$d$-Koszul algebras, 2-$d$ determined algebras and 2-$d$-Koszul algebras

TL;DR

This paper explores the conditions under which an algebra and its associated monomial algebra are both $d$-Koszul, introducing classes of 2-$d$-determined and 2-$d$-Koszul algebras and analyzing their structures.

Contribution

It establishes the equivalence of $d$-Koszul properties between an algebra with a homogeneous reduced Gröbner basis and its monomial algebra, and introduces the classes of 2-$d$-determined and 2-$d$-Koszul algebras.

Findings

An algebra with a homogeneous degree $d$ reduced Gröbner basis is $d$-Koszul iff its monomial algebra is $d$-Koszul.

2-$d$-determined monomial algebras are shown to be 2-$d$-Koszul.

The structure of the relations ideal in 2-$d$-determined algebras is fully characterized.

Abstract

The relationship between an algebra and its associated monomial algebra is investigated when at least one of the algebras is $d$-Koszul. It is shown that an algebra which has a reduced \grb basis that is composed of homogeneous elements of degree $d$ is $d$-Koszul if and only if its associated monomial algebra is $d$-Koszul. The class of 2-$d$-determined algebras and the class 2-$d$-Koszul algebras are introduced. In particular, it shown that 2-$d$-determined monomial algebras are 2-$d$-Koszul algebras and the structure of the ideal of relations of such an algebra is completely determined.