奈良教育大学学術リポジトリNEAR
On the Lyubeznik Number of Local Cohomology Modules
著者 KAWASAKI Ken‑ichiroh
journal or
publication title
奈良教育大学紀要. 自然科学
volume 49
number 2
page range 5‑7
year 2000‑11‑10
URL http://hdl.handle.net/10105/1415
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
On the Lyubeznik Number of Local Cohomology Modules
Ken‑ichiroh KAWASAKI '
(Depa血ent of Mathematics, Nara University of Education, Nara, 630‑8528, Japan)
(Received April 6, 2000)
Abstract
In this paper, we decide the heighest Lyubeznik number A 2,2 (A) by the number of the connected components of its punctured spectrum, where A is a complete noetherian local ring of dimension 2 containing a separably closed residue field, using the spectral sequence. This assertion gives a counterexample for Lyubeznik's question (cf. Question 4.5 of
L).
Key Words: local cohomology, spectral sequnce, Lyubeznik number, Bass number.
We assume that all rings are commutative and noetherian with identity throughout this paper.
Lyubeznik defines a numerical invariant of local rings with respect to local cohomology modules [L, Theorem‑Definition 4.1].
Definition 1. LetA be a local ring of dimension d which admits a surjective ring homomorphism n :R → A, where R is a regular local ring of dimension n containing a field. Set / = kerk and let m be the maximal ideal ofR. Then the Bass
number jip (m,畔 ‑(/?)) is finite and depends only onA, / and
p, but neither on R nor on n. We denote this invariant by ら(A), and we call this number the Lyubeznik number (or
the (/J, z)‑Lyubezmk number).
A complete local ring containing a field is always a sur‑
jective image ofa regular local ring containing a field. So, ifA is a local ring containing a field, but not necessarily a surjec‑
tive image of a regular local ring containing a field, one can set A >J(A)‑A v(Aつ, where A" is the completion ol A with respect to the maximal ideal.
Our aim in this paper is to prove the following:
Theorem 1. Let (R, m, K) be a complete regular local ring
containing a sept汀αbly closed residue field, and I an ideal ofdimension 2. Put X ‑ SpecA, where A‑R/I. Suppose that X is
5
equi‑dimensional. If the punctured spectrum ofX has t con‑
nected components, then X 2,2 (A)= t. In particular, the punc‑
tured spectrum ofX is connected if and only ifね2 (A)=l.
We recall some basic properties ofAp.i (cf. [L,(4.4, i, ii, Ill, IV, V印.
Theorem 2 (Lyubeznik). LetA be a local ring of dimension d containing afield, and I an ideal. Put X ‑ Speci? / /. Then the following assertions hold:
(i) Xp.i{A)‑¥)ifi>d;
(ii)ス ・(A)=0(/>>/;
(Ill)最d(A)≠0;
(iv) ifA is analyticallynomal, then X d,d(A) ‑ 1;
(v) ifA is complete intersection, then A <ij (A) = 1.
These results lead Lyubeznik to give the following ques‑
tion [L, Question 4.5],
Question 1 (Lyubeznik). Is it true that Xd.d {A)‑ 1 for allA?
Theorem 1 asserts that there are a lot of examples which yield a negative answer for the above question for a local ring of dimension 2.
Lemma 1. Let (R,m) be a regular local ring containi月g a field
* The author is partially supported by the Grants‑in‑Aid Scientific Research, The Ministry of Education, Science and Culture, Japan.
6 Ken‑ichiroh KAWASAKI
/an ideal ofR and I a non‑negath‑e integer. IfHj (R) = Oforj
≠Ithenku(R/I)=l.
Proof. Consider the spectral sequence:
HZHfiR) ⇒H,君q(R),
which degenarates by the assumption. Hence we have
irPTjn‑∫(*)‑#r w=o
forp =#/.ThereforeH,!,Hfn (R) ‑H"(R) ‑E時where
E(k) is the injective hull of k. The assertion follows from [L, Lemma 1.4 and Theorem 3.4a].
Proposition 1. Let (A, n) be a local ring containing afield of dimension d. Then thefoil帥・mg a∫sertions hold:
(i) //thedimension ofA is 1, thenれi(A)=l;
(ii) ifA has the embedding codimension 1 (that is, there
is a surjection R ‑*蝣A from a regular loco! ring with
dimR = d+1亡/蝣[HO】), then A dJバA)=l;
(ill) ifA is seトtheoretic complete intersection, then /Ud (A)=l.
Proof. After completing A with respect to n‑adic topology, we may assume that there is a surjection from a regular local ring R ofdimesion n to A. Let I be its kernel. Further we may assume that ∫ is a radical ideal.
