Non-Cohen-Macaulay unique factorization domains in small dimensions
In 1964 \textit{M. P. Murthy} [see Arch. Math. 15, 418--420 (1964; Zbl 0123.03401)] showed that a Cohen-Macaulay affine factorial domain is in fact a Gorenstein ring. Later, Samuel asked for the existence of a local unique factorization domain which is not a Cohen-Macaulay ring. The first example of a non-Cohen-Macaulay factorial domain was constructed by \textit{M.-J. Bertin} [see, C. R. Acad. Sci., Paris, Sér. A 264, 653--656 (1967; Zbl 0147.29503)]. In the paper under review the authors construct examples of non-Cohen-Macaulay unique factorization domains in small dimension. They find a unique factorization domain of dimension 3 which is not a Cohen-Macaulay ring. Moreover, there is an example of a five-dimensional affine ring \(S\) over a field \(k\) with the property that \(S\) is a non-Cohen-Macaulay unique factorization domain whenever Char \(k=2\), while it is a Gorenstein non-factorial ring for Char \(k\neq 2\).
scientific article; zbMATH DE number 5872536
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