Motivated by the local flavor of several well-known semi-orthogonal decompositions in algebraic geometry we introduce a technique called “conservative descent” in order to establish such decompositions locally. The decompositions we have in mind are those for projective bundles, blow-ups and root constructions. Our technique simplifies the proof of these decompositions and establishes them in greater generality. We also discuss semi-orthogonal decompositions for Brauer-Severi varieties. This is joint work with Daniel Bergh (Copenhagen).
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