Base Change for Semiorthogonal Decompositions

Let X be an algebraic variety over a base scheme S and φ : T → S a faithful base change. Given an admissible subcategory A in D(X), the bounded derived category of coherent sheaves on X, we construct an admissible subcategory AT in D b(X×ST ), called the base change of A, in such a way that the following base change theorem holds: if a semiorthogonal decomposition of D(X) is given then the base changes of its components form a semiorthogonal decomposition of D(X ×S T ). As an intermediate step we construct a compatible system of semiorthogonal decompositions of the unbounded derived category of countably coherent 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.