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 $\mathrm{\infty }$-category $\mathrm{Q}\mathrm{C}(X)$. For quasicompact, smooth, and separated $X$, we identify the corresponding map on tangent fibres with the dual HKR map $\mathrm{R}\mathrm{\Gamma }(X,{\mathrm{T}}_X)[1]\to {\mathrm{H}\mathrm{H}}^{\bullet }(X\mathrm{/}k)[2]$, 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 $\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}(k)\mathrm{\ne }2$), and exhibit cases where the entire (classical) deformation functor of $X$ is a derived invariant; this partially answers a question of Lieblich.