Noncommutative Koszul filtrations

TL;DR

This paper introduces the concept of Koszul filtrations in non-commutative graded algebras, exploring their properties, examples, and implications for algebraic structures with linear resolutions and rational Hilbert series.

Contribution

It extends the notion of Koszul filtrations from commutative to non-commutative algebras and investigates their connections with Koszul algebras and quadratic Groebner bases.

Findings

Algebras with Koszul filtration have rational Hilbert series.

Introduction of generalized Koszul (rate) filtrations.

Examples include Groebner flags, generic algebras, and single-relation algebras.

Abstract

We study associative graded algebras which have a ``complete flag'' of cyclic modules with linear free resolutions, i.e., algebras over which there is a cyclic Koszul module with every admissible number of relations (from zero up to the number of generators of the algebra). Commutative algebras with the same property has been studied in several papers by A. Conca and others. Here we present a non-commutative version. We introduce the concept of Koszul filtration in non-commutative algebras and study its connections with Koszul algebras and algebras with quadratic Groebner bases. Also, here are considered several examples, such as ``Groebner flags'', generic algebras, and algebras with one relation. A generalization of the concept Koszul filtration (generalized Koszul, or rate, filtration) leads to algebras with finite Backelin's rate and to coherent algebras. One of our main results is that every algebra with Koszul filtration (or with finite rate filtration) has rational Hilbert series.