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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

MSC:

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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