Poincaré Duality for Pro-Étale Cohomology of Stein Varieties

Definitions and mathematical statement

Let $p$ denote the prime implicit in the coefficients ${\mathbb{Q}}_p$, and let $X$ be a smooth, geometrically irreducible Stein variety of dimension $d$ over the base field $K$ appearing in the source. The supplied record does not specify the hypotheses on $K$, its relation to $p$, or the precise analytic category meant by "variety." These are unresolved hypotheses, and the statement must not be read as quantifying over arbitrary fields $K$.

For a Tate twist ${\mathbb{Q}}_p(j)$, write \[ H^i_{\mathrm{pro\acute et}}(X,{\mathbb{Q}}_p(j)) =H^i\left(R\Gamma_{\mathrm{pro\acute et}}(X,{\mathbb{Q}}_p(j))\right). \] The source specifies compact supports using an exhaustive affinoid covering $X=\bigcup_{n\in{\mathbb{N}}}U_n$ with $U_n\Subset U_{n+1}$, where $\Subset$ denotes relative compactness. It sets \[ \begin{aligned} R\Gamma_{\mathrm{pro\acute et}}(\partial X,{\mathbb{Q}}_p(j)) &:=\operatorname*{colim}_{n}R\Gamma_{\mathrm{pro\acute et}}(X\setminus U_n,{\mathbb{Q}}_p(j)),\\ R\Gamma_{\mathrm{pro\acute et},c}(X,{\mathbb{Q}}_p(j)) &:=\operatorname{fib}\!\left( R\Gamma_{\mathrm{pro\acute et}}(X,{\mathbb{Q}}_p(j)) \longrightarrow R\Gamma_{\mathrm{pro\acute et}}(\partial X,{\mathbb{Q}}_p(j)) \right),\\ H^i_{\mathrm{pro\acute et},c}(X,{\mathbb{Q}}_p(j)) &:=H^i\left(R\Gamma_{\mathrm{pro\acute et},c}(X,{\mathbb{Q}}_p(j))\right). \end{aligned} \] Here the fiber is a derived fiber. The notation $\partial X$ is specified through the displayed colimit; the excerpt does not identify it with a separately constructed boundary space or specify the category in which the colimit is taken.

Conjectural topological assertion. For such $X$, the source asks whether \[ H^i_{\mathrm{pro\acute et}}(X,{\mathbb{Q}}_p(j)) \quad\text{is nuclear Fr\'echet},\qquad H^i_{\mathrm{pro\acute et},c}(X,{\mathbb{Q}}_p(j)) \quad\text{is of compact type}. \] These are properties of topological ${\mathbb{Q}}_p$-vector spaces, not merely of their underlying vector spaces. A Fréchet space is a complete metrizable locally convex space. In the usual nonarchimedean conventions, nuclear Fréchet spaces admit countable projective presentations by Banach spaces with compact transition maps, and spaces of compact type admit countable locally convex inductive presentations by Banach spaces with injective compact transition maps. The excerpt does not specify the cohomology topologies or its precise functional-analytic conventions. The assertion is intended uniformly in cohomological degree and Tate twist, but explicit ranges for $i$ and $j$ are not supplied.

