Arithmetic Macaulayfications using ideals of dimension one
A desingularization of a Noetherian scheme \(Y\) is a regular scheme \(X\) together with a proper birational map from \(X\) to \(Y\). If \(X\) is only required to be (locally) Cohen-Macaulay then it is called a Macaulayfication of \(Y\). Hironaka has shown that in case \(Y = \text{Spec}(R)\) where \(R\) is a ring which is essentially of finite type over a field of characteristic zero, there is an ideal \(I\) in \(R\) such that \(\text{Proj}(R[It])\) is a desingularization of \(Y\). The Rees algebra \(R[It]\) is called an arithmetic Macaulayfication of \(Y\) if it is Cohen-Macaulay. This property is significantly stronger than the condition of being locally Cohen-Macaulay. The author shows that every local excellent normal domain \(R\) of dimension \(\geq 3\) has an arithmetic Macaulayfication if its non-\(F\)-rational locus is closed and has dimension \(\leq 1\). (Note that every regular ring is \(F\)-rational.) This generalizes the main result of \textit{I. M. Aberbach, C. Huneke} and \textit{K. E. Smith} in ``A tight closure approach to arithmetic Macaulayfication'', Ill. J. Math. 40, No. 2, 310-329 (1996) where the ring \(R\) is assumed to be \(F\)-rational on its punctured spectrum. The improvement is obtained by some modifications of the arguments given in the paper just mentioned.
scientific article; zbMATH DE number 942665