Explicit isomorphisms for a Herr-type complex over a metabelian extension

1. Problem Statement and Context

The Core Problem:
The paper addresses the computation of Galois cohomology for families of $p$-adic Galois representations over a $p$-adic field $K$. Specifically, it seeks to establish an explicit isomorphism between the Galois cohomology groups $H^i(G_K, V)$ and the cohomology of a specific algebraic complex constructed from $(\varphi, \tau)$-modules.

Context and Motivation:

• Classical Setting: In $p$-adic Hodge theory, Fontaine's $(\varphi, \Gamma)$-modules provide an equivalence between $p$-adic representations and modules over a period ring. Herr [Her98] constructed a complex using these modules to compute Galois cohomology for the cyclotomic extension.
• The Metabelian Extension: The authors focus on the extension $L = K_\infty \cdot K_{cycl}$, where $K_\infty$ is a Kummer tower (generated by $p^n$-th roots of a uniformizer $\pi$) and $K_{cycl}$ is the cyclotomic extension. The Galois group $G_\infty = \text{Gal}(L/K)$ is metabelian (an extension of $\mathbb{Z}_p$ by $\mathbb{Z}_p^\times$).
• Limitations of Previous Work:
  • Tavares Ribeiro [Tav11] constructed a complex for $(\varphi, \tau)$-modules to compute cohomology, but only for representations over finite extensions of $\mathbb{Q}_p$ (using dimension-shifting arguments).
  • Pottharst [Pot13] handled families of representations using $(\varphi, \Gamma)$-modules, but his method relied on the vanishing of $H^2(\Gamma, \cdot)$, which does not hold for the metabelian group $G_\infty$ (which has non-trivial $H^2$).
• The Gap: There was no known explicit complex using $(\varphi, \tau)$-modules that computes the cohomology of families of representations (defined over general $\mathbb{Q}_p$-affinoid algebras) over this specific metabelian extension, nor were there explicit maps realizing the isomorphism.

2. Methodology

The authors employ a combination of $p$-adic Hodge theory, spectral sequences, and explicit homological algebra.

Key Definitions and Setup:

• Families of Representations: They work with a free $S$-module $V$ of finite rank, where $S$ is a $\mathbb{Q}_p$-Banach algebra, equipped with a continuous $G_K$-action.
• $(\varphi, \tau)$-Modules: Using the theory of Fontaine-Wintenberger and results by Gao-Liu [GL20] and Gao-Poyeton [GP21], they associate a family of étale $(\varphi, \tau)$-modules, denoted $D^\dagger_{\tau, K}(V)$, to the representation $V$.
• Period Rings: They utilize overconvergent period rings $B^\dagger_{\tau, K}$ and their base changes to $L$, denoted $D^r_L$.

Strategic Approach:

1. Decomposition via Hochschild-Serre: They utilize the Hochschild-Serre spectral sequence for the extension $1 \to H_\infty \to G_K \to G_\infty \to 1$. This relates $H^*(G_K, V)$ to the cohomology of $H_\infty$ (the Kummer tower) and $G_\infty$.
2. Computing $H_\infty$-Cohomology: They show that the cohomology of the restriction of $V$ to $H_\infty$ can be computed using a simple two-term complex involving the Frobenius operator $\varphi$:
   $$0 \to D^r_L \xrightarrow{\varphi - 1} D^{pr}_L \to 0$$
   This generalizes Pottharst's result to the $(\varphi, \tau)$ setting.
3. Computing $G_\infty$-Cohomology: Since $G_\infty$ is not procyclic (it is generated by $\gamma$ and $\tau$), they construct a three-term complex to compute $H^*(G_\infty, M)$ for a $G_\infty$-module $M$. This is a generalization of the standard two-term complex used for cyclic groups.
4. Total Complex Construction: By combining the $H_\infty$-complex and the $G_\infty$-complex via the spectral sequence machinery, they construct a four-term complex (a total complex) made entirely of $(\varphi, \tau)$-modules.
5. Explicit Isomorphisms: The core technical contribution is constructing explicit maps ($h_1$ and $h_2$) between the Galois cohomology groups and the cohomology of this new complex. They use a "five-lemma" argument on a commutative diagram involving the inflation-restriction sequence and the complex's cohomology to prove these maps are isomorphisms.

