## 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

### Citations:

Zbl 0475.14001; Zbl 0526.14035
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### References:

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