Deformation theory for a morphism in the derived category with fixed lift of the codomain

TL;DR

This paper develops a deformation-obstruction framework for morphisms in derived categories with fixed codomain lifts, and applies it to prove the unique deformation of semiorthogonal decompositions in smooth proper families.

Contribution

It introduces a new deformation-obstruction calculus for morphisms with fixed codomain lifts in derived categories, providing tools for deformation theory in algebraic geometry.

Findings

Deformation-obstruction calculus for morphisms with fixed codomain in derived categories.

Proof of unique deformation of semiorthogonal decompositions in smooth proper families.

Application to deformations of complexes in derived categories.

Abstract

We develop the deformation-obstruction calculus for morphisms of complexes with a fixed lift of the codomain, to derived categories of flat nilpotent deformations of abelian categories. As an application, we give an alternative proof that semiorthogonal decompositions deform uniquely in smooth proper families of schemes.