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

Abstract. Toric face rings is a generalization of the concepts of affine monoid rings and Stanley-Reisner rings. We consider several properties which imply Koszulness for toric face rings over a field k. Generalizing works of Laudal, Sletsjøe and Herzog et al., graded Betti numbers of k over the toric face rings are computed, and a characterization of Koszul toric face rings is provided. We investigate a conjecture suggested by Römer about the sufficient condition for the Koszul property. The conjecture is inspired by Fröberg's theorem on the Koszulness of quadratic squarefree monomial ideals. Finally, it is proved that initially Koszul toric face rings are affine monoid rings.

No abstract

No abstract