Dualizing complex of a toric face ring

TL;DR

This paper describes the dualizing complex of a toric face ring under normality assumptions and explores how algebraic properties relate to the topology of its associated cell complex.

Contribution

It provides a concise description of the dualizing complex for toric face rings and links algebraic properties to topological features of cell complexes.

Findings

Dualizing complex characterized under normality.

Buchsbaum and Gorenstein* properties are topological.

Developed squarefree module theory over toric face rings.

Abstract

A "toric face ring", which generalizes both Stanley-Reisner rings and affine semigroup rings, is studied by Bruns, Roemer and their coauthors recently. In this paper, under the "normality" assumption, we describe a dualizing complex of a toric face ring $R$ in a very concise way. Since $R$ is not a graded ring in general, the proof is not straightforward. We also develop the squarefree module theory over $R$, and show that the Buchsbaum property and the Gorenstein* property of $R$ are topological properties of its associated cell complex.