[Paper Review] Hodge-Tate decomposition for non-smooth spaces

Haoyang Guo|arXiv (Cornell University)|Sep 22, 2019

Algebraic Geometry and Number Theory21 references4 citations

TL;DR

This paper generalizes the Hodge-Tate decomposition to proper rigid spaces over a complete algebraically closed non-archimedean field of characteristic 0, even when the space is not smooth. Using pro-étale cohomology, the étale-h topology, and resolution of singularities, the authors establish a spectral sequence that degenerates at $E_2$, extending the classical Hodge-Tate result from smooth to singular settings.

ABSTRACT

In this article, we generalize the Hodge-Tate decomposition of p-adic étale cohomology to non-smooth rigid spaces. Our strategy is to study pro-étale cohomology of rigid spaces introduced by Scholze, using the resolution of singularities and the simplicial method.

Motivation & Objective

• To extend the Hodge-Tate decomposition, originally valid for smooth proper rigid spaces, to the case of non-smooth proper rigid spaces over a complete algebraically closed non-archimedean field of characteristic 0.
• To develop a cohomological framework for singular rigid spaces by introducing the $\acute{\textrm{e}}$h-topology and its associated sheaves of differential forms.
• To establish a spectral sequence relating $p$-adic étale cohomology to coherent cohomology of differential forms on the $\acute{\textrm{e}}$h site, generalizing the smooth case.
• To prove degeneracy of the spectral sequence at $E_2$-page for proper rigid spaces, even when singular, using simplicial methods and resolution of singularities.
• To recover the canonical Galois-equivariant Hodge-Tate decomposition when the space is defined over a discretely valued subfield with perfect residue field.

Proposed method

• Introduce the $\acute{\textrm{e}}$h-topology on rigid spaces, generated by étale coverings, universal homeomorphisms, and blowup coverings, to allow local smoothness via resolution of singularities.
• Define the sheaf of $\acute{\textrm{e}}$h-differentials $\Omega^j_{\acute{\textrm{e}}h,/K}$ by sheafifying the continuous differential sheaf in the $\acute{\textrm{e}}$h-topology.
• Use pro-étale cohomology of rigid spaces, as developed by Scholze, to relate $p$-adic étale cohomology to the derived pushforward of the structure sheaf.
• Establish the (pro-étale)-$\acute{\textrm{e}}$h de Rham comparison isomorphism via the $\acute{\textrm{e}}$h-descent of differential forms and the degeneracy of the spectral sequence.
• Apply simplicial techniques and the strong liftability of the rigid space to show that the derived direct image $R\nu_*\widehat{\mathcal{O}}_X$ is quasi-isomorphic to a direct sum of twisted $\acute{\textrm{e}}$h-cohomology groups.
• Leverage the Degeneracy Theorem (Theorem 7.4.9) and finiteness results (Theorem 6.0.2) to prove the spectral sequence degenerates at $E_2$.

Experimental results

Research questions

• RQ1 Can the Hodge-Tate spectral sequence be extended to non-smooth proper rigid spaces over a complete algebraically closed non-archimedean field of characteristic 0?
• RQ2 How can the $\acute{\textrm{e}}$h-topology be used to define a coherent cohomology theory that captures the Hodge-Tate structure on singular spaces?
• RQ3 Does the spectral sequence relating $p$-adic étale cohomology to $\acute{\textrm{e}}$h-cohomology of differential forms degenerate at $E_2$ for singular rigid spaces?
• RQ4 Under what conditions does the Hodge-Tate decomposition admit a canonical Galois-equivariant splitting for non-smooth spaces?
• RQ5 Can the pro-étale cohomology of a rigid space be expressed as a direct sum of twisted $\acute{\textrm{e}}$h-cohomology groups of differential forms?

Key findings

• The Hodge-Tate spectral sequence for proper rigid spaces over a complete algebraically closed non-archimedean field of characteristic 0 degenerates at the $E_2$-page, even when the space is not smooth.
• For any proper rigid space $X$, there is a spectral sequence $E_2^{i,j} = \mathrm{H}^i(X_{\acute{\textrm{e}}h}, \Omega^j_{\acute{\textrm{e}}h,/K})(-j) \Longrightarrow \mathrm{H}^{i+j}(X_{\acute{\textrm{e}}t}, \mathbb{Q}_p) \otimes_{\mathbb{Q}_p} K$ that degenerates at $E_2$.
• When $X$ is smooth, $\mathrm{H}^i(X_{\acute{\textrm{e}}h}, \Omega^j_{\acute{\textrm{e}}h,/K})(-j)$ is isomorphic to $\mathrm{H}^i(X, \Omega^j_{X/K})(-j)$, recovering the classical Hodge-Tate spectral sequence.
• The cohomology groups $\mathrm{H}^i(X_{\acute{\textrm{e}}h}, \Omega^j_{\acute{\textrm{e}}h,/K})(-j)$ are finite-dimensional $K$-vector spaces, vanishing unless $0 \leq i,j \leq \dim X$.
• When $X$ is defined over a discretely valued subfield $K_0$ with perfect residue field, the spectral sequence is Galois-equivariant and splits canonically as $\mathrm{H}^n(X_{\acute{\textrm{e}}t}, \mathbb{Q}_p) \otimes K = \bigoplus_{i+j=n} \mathrm{H}^i(X_{\acute{\textrm{e}}h}, \Omega^j_{\acute{\textrm{e}}h,/K_0})(-j) \otimes_{K_0} K$.
• The degeneracy of the spectral sequence is established via the Degeneracy Theorem (Theorem 7.4.9), relying on strong liftability and the $\acute{\textrm{e}}$h-descent of differentials.