Derived functors and Hilbert polynomials over hypersurface rings

Abstract Let $(A,\mathfrak{m} )$ be a hypersurface local ring of dimension $d \geq 1$ and let I be an $\mathfrak{m} $ -primary ideal. We show that there is a integer rI $\geq\;-1$ (depending only on I) such that if M is any non-free maximal Cohen–Macaulay (= MCM) A-module the function $n \rightarrow \ell(\operatorname{Tor}^A_1(M, A/I^{n+1}))$ (which is of polynomial type) has degree rI. Analogous results hold for Hilbert polynomials associated to Ext-functors. Surprisingly, a key ingredient is the classification of thick subcategories of the stable category of MCM A-modules (obtained by Takahashi, see [11, 6.6]).