The period map from commutative to noncommutative deformations

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.