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Supernatural analogues of Beilinson monads

Part of: (Co)homology theory Commutative algebra: Homological methods

Published online by Cambridge University Press:  14 November 2016

Daniel Erman  and
Steven V Sam
Daniel Erman
Affiliation:
Department of Mathematics, University of Wisconsin, Madison, WI 53706-1325, USA email derman@math.wisc.edu
Steven V Sam
Affiliation:
Department of Mathematics, University of California, Berkeley, CA 94720-3840, USA Current address:Department of Mathematics, University of Wisconsin, Madison, WI 53706-1325, USA email svs@math.wisc.edu
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Abstract

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We use supernatural bundles to build $\mathbf{GL}$-equivariant resolutions supported on the diagonal of $\mathbb{P}^{n}\times \mathbb{P}^{n}$, in a way that extends Beilinson’s resolution of the diagonal. We thus obtain results about supernatural bundles that largely parallel known results about exceptional collections. We apply this construction to Boij–Söderberg decompositions of cohomology tables of vector bundles, yielding a proof of concept for the idea that those positive rational decompositions should admit meaningful categorifications.

Keywords

Sheaf cohomologymonadsBoij–Söderberg theory

MSC classification

Primary: 14F05: Sheaves, derived categories of sheaves and related constructions 13D02: Syzygies, resolutions, complexes
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 12 , December 2016 , pp. 2545 - 2562
Copyright
© The Authors 2016 

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