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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2512.24347 |
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| _version_ | 1866917177665257472 |
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| author | Moore, Samuel A. |
| author_facet | Moore, Samuel A. |
| contents | 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 $\mathrm{R}Γ(X, \mathrm{T}_X)[1] \to \mathrm{HH}^{\bullet}(X/k)[2]$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_24347 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The period map from commutative to noncommutative deformations Moore, Samuel A. Algebraic Geometry Algebraic Topology Number Theory 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 $\mathrm{R}Γ(X, \mathrm{T}_X)[1] \to \mathrm{HH}^{\bullet}(X/k)[2]$, 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. |
| title | The period map from commutative to noncommutative deformations |
| topic | Algebraic Geometry Algebraic Topology Number Theory |
| url | https://arxiv.org/abs/2512.24347 |