Asymptotic depth of Ext modules over 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 $depthEx{t}_A^{2i+l}(M,N/I^{n}N)$ 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 ${\mu }_j(\mathfrak{p},Ex{t}_A^{2i+l}(M,N/I^{n}N))$ has polynomial growth in $(n,i)$ with rational coefficients for all sufficiently large $(n,i)$.