Cohen-Macaulayness and computation of Newton graded toric rings

TL;DR

This paper explores when Cohen-Macaulay properties are preserved in Newton graded toric rings and face rings, providing examples, positive results, and an algorithm for computing Newton filtrations.

Contribution

It demonstrates that Cohen-Macaulayness is not always inherited but identifies conditions and provides an algorithm to compute Newton filtrations in toric rings.

Findings

Cohen-Macaulayness is not always inherited in Newton graded and face rings.

Existence of generating sets that preserve Cohen-Macaulayness.

Development of an algorithm to compute Newton filtrations.

Abstract

Let $H$ be a positive semigroup in $\mathbb{Z}^d$ generated by $A$, and let $K[H]$ be the associated semigroup ring over a field $K$. We investigate heredity of the Cohen-Macaulay property from $K[H]$ to both its $A$-Newton graded ring and to its face rings. We show by example that neither one inherits in general the Cohen-Macaulay property. On the positive side we show that for every $H$ there exist generating sets $A$ for which the Newton graduation preserves Cohen-Macaulayness. This gives an elementary proof for an important vanishing result on $A$-hypergeometric Euler-Koszul homology. As a tool for our investigations we develop an algorithm to compute algorithmically the Newton filtration on a toric ring.