Comparisons of Lie algebra cohomologies of $(\varphi,Γ)$-modules

TL;DR

This work develops a comprehensive framework to compare Lie algebra cohomologies of $(\varphi_L,\Gamma_L)$-modules over Robba rings, revealing how $L$-analytic and $\mathbb{Q}_p$-analytic cohomologies intertwine for $L/\mathbb{Q}_p$ of degree $d$. It proves a sharp decomposition $C^{\bullet}_{f,\mathbb{Q}_p\text{-Lie}}(M) \simeq \bigoplus_{n=0}^{d-1} C^{\bullet}_{f,L\text{-Lie}}(M)[-n]^{\binom{d-1}{n}}$ and a corresponding expression for $H^i_{cts}(M)$ in terms of $H^j_{an}(M)$, thereby clarifying and generalizing Fourquaux–Xie’s results. By introducing abstract Herr complexes for augmented algebras and employing Koszul- and Dolbeault-type constructions, the paper derives Euler-characteristic relations and cup-product phenomena that connect analytic and base-change cohomology, showing that for $L\neq \mathbb{Q}_p$ there exist étale $(\varphi_L,\Gamma_L)$-modules whose $\mathbb{Q}_p$-analytic cohomology is not obtained by Galois-base-change. It further develops a Dolbeault framework on analytic vectors to produce Frölicher-type spectral sequences linking $L$-Lie cohomology with twisted $\mathbb{Q}_p$-Lie cohomology, offering a robust toolbox for analyzing cohomological aspects of $p$-adic Galois representations in Lubin–Tate settings.

Abstract

We generalise a result of Fourquaux and Xie thereby completely determining the relationship between $\mathbb{Q}_p$ and $L$-analytic Lie algebra cohomology of analytic $(\varphi_L,Γ_L)$-modules. We use the results to conclude that for $L\neq \mathbb{Q}_p,$ there exist examples of étale $(\varphi_L,Γ_L)$-modules over Robba rings whose $\mathbb{Q}_p$-analytic cohomology does not arise as a base change of Galois cohomology.

Theorems & Definitions (66)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Definition 2.3
• Definition 3.1