Profinite and Solid Cohomology

TL;DR

This paper explores the properties of solid abelian groups introduced by Clausen and Scholze, focusing on their relation to profinite rings and modules, and demonstrating preservation of key algebraic structures.

Contribution

It establishes that the embedding of profinite modules into solid modules preserves Ext, tensor products, and analytic properties of profinite rings.

Findings

Embedding preserves Ext and tensor products.

Profinite rings are shown to be analytic.

Solid abelian groups have favorable categorical properties.

Abstract

Solid abelian groups, as introduced by Dustin Clausen and Peter Scholze, form a subcategory of all condensed abelian groups satisfying some ''completeness'' conditions and having favourable categorical properties. Given a profinite ring $R$, there is an associated condensed ring $\underline{R}$ which is solid. We show that the natural embedding of profinite $R$-modules into solid $\underline{R}$-modules preserves $\mathrm{Ext}$ and tensor products, as well as the fact that profinite rings are analytic.