Formal Integration of Derived Foliations

Lukas Brantner; Kirill Magidson; Joost Nuiten

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arXiv · 2502.05257

Formal Integration of Derived Foliations

Lukas Brantner·Kirill Magidson·Joost Nuiten

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Abstract

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 To\"{e}n-Vezzosi's infinitesimal derived foliations (under suitable finiteness hypotheses) are formally integrable.

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BibTeX RIS

Lukas Brantner, Kirill Magidson, Joost Nuiten. 2025-02-07. Formal Integration of Derived Foliations. https://arxiv.org/abs/2502.05257

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