## Bounding patterns for the cohomology of vector bundles

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- by Markus Brodmann, Andri Cathomen and Bernhard Keller PDF
- Proc. Amer. Math. Soc.
**142**(2014), 2327-2336 Request permission

## 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
- Received by editor(s): August 2, 2012
- Published electronically: March 19, 2014
- Communicated by: Irena Peeva
- © Copyright 2014 American Mathematical Society
- 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
- MathSciNet review: 3195757