# On the intersection of annihilator of the Valabrega–Valla module

### Tony J. Puthenpurakal

Indian Institute of Technology Bombay, Mumbai, India

## Abstract

Let $(A,m)$ be a Cohen–Macaulay local ring with an infinite residue field and let $I$ be an $m$-primary ideal. Let $x=x_{1},…,x_{r}$ be a $A$-superficial sequence \wrt \ $I$. Set

$V_{I}(x)=n≥1⨁ xI_{n}I_{n+1}∩(x) .$

A consequence of a theorem due to Valabrega and Valla is that $V_{I}(x)=0$ if and only if the initial forms $x_{1},…,x_{r}$ is a $G_{I}(A)$ regular sequence. Furthermore this holds if and only if depth $G_{I}(A)≥r$. We show that if depth $G_{I}(A)<r$ then

$a_{r}(I)=x=x_{1},…,x_{r}isa A−superficialsequencewithrespecttoI⋂ ann_{A}V_{I}(x)ism−primary.$

Suprisingly we also prove that under the same hypotheses,

$n≥1⋂ a_{r}(I_{n})isalsom−primary.$

## Cite this article

Tony J. Puthenpurakal, On the intersection of annihilator of the Valabrega–Valla module. Rend. Sem. Mat. Univ. Padova 135 (2016), pp. 21–37

DOI 10.4171/RSMUP/135-2