On the intersection of annihilator of the Valabrega–Valla module

Puthenpurakal, Tony

Rendiconti del Seminario Matematico della Università di Padova, Tome 135 (2016), pp. 21-37.

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

$${𝒱}_I\left(𝐱\right)=\underset{n\ge 1}{⨁}\frac{I^{n+1}\cap \left(𝐱\right)}{𝐱I^n}.$$

A consequence of a theorem due to Valabrega and Valla is that ${𝒱}_I\left(𝐱\right)=0$ if and only if the initial forms $x_1^{*},...,x_r^{*}$ is a $G_I\left(A\right)$ regular sequence. Furthermore this holds if and only if depth $G_I\left(A\right)\ge r$. We show that if depth $G_I\left(A\right)<r$ then

$${𝔞}_r\left(I\right)=\bigcap _{\begin{array}{c}𝐱=x_1,...,x_r\mathrm{is}\mathrm{a}\end{array}A-\mathrm{superficial}\mathrm{sequence}\mathrm{with}\mathrm{respect}\mathrm{to}I}{\mathrm{ann}}_A{𝒱}_I\left(𝐱\right)\mathrm{is}𝔪-\mathrm{primary}.$$

Suprisingly we also prove that under the same hypotheses,

$$\bigcap _{n\ge 1}{𝔞}_r\left(I^n\right)\mathrm{is}\mathrm{also}𝔪-\mathrm{primary}.$$