Bounds for the regularity of local cohomology of bigraded modules

TL;DR

This paper investigates bounds on the regularity of local cohomology modules of bigraded modules over polynomial rings, demonstrating linear bounds in several cases.

Contribution

It establishes linear bounds for the regularity of local cohomology components of bigraded modules, extending understanding of their algebraic properties.

Findings

Regularity of local cohomology modules is linearly bounded in certain cases.

The results apply to finitely generated bigraded modules over standard bigraded polynomial rings.

Abstract

Let $M$ be a finitely generated bigraded module over the standard bigraded polynomial ring $S=K[x_1,...,x_m, y_1,...,y_n]$, and let $Q=(y_1,...,y_n)$. The local cohomology modules $H^k_Q(M)$ are naturally bigraded, and the components $H^k_Q(M)_j=\Dirsum_iH^k_Q(M)_{(i,j)}$ are finitely generated graded $K[x_1,...,x_m]$-modules. In this paper we study the regularity of $H^k_Q(M)_j$, and show in several cases that $\reg H^k_Q(M)_j$ is linearly bounded as a function of $j$.