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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
DOI: https://doi.org/10.1090/S0002-9939-2014-12142-3
Published electronically: March 19, 2014
MathSciNet review: 3195757
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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 \[ \{h_{\mathcal {E}} \mid \mathcal {E}\in \mathcal {V}^t_K : \ h^i_{\mathcal {E}}(n) \leq h^{(i,n)} \mbox { for all} \ (i,n) \in \mathbb {S}\}\] 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 \[ \{(i,n_i)| \ i=0,\ldots ,t\} \mbox { with integers } n_0> n_1 > \cdots > n_t.\]


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Additional Information

Markus Brodmann
Affiliation: University of Zürich, Institute of Mathematics, Winterthurerstrasse 190, 8057 Zürich, Switzerland
MR Author ID: 41830
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
MR Author ID: 99940
ORCID: 0000-0002-4493-2040
Email: benikeller@access.uzh.ch

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