Castelnuovo-Mumford regularity and cohomological dimension

Abstract

Let $R={\oplus }_{i\in {\mathbb{N}}_0}{R}_n$ be a standard graded ring, $R_{+}:={\oplus }_{i\in \mathbb{N}}{R}_n$ 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 $\fa_0$.

arXiv comment9 pages, to appear in Communications in Algebra