( <j> )).
In this tornl, Vascoilcelos lemma is a即in generalized in KoszL1l complex. Letx=.批一. Xn be a sequence ot elements ofa ringA, M
a finitely generated A‑mixlule,こmd K.ixMl the Koszul complex
(cf.[BH, I ,ァ1.6」 or [R, Par・( One喜3.3] ).Theli we have
Theorem 1. Wit/川りtaion as abol・P we have the series of inclusions:
Supp(//o(x :M )) ⊃ Supp(H¥(x:M))⊃‑
⊃ Supp(H,,(x:M)).
/;; particular, ifHj ( X¥M ) is offinite length for somej, then Hi( x:M) is offinitピlPnghtftげallI≧;, andifH;(x¥M ) ‑0 f(u‑somej, then Hi(x:M ) ‑ Oforall I ≧j.
Proof. Set二X‑.Yi>,〜, Ah. We use the induction on the number of ele‑
ments.vi,‑, ‑v,,. T止e Lme prime p which is not contained in Supp (H,(x:M)) forsome / with O≦ /< n, so wehave Hi(x:M) p=0.
Consider the exact sequence obtained by the nlapping cone (cf.[R.
Part One §3.3]):
Ai ̲ . ̲̲ ̲̲ ̲ ̲ ̲̲ . .̲ .Vi
一一→Hi十¥{X':M) →H,+¥{∫:M) →H,(X':M) →Hiix':M) → Hi(x:M) ‑+H,‑i{x':M) →‑
Ai
where H,(x':M) →Hi(x':M) is the multiplication illapping by ¥i
(Case l主Ifai is contained in pAp, the exact sequence
,rl
Hi(x':M) →Hi ∫':M) →H,{x:M)p=O
implies that Hi ( x':M ) p‑ 0 by Nakayama's lcmilna. It follows from the induction's hyp*.、thesis thLit /// ( x':M ) p ‑ 0 for ally ≧ i. Hence wehaveHi (X:M I =O forallj≧/,Thereforepis not contained in Supp (H<+i(X¥M )).
(Case 2). If.vi is not contained in pzip, thelLVi is a unit in Ap. So the muhtiphcation mappings
Ai
Hi(x':M )p→H/(X':M lp
* Thc・ plでsent address is Department of Mathematics, School of Education, Waseda University, Tokyo.
蝣)
JL...A Yukitosln HINOHARA. Ken‑ichiroh KAWASAKI anL Kazuyoshi TAKAHASIII
are isomol‑phisms for allj ≧ 0. Hence, we have Hj(X:M)p= Q for all./ ≧ 0 by the above long exact sequence obtained by the mapping cone. In this case, we also have that p is not coiレ tailled in Supp (Hi+i ( x:M )L
It follows froili (Case 1) and (Case 21 that ¥i{H,(x:M ))p
‑0, thenH,+i(x:M )p = 0, thatis Supp (H, (x:M)) ⊃ Supp
(Hi十i ( x:M )).Therefore we have the series ot inclusions ab in
the theorenl.
IfH′ x:M ) isoffinitelength,wehとIvethaいorall ≧j、
Supp (Hi ( x:M )) iscontained in Supp (Hi (X:M )) which con‑
sists of a tinite number of maxitlnal ideals. Therefore Hi (x;M) is offinite length.
IfHj (x:M) = 0 for some./, then Supp(H,(x:M)) =玖Hence
wehave Supp(///(x:M)) = 0 forall /≧j. that is Hi {x:M) = 0.
In [S, Appendice II]. the partial Eulcr chこ11・actei・IStlCS IS
defined and its 一ion‑negativity is proved under the hyjコothesis
thatx is a system ofpararlleters or a local ring (see also [BH.
Relllark 4.6.12]). By virtue of Theorem 1, the definition is slightly generalized as follows:
Definition. IF H, (x:M) is of finite length tor s川11ej,we
define the partia一 Eul亡r charactristies:
x,(x:M)‑≡ ‑I)'‑‑'?(H,{x:M))・
J≧/
where 2 denotes the length ofamodule.
Theorem 2. LetA br ′ing , Mafi〃itely g川…JledA一間rJdidP. X
‑ A'i,〜, X,i C=equence ofど/ピmeats oj A. IfH, {x:M) i・v offinite l川gthforsomej≧0, thenXj(X:M )=∑J≧′ ‑¥)'‑>2(Hi(x:M H
is nofinPgativtノ.
proof. We use the mduCtion on the numbern ofelementsx ‑
A‑i. ‑‥Xn. We can easily see the thec〕1‑e】1i in the case ofn = 1
(see, for exailnple. [F, Examp一e A.2.2」 So we assume n > 1.
