Quadratic algebras with Ext algebras generated in two degrees

TL;DR

This paper constructs non-Koszul quadratic algebras that mimic Koszul properties up to any finite cohomological degree, providing new examples that challenge existing classifications.

Contribution

It introduces explicit non-commutative quadratic algebras with bigraded Yoneda algebras generated in two specific degrees, answering a question by Green and Marcos.

Findings

Existence of non-Koszul algebras appearing Koszul up to any degree

Construction of algebras with Yoneda algebra generated in degrees (1,1) and (m,m+1)

Examples of m-Koszul algebras that are not Koszul

Abstract

We show that there exist non-Koszul graded algebras that appear to be Koszul up to any given cohomological degree. For any integer m>2 we exhibit a non-commutative quadratic algebra for which the corresponding bigraded Yoneda algebra is generated in degrees (1,1) and (m,m+1). The algebra is therefore not Koszul but is m-Koszul (in the sense of Backelin). These examples answer a question of Green and Marcos.