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

Brodmann, M.

J. Algebra 143, No. 2, 401-435 (1991).

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)

MSC:

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

Keywords:

pull back of standard blowing up; Rees-modification of a coherent sheaf; birational model; Macaulayfication; birational models; Macaulay fications

References:

[1]	Marca, G.a, Lokale Kohomologie und Komplemente affiner Hyperflaechen (1988), Diplomarbeit, Universitaet Zuerich
[2]	Brodmann, M., Kohomologische Eigenschaften von Aufblasungen an lokal vollstaendigen Durchschnitten (1980), Habilitationsschrift: Habilitationsschrift Muenster
[3]	Brodmann, M., Finiteness of ideal transforms, J. Algebra, 63, 162-185 (1980) · Zbl 0396.13018
[4]	Brodmann, M., Local cohomology of certain Rees- and form-rings, I, J. Algebra, 81, 29-57 (1983) · Zbl 0475.14001
[5]	Brodmann, M., Local cohomology of certain Rees- and form-rings, II, J. Algebra, 86, 457-493 (1984) · Zbl 0543.14003
[6]	Brodmann, M., Two types of birational models, Comment. Math. Helv., 58, 388-415 (1983) · Zbl 0526.14035
[7]	Brodmann, M., A few remarks on blowing-up and connectedness, J. Reine Angew. Math., 370, 52-60 (1986) · Zbl 0578.14020
[9]	Cottini, G., Lokale Kohomologie, Stein-Grothendieck-Faktoren und Aufblasungen, (Diplomarbeit (1988), Universitaet Zuerich)
[10]	Faltings, G., Ueber Macaulayfizierung, Math. Ann., 238, 175-192 (1978) · Zbl 0398.14002
[11]	Faltings, G., Der Endlichkeitssatz in der lokalen Kohomologie, Math. Ann., 255, 45-56 (1981) · Zbl 0451.13008
[12]	Goto, S., Blowing-up of Buchsbaum rings, (Commutative Algebra. Commutative Algebra, London Mathematical Society Lecture Note Series, Vol. 72 (1981), Cambridge Univ. Press: Cambridge Univ. Press London/New York), 140-162
[13]	Goto, S., A note on standard systems of parameters for generalized CM modules, (Proceedings, Symp. (1982), Kyoto University: Kyoto University Kyoto), 181-182
[14]	Goto, S.; Shimoda, Y., On Rees-algebras over Buchsbaum rings, J. Math. Kyoto Univ., 20, 691-708 (1980) · Zbl 0473.13010
[16]	Grothendieck, A., EGA, III, Publ. Math. IHES, 11 (1961)
[17]	Grothendieck, A., EGA, IV, Publ. Math. IHES, 24 (1969)
[18]	Grothendieck, A., SGA, II (1968), Paris
[19]	Hartshorne, R., Algebraic Geometry (1977), Springer: Springer New York · Zbl 0367.14001
[20]	Hartshorne, R., Cohomological dimension of algebraic varieties, Ann. of Math., 99, 403-450 (1968) · Zbl 0169.23302
[21]	Matsumura, H., Commutative Algebra (1980), Benjamin: Benjamin London · Zbl 0211.06501
[22]	Rees, D., On a problem of Zariski, Illinois J. Math., 2, 145-149 (1968) · Zbl 0078.02702
[23]	Schenzel, P., Standard systems of parameters and their blowing-up rings, J. Reine Angew. Math., 344, 201-220 (1983) · Zbl 0497.13012
[24]	Schenzel, P.; Cuong, N. T.; Trung, N. V., Verallgemeinerte Cohen-Macaulay-Moduln, Math. Nachr., 35, 57-73 (1978) · Zbl 0398.13014
[25]	Schneider, M.; Spindler, H.; Okonek, C., Vector bundles on complex projective space, (Progr. Math., Vol. 3 (1980), Birkhäuser: Birkhäuser Basel)
[26]	Serre, J. P., Faisceaux algébriques cohérents, Ann. of Math., 197-278 (1955) · Zbl 0067.16201
[27]	Stueckrad, J.; Vogel, W., Buchsbaum Rings and Applications (1986), VEB Deutscher Verlag der Wissenschaften: VEB Deutscher Verlag der Wissenschaften Berlin
[28]	Trung, N. V., Standard-system of parameters of generalized CM-modules, (Rep. Kariusawa Symp. Comm. Alg. (1982)), 1-17
[29]	Trung, N. V., Zur Theorie der Buchsbaum-Ringe (1983), Halle