Filtered formal groups, Cartier duality, and derived algebraic geometry

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 $\hat{\mathbb{G}}$, this provides a ${\mathbb{G}}_m$-equivariant degeneration of $\hat{\mathbb{G}}$ 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 $\hat{\mathbb{G}}$. 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 $\hat{\mathbb{G}}$-Hochschild homology set out in loc. cit., and describe "lifts" of these invariants to the setting of spectral algebraic geometry.

Publication version

References in corpus (5)

• Higher Topos Theory
• Completions and derived de Rham cohomology
• Perfectoid rings as Thom spectra
• Classes caractéristiques des schémas feuilletés
• Prismatization