Filtered formal groups, Cartier duality, and derived algebraic geometry

TL;DR

This paper introduces filtered formal groups, explores their duality with filtered Hopf algebras, and applies derived algebraic geometry techniques to study degenerations and invariants like Hochschild homology.

Contribution

It develops a new notion of filtered formal groups, establishes a duality with filtered Hopf algebras, and connects these to derived algebraic geometry and spectral invariants.

Findings

Established a duality between filtered formal groups and filtered Hopf algebras.

Constructed a $ ext{G}_m$-equivariant degeneration of formal groups to their tangent Lie algebra.

Identified the filtration on the deformation's coordinate algebra with the adic filtration.

Abstract

We develop a notion of formal groups in the filtered setting and describe a duality relating these to a specified class of filtered Hopf algebras. We then study a deformation to the normal cone construction in the setting of derived algebraic geometry. Applied to the unit section of a formal group $\widehat{\mathbb{G}}$, this provides a $\mathbb{G}_m$-equivariant degeneration of $\widehat{\mathbb{G}}$ to its tangent Lie algebra. We prove a unicity result on complete filtrations, which, in particular, identifies the resulting filtration on the coordinate algebra of this deformation with the adic filtration on the coordinate algebra of $\widehat{\mathbb{G}}$. We use this in a special case, together with the aforementioned notion of Cartier duality, to recover the filtration on the filtered circle of [MRT19]. Finally, we investigate some properties of $\widehat{\mathbb{G}}$-Hochschild homology set out in loc. cit., and describe "lifts" of these invariants to the setting of spectral algebraic geometry.