Seminormality and local cohomology of toric face rings
type: Item
instance of: scholarly article
title: Seminormality and local cohomology of toric face rings
zbMATH Open document ID: 1285.13022
author: Dang Hop Nguyen
published in: Journal of Algebra
publication date: 2013-05-27T00:00:00Z
full work available at URL: https://arxiv.org/abs/1112.0375
review text: 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;…
reviewed by: Ahad Rahimi
Mathematics Subject Classification ID: 13D45
zbMATH DE Number: 6166673
zbMATH Keywords: seminormality
zbMATH Keywords: local cohomology
zbMATH Keywords: toric face rings
zbMATH Keywords: monoid rings
zbMATH Keywords: \(F\)-injectivity
zbMATH Keywords: \(F\)-purity
zbMATH Keywords: \(F\)-regularity
MaRDI profile type: Publication
OpenAlex ID: W2160882823
arXiv ID: 1112.0375
cites work: Cohomology of partially ordered sets and local cohomology of section rings
cites work: Subdivisions of toric complexes
cites work: On algebras associated to partially ordered sets
cites work: Polytopes, Rings, and K-Theory
cites work: Q3843140
cites work: On seminormal monoid rings
cites work: Gröbner bases and Betti numbers of monoidal complexes
cites work: Free resolutions of simplicial posets
cites work: F-Purity and Rational Singularity
cites work: Quotient algebras of Stanley-Reisner rings and local cohomology
cites work: Rings of Invariants of Tori, Cohen-Macaulay Rings Generated by Monomials, and Polytopes
cites work: Tight Closure, Invariant Theory, and the Briancon-Skoda Theorem
cites work: The purity of the Frobenius and local cohomology
cites work: On toric face rings
cites work: Topological results in combinatorics
cites work: Dualizing Complex of a Toric Face Ring
cites work: Monomial subrings in arbitrary dimension
cites work: F-injective singularities are Du Bois
cites work: On the Cohen-Macaulay property in commutative algebra and simplicial topology
cites work: The Upper Bound Conjecture and Cohen-Macaulay Rings
cites work: Q3797807
cites work: On seminormality
cites work: Q5605674
cites work: On monotonicity of F-blowup sequences
DOI: 10.1016/J.JALGEBRA.2012.08.017
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description: scientific article