Base change for semiorthogonal decompositions

Let X be an algebraic variety over a base scheme S and ϕ: T → S a base change. Given an admissible subcategory 𝒜 in 𝒟 b ( X ), the bounded derived category of coherent sheaves on X , we construct under some technical conditions an admissible subcategory 𝒜 T in 𝒟 b ( X × S T ), called the base change of 𝒜, in such a way that the following base change theorem holds: if a semiorthogonal decomposition of 𝒟 b ( X ) is given, then the base changes of its components form a semiorthogonal decomposition of 𝒟 b ( X × S T ) . 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.