(i). Sincethe grade of/is n‑1, Hf(R) = Oforq <n‑l. SinceR is a complete domain, H/ (R) = 0 for q > n‑1 by the local Hartshorne‑Lichtenbaum vanishing theorem. Hence we have Hf(R) = Oforj≠w‑1.
(ii). By the Mayer‑Vietoris sequence, we may assume that / has the intersection of only height one prime. Since R is nor‑
mal, /is principal up to radical. Hence we have /// (/?) = 0 for Kサ
(iii). Since / is generated by a regular sequence for R up to radical, we have Hi (月) = 0 for./≠n‑d.
Therefore the assertion fol一ows from Lemma 1.
ProofofTheorem 1.
(Step 1). We supposethatJ= 1. OnehandH (R)=O fori>2by [L,4.4i)], ontheotherhand,H, (R) =Ofor;>n‑2, since the punctured spectrum of X is connected (cf. [HL.
Theorem 2.9]). Therefore we have HJ(R) = 0 forj^n‑2. It follows from Lemma 1 that A 2,2(A)=l.
(Step 2). We suppose that t > 1. Let Spec/? IJ＼ mibe the union of the first (ト1) connected commponents of X＼ mt and let SpecR/Q ＼ mi be the last component. The Mayer‑Vietoris sequence for local cohomology modules gives an isomor‑
phism:
Hf ‑(R) 0 Hf‑(R) ‑ H,n ‑ (R).
Note that H,f,H" ‑(/?) ≠O and H′gHq ¥R) 辛 O by [L, 4.4iii)].
The assertion of Theorem 1 follows from the induction on t applying the additive functor Hm (‑) to the above isomorphism.
Example 1. We can give an example for Theorem 1. For example, let R be the localization ot the polynomial ring k [xi.
x‑2,'‑, X2,} in 2r variables by the irrelevant ideal (jti,九 一r2′),
where A: is a field and x are indetermmates. And let / be an ideal ofR of the following form:
(Jci,Xi,X‑ . X*)nO¥‑i, X2, X‑.ふxs,・蝣・, X‑n)∩ ‑ ∩(Jfl,‑, X‑U‑2,ふ1, x‑i.). Here the notation xi means omit xi. Note that the ideal (.vi, ,xi,X:C‑, Xz) + (Xi‥X>,X¥XiXa, ‑,**)∩ ‑∩(*.,‑‑,X2,‑l,茄,‑ ,‑fe) is the maximal ideal ofR, so it follows from the Mayer‑
Vietons sequence that A 2,2 (/? // ) =? and further we see that the punctured spectrum of SpecR/I has f‑connected compo‑
nents.
Remark 1. Under the condition of the above theorem, we have the value Ap,< (A) as follows:
〝＼ヾ 2 1 0
2 t 0 0
1 0 0 0
0 0 f ‑ 1 0
by[HK, Theorem 3.6].
Remark 2. The condition equi‑dimensionalH ofTheorm 1 is
ll
essential. Suppose X is not equidimensional and decompose X to XiUX2, where Xi is pure dimension 1 and X2 is pure dimen‑
sion 2. If the punctured spectrum of X2 has t connected com‑
ponents, then A 2.2 (A) = t. Indeed, it easily follows from the Mayer‑Vietons sequence for local cohomolgy modules with respect to X, Xi, X2.
Acknowledgment. The author would like to thank Professor Gennady Lyubeznik and Doctor Hans Ulnch Walther great ful‑
1y for their comments. Doctor Walther told resent results to the
author as in the note at the end of this paper, and sent his
preprint. And the author would like also to thank Professor
Yukitoshi Hinohara, Professor Sadao Tachibana, Tadashi
Kanzo and Kazufumi Eto for their valuable suggestions and
helpful discussions with them during work on this paper.
On the Lyubeznik number of local cohomology modules
References
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[HLJ C.Huneke and G.Lyubeznik, On the vanishing of local cohomol‑
ogy modules', Invent, Math. 102(1990)73‑93.
[HK] C. Huneke and J. Koh, Co finiteness and vanishing of local coho‑
mology modules , Math. ProC Cambridge Philos. Soc. 110 (1991) 421‑429.
「L] G. Lyubeznik. Finiteness properties of local cohomology modules (an application of D‑modules to commutative algebra) , Invent.
Math. 102(1993)41‑55.
[M] H. Matsumura. Commutative Algebra. Benjamin / Cummings, Reading, MA, 2nd ed., 1980.
7
Note added in the proof of Theorem 1.
After the author sent this preprint to Professor Lyubeznik, Doctor
Walther sent his comments to the author by the e‑mai一 on dated
