Seminormality and local cohomology of toric face rings

TL;DR

This paper characterizes seminormal and normal toric face rings, extends local cohomology results, provides a combinatorial formula generalizing Hochster's, and explores properties like F-purity and F-regularity.

Contribution

It introduces new characterizations of seminormal and normal toric face rings, extends local cohomology vanishing results, and generalizes Hochster's formula for these rings.

Findings

Characterization of seminormal and normal toric face rings.

Vanishing of certain local cohomology graded parts.

A combinatorial formula generalizing Hochster's formula.

Abstract

We characterize the toric face rings that are normal (respectively seminormal). Extending results about local cohomology of Brun, Bruns, Ichim, Li and R\"omer of seminormal monoid rings and Stanley toric face rings, we prove the vanishing of certain graded parts of local cohomology of seminormal toric face rings. The combinatorial formula we obtain generalizes Hochster's formula. We also characterize all (necessarily seminormal) toric face rings that are $F$-pure or $F$-split over a field of characteristic $p>0$. An example is given to show that $F$-injectivity does not behave well with respect to face projections of toric face rings. Finally, it is shown that weakly $F$-regular toric face rings are normal affine monoid rings.