Motivated by the local flavor of several well-known semi-orthogonal
decompositions in algebraic geometry, we introduce a technique called
conservative descent, which shows that it is enough to establish these
decompositions locally. The decompositions we have in mind are those for
projectivized vector bundles and blow-ups, due to Orlov, and root stacks, due
to Ishii and Ueda. Our technique simplifies the proofs of these decompositions
and establishes them in greater generality for arbitrary algebraic stacks.Comment: Final versio
