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Base change for semiorthogonal decompositions

Part of: Abelian categories Algebraic geometry: Foundations

Published online by Cambridge University Press:  15 February 2011

Alexander Kuznetsov
Alexander Kuznetsov*
Affiliation:
Algebra Section, Steklov Mathematical Institute, 8 Gubkin str., Moscow 119991, Russia (email: akuznet@mi.ras.ru) The Poncelet Laboratory, Independent University of Moscow, 119002, Bolshoy, Vlasyevskiy Pereulok 11, Moscow, Russia
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Abstract

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Let X be an algebraic variety over a base scheme S and ϕ:T→S a base change. Given an admissible subcategory 𝒜 in 𝒟b(X), the bounded derived category of coherent sheaves on X, we construct under some technical conditions an admissible subcategory 𝒜T in 𝒟b(X×ST), called the base change of 𝒜, in such a way that the following base change theorem holds: if a semiorthogonal decomposition of 𝒟b (X)is given, then the base changes of its components form a semiorthogonal decomposition of 𝒟b (X×ST) . As an intermediate step, we construct a compatible system of semiorthogonal decompositions of the unbounded derived category of quasicoherent sheaves on X and of the category of perfect complexes on X. As an application, we prove that the projection functors of a semiorthogonal decomposition are kernel functors.

Keywords

base changesemiorthogonal decomposition

MSC classification

Secondary: 14A22: Noncommutative algebraic geometry 18E30: Derived categories, triangulated categories
Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 3 , May 2011 , pp. 852 - 876
Copyright
Copyright © Foundation Compositio Mathematica 2011

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