For a module N of finit亡Iength, we have @{N)‑
≡ H (N m,), where m* are lllaximal ideals contained in Supp(AO.
Therefore, we may assume thatA is a locこIl ring (A, n叶
Ifthe ideal /= LVi,・・・.Xn) isequal toA. then H, (x:M) ‑0
for all / ≧ 0 by [B.AX 158,帆Proposition 6]. Therefore we alwayshaveXi(x.M)=0. Sowemayassumethat/⊂ m.
Set二rr = 、1‑ Xn. Consider the long exact sequence
obtained by the mapping cone:
lll+l . ill
Hi+¥(x':M) →FIl十i(x:M) →Hi(r:M) →Hi(x':M) →
Ht (x二M) ‑→Hn(X'・M),
where di, ili+i are the multiplicatu‑n by a‑. From this exact sequence, we have shor t exact sequences:
0‑‑ Coker(di) → Hi(x:M) ‑→ KerU//‑ll → 0,
where Coker (ch) and Ker (ch‑¥) are of finite length ior all /≧j.
Theretore, we have
HHi{x:M )) =?(Coker(J/)) + HKer(c!i一川 X,{X¥M)= e(Ker(d一川
+ ∑!≧′(‑1)/Jj
( e(Coker (di)) ‑ 」 (KeiつtI用L From the exact sequence
O^Ka・(di) ‑Hi(x'こM)
Ai
→Hi(X; M)一・Coker(di) ‑→ 0,
we have diill SLipp (Hi (XこM)) ≦ 1 forall /≧./ ‑ SI】ice Ker(<//) and Coker (di) are offinite length.
If th亡dimenSion of Supp {Hi (∫'こM)) equals zero, then
we havedim Supp (Hi (x:Mり= Oforal日≧j ,ニind ^(Coker (di)) ‑ ^(Kei・ (〔//)) =O by [F. LemmaA.2.1」. Hence it follows that we haveX,(x:M ) =2(Ker (ch‑i)) ≧ 0. Sc‑ we assume that the dime)‑sion ofSupp (/// (x ; M)) equals one.
Let p be any minimal prime of Supp (//; (x; M仕thei‑
we I‑ュve‑r.瑳p. Indeed, suppose thとItAi ∈ p. Since p ∈Ass
(Hi(x: M)).there is an clement ¥v ∈ H,(x'こM) such thatA/p
三Air ⊂ Hjix'; M), andAw ⊂ Ker((//).Now sinceA is local
and the dimension of Supp (Hi (x: M)) equals on亡 we have dim A/p = I. which contradicts the assumption that Ker (dj) is of finite length.
Therefore it fol一ows from 「F. Leillnla A.2.71. that
(1) P.(.Y.,Hi(XこM)1
‑ ∑e(ffi(x'こM)p) e(A/(p+・viA)).
p
where p is the millimal prime ofSupp (//, (x; M)'. The above
equation (1) ho一ds forこill / ≧j , hence we have (he inequality
X,(x':MP)‑≡ ‑!)'‑> e(Hi(x'ニM)pl≧0.
.:'≧!
by the induction's hypothesis. Therefore we have
∑ ‑1)' i(e(Cuker(di))‑e{Ker(di)))
∴l
=∑(‑1)′lpe心,Hi(X';M))
!＼.1・
=∑ ‑1)∫‑j∑ HH/(x'こM)p)^(A/(p+‑viAH
/≧!
=∑p(≡ ‑i)'‑jeiHi(x';A/)p))」(A/(p+‑v,/A))
・'̀∴′
=∑pxi(x':Mp)eiA/(v+xiA))>Q.
Therefore we coiljlude th之it Xi(∫:M) ≧ O.
References
[B] Nicolas Bourbaki, A圧〜!?re Chap. 10. Alg帥re lìwwlogiquc, Masson 1980.
LBHJ Win fried Brims and Jiir^cil Her/0g. ('ohen‑Mctcaulax Rin吋s.
An Application of Vasconcelos'Lemma
Cambridge University Press 39, Cambridge, 1998. [S] Jean‑Pierre Serre. Algebi・ピ1ocale. Multiplicite. Lecture Note in [FT William Fulton, Intersection Th紺 Springer‑Verlag, Ber一in, Math. ll. Springeì、 1965.
1984. [VJ Wolmer. V. Vasconcelos, On finitely押了erated flat modules, [R] Paul C. Roberts, Multiplicities and Cliern Classes in hoc〟 Trans. Amer. Math. Soc. 138, 1969, 505‑12.
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