Smooth profinite groups, II: the Uplifting Pattern

TL;DR

This paper develops a scheme-theoretic framework for smooth profinite groups, introducing the Uplifting Pattern to lift extensions of vector bundles, which is crucial for proving smoothness properties.

Contribution

It introduces the Uplifting Pattern, a novel method for lifting equivariant vector bundle extensions in the context of smooth profinite groups.

Findings

The Uplifting Pattern enables lifting of vector bundle extensions to $ extbf{W}_2$-counterparts.

The scheme-theoretic approach enhances the understanding of smooth profinite groups.

Key technical tools include Hochschild cohomology and Frobenius lifting.

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'.