Castelnuovo-Mumford regularity of deficiency modules

TL;DR

This paper establishes bounds on the Castelnuovo-Mumford regularity of deficiency modules of graded modules over Noetherian rings, relating it to initial degrees and specific module invariants, with implications for algebraic geometry and commutative algebra.

Contribution

It provides new bounds for the regularity of deficiency modules based on initial degrees and diagonal invariants, advancing understanding of module complexity.

Findings

Regularity of deficiency modules is bounded by initial degrees and diagonal values.

Derived several bounds for the regularity of deficiency modules.

Results applicable to modules over Noetherian homogeneous rings.

Abstract

Let $d \in \N$ and let $M$ be a finitely generated graded module of dimension $\leq d$ 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 $i \in \N_0$ and $n \in Z$, let $d^i_M(n)$ denote the $R_0$-length of the $n$-th graded component of the $i$-th $R_+$-transform module $D^i_{R_+}(M)$ 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" $d^j_M(-j)$ with $j = 0,..., d-1$. As an application of this we get a number of further bounding results for $\reg(K^i(M))$.