Cohomology of standard blowing-up. (English) Zbl 0751.14010

This paper defines and studies the Rees-modification \(\tilde{\mathcal F}\) of a coherent sheaf \({\mathcal F}\) over a noetherian affine scheme \(X=\text{Spec} R\), under the hypothesis that the local cohomology modules \(H^ i_ p({\mathcal F})\) are finitely generated for all \(i<\dim_ p(X)\), where \(p\in X\) is a closed point. In this situation \(X\) admits the socalled standard blowing up \(\pi:\tilde X\to X\), with respect to \({\mathcal F}\) centered at \(p\) and the author expresses the total cohomology modules \(H^ i_ *(\tilde X,\tilde{\mathcal F})\) and the length of the module \(H^{d-1}(\tilde X,{\mathcal F}(n))\) in terms of \(H^ i_ p({\mathcal F})\) \((1<i<d)\) and \(\Gamma(X-\{p\}),{\mathcal F})\). To do that, he studies the exceptional fibre \(\tilde{\mathcal F}/E\), by computing the cohomology of the exceptional fibre \(E\) with coefficients in the exceptional sheaf and the extremal cohomological Hilbert functions of the exceptional sheaf. Finally, he compares the Rees modification and the pull back \(\pi^*{\mathcal F}\). The results of this paper play an interesting role in the problem of searching certain birational models, the so called Macaulayfications.
Reviewer: C.Massaza (Torino)


14F25 Classical real and complex (co)homology in algebraic geometry
14E05 Rational and birational maps
14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)
14B05 Singularities in algebraic geometry
Full Text: DOI


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