Divisor class groups of affine semigroup rings associated with distributive lattices
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 \(P\).
scientific article; zbMATH DE number 63616
scientific article