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Purity and decomposition theorems for staggered sheaves

Part of: Categories and geometry (Co)homology theory

Published online by Cambridge University Press:  02 April 2012

Pramod N. Achar  and
David Treumann
Pramod N. Achar
Affiliation:
Department of Mathematics, Louisiana State University, 303 Lockett Hall, Baton Rouge, LA 70803–4918,USA (pramod@math.lsu.edu)
David Treumann
Affiliation:
Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208–2730,USA (treumann@math.northwestern.edu)
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Abstract

Two major results in the theory of ℓ-adic mixed constructible sheaves are the purity theorem (every simple perverse sheaf is pure) and the decomposition theorem (every pure object in the derived category is a direct sum of shifts of simple perverse sheaves). In this paper, we prove analogues of these results for coherent sheaves. Specifically, we work with staggered sheaves, which form the heart of a certain t-structure on the derived category of equivariant coherent sheaves. We prove, under some reasonable hypotheses, that every simple staggered sheaf is pure, and that every pure complex of coherent sheaves is a direct sum of shifts of simple staggered sheaves.

Keywords

coherent sheavesderived categoriespuritydecomposition theorem

MSC classification

Secondary: 14F05: Sheaves, derived categories of sheaves and related constructions 18F20: Presheaves and sheaves
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 11 , Issue 4 , October 2012 , pp. 695 - 745
Copyright
Copyright © Cambridge University Press 2012

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