## 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)

### Citations:

Zbl 0398.14002; Zbl 0526.14035