Let X be a Noetherian scheme. A birational proper morphism Y → X of schemes is said to be a Macaulayfication of X if Y is a Cohen-Macaulay scheme. This notion was introduced by Faltings and he established that there exists a Macaulayfication of a quasi-projective scheme over a Noetherian ring possessing a dualizing- complex if its non-Cohen-Macaulay locus is of dimension 0 or 1. Of course, a desingularization is a Macaulayfication and Hironaka gave a desingularization of arbitrary algebraic variety over a field of characteristic 0. But Faltings' method to construct a Macaulayfication is independent of the characteristic of a scheme. Furthermore, several authors are interested in a Macaulayfication.
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