On the intersection of Annihilator of the Valabrega-Valla module

Abstract

Let $(A,\m)$ be a \CM \ local ring with an infinite residue field and let $I$ be an $\m$-primary ideal. Let $\bx = x_1,\ldots,x_r$ be a $A$-superficial sequence \wrt \ $I$. Set $$\Vc_I(\bx) = \bigoplus_{n\geq 1} \frac{I^{n+1}\cap (\bx)}{\bx I^n}. $$ A consequence of a theorem due to Valabrega and Valla is that $\Vc_I(\bx) = 0$ \ff \ the initial forms $x_1^*,\ldots,x_r^*$ 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 \[ \af_r(I)= \bigcap_{\substack{\text{$\bx = x_1,\ldots,x_r$ is a} \\ \text{$A$-superficial sequence w.r.t $I$}}} \ann_A \Vc_I(\bx) \quad \ \text{is} \ \m\text{-primary}. \] Suprisingly we also prove that under the same hypotheses, \[ \bigcap_{n\geq 1} \af_r(I^n) \quad \ \text{is also} \ \m\text{-primary}. \]