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

Pieter Belmans, Wendy Lowen, Shinnosuke Okawa, Andrea T. Ricolfi

TL;DR

The paper develops a deformation-obstruction calculus for lifting morphisms of complexes with a fixed lift of the codomain to derived categories under flat nilpotent deformations of abelian categories. It defines obstruction classes living in Ext-type groups obtained from derived Hom complexes and shows when lifts exist and are unique up to torsor actions, first in the homotopy category of injectives and then in the derived category via a left-adjoint restriction framework. The main geometric payoff is a new, self-contained proof of the uniqueness of deformations of semiorthogonal decompositions in smooth proper families, achieved by a deformation-then-algebraization strategy for decomposition-triangles and their associated Fourier-Mukai kernels. The approach relies on coderived-model techniques, Artinian/base-change arguments, and formal-geometry comparison theorems to translate local lifting data into global, algebraizable deformations, with explicit obstruction formulae and torsor structures guiding the deformation space.

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.

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

TL;DR

The paper develops a deformation-obstruction calculus for lifting morphisms of complexes with a fixed lift of the codomain to derived categories under flat nilpotent deformations of abelian categories. It defines obstruction classes living in Ext-type groups obtained from derived Hom complexes and shows when lifts exist and are unique up to torsor actions, first in the homotopy category of injectives and then in the derived category via a left-adjoint restriction framework. The main geometric payoff is a new, self-contained proof of the uniqueness of deformations of semiorthogonal decompositions in smooth proper families, achieved by a deformation-then-algebraization strategy for decomposition-triangles and their associated Fourier-Mukai kernels. The approach relies on coderived-model techniques, Artinian/base-change arguments, and formal-geometry comparison theorems to translate local lifting data into global, algebraizable deformations, with explicit obstruction formulae and torsor structures guiding the deformation space.

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.

Paper Structure

This paper contains 9 sections, 12 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Setup and notation
  3. Deforming morphisms in the homotopy category of injectives
  4. Deforming morphisms in the derived category
  5. Deforming semiorthogonal decompositions
  6. Acknowledgements
  7. Deformation theory for morphisms in the homotopy category of injectives
  8. Restriction to the derived category
  9. Uniqueness of deformations of semiorthogonal decompositions

Key Result

lemma 2.1

Let $\mathcal{C}$ be a Grothendieck category, and $s\colon F\to G$ a morphism in $\operatorname{Ch}(\mathop{\mathrm{Inj}}\nolimits\mathcal{C})$. There exists a homotopy equivalence $h\colon G \to G'$ with $G'\in \operatorname{Ch}(\mathop{\mathrm{Inj}}\nolimits\mathcal{C})$ such that the composition

Theorems & Definitions (31)

  • example 1.1
  • remark 1.2
  • lemma 2.1
  • proof
  • lemma 2.2
  • proof
  • lemma 2.3
  • proof
  • lemma 2.4
  • lemma 2.5
  • ...and 21 more