Derived functors and Hilbert polynomials over hypersurface rings-II

Let $(A,\mathfrak{m})$ be a hypersurface local ring of dimension $d\ge 1$, $N$ a perfect $A$-module and let $I$ be an ideal in $A$ with $\mathrm{\ell }(N\mathrm{/}IN)$ finite. We show that there is a integer $r_I\ge -1$ (depending only on $I$ and $N$) such that if $M$ is any non-free maximal \CM \ (= MCM) $A$-module the functions $n\to \mathrm{\ell }({\text{Tor}}_1^A(M,N\mathrm{/}{I}^{n+1}N))$, $n\to \mathrm{\ell }({\text{Ext}}_A^1(M,N\mathrm{/}{I}^{n+1}N))$ and $n\to \mathrm{\ell }({\text{Ext}}^{d+1}(N\mathrm{/}{I}^{n+1}N,M))$ (which are all of polynomial type) has degree $r_I$. Surprisingly a key ingredient is the classification of thick subcategories of the stable category of MCM $A$-modules (obtained by Takahashi, see \cite[6.6]{T}).