Formal Integration of Derived Foliations
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909946921091072 |
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| author | Brantner, Lukas Magidson, Kirill Nuiten, Joost |
| author_facet | Brantner, Lukas Magidson, Kirill Nuiten, Joost |
| contents | Frobenius' theorem in differential geometry asserts that every involutive subbundle of the tangent bundle of a manifold $M$ integrates to a decomposition of $M$ into smooth leaves. We prove an infinitesimal analogue of this result for locally coherent qcqs schemes $X$ over coherent rings. More precisely, we integrate partition Lie algebroids on $X$ to formal moduli stacks $X \rightarrow S$ where $S$ is the formal leaf space and the fibres of $X \rightarrow S$ are the formal leaves. We deduce that deformations of $X$-families of algebro-geometric objects are controlled by partition Lie algebroids on $X$. Combining our integration equivalence with a result of Fu, we deduce that Toën-Vezzosi's infinitesimal derived foliations (under suitable finiteness hypotheses) are formally integrable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_05257 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Formal Integration of Derived Foliations Brantner, Lukas Magidson, Kirill Nuiten, Joost Algebraic Geometry Algebraic Topology 14B12, 18M70, 17B66, 14D15, 57R30, 57R32 Frobenius' theorem in differential geometry asserts that every involutive subbundle of the tangent bundle of a manifold $M$ integrates to a decomposition of $M$ into smooth leaves. We prove an infinitesimal analogue of this result for locally coherent qcqs schemes $X$ over coherent rings. More precisely, we integrate partition Lie algebroids on $X$ to formal moduli stacks $X \rightarrow S$ where $S$ is the formal leaf space and the fibres of $X \rightarrow S$ are the formal leaves. We deduce that deformations of $X$-families of algebro-geometric objects are controlled by partition Lie algebroids on $X$. Combining our integration equivalence with a result of Fu, we deduce that Toën-Vezzosi's infinitesimal derived foliations (under suitable finiteness hypotheses) are formally integrable. |
| title | Formal Integration of Derived Foliations |
| topic | Algebraic Geometry Algebraic Topology 14B12, 18M70, 17B66, 14D15, 57R30, 57R32 |
| url | https://arxiv.org/abs/2502.05257 |