Dualizing Complex of a Toric Face Ring II: Non-normal Case

TL;DR

This paper extends the theory of dualizing complexes to non-normal toric face rings, showing that a graded Matlis dual of a Cech complex provides a dualizing complex even in the non-graded setting.

Contribution

It demonstrates that the dualizing complex can be obtained via the graded Matlis dual of a Cech complex for non-normal toric face rings, broadening previous results.

Findings

Dualizing complex constructed via Matlis dual of Cech complex

Applicable to non-normal toric face rings

Generalizes known results for normal cases

Abstract

The notion of "toric face rings" generalizes both Stanley-Reisner rings and affine semigroup rings, and has been studied by Bruns, Romer, et.al. Here, we will show that, for a toric face ring $R$, the "graded" Matlis dual of a Cech complex gives a dualizing complex. In the most general setting, $R$ is not a graded ring in the usual sense. Hence technical argument is required.