Base change for semiorthogonal decompositions

By Alexander Kuznetsov. Compositio Mathematica, 2007

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Abstract 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.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/cea43130980e6f3d4951a9faf32d6bbc60fdc50c

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Citations in the corpus

New derived symmetries of some hyperkähler varieties (2011, background): “tibilities among units and counits hold at the level of kernels [7, Appendix]. The same is true if A and Bare admissible subcategories of these, because the projection functors are induced by kernels [25].3 It is also possible to do business with derived categories of non-compact and singular variet”
Semiorthogonal decompositions in algebraic geometry (2014, background): “near. The advantage of linear semiorthogonal decompositions lies in the fact that they obey a base change result. For a base change T → S denote by π : X ×S T → X the induced projection. Theorem 1.8 ([K11]). If X is an algebraic variety over S and (15) is an S-linear semiorthogonal decomposition the”
A category of kernels for equivariant factorizations and its implications for Hodge theory (2011, background): “e have to replace S(Z) with the graded algebra M i≥0 HomDb(cohZ×Z)(P,P∗{i}∗P) where P= Φ ◦Φ! is the kernel associated to the inclusion of Dabs[fact(An,G m,w)] → Db(cohZ) as an admissible subcategory, [Kuz11]. In the general type case, we have diﬀerent kernels, Ki= Ψ!◦(i)◦Ψ, for each i. The natural r”
The K3 category of a cubic fourfold (2015, background): “First note that by results of Kuznetsov [Kuz09] one has HH ∗(AX′) ' Ext∗X′×X(P, P ) ' HH ∗(AX).”
Fibrations in complete intersections of quadrics, Clifford algebras, derived categories, and rationality problems (2011, background): “A theory of base change for triangulated category has been developed in [65].”