$d$-Koszul algebras, 2-$d$ determined algebras and 2-$d$-Koszul algebras
Edward L. Green, Eduardo do N. Marcos

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 -Koszul. It is shown that an algebra which has a reduced \grb basis that is composed of homogeneous elements of degree is -Koszul if and only if its associated monomial algebra is -Koszul. The class of 2--determined algebras and the class 2--Koszul algebras are introduced. In particular, it shown that 2--determined monomial algebras are 2--Koszul algebras and the structure of the ideal of relations of such an algebra is completely determined.
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Commutative Algebra and Its Applications · Homotopy and Cohomology in Algebraic Topology
