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Bounds for cohomological deficiency functions of projective schemes over Artinian rings


Brodmann, M; Matteotti, C; Minh, N (2003). Bounds for cohomological deficiency functions of projective schemes over Artinian rings. Vietnam Journal of Mathematics, 31(1):71-113.

Abstract

Let $X$ be a projective scheme over an artinian commutative ring $R_0$ and let $\Cal{F}$ be a coherent sheaf of $\Cal{O}_X$-modules. We give bounds on the so called cohomological deficiency functions $\Delta^i_{X, \Cal{F}}$ and the cohomological postulation numbers $\nu^i_{X, \Cal{F}}$ of the pair $(X, \Cal{F}).$ As bounding invariants we use the "cohomology diagonal" $ \big( h^j_{X, \Cal{F}}(-j) \big)_{j\le i}$ at and below level $i$ and the $i$-th "cohomological Hilbert polynomial" $p^i_{X, \Cal{F}}$ of the pair $(X, \Cal{F}).$ Our bounds present themselves as a quantitative and extended version of the vanishing theorem of Severi\,--\,Enriques\,--\,Zariski\,--\,Serre.

Abstract

Let $X$ be a projective scheme over an artinian commutative ring $R_0$ and let $\Cal{F}$ be a coherent sheaf of $\Cal{O}_X$-modules. We give bounds on the so called cohomological deficiency functions $\Delta^i_{X, \Cal{F}}$ and the cohomological postulation numbers $\nu^i_{X, \Cal{F}}$ of the pair $(X, \Cal{F}).$ As bounding invariants we use the "cohomology diagonal" $ \big( h^j_{X, \Cal{F}}(-j) \big)_{j\le i}$ at and below level $i$ and the $i$-th "cohomological Hilbert polynomial" $p^i_{X, \Cal{F}}$ of the pair $(X, \Cal{F}).$ Our bounds present themselves as a quantitative and extended version of the vanishing theorem of Severi\,--\,Enriques\,--\,Zariski\,--\,Serre.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:2003
Deposited On:27 May 2010 08:47
Last Modified:19 Feb 2018 21:13
Publisher:Springer
ISSN:2305-221X
OA Status:Green
Official URL:http://www.math.ac.vn/publications/vjm/vjm_31/71.html

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