Koszulity of algebras with non-pure resolutions

TL;DR

This paper investigates the homological properties of graded algebras with non-pure resolutions, including Artin-Schelter regular algebras, and explores module decompositions under specific conditions.

Contribution

It introduces conditions under which modules with non-pure resolutions can be decomposed into extensions of modules with pure resolutions, advancing understanding of algebra homology.

Findings

Includes Artin-Schelter regular algebras of types (12221) and (13431)

Provides a decomposition method for modules with non-pure resolutions

Establishes conditions for module extension by pure resolution modules

Abstract

We discuss certain homological properties of graded algebras whose trivial modules admit non-pure resolutions. Such algebras include both of Artin-Schelter regular algebras of types (12221) and (13431). Under certain conditions, a module with non-pure resolution is decomposed to form an extension by two modules with pure resolutions.