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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Sheaf cohomology and free resolutions over exterior algebras


Authors: David Eisenbud, Gunnar Fløystad and Frank-Olaf Schreyer
Journal: Trans. Amer. Math. Soc. 355 (2003), 4397-4426
MSC (2000): Primary 14F05, 14Q20, 16E05
Published electronically: July 10, 2003
MathSciNet review: 1990756
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Abstract: We derive an explicit version of the Bernstein-Gel'fand-Gel'fand (BGG) correspondence between bounded complexes of coherent sheaves on projective space and minimal doubly infinite free resolutions over its ``Koszul dual'' exterior algebra. Among the facts about the BGG correspondence that we derive is that taking homology of a complex of sheaves corresponds to taking the ``linear part'' of a resolution over the exterior algebra.

We explore the structure of free resolutions over an exterior algebra. For example, we show that such resolutions are eventually dominated by their ``linear parts" in the sense that erasing all terms of degree $>1$ in the complex yields a new complex which is eventually exact.

As applications we give a construction of the Beilinson monad which expresses a sheaf on projective space in terms of its cohomology by using sheaves of differential forms. The explicitness of our version allows us to prove two conjectures about the morphisms in the monad, and we get an efficient method for machine computation of the cohomology of sheaves. We also construct all the monads for a sheaf that can be built from sums of line bundles, and show that they are often characterized by numerical data.


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Additional Information

David Eisenbud
Affiliation: Department of Mathematics, University of California Berkeley, Berkeley, California 94720
Email: eisenbud@math.berkeley.edu

Gunnar Fløystad
Affiliation: Mathematisk Institutt, Johs. Brunsgt. 12, N-5008 Bergen, Norway
Email: gunnar@mi.uib.no

Frank-Olaf Schreyer
Affiliation: FB Mathematik, Universität Bayreuth D-95440 Bayreuth, Germany
Email: schreyer@btm8x5.mat.uni-bayreuth.de

DOI: http://dx.doi.org/10.1090/S0002-9947-03-03291-4
PII: S 0002-9947(03)03291-4
Received by editor(s): December 1, 2001
Published electronically: July 10, 2003
Additional Notes: The first and third authors are grateful to the NSF for partial support during the preparation of this paper. The third author wishes to thank MSRI for its hospitality.
Article copyright: © Copyright 2003 American Mathematical Society