Castelnuovo-Mumford Regularity in Biprojective Spaces

TL;DR

This paper extends the concept of Castelnuovo-Mumford regularity to bigraded modules over biprojective spaces, providing foundational results analogous to classical regularity theory in a new multi-graded context.

Contribution

It introduces a definition of regularity for bigraded modules and proves key analogs of classical results in this setting, advancing the understanding of multi-graded algebraic structures.

Findings

Defined regularity for bigraded modules and rings

Proved analogs of classical regularity results in biprojective spaces

Established foundational properties for multi-graded algebraic geometry

Abstract

We define the concept of regularity for bigraded modules and bigraded polynomial ring. In this setting we prove analogs of some of the classical results on $m$-regularity for graded modules over polynomial algebras.