Conservative descent for semi-orthogonal decompositions

TL;DR

This paper introduces conservative descent, a technique that simplifies and generalizes the proof of semi-orthogonal decompositions in algebraic geometry, applicable to various geometric constructions.

Contribution

The paper presents conservative descent, a new method that reduces the need for global arguments by establishing semi-orthogonal decompositions locally, extending known results to broader contexts.

Findings

Simplifies proofs of semi-orthogonal decompositions

Extends decompositions to arbitrary algebraic stacks

Provides a unified local approach for various geometric cases

Abstract

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.