Smooth profinite groups, II: the Uplifting Pattern

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'.