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Finiteness and duality of cohomology of $(\varphi,Γ)$-modules and the 6-functor formalism of locally analytic representations

Yutaro Mikami

TL;DR

This work extends finiteness and duality results for cohomology of $(\varphi,\Gamma)$-modules from rigid-analytic base algebras to general condensed coefficient rings by employing solid $\mathcal{D}$-stacks and the Clausen–Scholze/Heyer–Mann 6-functor framework. The authors show that the cohomology $\Gamma_{\varphi,\Gamma_K}(\mathcal{M})$ of a $(\varphi,\Gamma_K)$-module over $B_{K,\infty,A}$ is equivalent to the pushforward $(g_A)_*\mathcal{M}$, and that the morphism $g: X_K^{\mathrm{la}}/\Gamma_K^{\mathrm{la}} \to \mathrm{Spec}\, \mathbb{Q}_{p,\square}$ is weakly $\mathcal{D}$-proper and $\mathcal{D}$-smooth with dualizing complex $\mathcal{O}_{X_K^{\mathrm{la}}}\cdot\chi[2]$, yielding Tate–duality-type identifications. This setup furnishes finiteness, duality, and Euler-characteristic control for families of $(\varphi,\Gamma)$-modules across broad coefficient rings, and provides a new proof path for known KPX cases while aligning with Emerton–Gee–Hellmann’s categorical local Langlands program. The results have implications for eigenvarieties and $p$-adic Langlands correspondences by enabling robust six-functor formalism in a flexible analytic-stack context.

Abstract

Finiteness and duality of cohomology of families of $(\varphi,Γ)$-modules were proved by Kedlaya-Pottharst-Xiao. In this paper, we study solid locally analytic representations introduced by Rodrigues Jacinto-Rodríguez Camargo in terms of analytic stacks and 6-functor formalisms, which are developed by Clausen-Scholze, Heyer-Mann, respectively. By using this, we will provide a generalization of the result of Kedlaya-Pottharst-Xiao, giving a new proof for cases already proved there.

Finiteness and duality of cohomology of $(\varphi,Γ)$-modules and the 6-functor formalism of locally analytic representations

TL;DR

This work extends finiteness and duality results for cohomology of -modules from rigid-analytic base algebras to general condensed coefficient rings by employing solid -stacks and the Clausen–Scholze/Heyer–Mann 6-functor framework. The authors show that the cohomology of a -module over is equivalent to the pushforward , and that the morphism is weakly -proper and -smooth with dualizing complex , yielding Tate–duality-type identifications. This setup furnishes finiteness, duality, and Euler-characteristic control for families of -modules across broad coefficient rings, and provides a new proof path for known KPX cases while aligning with Emerton–Gee–Hellmann’s categorical local Langlands program. The results have implications for eigenvarieties and -adic Langlands correspondences by enabling robust six-functor formalism in a flexible analytic-stack context.

Abstract

Finiteness and duality of cohomology of families of -modules were proved by Kedlaya-Pottharst-Xiao. In this paper, we study solid locally analytic representations introduced by Rodrigues Jacinto-Rodríguez Camargo in terms of analytic stacks and 6-functor formalisms, which are developed by Clausen-Scholze, Heyer-Mann, respectively. By using this, we will provide a generalization of the result of Kedlaya-Pottharst-Xiao, giving a new proof for cases already proved there.

Paper Structure

This paper contains 18 sections, 86 theorems, 231 equations.

Table of Contents

  1. Background
  2. Statements of the main result
  3. Outline of the paper
  4. Convention
  5. Convention and notation about condensed mathematics
  6. Solid $\mathcal{D}$-stacks and 6-functor formalisms
  7. Solid $\mathcal{D}$-stacks
  8. Suave and prim objects and morphisms
  9. Examples
  10. Dagger groups
  11. Locally analytic representations and $h\dagger$-analytic representations
  12. Locally analytic representations and six functors
  13. Semilinear representations
  14. Cohomology of $(\varphi,\Gamma)$-modules
  15. Locally analytic actions and suaveness
  16. ...and 3 more sections

Key Result

Proposition 1

For a $(\varphi,\Gamma_K)$-module $\mathcal{M}$ over $B_{K,\infty,A}$, the $(\varphi,\Gamma_K)$-cohomology $\Gamma_{\varphi,\Gamma_K}(\mathcal{M})\in \mathcal{D}(A_{\square})$ defined in Mikami24 is equivalent to $(g_{A_{\square}})_*\mathcal{M} \in \mathcal{D}(A_{\square})$, where $g_{A_{\square}}\c

Theorems & Definitions (222)

  • Proposition 1: Proposition \ref{['prop:comparison cohomology']}
  • Theorem 2: Corollary \ref{['cor:X D-smooth']}, Corollary \ref{['cor:X weakly proper']}, Theorem \ref{['thm:Local Tate duality']}
  • Definition 3
  • Remark 4
  • Proposition 5: Proposition \ref{['prop:geometric interpretation']}, Proposition \ref{['prop:6-ff classifying stack']}
  • Remark 6
  • Remark 7
  • Definition 1.1: RC24
  • Remark 1.2
  • Definition 1.3: RC24
  • ...and 212 more