On the intersection of annihilator of the Valabrega–Valla module

Tony J. Puthenpurakal

doi:10.4171/rsmup/135-2

JournalsrsmupVol. 135pp. 21–37

On the intersection of annihilator of the Valabrega–Valla module

Tony J. Puthenpurakal
Indian Institute of Technology Bombay, Mumbai, India

Download PDF

Abstract

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

V_{I} (x) = n \geq 1 ⨁ \frac{I ^{n + 1} \cap ( x )}{x I ^{n}} .

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

a_{r} (I) = x = x_{1}, \dots, x_{r} is a A - superficial sequence with respect to I ⋂ ann_{A} V_{I} (x) is m - primary .

Suprisingly we also prove that under the same hypotheses,

n \geq 1 ⋂ a_{r} (I^{n}) is also m - 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