The Ext algebra and a new generalisation of D-Koszul algebras

TL;DR

This paper introduces (D,A)-stacked algebras, a broad generalization of Koszul and D-Koszul algebras, and explores their Ext algebra structure, including explicit descriptions and regradings to Koszul algebras.

Contribution

It defines (D,A)-stacked algebras, characterizes their properties, and provides explicit descriptions of their Ext algebras, including quadratic Gr"obner bases and regradings.

Findings

Ext algebra is finitely generated in degrees 0, 1, 2, 3

Explicit quiver and relations for monomial case

Regrading yields a Koszul algebra

Abstract

We generalise Koszul and D-Koszul algebras by introducing a class of graded algebras called (D,A)-stacked algebras. We give a characterisation of (D,A)-stacked algebras and show that their Ext algebra is finitely generated as an algebra in degrees 0, 1, 2 and 3. In the monomial case, we give an explicit description of the Ext algebra by quiver and relations, and show that the ideal of relations has a quadratic Gr\"obner basis; this enables us to give a regrading of the Ext algebra under which the regraded Ext algebra is a Koszul algebra.