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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Bounding patterns for the cohomology of vector bundles


Authors: Markus Brodmann, Andri Cathomen and Bernhard Keller
Journal: Proc. Amer. Math. Soc. 142 (2014), 2327-2336
MSC (2010): Primary 13D45, 13D07; Secondary 14B15
Published electronically: March 19, 2014
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Abstract: Let $ t \in \mathbb{N}$, let $ K$ be a field and let $ \mathcal {V}^t_K$ denote the class of all algebraic vector bundles over the projective space $ \mathbb{P}^t_K$.

The cohomology table of a bundle $ \mathcal {E} \in \mathcal {V}^t_K$ is defined as the family of non-negative integers $ h_{\mathcal {E}}:= \big (h^i(\mathbb{P}^t_K,\mathcal {E}(n))\big )_{(i,n) \in \mathbb{N}_0 \times \mathbb{Z}}$.

A set $ \mathbb{S} \subseteq \{0,\ldots ,t\}\times \mathbb{Z}$ is said to be a bounding pattern for the cohomology of vector bundles over $ \mathbb{P}^t_K$ if for each family $ (h^{(i,n)})_{(i,n) \in \mathbb{S}}$ of non-negative integers, the set of cohomology tables

$\displaystyle \{h_{\mathcal {E}} \mid \mathcal {E}\in \mathcal {V}^t_K : \ h^i_{\mathcal {E}}(n) \leq h^{(i,n)}$$\displaystyle \mbox { for all} \ (i,n) \in \mathbb{S}\}$

is finite. Our main result says that this is the case if and only if $ \mathbb{S}$ contains a quasi-diagonal of width $ t$, that is, a set of the form

$\displaystyle \{(i,n_i)\vert \ i=0,\ldots ,t\}$$\displaystyle \mbox { with integers } n_0> n_1 > \cdots > n_t.$


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Additional Information

Markus Brodmann
Affiliation: University of Zürich, Institute of Mathematics, Winterthurerstrasse 190, 8057 Zürich, Switzerland
Email: brodmann@math.uzh.ch

Andri Cathomen
Affiliation: University of Zürich, Institute of Mathematics, Winterthurerstrasse 190, 8057 Zürich, Switzerland
Email: a.cathomen@gmail.com

Bernhard Keller
Affiliation: University of Zürich, Institute of Mathematics, Winterthurerstrasse 190, 8057 Zürich, Switzerland
Email: benikeller@access.uzh.ch

DOI: http://dx.doi.org/10.1090/S0002-9939-2014-12142-3
PII: S 0002-9939(2014)12142-3
Keywords: Algebraic vector bundles, boundedness of cohomology, cohomology table
Received by editor(s): August 2, 2012
Published electronically: March 19, 2014
Communicated by: Irena Peeva
Article copyright: © Copyright 2014 American Mathematical Society