The period map from commutative to noncommutative deformations

TL;DR

This paper investigates the relationship between classical and noncommutative deformations of schemes via the period map, establishing conditions for invariance and injectivity, and exploring derived invariants in deformation theory.

Contribution

It identifies the period map on tangent fibers with the dual HKR map for smooth schemes and proves liftability and deformation invariance results in derived settings.

Findings

The tangent map corresponds to the dual HKR map.

Liftability along square-zero extensions is a derived invariant.

Certain classical deformation functors are also derived invariants.

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 $\mathrm{R}\Gamma(X, \mathrm{T}_X)[1] \to \mathrm{HH}^{\bullet}(X/k)[2]$, 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.