Koszul incidence algebras, affine semigroups, and Stanley-Reisner ideals

TL;DR

This paper unifies key results in combinatorial homological and commutative algebra by characterizing the Koszul property and linear resolutions in incidence algebras, affine semigroup rings, and monomial ideals through Cohen-Macaulay properties.

Contribution

It provides a unifying theorem linking the Koszul property with Cohen-Macaulay conditions across various algebraic structures, extending previous isolated results.

Findings

Characterization of Koszul property via Cohen-Macaulayness in graded cases.

Extension to sequential Cohen-Macaulay property in nongraded settings.

Unified framework for incidence algebras, affine semigroup rings, and monomial ideals.

Abstract

We prove a theorem unifying three results from combinatorial homological and commutative algebra, characterizing the Koszul property for incidence algebras of posets and affine semigroup rings, and characterizing linear resolutions of squarefree monomial ideals. The characterization in the graded setting is via the Cohen-Macaulay property of certain posets or simplicial complexes, and in the more general nongraded setting, via the sequential Cohen-Macaulay property.