Linearity defects of modules over commutative rings

TL;DR

This paper investigates the linearity defect of modules over commutative rings, exploring its behavior under algebraic operations and establishing connections with ring properties like being Koszul, while also introducing an injective analogue.

Contribution

It introduces an injective analogue of the linearity defect and relates it to existing invariants, advancing understanding of module resolutions over commutative rings.

Findings

Linearity defect tracks the acyclicity point in minimal resolutions.

A local ring is Koszul iff it has a Cohen-Macaulay Koszul module of minimal degree.

An injective analogue of the linearity defect is defined and related to other invariants.

Abstract

This article concerns linear parts of minimal resolutions of finitely generated modules over commutative local, or graded rings. The focus is on the linearity defect of a module, which marks the point after which the linear part of its minimal resolution is acyclic. The results established track the change in this invariant under some standard operations in commutative algebra. As one of the applications, it is proved that a local ring is Koszul if and only if it admits a Koszul module that is Cohen-Macaulay of minimal degree. An injective analogue of the linearity defect is introduced and studied. The main results express this new invariant in terms of linearity defects of free resolutions, and relate it to other ring theoretic and homological invariants of the module.