Derived functors and Hilbert polynomials over hypersurface rings

TL;DR

This paper investigates the behavior of Tor and Ext functors over hypersurface rings, revealing a polynomial degree bound related to Cohen-Macaulay modules and utilizing classification of thick subcategories.

Contribution

It establishes a bound on the polynomial degree of Tor and Ext functions for MCM modules over hypersurface rings, linking to thick subcategory classification.

Findings

Degree of Tor^A_1(M, A/I^{n+1}) is bounded by r_I

Similar bounds hold for Hilbert polynomials of Ext-functors

Key role of thick subcategory classification in proofs

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 non-negative integer $r_I$ (depending only on $I$) such that if $M$ is any non-free maximal Cohen-Macaulay $A$-module the function $n \rightarrow \ell(Tor^A_1(M, A/I^{n+1}))$ (which is of polynomial type) has degree $r_I$. 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).