On the finite generation of a family of Ext modules

TL;DR

This paper proves the finite generation of a bi-graded module formed by Ext modules over certain quotient rings, with applications to local complete intersection rings.

Contribution

It establishes the finite generation of a bi-graded Ext module over a polynomial extension of the Rees algebra, a novel result in the context of complete intersection rings.

Findings

The Ext module is finitely generated over the polynomial extension of the Rees algebra.

Provides new tools for studying local complete intersection rings.

Demonstrates applications of the main theorem to specific algebraic structures.

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 $\R = \bigoplus_{n\geq 0}I^n$, the Rees-Algebra of $I$. Let $N = \bigoplus_{j \geq 0}N_j$ be a finitely generated graded $\R$-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.