The period map from commutative to noncommutative deformations

TL;DR

This work constructs and analyzes a commutative-to-noncommutative period map that sends deformations of a smooth qcqs scheme $X$ to deformations of the derived category $\mathrm{QC}(X)$. It identifies the tangent map with the dual HKR map and provides sharp injectivity criteria across characteristics, along with criteria for HKR degeneration. By embedding deformations into the derived formalism and leveraging derived Atiyah transformations, the authors obtain derived-invariance results for liftability along square-zero extensions and establish conditions under which classical deformation functors are preserved under derived equivalence. The framework connects commutative deformation theory, Hochschild (co)homology, and partition Lie algebras, with applications to Calabi–Yau settings and to questions raised by Lieblich, contributing tools for comparing commutative and noncommutative deformation theories in positive and mixed characteristic.

Abstract

We study the period map from infinitesimal deformations of a scheme $X$ over a perfect field $k$ to those of the associated $k$-linear $\infty$-category $\mathrm{QC}(X)$. For quasicompact, smooth, and separated $X$, we identify the corresponding map on tangent fibres with the dual HKR map $\mathrm{R}Γ(X, \mathrm{T}_X)[1] \to \mathrm{HH}^{\bullet}(X/k)[2]$, and give conditions for injectivity on homotopy groups. As applications, we prove liftability along square-zero extensions to be a derived invariant (at least when $\mathrm{char}(k) \ne 2$), and exhibit cases where the entire (classical) deformation functor of $X$ is a derived invariant; this partially answers a question of Lieblich.

Figures (1)

• Figure 1: The 'Hodge quartet'; borrowed from AntieauBragg.

Theorems & Definitions (75)

• Theorem A
• Proposition 1.1
• Theorem B
• Corollary 1.2
• Definition 2.2
• Definition 2.3
• Remark 2.4
• Definition 2.5
• Proposition 2.6
• proof