Derived functors and Hilbert polynomials over hypersurface rings-II

TL;DR

This paper investigates the polynomial growth of certain homological functions over hypersurface rings, revealing a uniform degree related to ideal and module properties, and utilizes classification of thick subcategories in MCM modules.

Contribution

It establishes a uniform degree for polynomial functions involving Tor and Ext over hypersurface rings, linking it to ideal and module invariants, and employs classification of thick subcategories.

Findings

Polynomial degree $r_I$ depends only on $I$ and $N$.

Functions involving Tor and Ext have polynomial type with degree $r_I$.

Classification of thick subcategories is key to the results.

Abstract

Let $(A,\mathfrak{m})$ be a hypersurface local ring of dimension $d \geq 1$, $N$ a perfect $A$-module and let $I$ be an ideal in $A$ with $\ell(N/IN)$ finite. We show that there is a integer $r_I \geq -1$ (depending only on $I$ and $N$) such that if $M$ is any non-free maximal \CM \ (= MCM) $A$-module the functions $n \rightarrow \ell(\text{Tor}^A_1(M, N/I^{n+1}N))$, $n \rightarrow \ell(\text{Ext}^1_A(M, N/I^{n+1}N))$ and $n \rightarrow \ell(\text{Ext}^{d+1}(N/I^{n+1}N, M))$ (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}).