Smooth profinite groups, II: the Uplifting Pattern
Mathieu Florence
TL;DR
This work advances the scheme-theoretic study of smooth profinite groups by introducing the Uplifting Pattern, a two-step process (base-change and group-change) that lifts Γ-equivariant vector-bundle extensions to Γ-W_2-bundles, with an eye toward the Smoothness Theorem. It builds a robust toolkit around Witt vectors, divided powers, and Frobenius actions, including the construction of WTF data (Witt–Teichmüller–Frobenius) and the universal uplifting schemes, to manage integral liftings and obstructions. Central technical pillars are Hochschild cohomology for equivariant extensions, the notion of splitting and good filtrations, and the development of splitting schemes that organize liftings and obstructions coherently. Together these contribute a concrete framework for lifting problems in the integral setting, with implications for cyclotomic pairs, Hochschild-vanishing results, and the broader objective of proving smoothness results in the trilogy. The methods provide a path to control obstructions via $p$-torsion-free bases and to relate mod $p$ phenomena to integral liftings through Witt-theoretic constructions, enabling deeper structural insights into smooth profinite groups and their representations in a scheme-theoretic context.
Abstract
This text presents a scheme-theoretic enhancement of the theory of smooth profinite groups and cyclotomic pairs, introduced in the paper `Smooth profinite groups, I'. To do so, our main technical tools are Hochschild cohomology of affine group schemes and lifting frobenius of vector bundles. The main contribution of this work is the Uplifting Pattern. It is a natural process, to lift a given equivariant extension of vector bundles, to its $\mathbf W_2$-counterpart, upon a `reasonable' combination of base-change and group-change. This is the key ingredient to prove the Smoothness Theorem, in the paper `Smooth profinite groups, III'.