Abstract
We describe the Tate resolution of a coherent sheaf or complex of coherent sheaves on a product of projective spaces. Such a resolution makes explicit all the cohomology of all twists of the sheaf, including, for example, the multigraded module of twisted global sections, and also the Beilinson monads of all twists. Although the Tate resolution is highly infinite, any finite number of components can be computed efficiently, starting either from a Beilinson monad or from a multigraded module.
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Acknowledgments
This paper reports on work started during the Commutative Algebra Program, 2012-13, at the Mathematical Sciences Research Institute in Berkeley, and continued at the Mathematisches Forschungsinstitut Oberwolfach. We are grateful to these institutes for providing a beautiful and exciting environment for this work. The first and second authors are grateful to the National Science Foundation, and the second and third authors are grateful to the Simons Foundation for partial support during this period. This material is based upon work supported by the National Science Foundation under Grant No. 0932078 000, while the authors were in residence at the Mathematical Science Research Institute in Berkeley, California in 2012–2013.
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Dedicated to Ngo Viet Trung on the occasion of his sixtieth birthday
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Eisenbud, D., Erman, D. & Schreyer, FO. Tate Resolutions for Products of Projective Spaces. Acta Math Vietnam 40, 5–36 (2015). https://doi.org/10.1007/s40306-015-0126-z
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DOI: https://doi.org/10.1007/s40306-015-0126-z