Generalized Koszul properties for augmented algebras

TL;DR

This paper explores the relationship between filtrations on augmented algebras and their Yoneda algebras, establishing a monomorphism that helps identify the K_2 property in graded cases.

Contribution

It introduces a bigraded algebra monomorphism linking filtrations on augmented algebras to their Yoneda algebras, aiding in the analysis of the K_2 property.

Findings

Established a monomorphism from gr E(A) to E_Gr(gr A)

Applied the monomorphism to connected graded algebras

Provided criteria for the K_2 property in graded algebras

Abstract

Under certain conditions, a filtration on an augmented algebra A admits a related filtration on the Yoneda algebra E(A) := Ext_A(K, K). We show that there exists a bigraded algebra monomorphism from gr E(A) to E_Gr(gr A), where E_Gr(gr A) is the graded Yoneda algebra of gr A. This monomorphism can be applied in the case where A is connected graded to determine that A has the K_2 property recently introduced by Cassidy and Shelton.