Rendiconti del Seminario Matematico della Università di Padova

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Volume 135, 2016, pp. 21–37
DOI: 10.4171/RSMUP/135-2

On the intersection of annihilator of the Valabrega–Valla module

Tony J. Puthenpurakal[1]

(1) Indian Institute of Technology Bombay, Mumbai, India

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, $$\bigcap_{n \geq 1} \mathfrak a_r(I^n) \: \: \mathrm {is \: also} \: \mathfrak m \mathrm{-primary}.$$

Keywords: Blow-up algebras, multiplicity theory, core

Puthenpurakal Tony: On the intersection of annihilator of the Valabrega–Valla module. Rend. Sem. Mat. Univ. Padova 135 (2016), 21-37. doi: 10.4171/RSMUP/135-2