On Macaulayfication obtained by a blow-up whose center is an equi-multiple ideal. (With an appendix by Kikumichi Yamagishi)
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 strong \(d\)-sequence which means that any sequence of powers of the given elements in any order forms a \(d\)-sequence. The first theorem gives information on the graded vanishing of these local cohomology modules. The second theorem gives explicit formulas for \(H_N^i(R(L))\) and \(H_N^i(G(L))\) in terms of the local cohomology modules \(H_L^i(A)\) when the reduction number of \(L\) is one. Note that the case \(L = m\) has been already studied by \textit{S. Goto} [J. Algebra 76, 383-399 (1982; Zbl 0482.13012)]\ and \textit{M. Brodmann} [``Kohomologische Eigenschaften von Aufblasungen an lokal vollständigen Durchschnitten'', Habilitationsschrift (Münster 1980)] and the case of an \(m\)-primary ideal by the reviewer [Ngô Viêt Trung, Nagoya Math. J. 102, 1-49 (1986; Zbl 0649.13008)], where the existence of such an ideal \(L\) can be expressed solely in a numerical relation involving the length \(\ell(L/L^2)\) and the multiplicity \(e(L)\).
scientific article; zbMATH DE number 1014004
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