Local cohomology of certain Rees- and form-rings. II. (English) Zbl 0543.14003

Let \((\mathfrak R,\mathfrak m)\) be a local noetherian ring and let \(I\subseteq R\) be an ideal. By \(\mathfrak R(I)\) we denote the Rees-ring of \(I\), which is defined as \(\mathfrak R(I)=\oplus_{n\ge 0}I^n\). The form ring \(Gr(I)\) of \(I\) is defined as \(Gr(I)=\mathfrak R(I)/I\mathfrak R(I)=\oplus_{n\ge 0}I^n/I^{n+1}\). Recall that the morphism \(\pi: \operatorname{Proj}(\mathfrak R(I))\to \operatorname{Spec}(R)\) defines the blow-up of \(\operatorname{Spec}(R)\) at \(\operatorname{Spec}(R/I)\) and that \(\operatorname{Proj}(Gr(I))\) is the exceptional fiber of this blow-up. This geometric interpretation gives the motivation to calculate the local cohomology modules \(H^i_{\mathfrak m(I)}(\mathfrak R(I))\) and \(H^i_{\mathfrak m(I)}(Gr(I))\) (in a certain range of \(i)\), where \(\mathfrak m(I)\) is the homogeneous maximal ideal \(\mathfrak m\oplus I\oplus I^2\oplus\cdots\) of \(\mathfrak R(I)\) and where \(I\) belongs to a particular class of ideals. This continues a corresponding investigation, begun in the first part of the paper [ibid. 81, 29–57 (1983; Zbl 0475.14001)] .
In part II we are mainly concerned with a class of ideals \(I\), whose Rees-rings may be understood as so called symbolic Rees-rings. In particular, our ideals are normally torsion-free in some important cases. We also investigate the behaviour of our blow-up under Veronese transforms. This leads to a characterization of some special (Buchsbaum-)singularities.
Our results also give the local background for an improved version of Faltings’ “Macaulayfication” [the author, Comment. Math. Helv. 58, 388–415 (1983; Zbl 0526.14035)].
Reviewer: Markus Brodmann


14B15 Local cohomology and algebraic geometry
14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
14B05 Singularities in algebraic geometry
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[1] Brodmann, M, Finiteness of ideal transforms, J. algebra, 63, 162-185, (1980) · Zbl 0396.13018
[2] Brodmann, M, Kohomologische eigenschaften von aufblasungen an lokal vollständingen durchschnitten, (1980), Habilitationsschrift Münster
[3] Brodmann, M, Local cohomology of certain Rees- and form-rings, (1980), I, preprint
[4] Brodmann, M, Blow-up and asymptotic depth of higher conormal modules, (1981), preprint
[5] Faltings, G, Ueber macaulayfixierung, Math. ann., 238, 175-192, (1978)
[6] Flenner, H, Die Sätze von Bertini für lokale ringe, Math. ann., 229, 97-111, (1977) · Zbl 0398.13013
[7] Goto, S; Shimoda, Y, On Rees-algebras over Buchsbaum-rings, J. math. Kyoto univ., 20, 691-708, (1980) · Zbl 0473.13010
[8] Grothendieck, A, Ega iv, Publ. math. IHES, 24, (1965)
[9] Grothendieck, A, Sga ii, (1968), Noth-Holland Amsterdam
[10] Grothendieck, A, Local cohomology, () · Zbl 0185.49202
[11] Hartshorne, R, Residues and duality, () · Zbl 0196.24301
[12] Hartshorne, R, Algebraic geometry, (1977), Springer Pub New York · Zbl 0367.14001
[13] Herzog, J; Simis, A; Vasconcelos, W.V, Approximation complexes of blowing-up rings, (1980), preprint · Zbl 0484.13006
[14] {\scC. Huneke}, The theory of d-sequences and powers of ideals, Adv. in Math., in press. · Zbl 0505.13004
[15] Huneke, C, On the symmetric and Rees algebra of an ideal generated by a d-sequence, J. algebra, 62, 268-275, (1980) · Zbl 0439.13001
[16] Matsumura, H, Commutative algebra, (1970), Benjamin New York · Zbl 0211.06501
[17] Schenzel, P, Regular sequences in Rees- and symmetric algebras, (1981), preprint · Zbl 0483.13009
[18] Schenzel, P; Stückrad, J; Vogel, W, Theorie der Buchsbaum-moduln, (1979), preprint 13/24, Halle
[19] Rees, D, The grade of an ideal or module, (), 28-42 · Zbl 0079.26602
[20] Valla, G, Certain graded algebras are always Cohen-Macaulay, J. algebra, 42, 537-548, (1976) · Zbl 0338.13013
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