Linearity Defect and Regularity over a Koszul Algebra

TL;DR

This paper investigates the properties of linearity defect and regularity over Koszul algebras, establishing finiteness and uniform bounds for modules over certain Koszul duals, advancing understanding of their homological invariants.

Contribution

It proves that for Koszul complete intersections, both the regularity and linearity defect are finite for all modules, and establishes a uniform bound for linearity defect of graded ideals in exterior algebras.

Findings

Finiteness of reg and ld for modules over Koszul complete intersections.

Existence of a uniform bound for ld of graded ideals in exterior algebras.

Extension of previous results on linearity defect in Koszul algebras.

Abstract

Let A be a Koszul algebra, and $mod A$ the category of finitely generated graded left A-modules. The "linearity defect" ld_A(M) of $M \in mod A$ is an invariant defined by Herzog and Iyengar. An exterior algebra E is a Koszul algebra which is the Koszul dual S^! of a polynomial ring S. Eisenbud et al. showed that $ld_E(M) < \infty$ for all $M \in mod E$. Improving their result, we show the following (and many other facts): (*) If A is a Koszul complete intersection, then $reg_{A^!} (M) < \infty$ and $ld_{A^!} (M) < \infty$ for all $M \in mod A^!$. (**) There is a uniform bound of $ld(J)$, where J is a graded ideal of E.