Moduli spaces of semiorthogonal decompositions in families

For a smooth projective family of schemes we prove that a semiorthogonal decomposition of the bounded derived category of coherent sheaves of the fiber of a point uniquely deforms over an \'etale neighborhood of the point. We do this using a comparison theorem between semiorthogonal decompositions and decomposition triangles of the structure sheaf of the diagonal. We then apply to the latter a deformation theory for morphisms with a fixed lift of the target, which is developed in the appendix. Using this as a key ingredient we introduce a moduli space which classifies semiorthogonal decompositions of the category of perfect complexes of a smooth projective family. We show that this is an \'etale algebraic space over the base scheme of the family, which can be non-quasicompact and non-separated. This is done using Artin's criterion for a functor to be an \'etale algebraic space over the base. We generalize this to families of geometric noncommutative schemes in the sense of Orlov. We also define the open subspace classifying non-trivial semiorthogonal decompositions only, which is used in a companion paper to study indecomposability of derived categories in various examples.