On the intersection of annihilator of the Valabrega–Valla module

  • Tony J. Puthenpurakal

    Indian Institute of Technology Bombay, Mumbai, India


Let (A,m)(A,\mathfrak m) be a Cohen–Macaulay local ring with an infinite residue field and let II be an m\mathfrak m-primary ideal. Let x=x1,,xr\mathbf x = x_1, \ldots, x_r be a AA-superficial sequence \wrt \ II. Set

VI(x)=n1In+1(x)xIn.\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 VI(x)=0\mathcal V_I(\mathbf x) = 0 if and only if the initial forms x1,,xrx_1^*, \ldots, x_r^* is a GI(A)G_I (A) regular sequence. Furthermore this holds if and only if depth GI(A)rG_I(A) \geq r. We show that if depth GI(A)<rG_I(A) < r then

ar(I)=x=x1,,xrisaAsuperficialsequencewithrespecttoIannAVI(x)ismprimary.\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,

n1ar(In)isalsomprimary.\bigcap_{n \geq 1} \mathfrak a_r(I^n) \: \: \mathrm {is \: also} \: \mathfrak m \mathrm{-primary}.

Cite this article

Tony J. Puthenpurakal, On the intersection of annihilator of the Valabrega–Valla module. Rend. Sem. Mat. Univ. Padova 135 (2016), pp. 21–37

DOI 10.4171/RSMUP/135-2