×

A few remarks on “Macaulayfication” of sheaves. (English) Zbl 0822.14022

Bruns, Winfried (ed.) et al., Commutative algebra: Extended abstracts of an international conference, July 27 - August 1, 1994, Vechta, Germany. Cloppenburg: Runge. Vechtaer Universitätsschriften. 13, 29-32 (1994).
Let \(X\) be a quasiprojective scheme over a noetherian regular ring \(K\). A Macaulayfication of \(X\) is a proper birational morphism \(\pi : \widetilde X \to X\) such that \(\widetilde X\) is a Cohen-Macaulay (CM) scheme. In Math. Ann. 238, 175-192 (1978; Zbl 0398.14002), G. Faltings gave a method for constructing such a Macaulayfication under the assumption that \(\dim \text{NCM} (X) \leq 1\), where \(\text{NCM} (X)\) is the locus of the non-CM-points of \(X\). The author of the present paper also gave a procedure of construction of a Macaulayfication [in Comment. Math. Helv. 58, 388-415 (1983; Zbl 0526.14035)], under the same hypothesis: \(\dim \text{NCM} (X) \leq 1\). In this article the latter method is extended to coherent sheaves over \(X\). The idea is to use the \(Z^{(2)}\)-closure \({\mathcal F}'\) of a torsion-free coherent sheaf \({\mathcal F}\) of \({\mathcal O}_ x\)-modules (defined by Grothendieck) and the scheme \(X' = \text{Spec} ({\mathcal O}_ x)'\) which is the least finite birational \(S_ 2\)-scheme over \(X\). Let \(\pi : Y \to X\) be a proper birational morphism of \(K\)- schemes and \({\mathcal F}\) be a coherent sheaf of \({\mathcal O}_ x\)-modules. Then the modification \(\overline \pi {\mathcal F}\) of \({\mathcal F}\) is defined by \(\overline \pi {\mathcal F} = \pi^* {\mathcal F}/{\mathcal G}\) where \({\mathcal G} \subset \pi^* {\mathcal F}\) is the subsheaf of all sections which vanish at the exceptional locus of \(\pi\). Moreover, \(\overline \pi {\mathcal F}\) is torsion free. Then \((\overline \pi {\mathcal F})'\) can be defined as a sheaf over the \(S_ 2\)-scheme \(Y'\) over \(Y\). Main result:
Theorem 1: If the locus \(\text{NCM} ({\mathcal F})\) of non-CM points of \({\mathcal F}\) has dimension \(\leq 1\) then there exists a proper birational morphism \(\pi : Y \to X\) such that \((\overline \pi {\mathcal F})'\) is a sheaf of CM-modules over \(Y'\). The scheme \(Y\) is constructed as a blowing-up morphism of \(X\) with respect to a sheaf of ideals \(I\) on \(X\).
For the entire collection see [Zbl 0799.00021].

MSC:

14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
14E05 Rational and birational maps
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
PDF BibTeX XML Cite