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Cohomological patterns of coherent sheaves over projective schemes

✍ Scribed by M. Brodmann; M. Hellus

Publisher
Elsevier Science
Year
2002
Tongue
English
Weight
181 KB
Volume
172
Category
Article
ISSN
0022-4049

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✦ Synopsis

We study the sets P(X; F) = {(i; n) ∈ N0 Γ— Z | H i (X; F(n)) = 0}, where X is a projective scheme over a noetherian ring R0 and where F is a coherent sheaf of OX -modules. In particular we show that P(X; F) is a so called tame combinatorial pattern if the base ring R0 is semilocal and of dimension 6 1. If X = P d R 0 is a projective space over such a base ring R0, the possible sets P(X; F) are shown to be precisely all tame combinatorial patterns of width 6 d. We also discuss the "tameness problem" for arbitrary noetherian base rings R0 and prove some stability results for the R0-associated primes of the R0-modules H i (X; F(n)).

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A diagonal bound for cohomological postu
A diagonal bound for cohomological postulation numbers of projective schemes
✍ M. Brodmann; A.F. Lashgari πŸ“‚ Article πŸ“… 2003 πŸ› Elsevier Science 🌐 English βš– 197 KB

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 Ξ½ i F of F, e.g., the ultimate places at which the cohomological Hilbert functions n β†’ dim K (H i (X, F(n))) =: h i F (n) start to be polynomial for n 0, are