Castelnuovo-Mumford regularity of deficiency modules

Abstract

Let $d\in \mathbb{N}$ and let $M$ be a finitely generated graded module of dimension $\le d$ over a Noetherian homogeneous ring $R$ with local Artinian base ring $R_0$. Let $\beg(M)$, $\gendeg(M)$ and $\reg(M)$ respectively denote the beginning, the generating degree and the Castelnuovo-Mumford regularity of $M$. If $i\in {\mathbb{N}}_0$ and $n\in Z$, let $d_M^i(n)$ denote the $R_0$-length of the $n$-th graded component of the $i$-th $R_{+}$-transform module $D_{R_{+}}^i(M)$ of $M$ and let $K^i(M)$ denote the $i$-th deficiency module of $M$. Our main result says, that $\reg(K^i(M))$ is bounded in terms of $\beg(M)$ and the "diagonal values" $d_M^j(-j)$ with $j=0,...,d-1$. As an application of this we get a number of further bounding results for $\reg(K^i(M))$.

25 pages, the previous version divided in two parts