The period map from commutative to noncommutative deformations

Samuel A. Moore

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arXiv · 2512.24347

The period map from commutative to noncommutative deformations

Samuel A. Moore

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

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BibTeX RIS

Samuel A. Moore. 2025-12-30. The period map from commutative to noncommutative deformations. https://arxiv.org/abs/2512.24347

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