Base change for semiorthogonal decompositions

TL;DR

This paper develops a method for base changing semiorthogonal decompositions in derived categories of algebraic varieties, ensuring their stability under base change and relating projection functors to kernel functors.

Contribution

It introduces a construction for the base change of admissible subcategories in derived categories, preserving semiorthogonal decompositions under base change.

Findings

Base change of semiorthogonal decompositions is well-defined and compatible with fiber products.

Projection functors in semiorthogonal decompositions are representable as kernel functors.

Constructs compatible systems of semiorthogonal decompositions for unbounded derived categories.

Abstract

Consider an algebraic variety $X$ over a base scheme $S$ and a faithful base change $T \to S$. Given an admissible subcategory $\CA$ in the bounded derived category of coherent sheaves on $X$, we construct an admissible subcategory in the bounded derived category of coherent sheaves on the fiber product $X\times_S T$, called the base change of $\CA$, in such a way that the following base change theorem holds: if a semiorthogonal decomposition of the bounded derived category of $X$ is given then the base changes of its components form a semiorthogonal decomposition of the bounded derived category of the fiber product. As an intermediate step we construct a compatible system of semiorthogonal decompositions of the unbounded derived category of quasicoherent sheaves on $X$ and of the category of perfect complexes on $X$. As an application we prove that the projection functors of a semiorthogonal decomposition are kernel functors.