A duality between Lie algebroids and infinitesimal foliations

TL;DR

This paper establishes an equivalence between partition Lie algebroids and infinitesimal derived foliations in derived algebraic geometry, extending the understanding of algebraic foliations across different characteristics.

Contribution

It introduces a new equivalence between two notions of algebraic foliations, refining PD Koszul duality with the Hodge filtration in derived algebraic geometry.

Findings

Established an equivalence between partition Lie algebroids and infinitesimal derived foliations.

Refined PD Koszul duality using the completed Hodge filtration.

Applicable in general characteristics with finiteness conditions.

Abstract

There are two natural analogues of algebraic foliations in derived algebraic geometry, called partition Lie algebroids and infinitesimal derived foliations, and both make sense in general characteristics. We construct an equivalence between these two notions under some finiteness conditions. Our method is refining the PD Koszul duality in \cite{BM}\cite{BCN} using the (completed) Hodge filtration.