A finiteness result for associated primes of certain ext - modules

A FINITENESS RESULT FOR ASSOCIATED PRIMES

OF CERTAIN EXT-MODULES

MARKUS BRODMANN and LE THANH NHAN

Abstract 1

. Using some properties of unconditioned M−sequences in dimension > s, we

give a finiteness result for the set [

n∈N

AssR(Exti

R(R/In

, M)).

1 Introduction

Throughout this paper, let R be a Noetherian commutative ring, let M be a finitely generated

R−module, and A an Artinian R−module.

For an ideal I of R, it was shown in [B] that the two sequences of associated primes

AssR(M/InM) and AssR(I

nM/In+1M), n = 1, 2, . . .

eventually become constant for large n. Sharp [Sh] proved the dual result for Artinian mod￾ules: AttR(0 :A I

n

) and AttR

￾

(0 :A I

n

)



(0 :A I

n−1

)

do not depend on n for n large. Starting

from the observation that M/InM ∼= TorR

0

(R/In

, M) and 0 :A I

n ∼= Ext0

R(R/In

, A) for any

n, Melkersson and Schenzel [MS] extended the above results as follows: For any given integer

i ≥ 0, the sequences

AssR

￾

TorR

i

(R/In

, M)

and AttR

￾

Exti

R(R/In

, A)

, n = 1, 2, . . .

become independent of n for large n. Melkersson and Schenzel [MS] also asked whether the

set AssR

￾

Exti

R(R/In

, M)

is independent of n for large n.

In fact, [

n∈N

AssR

￾

Exti

R(R/In

, M)

is not a finite set in general, and therefore the set

AssR

￾

Exti

R(R/In

, M)

depends on n for n large. Indeed, Katzman [Ka, Corollary 1.3]

gave an example of a Noetherian local ring (R, m) with two elements x, y ∈ m such that

AssR

￾

H2

(x,y)R

(R)

is an infinite set. Therefore the set [

n∈N

AssR

￾

Ext2

R(R/(x, y)

n

, R)

is infi￾nite.

For convenience, for a subset T of Spec R and an integer i ≥ 0, we set

(T)i

: = {p ∈ T : dim R/ p = i};

(T)≥i

: = {p ∈ T : dim R/ p ≥ i}.

1Key words and phrases: Supports of local cohomology modules, associated primes, filter regular sequences, M−sequences

in dimension > s.

2000 Subject Classification: 13D45, 13E05.

The second author was partially supported by the Swiss National Science Foundation (Project No 20-10349