On the intersection of annihilator of the Valabrega–Valla module

Abstract

Let $(A,\mathfrak m)$ be a Cohen–Macaulay local ring with an infinite residue field and let $I$ be an $\mathfrak m$-primary ideal. Let $\mathbf x = x\_1, \ldots, x\_r$ be a $A$-superficial sequence \wrt \ $I$. Set $$ \mathcal V\_I(\mathbf x) = \bigoplus\_{n \geq 1} \frac{I^{n+1} \cap (\mathbf x) }{\mathbf x I^n}. $$ A consequence of a theorem due to Valabrega and Valla is that $\mathcal V\_I(\mathbf x) = 0$ if and only if 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 $$ \mathfrak a\_r(I)= \bigcap\_{\substack{\mathbf x = x\_1, \ldots, x\_r : \mathrm {is : a}} : \ {A-\mathrm {superficial : sequence : with : respect : to} : I}} \mathrm {ann}\_A \mathcal V\_I(\mathbf x) : : : \mathrm {is} : \mathfrak m \mathrm {-primary}. $$ Suprisingly we also prove that under the same hypotheses, $$ \bigcap\_{n \geq 1} \mathfrak a\_r(I^n) : : \mathrm {is : also} : \mathfrak m \mathrm{-primary}. $$