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 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 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 Poincaré series. As an application, we show that the Poincaré series of all finitely generated modules over a stretched Cohen-Macaulay local ring are rational, sharing a common denominator.

Koszul cycles and Golod rings

“…The result that R is Golod, if (iii) is satisfied, has been shown in the recent paper [4]. Our proof of this case can be deduced without any big efforts from the case that R is standard graded.…”

“…Very recently in [4] it was shown that all finitely generated modules over a stretched Artinian local ring R have a rational Poincaré series with a common denominator by studying the algebra structure of the Koszul homology of R. Among other results they proved in [4,Theorem 5.4] that R is Golod, if τ = h. By using our methods we give an alternative proof of the result and show Theorem 3.5. Let (R, m, K) be a stretched local ring or a stretched standard graded Kalgebra.…”

A survey on the Koszul homology algebra

“…However, the authors of [23] provide stronger conditions on the algebra structure of H(R) that are enough to imply Koszulness of R. They prove that if H(R) is generated by either a single element of bidegree (1, 2), or by a special set of elements in the linear strand, then R is Koszul. More precisely, their result is as follows.…”

Poincaré series of compressed local Artinian rings with odd top socle degree

“…Theorem 6.5. [17] Let (R, m,k k k) be an Artinian local ring with top socle degree s, K R be the Koszul complex on a minimal generating set of m, and τ and b be integers with s − τ ≤ b ≤ s − 1 and 2 ≤ τ + 1 ≤ v(R). If there exists a cycle z 1 in Z(m τ K R ) with z 2 1 = 0 and (6.5.1)…”