Seminormality and local cohomology of toric face rings
Let \(K\) be a field and \(S=K[x_1, \dots, x_n]\) be the polynomial ring in the variables \(x_1,\ldots,x_n\). Denote by \(K[\mathcal{M}]=S/I_{\mathcal{M}}\) the toric face ring of \(\mathcal{M}\) over \(K\) where \(I_{\mathcal{M}}\) is the toric face ideal. The toric face rings are a natural generalization of Stanley-Reisner rings and affine monoid rings. As the toric face rings are always reduced. The author uses this fact and the Hochster's theorem showing that: if \(K[\mathcal{M}]\) is normal, then \(K[\mathcal{M}]\) is a normal affine monoid ring. In particular, \(K[\mathcal{M}]\) is Cohen-Macaulay. Next the author deals with a generalization of the result of Bruns, Ichim, Li and Römer [\textit{B. Ichim} and \textit{T. Römer}, J. Pure Appl. Algebra 210, No. 1, 249--266 (2007; Zbl 1117.05113); \textit{W. Bruns} et al., J. Algebra 302, No. 1, 361--386 (2006; Zbl 1112.13024)] concerning local cohomology of seminormal monoid rings as follows: Let \(\mathcal{\sum}\) be a rational pointed fan in \(\mathbb{R}^d\) where \(d\geq 1\) and \(\mathcal{M}\) a seminormal monoidal complex supported on \(\sum\). If \(H^i_{\mathfrak m}(K[\mathcal{M}])_a\neq 0\) for some \(a \in \mathbb{Z}^d\), then \(a\in -\overline{M_C}\) for a cone \(C \in \sum\) of dimension less that or equal \(i\). In particular, if \(a \not \in \bigcup_{C\in \sum}(-\overline{M_C})\), then \(H^i_{\mathfrak m}(K[\mathcal{M}])_a=0\). Here \(\mathfrak m\) is the unique graded maximal ideal of \(K[\mathcal{M}]\) and \(\overline{M_C}\) is the normalization of \(M_C=C \cap \mathbb{Z}^d\) in \(\mathbb{Z}M_C\). Moreover, a combinatorial formula is obtained to generalize Hochster's formula for local cohomology of Stanley-Reisner rings. Finally, the author characterizes the \(F\)-purity and \(F\)-splitness of the toric face ring \(K[\mathcal{M}]\) with \(\mathrm{char} K=p>0.\) As a consequence one obtains the following well-known result of \textit{M. Hochster} and \textit{J. L. Roberts} [Adv. Math. 21, 117--172 (1976; Zbl 0348.13007)] which says: If \(I\subseteq S\) be a square-free monomial ideal with \(\mathrm{char} K=p>0\), then \(S/I\) is \(F\)-pure.
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