Castelnuovo-Mumford regularity of Ext modules and homological degree

TL;DR

This paper establishes bounds on the Castelnuovo-Mumford regularity of Ext modules over polynomial rings, relates it to homological degree, and refines these bounds in specific cases, advancing understanding of module complexity.

Contribution

It provides new bounds for the regularity of Ext modules and relates homological degree to regularity, addressing a question posed by Vasconcelos.

Findings

Bounds for regularity of Ext modules are established.

Refined bounds when the second module is the ring.

A positive answer to Vasconcelos's question on homological degree.

Abstract

Bounds for the Castelnuovo-Mumford regularity of Ext modules, over a polynomial ring over a field, are given in terms of the initial degrees, Castelnuovo-Mumford regularities and number of generators of the two graded modules involved. These general bounds are refined in the case the second module is the ring. Other estimates, for instance on the size of graded pieces of these modules, are given. We also derive a bound on the homological degree in terms of the Castelnuovo-Mumford regularity. This answers positively a question raised by Vasconcelos.