On Macaulayfication of sheaves
Let \(X\) be a Noetherian scheme. We denote by \(\text{CM}(X)\) the Cohen-Macaulay locus of \(X\). A Macaulayfication of \(X\) is a proper birational morphism \(f\) from a Cohen-Macaulay Noetherian scheme \(Y\) to \(X\) such that the restriction of \(f\) on \(f^{-1}(\text{CM}(X))\) is an isomorphism. \textit{G. Faltings} [Math. Ann. 238, 175-192 (1978; Zbl 0398.14002)] constructed Macaulayfications for quasi-projective schemes \(X\) over a regular ring with \(\dim(X \setminus \text{CM}(X)) \leq 1\). The paper under review proves a similar result on Macaulayfication of coherent sheaves whose non-Cohen-Macaulay locus has dimension at most one. The proof follows Brodman's method of Macaulayfication which preserves the regular locus [\textit{M. Brodman}, Comment. Math. Helv. 58, 388-415 (1983; Zbl 0526.14035)]. The proper birational morphism that resolves the non-Cohen-Macaulay locus of a sheaf is obtained by blowing up a sheaf of ideals which is locally generated by standard sequences. Standard sequences were introduced also by Brodmann and they were systematically studied by the reviewer [\textit{Ngo Viet Trung}, ``Standard systems of parameters of generalized Cohen-Macaulay modules'', in: Commutative Algebra, Proc., 4-th Japan Symposium on Commutative Algebra, Karuizawa 1982, 164-180 (1982); see also Nagoya Math.~J. 102, 1-49 (1986; Zbl 0637.13013)].
scientific article; zbMATH DE number 1339446