Asymptotic depth of Ext modules over complete intersection rings

Abstract

Let $\left(A,\mathfrak{𝔪}\right)$ 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 $depth{Ext}_A^{2i+l}\left(M,N∕I^{n}N\right)$ 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 ${\mu }_j\left(\mathfrak{𝔭},{Ext}_A^{2i+l}\left(M,N∕I^{n}N\right)\right)$ have polynomial growth in $\left(n,i\right)$ with rational coefficients for all sufficiently large $\left(n,i\right)$.