3. Key Contributions

1. Construction of the Complex: The authors define a specific four-term complex $C^\bullet$ associated with the $(\varphi, \tau)$-module $D^r_L$:
   $$0 \to D^r_L \xrightarrow{d_0} D^{pr}_L \oplus D^r_L \oplus D^r_L \xrightarrow{d_1} D^{pr}_L \oplus D^{pr}_L \oplus D^r_L \xrightarrow{d_2} D^{pr}_L \to 0$$
   The differentials $d_0, d_1, d_2$ are explicitly defined using matrices involving $\varphi-1$, $\gamma-1$, $\tau-1$, and a specific operator $\delta = \tau^{\chi(\gamma)-1}\tau^{-1}$.

2. Explicit Isomorphisms for Families: Unlike previous works that established existence or worked only for single representations, this paper provides explicit $S$-linear homomorphisms:

   • $h_1: H^1(G_K, V) \to H^1(C^\bullet)$
   • $h_2: H^2(G_K, V) \to H^2(C^\bullet)$
     These maps are constructed using continuous sections and specific cocycle representatives, ensuring they are well-defined and independent of choices.

3. Handling Non-Vanishing $H^2$: The paper successfully navigates the technical difficulty that $H^2(G_\infty, V^{H_\infty}) \neq 0$. This prevents the "free" isomorphism for $H^2$ seen in the $(\varphi, \Gamma)$ case. The authors explicitly construct the map $h_2$ by lifting cocycles and verifying their properties, a significant technical hurdle overcome in this work.

4. Generalization to Banach Algebras: The results hold for families of representations over arbitrary $\mathbb{Q}_p$-Banach algebras (affinoid algebras), not just finite extensions of $\mathbb{Q}_p$.

4. Main Results

Theorem 1 (Theorem 26 in the paper):
Let $V$ be a family of representations of $G_K$ over a $\mathbb{Q}_p$-Banach algebra $S$. Let $D^\dagger_{\tau, K}(V)$ be the associated family of $(\varphi, \tau)$-modules. Then, the cohomology of the complex $C^\bullet$ defined above is isomorphic to the Galois cohomology of $V$:
$$H^i(C^\bullet) \cong H^i(G_K, V) \quad \text{for } i = 0, 1, 2.$$
Furthermore, the authors provide explicit formulas for the isomorphisms $h_1$ and $h_2$.

Corollary:
The result specializes to recover Tavares Ribeiro's theorem [Tav11] when $S$ is a finite extension of $\mathbb{Q}_p$, providing a new proof via the family framework.

5. Significance

• Foundational for $p$-adic Hodge Theory: This work establishes the foundational properties of cohomology for families of $(\varphi, \tau)$-modules, analogous to the well-established theory for $(\varphi, \Gamma)$-modules.
• Applications to Deformation Theory: Since $(\varphi, \tau)$-modules are crucial for studying Galois deformation rings (especially for semi-stable representations), having an explicit complex for cohomology allows for the computation of tangent spaces and obstruction spaces in deformation problems over families.
• Arithmetic Geometry: The explicit isomorphisms are essential for applications in Iwasawa theory and the study of $p$-adic $L$-functions, where one often needs to relate Galois cohomology classes to algebraic objects in a computable way.
• Technical Advancement: The paper demonstrates how to handle metabelian extensions with non-trivial higher cohomology in the context of families, a scenario where previous spectral sequence arguments (like Pottharst's) fail or require significant modification.

In summary, the paper bridges a critical gap in $p$-adic Hodge theory by providing a computable, explicit complex for Galois cohomology in the setting of metabelian extensions and families of representations, complete with rigorous proofs of the isomorphisms.