Divisor class groups of affine semigroup rings associated with distributive lattices
type: Item
instance of: scholarly article
title: Divisor class groups of affine semigroup rings associated with distributive lattices
zbMATH Open document ID: 0759.13009
DOI: 10.1016/0021-8693(92)90021-D
author: Mitsuyasu Hashimoto
author: Takayuki Hibi
author: Atsushi Noma
published in: Journal of Algebra
publication date: 1992-09-27T00:00:00Z
review text: Let \(k\) be a field and let \(L\) be a finite lattice. Let \(R=k[\{X_ \alpha;\alpha\in L\}]\) denote the polynomial ring over \(k\) whose indeterminates \(X_ \alpha\) are labelled by the elements \(\alpha\) of \(L\). Let \({\mathcal I}(L)\) denote the ideal of \(R\) generated by the elements \(X_ \alpha X_ \beta-X_{\alpha\wedge\beta}X_{\alpha\lor\beta}\) and let \(A[L]\) denote the \(k\)-algebra \(R/{\mathcal I}(L)\). The ring \(A[L]\) is an integral domain if and only if the lattice \(L\) is distributive, and in this case \(A[L]\) is normal and Cohen-Macaulay [see, for example, \textit{M. Hashimoto}, J. Pure Appl. Algebra 71, No. 2/3, 157-173 (1991; Zbl 0748.17011)]. Now suppose that \(L\) is the distributive lattice consisting of all order ideals of a finite poset \(P\), ordered by inclusion. In this case, it is proved that the divisor class group of \(A[L]\) is free of rank \(p+q+e-d-1\) where \(p\) (resp. \(q\)) is the number of maximal (resp. minimal) elements of \(P\), \(e\) is the number of edges in the Hasse diagram of \(P\), and \(d\) is the cardinality of \(P\).
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 06D05
Mathematics Subject Classification ID: 13C20
Mathematics Subject Classification ID: 13G05
Mathematics Subject Classification ID: 20M25
zbMATH DE Number: 63616
zbMATH Keywords: affine semigroup rings
zbMATH Keywords: distributive lattice
zbMATH Keywords: divisor class group
reviewed by: Patrick F. Smith
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 63616