The Koszul property as a topological invariant and a measure of singularities

TL;DR

This paper demonstrates that the Koszul property of a certain algebra associated with a cell complex is a topological invariant, linking algebraic properties to topological features and singularities.

Contribution

It proves that the Koszul property of R(X) is a topological invariant and establishes conditions on local singularities affecting this property.

Findings

Koszul property is a topological invariant for cell complexes.

Conditions on local singularities influence the Koszul property.

R(X) being Koszul relates to the topology of the cell complex.

Abstract

Cassidy, Phan and Shelton associate to any regular cell complex X a quadratic K-algebra R(X). They give a combinatorial solution to the question of when this algebra is Koszul. The algebra R(X) is a combinatorial invariant but not a topological invariant. We show that nevertheless, the property that R(x) be Koszul is a topological invariant. In the process we establish some conditions on the types of local singular- ities that can occur in cell complexes X such that R(X) is Koszul, and more generally in cell complexes that are pure and connected by codimension one faces.