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

Abstract

Let k be a field and R a standard graded k-algebra. We denote by HR 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 HR 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 Poincare 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.