On Macaulayfication of Noetherian schemes

Abstract

The Macaulayfication of a Noetherian scheme X is a birational proper morphism from a Cohen-Macaulay scheme to X. In 1978 Faltings gave a Macaulayfication of a quasi-projective scheme if its non-Cohen-Macaulay locus is of dimension 0 or 1. In the present article, we construct a Macaulayfication of Noetherian schemes without any assumption on the non-Cohen-Macaulay locus. Of course, a desingularization is a Macaulayfication and, in 1964, Hironaka already gave a desingularization of an algebraic variety over a field of characteristic 0. Our method, however, to construct a Macaulayfication is independent of the characteristic.