Moduli spaces of semiorthogonal decompositions in families

To a smooth and proper morphism $\mathcal{X}\to U$ with quasicompact semiseparated target we associate a sheaf in the 'etale topology, which takes an affine $U$-scheme $V$ to the set of $V$-linear semiorthogonal decompositions (of fixed length) of the category $\mathrm{Perf}{\mathcal{X}}_V$. We use Artin’s criterion to prove that, when $U$ is excellent, this is in fact an algebraic space which is moreover 'etale (though in general non-quasicompact and non-separated) over $U$. We moreover generalise the construction of the sheaf to families of geometric noncommutative schemes in the sense of Orlov. We also define a subfunctor classifying nontrivial semiorthogonal decompositions, and conjecture it is an open and closed subspace. Along the way, we prove that for a smooth and proper family of schemes, a semiorthogonal decomposition of the bounded derived category of coherent sheaves of a fibre uniquely deforms over an 'etale neighbourhood of the point.