Cohomology of $(\varphi, τ)$-modules

TL;DR

This work constructs and compares cohomology theories for $({\varphi},\tau)$-modules, extending the classical Herr cohomology of $({\varphi},\Gamma)$-modules to a setting that better captures integral and analytic phenomena. The authors develop three complementary cohomological frameworks: (i) $({\varphi},\Gamma_K)$–type cohomology via $\mathrm{R}\Gamma(\Gamma_K,-)^{\varphi=1}$, (ii) a $({\varphi},\hat{G})$–based approach incorporating the full $\hat{G}$-action to stabilize $\tau$, and (iii) a $({\varphi},\mathrm{Lie}\hat{G})$-Lie algebra method using $N_\nabla$ to realize monodromy and derive invariants. They show that, under suitable axioms (notably TS-1 and its Fréchet refinement) and descent arguments, these cohomologies recover the expected Galois cohomology for representations $V$ and agree across étale, overconvergent, and rigid-overconvergent theories. The paper also establishes equivalences of module categories, constructs a differential operator compatible with the $\tau$-action, and situates $B$-pairs within this unified cohomological picture. Overall, the results provide a conceptual, axiomatic framework that unifies and extends cohomology theories in $p$-adic Hodge theory and integral $p$-adic representations.

Abstract

We construct cohomology theories for $(\varphi, τ)$-modules, and study their relation with cohomology of $(\varphi, Γ)$-modules, as well as Galois cohomology. Our method is axiomatic, and can treat the étale case, the overconvergent case, and the rigid-overconvergent case simultaneously. We use recent advances in locally analytic cohomology as a key ingredient.

Theorems & Definitions (115)

• Theorem 1.3: cf. § \ref{['sec:equiv cats']}
• Remark 1.5
• Theorem 1.6: cf. Thm. \ref{['thmcohononetale']}
• Remark 1.7
• Remark 1.8
• Theorem 1.9: cf. Thm. \ref{['thmcohononetale']}
• proof : Sketch of main ideas.
• Remark 1.10
• Theorem 1.11
• proof : Sketch of main ideas.