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Local cohomology annihilators and Macaulayfication. (English) Zbl 1365.13038

A Macaulayfication of a Noetherian scheme \(X\) is a pair \((Y, \pi)\) consisting of a Cohen-Macaulay scheme \(Y\) and of a birational proper morphism \(\pi: Y\rightarrow X\). T. Kawasaki [Trans. Am. Math. Soc. 354, No. 1, 123–149 (2002; Zbl 1087.13502)] constructed a Macaulayfication for any quasi-projective scheme over a Noetherian ring provided the ground ring admits a dualizing complex by using the notion of \(p\)-standard systems of parameters, introduced by by the first author [Math. Proc. Camb. Philos. Soc. 109, No. 3, 479–488 (1991; Zbl 0732.13005)]. Note that local cohomology annihilators appear through the notion of \(p\)-standard system of parameters.
For an ideal \(I \subset R\) of a Noetherian ring the Rees ring \(\mathcal{R}(I)\) gives rise to a blowing up \(Y = \text{Proj } \mathcal{R}(I) \overset{\pi}{\text{Spec }} R\). If \(\mathcal{R}(I)\) is a Cohen-Macaulay ring, the Macaulayfication \(Y\) is called an arithmetic Macaulayfication. This concept is extended to \(R\)-modules \(M\) and the form module \(\mathcal{R}(M,I)\). Then it is shown that the following are equivalent: (1) \(M\) admits an arithmetic Macaulayfication. (2) \(M\) admits a \(p\)-standard system of parameters. (3) \(R/\text{Ann}_R M\) admits a \(p\)-standard system of parameters.
Moreover the authors prove the equivalence of the following: (a) \(R\) admits a \(p\)-standard system of parameters. (b) \(R\) is universally catenary and for any quotient \(S\) of \(R\), \(\text{Spec} (S)\) has a Macaulayfication. (c) All essentially of finite type \(R\)-algebras verify Faltings’ Annihilator Theorem. – As an application this shows the existance of a Macaulayfication of the spectrum of a local ring that is the quotient of Cohen-Macaulay ring.

MSC:

13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
13D45 Local cohomology and commutative rings
14B05 Singularities in algebraic geometry
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