A diagonal bound for cohomological postulation numbers of projective schemes
✍ Scribed by M. Brodmann; A.F. Lashgari
- Publisher
- Elsevier Science
- Year
- 2003
- Tongue
- English
- Weight
- 197 KB
- Volume
- 265
- Category
- Article
- ISSN
- 0021-8693
No coin nor oath required. For personal study only.
✦ Synopsis
Let X be a projective scheme over a field K and let F be a coherent sheaf of O X -modules. We show that the cohomological postulation numbers ν i F of F, e.g., the ultimate places at which the cohomological Hilbert functions n → dim K (H i (X, F(n))) =: h i F (n) start to be polynomial for n 0, are (polynomially) bounded in terms of the cohomology diagonal
of F. As a consequence, we obtain that there are only finitely many different cohomological Hilbert functions h i F if F runs through all coherent sheaves of O X -modules with fixed cohomology diagonal. In order to prove these results, we extend the regularity bound of Bayer and Mumford [Computational Algebraic Geometry and Commutative Algebra, Proc. Cortona, 1991, Cambridge Univ. Press, 1993, pp. 1-48] from graded ideals to graded modules. Moreover, we prove that the Castelnuovo-Mumford regularity of the dual F ∨ of a coherent sheaf of O P r K -modules F is (polynomially) bounded in terms of the cohomology diagonal of F.