Relative Cohen--Macaulayness of bigraded modules

TL;DR

This paper investigates the local cohomology properties of finitely generated bigraded modules over a standard bigraded polynomial ring, focusing on modules with a unique nonvanishing local cohomology with respect to one irrelevant ideal.

Contribution

It characterizes the relative Cohen--Macaulayness of bigraded modules based on their local cohomology behavior, extending existing theories in multigraded algebra.

Findings

Identifies conditions for modules to be relative Cohen--Macaulay

Provides criteria for the vanishing of local cohomology in bigraded modules

Enhances understanding of the structure of bigraded modules in algebraic geometry

Abstract

In this paper we study the local cohomology of all finitely generated bigraded modules over a standard bigraded polynomial ring which have only one nonvanishing local cohomology with respect to one of the irrelevant bigraded ideals.