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We develop a notion of formal groups in the filtered setting and describe aduality relating these to a specified class of filtered Hopf algebras. We thenstudy a deformation to the normal cone construction in the setting of derivedalgebraic geometry. Applied to the unit section of a formal group$\widehat{\mathbb{G}}$, this provides a $\mathbb{G}_m$-equivariant degenerationof $\widehat{\mathbb{G}}$ to its tangent Lie algebra. We prove a unicity resulton complete filtrations, which, in particular, identifies the resultingfiltration on the coordinate algebra of this deformation with the adicfiltration on the coordinate algebra of $\widehat{\mathbb{G}}$. We use this ina special case, together with the aforementioned notion of Cartier duality, torecover the filtration on the filtered circle of [MRT19]. Finally, weinvestigate some properties of $\widehat{\mathbb{G}}$-Hochschild homology setout in loc. cit., and describe "lifts" of these invariants to the setting ofspectral algebraic geometry.

Filtered formal groups, Cartier duality, and derived algebraic geometry