Seminormality and local cohomology of toric face rings

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.Comment: Final version. Replace the old proof of Lemma 4.6(ii) by an accurate one. Some minor errors are corrected. This is part of the author's thesis. To appear in Journal of Algebr

Dualizing complexes of seminormal affine semigroup rings and toric face rings

“…In some sense, Proposition 6.5 generalizes and refines the results and the problem in §4 of Nguyen [12] (especially, [12,Theorem 4.3]). However, the toric face rings in [12] are assumed to have nice multigradings, while the "L-grading" of our k[M] is not the grading in the usual sense. Corollary 6.7.…”

“…Recently, Nguyen [12] studied seminormal toric face rings mainly focusing on the local cohomology modules, but he also remarked that k[M] is seminormal if and only if k[M σ ] is seminormal for all σ. In this sense, the seminormality is a natural condition for toric face rings.…”