Formal Integration of Derived Foliations

TL;DR

This paper extends Frobenius' theorem to the realm of algebraic geometry, showing how certain Lie algebroids on schemes can be integrated into formal moduli stacks, linking deformations to formal foliations.

Contribution

It introduces a method to integrate partition Lie algebroids on schemes into formal moduli stacks, establishing a bridge between infinitesimal structures and formal geometric objects.

Findings

Integration of partition Lie algebroids into formal moduli stacks.

Deformations of algebraic objects are governed by these Lie algebroids.

Infinitesimal derived foliations are shown to be formally integrable.

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.