Colloquium abstracts

Tony J. Puthenpurakal
Indian Institute of Technology, Mumbai
March 31, 2011

On the intersection of annihilator of the Valabrega-valla module:  Let $(A,\mathfrak{m})$ be a Cohen-Macaulay local ring with an infinite residue field and let $I$ be an $\mathfrak{m}$-primary ideal. If $J = (x_1,\ldots,x_s)$ is a minimal reduction of $I$ then consider the $A$-module $${\mathcal V}c_I(J) = \bigoplus_{n\geq 1} \frac{I^{n+1} \cap J}{JI^n}.$$ A consequence of a theorem due to Valabrega and Valla is that $\mathcal{V}_I(J) = 0$ if and only if $G_I(A)$ is Cohen-Macaulay. We show that if $G_I(A)$ is not Cohen-Macaulay then $$\bigcap_{\substack{\text{$J$ minimal } \\ \text{ reduction of $I$}}} $\operatorname{ann}_A \mathcal{V}_I(J) \quad \text{is} \ \m\text{-primary}.$$