Castelnuovo-Mumford regularity and cohomological dimension

TL;DR

This paper introduces a generalized notion of Castelnuovo-Mumford regularity for graded modules over standard rings, explores its relation to classical regularity, and connects it with cohomological dimensions and free resolutions.

Contribution

It defines a new regularity invariant with respect to an ideal sum, relates it to existing invariants, and investigates its properties and independence in certain cases.

Findings

The new regularity can be expressed via minimal free resolutions.

In some cases, the invariant is independent of the choice of the ideal.

Relations between the new invariant and cohomological dimension are established.

Abstract

Let $R=\oplus_{i\in \N_0}R_n$ be a standard graded ring, $R_+ :=\oplus_{i\in \N}R_n$ be the irrelevant ideal of $R$ and $\fa_0$ be an ideal of $R_0$. In this paper, as a generalization of the concept of Castelnouvo-Mumford regularity $\reg(M)$ of a finitely generated graded $R$-module $M$, we define the regularity of $M$ with respect to $\fa_0+ R_+$, say $\reg_{\fa_0+ R_+}(M)$. And we study some relations of this new invariant with the classic one. To this end, we need to consider the cohomological dimension of some finitely generated $R_0$-modules. Also, we will express $\reg_{\fa_0+ R_+}(M)$ in terms of some invariants of the minimal graded free resolution of $M$ and see that in a special case this invariant is independent of the choice of $\fa_0$.