$\mathbb{G}_m$-cohomology of $p$-adic Stein spaces

Technical Summary: $G_m$-Cohomology of $p$-adic Stein Spaces

Problem Statement
The paper addresses the computation of the étale cohomology groups $H^i_{\text{ét}}(X, \mathbb{G}_m)$ for smooth Stein spaces $X$ over a complete algebraically closed field $C$ of characteristic 0 (containing $\mathbb{Q}_p$). While the cohomology of $\mathbb{G}_m$ is well-understood for algebraic schemes (where it is torsion in degrees $i \ge 2$ by Grothendieck's theorem), the analytic case presents significant challenges. In the analytic setting, the cohomology groups are not necessarily torsion, and standard algebraic strategies (such as relating $\mathbb{G}_m$ to pushforwards from points) fail. The authors aim to describe these groups for classical rigid analytic spaces, specifically the affine space $\mathbb{A}^d$, the torus $\mathbb{G}_m^d$, and the Drinfeld symmetric space $\mathbb{H}^d$, by establishing a structural filtration and computing its graded pieces.

Methodology
The authors employ a multi-step strategy combining $p$-adic Hodge theory, limit arguments over Stein coverings, and combinatorial methods on the Bruhat-Tits building.

1. Filtration via Principal Units: The core strategy involves decomposing $\mathbb{G}_m$ using the subgroup of principal units $U = 1 + \mathfrak{m}\mathcal{O}^+$. This yields an exact sequence $1 \to U \to \mathbb{G}_m \to \mathcal{O}^\times \to 1$, where $\mathcal{O}^\times = \mathbb{G}_m/U$. This induces a filtration on $H^i_{\text{ét}}(X, \mathbb{G}_m)$ with graded pieces related to the cohomology of $U$ and $\mathcal{O}^\times$.
2. Separation of Torsion Parts: The authors distinguish between the $p$-part and the prime-to-$p$ ($p'$) part of the cohomology.
   • The $p$-part is analyzed via the logarithm and exponential maps, relating $U$ to the structure sheaf $\mathcal{O}$. This connects the problem to coherent cohomology and the Hodge-Tate logarithm map.
   • The $p'$-part is analyzed via the Kummer exact sequence, relating $\mathcal{O}^\times$ to $\ell$-adic cohomology for primes $\ell \neq p$.
3. Pro-étale Site and Hodge-Tate Isomorphism: To handle the $p$-part, the authors utilize the pro-étale site $X_{\text{proét}}$ and the canonical projection $\nu: X_{\text{proét}} \to X_{\text{ét}}$. They employ the Hodge-Tate isomorphism (Scholze) and the Bloch-Kato exponential map (Colmez-Dospinescu-Nizioł) to relate pro-étale cohomology to differential forms $\Omega^i_X$.
4. Limit Arguments and Flatness: Since Stein spaces are limits of affinoid spaces $X_n$, the authors use derived limit formalism ($R\lim$) to pass from the cohomology of the affinoids to the Stein space. A crucial technical requirement is the flatness of the transition maps in the projective system of $\ell$-adic cohomology groups for the Stein covering.
5. Combinatorial Geometry: For the Drinfeld symmetric space, the authors adapt arguments from de Shalit and Schneider-Stuhler. They utilize the geometry of the Bruhat-Tits building, specifically the convexity of certain subcomplexes (balls in the combinatorial metric), to prove the necessary flatness and acyclicity properties for coefficient systems.

Key Contributions and Results

• Construction of a Filtration: The paper constructs a filtration $F^{(i)}_{\text{ét}, \mathbb{G}_m}(X)$ for the cohomology of any smooth Stein space $X$. The graded pieces are identified as:

  • $gr_0$: Related to the cokernel of the Hodge-Tate logarithm, identified with $\Omega^{i-1}(X)(1-i)/\text{Ker}(d)$.
  • $gr_1$: Related to the kernel of the Hodge-Tate logarithm, identified with a quotient involving Hyodo-Kato cohomology $HK^i(X)$ and pro-étale cohomology.
  • $gr_2$: Related to the $p'$-torsion, identified with a completed direct sum of $\ell$-adic cohomology groups for $\ell \neq p$.

• Explicit Computations for Classical Spaces: Under specific technical hypotheses (torsion of $\mathcal{O}^\times$-cohomology on affinoids, flatness of $\ell$-adic cohomology, and injectivity of the map to Hyodo-Kato cohomology), the authors compute the graded pieces for:

  • Affine Space $\mathbb{A}^d$: The cohomology is described in terms of differential forms.
  • Torus $\mathbb{G}_m^d$: The graded pieces involve exterior powers of $\mathbb{Q}_p$ and $\mathbb{Q}_\ell$.
  • Drinfeld Symmetric Space $\mathbb{H}^d$: The graded pieces are expressed in terms of generalized Steinberg representations $Sp_i(\cdot)^*$ and their duals. Specifically, $gr_1$ involves $Sp_i(\mathbb{Q}_p)^*/Sp_i(\mathbb{Z}_p)^*$ and $gr_2$ involves a sum over $\ell \neq p$ of similar structures.

• Torsion and Flatness Results:

  • The authors prove that for closed balls, closed annuli, and the standard Stein covering opens $U_n$ of the Drinfeld space, the groups $H^i_{\text{ét}}(X, \mathcal{O}^\times)$ are torsion for $i \ge 2$.
  • They establish the flatness of $H^i_{\text{ét}}(U_n, \mathbb{Z}/\ell^n\mathbb{Z})$ over $\mathbb{Z}/\ell^n\mathbb{Z}$ for $\ell \neq p$, a non-trivial result for the Drinfeld space that relies on the acyclicity of coefficient systems on the Bruhat-Tits building.

Significance and Claims
The paper claims to provide a complete description of the $\mathbb{G}_m$-cohomology for a significant class of $p$-adic Stein spaces, extending known results from the algebraic case to the analytic setting. The authors emphasize that their strategy unifies the treatment of $p$-torsion (via $p$-adic Hodge theory) and prime-to-$p$ torsion (via $\ell$-adic cohomology).

The work generalizes a previous computation for the affine space (by Ertl, Gilles, and Nizioł) to the torus and the Drinfeld symmetric space. A key claim is that the Drinfeld symmetric space, despite its complexity, satisfies the necessary technical conditions (torsion of $\mathcal{O}^\times$ and flatness of $\ell$-adic cohomology) to allow for this explicit filtration. The authors note that the results rely on deep properties of the Bruhat-Tits building and the acyclicity of specific coefficient systems, bridging rigid analytic geometry with representation theory.

The paper does not claim to solve the problem for all smooth Stein spaces but rather identifies a "sufficiently large class" (including the main examples of interest) where the cohomology is torsion and the filtration is explicit. The authors acknowledge that the general case requires further conditions to avoid pathologies in the limit argument, which are verified for their specific examples.