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Profinite and Solid Cohomology

Jiacheng Tang

TL;DR

This work embeds the category of profinite $R$-modules into the abelian and well-behaved category of solid $\underline{R}$-modules, preserving Ext and tensor products and enabling a robust solid/cohomology theory for profinite objects. By developing the solid framework (solid abelian groups, solid modules, and their tensor structure) and connecting it to condensed rings and group rings, the authors show that solid cohomology extends classical profinite cohomology via natural isomorphisms on Ext and completed tensor products. They also establish that profinite rings are analytic and that condensed group rings provide a unified approach to group actions and cohomology within the solid/condensed setting. These results yield a coherent, categorical foundation for studying profinite modules and their cohomology with tools from condensed mathematics, with potential applications to analytic and topological algebraic structures.

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.

Profinite and Solid Cohomology

TL;DR

This work embeds the category of profinite -modules into the abelian and well-behaved category of solid -modules, preserving Ext and tensor products and enabling a robust solid/cohomology theory for profinite objects. By developing the solid framework (solid abelian groups, solid modules, and their tensor structure) and connecting it to condensed rings and group rings, the authors show that solid cohomology extends classical profinite cohomology via natural isomorphisms on Ext and completed tensor products. They also establish that profinite rings are analytic and that condensed group rings provide a unified approach to group actions and cohomology within the solid/condensed setting. These results yield a coherent, categorical foundation for studying profinite modules and their cohomology with tools from condensed mathematics, with potential applications to analytic and topological algebraic structures.

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 , there is an associated condensed ring which is solid. We show that the natural embedding of profinite -modules into solid -modules preserves and tensor products, as well as the fact that profinite rings are analytic.

Paper Structure

This paper contains 12 sections, 42 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Solid Modules and (Co)homology
  4. Embedding Topological Modules into Condensed Modules
  5. Solid Modules
  6. Solid Modules over Profinite Rings
  7. Preservation of $\mathop{\mathrm{Hom}}\nolimits$ and $\mathop{\mathrm{Ext}}\nolimits$
  8. Preservation of Tensor Products
  9. Condensed Group Rings
  10. Recognising Profinite Modules
  11. Analytic Rings
  12. Basics of Condensed Modules

Key Result

Theorem 1.1

Let $R$ be a profinite ring and $M$ be a profinite $R$-module.

Theorems & Definitions (89)

  • Theorem 1.1: Theorem \ref{['thm5']}(ii), (iii) and Proposition \ref{['rtensors']}(iii)
  • Definition 2.1
  • Theorem 2.2: Theorem \ref{['thm1a']}
  • proof
  • Theorem 2.3: Theorem \ref{['thm2a']}
  • proof
  • Theorem 2.4: Theorem \ref{['thm3a']}
  • Theorem 2.5: Theorem \ref{['thm4a']}
  • Proposition 2.6: Proposition \ref{['homtensora']}
  • Proposition 3.1
  • ...and 79 more