Formal Integration of Derived Foliations

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.