Filtered formal groups, Cartier duality, and derived algebraic geometry

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 G, this provides a Gₘ-equivariant degeneration of 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 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 G-Hochschild homology set out in loc. cit. , and describe "lifts" of these invariants to the setting of spectral algebraic geometry.