Gr\"obner bases and Betti numbers of monoidal complexes

TL;DR

This paper studies monoidal complexes and their associated toric face rings, computing initial ideals and Betti numbers, thereby extending classical theorems in combinatorial algebra to a broader context.

Contribution

It introduces a method to compute initial ideals and Betti numbers for toric face rings, generalizing Hochster's theorems to monoidal complexes.

Findings

Computed initial ideals of presentation ideals.

Determined graded Betti numbers of toric face rings.

Generalized Hochster's theorems to monoidal complexes.

Abstract

In this note we consider monoidal complexes and their associated algebras, called toric face rings. These rings generalize Stanley-Reisner rings and affine monoid algebras. We compute initial ideals of the presentation ideal of a toric face ring, and determine its graded Betti numbers. Our results generalize celebrated theorems of Hochster in combinatorial commutative algebra.