On the Koszul property of toric face rings

TL;DR

This paper investigates the conditions under which toric face rings exhibit the Koszul property, providing a characterization, computing Betti numbers, and exploring related conjectures and special cases.

Contribution

It generalizes previous work to characterize Koszulness in toric face rings, computes their graded Betti numbers, and proves that initially Koszul rings are affine monoid rings.

Findings

Characterization of Koszul toric face rings.

Computation of graded Betti numbers for these rings.

Proof that initially Koszul rings are affine monoid rings.

Abstract

Toric face rings is a generalization of the concepts of affine monoid rings and Stanley-Reisner rings. We consider several properties which imply Koszulness for toric face rings over a field $k$. Generalizing works of Laudal, Sletsj\o{}e and Herzog et al., graded Betti numbers of $k$ over the toric face rings are computed, and a characterization of Koszul toric face rings is provided. We investigate a conjecture suggested by R\"{o}mer about the sufficient condition for the Koszul property. The conjecture is inspired by Fr\"{o}berg's theorem on the Koszulness of quadratic squarefree monomial ideals. Finally, it is proved that initially Koszul toric face rings are affine monoid rings.