Deformation Theory and Partition Lie Algebras
Lukas Brantner, Akhil Mathew
TL;DR
The paper extends the classical Lurie–Pridham equivalence between formal moduli problems and differential graded Lie algebras from characteristic zero to finite and mixed characteristics. It introduces partition Lie algebras, a homotopical generalisation encoded by a Koszul-dual framework built from monads and leveraging the topology of partition complexes, to classify infinitesimal deformations over arbitrary fields and complete local bases. A key achievement is computing homotopy groups of free partition Lie algebras and establishing a precise equivalence Moduli_{k,Δ} ≃ Alg_{Lie^{π}_{k,Δ}} (and their spectral analogues), thereby providing Lie-algebraic control of formal moduli in nonzero characteristics. The approach unifies E_∞-algebras, simplicial commutative rings, and operadic contexts under a single axiomatic deformation theory, with robust descent and completion techniques that extend to mixed characteristic bases. These results yield concrete tools for understanding deformations in arithmetic geometry and related areas, and reveal new algebraic structures governing formal moduli in broad settings.
Abstract
A theorem of Lurie and Pridham establishes a correspondence between formal moduli problems and differential graded Lie algebras in characteristic zero, thereby formalising a well-known principle in deformation theory. We introduce a variant of differential graded Lie algebras, called partition Lie algebras, in arbitrary characteristic. We then explicitly compute the homotopy groups of free algebras, which parametrise operations. Finally, we prove generalisations of the Lurie-Pridham correspondence classifying formal moduli problems via partition Lie algebras over an arbitrary field, as well as over a complete local base.