On the finite generation of a family of Ext modules

Abstract

Let $Q$ be a Noetherian ring with finite Krull dimension and let $\mathbf{f}=f_1,...f_c$ be a regular sequence in $Q$. Set $A=Q/(\mathbf{f})$. Let $I$ be an ideal in $A$, and let $M$ be a finitely generated $A$-module with $\projdim_Q M$ finite. Set $\mathbb{R}={⨁}_{n\ge 0}{I}^n$, the Rees-Algebra of $I$. Let $N={⨁}_{j\ge 0}{N}_j$ be a finitely generated graded $\mathbb{R}$-module. We show that \[\bigoplus_{j\geq 0}\bigoplus_{i\geq 0} \Ext^{i}_{A}(M,N_j) \] is a finitely generated bi-graded module over $\Sc = \R[t_1,...,t_c]$. We give two applications of this result to local complete intersection rings.