Cohen-Macaulay seminormalizations of unions of linear subspaces
Let \(R\) be the coordinate ring of a finite union of subspaces of an affine space and \(S\) be its seminormalization. If \(X\) is the corresponding intersection poset and \(\mathcal A\) the sheaf of coordinate rings on \(X\), then the author shows that \(S\) is Cohen-Macaulay if and only if \(\mathcal A\) is ``gradual'' and for every \(x\in X\), \(\tilde H^i(X^x,\mathcal A)=0\) for \(i< \mathrm{rk}(x)-2\), where \(X^x:=\{y\in X\mid y<x\}\). The main idea is to use a result of Orecchia that expresses \(S\) as \(\Gamma(\mathcal A)\), then to estimate the depth of \(S\) by constructing a resolution by modules of known depth. The result generalizes an equivalent version of Reisner's theorem on the Cohen-Macaulay property of face rings of simplicial posets.
scientific article; zbMATH DE number 4156609