On Macaulayfication obtained by a blow-up whose center is an equi-multiple ideal. (With an appendix by Kikumichi Yamagishi)
type: Item
instance of: scholarly article
title: On Macaulayfication obtained by a blow-up whose center is an equi-multiple ideal. (With an appendix by Kikumichi Yamagishi)
zbMATH Open document ID: 0876.13010
author: Kazuhiko Kurano
published in: Journal of Algebra
publication date: 1997-11-18T00:00:00Z
review text: Let \((A,m)\) be a local ring and \(L\) an ideal of \(A\). The blow-up of Spec\((A)\) with center in \(L\) is the projective spectrum Proj\((R(L))\) of the Rees algebra \(R(L)\). The map Proj\((R(L))\to\) Spec\((A)\) is called a Macaulayfication of Spec\((A)\) if Proj\((R(L))\) is a Cohen-Macaulay scheme. \textit{G. Faltings} [Math. Ann. 238, 175-192 (1978; Zbl 0398.14002)]\ has shown that Macaulayfication does exist if the non-Cohen-Macaulay locus of Spec\((A)\) is of dimension \(\leq 1\). Due to a recent work of \textit{Kawasaki} [``On the Macaulayfication of certain quasi-projective schemes'' (preprint)]\ we know in the above case that the Macaulayfication can be obtained by an equimultiple ideal \(L\), that is, the dimension of the fiber ring \(G(L)\otimes (A/m)\) is equal to the height of \(L\), where \(G(L)\) denotes the associated graded ring of \(A\) with respect to \(L\). The first main theorem of this paper gives a necessary and sufficient condition for Proj\((R(L))\) to be Cohen-Macaulay when \(L\) is an arbitrary equimultiple ideal. Inspired by an interesting result of \textit{I. M. Aberbach}, \textit{C. Huneke}, and \textit{K. E. Smith} [``A tight closure approach to arithmetic Macaulayfication'', Ill. J. Math. 40, No. 2, 310-329 (1996)] the author goes on to study the case \(L= (y_1,\ldots,y_d)^*\), where \(y_1,\ldots,y_d\) is a subsystem of parameters of \(A\) with each \(y_i\) being a test element and the symbol \(^*\) means the tight closure. In the case \(\dim(A/(y_1,\ldots,y_d)) \leq 1\) he is able to compute the local cohomology modules \(H_M^i(R(L))\) and \(H_M^i(G(L))\) explicitly in terms of the local cohomology modules \(H_m^i(A)\), where \(M\) is the maximal graded ideal of \(R(L)\). The appendix of \textit{K. Yamagishi} deals with the computation of the local cohomology modules \(H_N^i(R(L))\) and \(H_N^i(G(L))\), \(N = LR(L)+R(L)_+\), in a more general settting, namely, when \(L\) has a minimal reduction generated by an unconditioned stron…
reviewed by: Trung Ngo Viet
Mathematics Subject Classification ID: 13H10
zbMATH DE Number: 1014004
zbMATH Keywords: Rees algebra
zbMATH Keywords: associated graded ring
zbMATH Keywords: equimultiple ideal
zbMATH Keywords: Macaulayfication
zbMATH Keywords: \(d\)-sequence
zbMATH Keywords: local cohomology
zbMATH Keywords: tight closure
MaRDI profile type: Publication
full work available at URL: jabr.1996.6904
OpenAlex ID: W2034295427
cites work: A tight closure approach to arithmetic Macaulayfication
cites work: Arithmetic Macaulayfications using ideals of dimension one
cites work: Two types of birational models
cites work: Über Macaulayfizierung. (On macaulayfication)
cites work: The Cohen-Macaulay symbolic Rees algebras for curve singularities
cites work: On graded rings. I
cites work: Local cohomology. A seminar given by A. Grothendieck, Harvard University, Fall 1961. Notes by R. Hartshorne
cites work: Resolution of singularities of an algebraic variety over a field of characteristic zero. I
cites work: Tight Closure, Invariant Theory, and the Briancon-Skoda Theorem
cites work: On Macaulayfication of certain quasi-projective schemes
cites work: Inequalities related to certain couples of local rings
cites work: Idealizations of maximal Buchsbaum modules over a Buchsbaum ring
DOI: 10.1006/JABR.1996.6904
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description: scientific article; zbMATH DE number 1014004