The period map from commutative to noncommutative deformations

Abstract

We study the period map from infinitesimal deformations of a qcqs derived scheme $X$ over a field $k$ to those of the associated $k$-linear $\infty$-category $\mathrm{QC}(X)$. We identify the corresponding map on tangent fibres with the dual HKR map $\mathrm{R}\Gamma (X,{\mathbf{T}}_{X/k})[1]⟶{\mathrm{HH}}^{\bullet }(X/k)[2]$ and study its injectivity on homotopy groups. As applications, we show liftability of qcqs schemes 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.

33 pages. v2: main result generalised to qcqs derived schemes; statement of Corollary 1.4 corrected. Comments welcome!