$\mathcal K_2$ factors of Koszul algebras and applications to face rings

TL;DR

This paper introduces the concept of K2 algebras, a generalization of Koszul algebras, and demonstrates their properties and applications to face rings of simplicial complexes, linking algebraic and topological conditions.

Contribution

It proves a strong theorem about K2 factor algebras of Koszul algebras and applies it to show face rings are K2 when the Alexander dual is Cohen-Macaulay.

Findings

Stanley-Reisner face rings are K2 under certain conditions

K2 algebras generalize Koszul algebras with specific cohomology generation

Theorem links algebraic properties to topological Cohen-Macaulay conditions

Abstract

Generalizing the notion of a Koszul algebra, a graded k-algebra A is K2 if its Yoneda algebra is generated as an algebra in cohomology degrees 1 and 2. We prove a strong theorem about K2 factor algebras of Koszul algebras and use that theorem to show the Stanley-Reisner face ring of a simplicial complex is K2 whenever the Alexander dual simplicial complex is (sequentially) Cohen-Macaulay.