Ext functors, support varieties and Hilbert polynomials over complete intersection rings

Let $(A,\mathfrak{m})$ be a complete intersection of dimension $d\ge 1$ and codimension $c\ge 1$. Let $I$ be an $\mathfrak{m}$-primary ideal and let $M$ be a finitely generated $A$-module. For $i\ge 1$ let ${\psi }_i^I(M)$ be the degree of the polynomial type function $n\to \mathrm{\ell }(Ex{t}_A^i(M,A\mathrm{/}{I}^n))$. We show that for $j=0,1$ and for all $i\gg 0$ we have ${\psi }_{2i+j}^I(M)$ is a constant and let $r_0^I(M)$ and $r_1^I(M)$ denote these constant values. Set $r^I(M)=\max \{r_0^I(M),r_1^I(M)\}$. We show that $r^I(M)$ is an invariant of $I,A$ and the support variety of $M$. We set the degree of the zero polynomial to be $-\mathrm{\infty }$. If $r^I(M)\le 0$ then we show that $reg{G}_I({\mathrm{\Omega }}^i(M))$ for $i\ge 0$ is bounded. We give an application of this result to syzgetic Artin-Rees property of $M$. We also give several examples which illustrate our results.