Asymptotic depth of Ext modules over complete intersection rings

TL;DR

This paper investigates the asymptotic behavior of Ext modules over complete intersection rings, showing stability of depth and polynomial growth of Bass numbers in large parameters.

Contribution

It establishes the stability of depth and polynomial growth of Bass numbers for Ext modules over complete intersection rings as parameters grow large.

Findings

Depth of Ext modules stabilizes for large indices and powers.

Bass numbers grow polynomially with rational coefficients in large parameters.

Results apply to modules over local complete intersection rings.

Abstract

Let $(A,\mathfrak{m})$ be a local complete intersection ring and let $I$ be an ideal in $A$. Let $M, N$ be finitely generated $A$-modules. Then for $l = 0,1$, the values $depth \ Ext^{2i+l}_A(M, N/I^nN)$ become independent of $i, n$ for $i,n \gg 0$. We also show that if $\mathfrak{p}$ is a prime ideal in $A$ then the $j^{th}$ Bass numbers $\mu_j\big(\mathfrak{p},\ Ext^{2i+l}_A(M,N/{I^nN})\big)$ has polynomial growth in $(n,i)$ with rational coefficients for all sufficiently large $(n,i)$.