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

Finiteness and duality of cohomology of families of $(φ,Γ)$-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 provide a generalization of the result of Kedlaya-Pottharst-Xiao, giving a new proof for cases already proved there.

💡 Research Summary

The paper revisits the finiteness and duality results for the cohomology of (φ,Γ)-modules, originally proved by Kedlaya‑Pottharst‑Xiao (KPX) for families over affinoid Qₚ‑algebras, and extends them to a much broader class of coefficient algebras using the modern machinery of solid mathematics.

First, the author adopts the notion of solid locally analytic representations introduced by Rodrigues Jacinto‑Rodríguez Camargo and interprets them as quasi‑coherent sheaves on solid D‑stacks, a concept developed by Clausen‑Scholze. The six‑functor formalism on solid D‑stacks, as established by Heyer‑Mann, provides the functors (f^, f_, f_!, f^!, ⊗, Hom) together with the crucial notions of “weakly D‑proper” and “D‑smooth” morphisms. These properties guarantee that push‑forward and pull‑back functors are mutually adjoint in a way that respects tensor products, which is essential for duality arguments.

The geometric object at the heart of the study is the Fargues‑Fontaine curve X_{K,∞} associated to a finite extension K/ℚₚ. After taking the Γ_K‑locally analytic quotient, one obtains the analytic stack X_{K,la}=X_{K,∞}^{Γ_K‑la}. A (φ,Γ)-module M over a coefficient algebra A is realized as a solid quasi‑coherent sheaf on the stack X_{K,la,A}=X_{K,la}×{Sp ℚₚ}Sp A. Proposition 0.1 shows that the usual (φ,Γ)-cohomology Γ{φ,Γ}(M) coincides with the derived push‑forward (g_A)^*M, where g_A : X_{K,la,A} → Sp A is the natural projection.

The main technical achievement is Theorem 0.2, which proves that the morphism
 g : X_{K,la}/Γ_{K,la} → AnSpec ℚₚ♯
is both weakly D‑proper and D‑smooth, with dualizing complex 𝒪_{X_{K,la}}·χ