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Relative Beilinson monad and direct image for families of coherent sheaves

Authors: David Eisenbud and Frank-Olaf Schreyer
Journal: Trans. Amer. Math. Soc. 360 (2008), 5367-5396
MSC (2000): Primary 14F05, 13D02
Published electronically: April 17, 2008
MathSciNet review: 2415078
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Abstract: The higher direct image complex of a coherent sheaf (or finite complex of coherent sheaves) under a projective morphism is a fundamental construction that can be defined via a Čech complex or an injective resolution, both inherently infinite constructions. Using free resolutions it can be defined in finite terms. Using exterior algebras and relative versions of theorems of Beilinson and Bernstein-Gel$ '$fand-Gel$ '$fand, we give an alternate and generally more efficient description in finite terms.

Using this exterior algebra description we can characterize the generic finite free complex of a given shape as the direct image of an easily-described vector bundle. We can also give explicit descriptions of the loci in the base spaces of flat families of sheaves in which some cohomological conditions are satisfied: for example, the loci where vector bundles on projective space split in a certain way, or the loci where a projective morphism has higher dimensional fibers.

Our approach is so explicit that it yields an algorithm suited for computer algebra systems.

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Additional Information

David Eisenbud
Affiliation: Department of Mathematics, University of California, Berkeley, Berkeley, California 94720

Frank-Olaf Schreyer
Affiliation: Mathematik und Informatik, Campus E 2.4, Universität des Saarlandes, D-66123 Saar- brücken, Germany

Received by editor(s): July 31, 2005
Received by editor(s) in revised form: September 28, 2006
Published electronically: April 17, 2008
Article copyright: © Copyright 2008 American Mathematical Society

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