Detecting Koszulness and related homological properties from the algebra structure of Koszul homology

TL;DR

This paper explores how the algebraic structure of Koszul homology reflects whether a ring is Koszul, and demonstrates that certain multiplicative conditions imply strong homological properties, including rational Poincaré series for modules.

Contribution

It establishes new links between Koszul homology structure and homological properties of rings, extending results to local rings and providing explicit computations.

Findings

Conditions on Koszul homology imply Golod homomorphisms

Poincaré series of modules over stretched Cohen-Macaulay rings are rational

Explicit formulas for Poincaré series are derived

Abstract

Let $k$ be a field and $R$ a standard graded $k$-algebra. We denote by $\operatorname{H}^R$ the homology algebra of the Koszul complex on a minimal set of generators of the irrelevant ideal of $R$. We discuss the relationship between the multiplicative structure of $\operatorname{H}^R$ and the property that $R$ is a Koszul algebra. More generally, we work in the setting of local rings and we show that certain conditions on the multiplicative structure of Koszul homology imply strong homological properties, such as existence of certain Golod homomorphisms, leading to explicit computations of Poincar\'e series. As an application, we show that the Poincar\'e series of all finitely generated modules over a stretched Cohen-Macaulay local ring are rational, sharing a common denominator.