Conjectural duality assertion and unresolved dimension dependence. The clean statement asks whether cup product gives perfect duality between ordinary and compactly supported pro-étale cohomology. The extracted original text, despite introducing arbitrary dimension $d$, displays the numerical formulas \[ \begin{aligned} \gamma_{X,i}:H^i_{\mathrm{pro\acute et}}(X,{\mathbb{Q}}_p(j)) &\xrightarrow{\sim} \left(H^{4-i}_{\mathrm{pro\acute et},c}(X,{\mathbb{Q}}_p(2-j))\right)^*,\\ \gamma^c_{X,i}:H^i_{\mathrm{pro\acute et},c}(X,{\mathbb{Q}}_p(j)) &\xrightarrow{\sim} \left(H^{4-i}_{\mathrm{pro\acute et}}(X,{\mathbb{Q}}_p(2-j))\right)^*, \end{aligned} \qquad (-)^*:=\operatorname{Hom}_{{\mathbb{Q}}_p}(-,{\mathbb{Q}}_p). \] The extraction is visibly incomplete and interleaves these formulas with a higher-dimensional conjecture. Consequently, it does not establish that the constants $4$ and $2$ are the intended complementary degree and twist in arbitrary dimension. No replacement involving $d$ is inferred here. The displayed constants would require a cup-product pairing followed by a trace of the form \[ H^i_{\mathrm{pro\acute et}}(X,{\mathbb{Q}}_p(j)) \times H^{4-i}_{\mathrm{pro\acute et},c}(X,{\mathbb{Q}}_p(2-j)) \longrightarrow H^4_{\mathrm{pro\acute et},c}(X,{\mathbb{Q}}_p(2)) \longrightarrow{\mathbb{Q}}_p, \] but the trace and its hypotheses are absent from the supplied extract. The notation $\operatorname{Hom}_{{\mathbb{Q}}_p}$ also leaves unresolved whether continuous maps are intended, which topology is placed on the dual, and whether the displayed isomorphisms are topological. These omissions prevent a fully specified higher-dimensional perfect-duality statement.

Qualifications retained from the surrounding source. The excerpt states that in dimension $1$, its compact-support complex with coefficients ${\mathbb{Q}}_p$ agrees with Huber's \’etale cohomology with compact support, and expresses the belief that this agreement holds in every dimension; the latter is not presented as established. For smooth dagger varieties over $K$ of dimension $1$, it describes proper cohomology groups as "finite," asserts the nuclear Fréchet/compact-type distinction above for Stein varieties, and says that the roles are reversed for dagger affinoids. The meaning of "finite" is not clarified in the extract.

The surrounding discussion also reports a derived duality for Stein $X$, written \[ R\Gamma_{\mathrm{pro\acute et}}(X,{\mathbb{Q}}_p(j)) \xrightarrow{\sim} D\!\left(R\Gamma_{\mathrm{pro\acute et},c}(X,{\mathbb{Q}}_p(2-j))[4]\right), \qquad D(-):=R\operatorname{Hom}_{{\mathbb{Q}}_p}(-,{\mathbb{Q}}_p). \] Its ambient derived-category notation is truncated, and the scope of this assertion relative to the surrounding curve discussion is unresolved. The preceding argument invokes vanishing of positive $\operatorname{Ext}$ groups for compactly supported cohomology of compact type, describes such spaces as colimits of Smith spaces that are projective solid objects, and invokes classical reflexivity to obtain the reverse duality. These are retained as contextual source assertions with incomplete categorical hypotheses, not as a proof of the higher-dimensional conjecture.

Short English statement

For a smooth geometrically irreducible Stein variety, the problem asks whether ordinary pro-étale cohomology is nuclear Fréchet, compactly supported cohomology is of compact type, and cup product gives perfect duality. The supplied extract leaves the base-field hypotheses, topologies, and higher-dimensional complementary degrees and twists unresolved. Its displayed $4-i$ and $2-j$ formulas are preserved without extrapolation. This is a statement-only draft requiring mathematical review.

Sources

• Local UnsolvedMath registry, supplied complete record, permanent ID $30005575$; exact source problem code catalogue entry; title: "Poincaré Duality for Pro-Étale Cohomology of Stein Varieties." The record assigns the problem to the Oberwolfach Reports Open Problems collection and gives proposed year $2023$; no individual proposer is supplied.
• Supplied report citation: Algebraische Zahlentheorie (2023), Oberwolfach Reports, pp. 1543--1606, DOI 10.4171/owr/2023/28. https://doi.org/10.4171/owr/2023/28. The supplied background describes these reports as records of mathematical workshops and their open-problem discussions. No exact problem page or contribution author is supplied.
• Supplied literature source, labeled "Achinger--Gilles--Nizioł": https://arxiv.org/abs/2501.13651.
• Supplied literature source, labeled "arithmetic duality": https://arxiv.org/abs/2412.11786.
• Supplied literature source, labeled "original OWR report": https://doi.org/10.4171/owr/2023/28.
• The exact supplied literature-check date is 2026-08-22. The supplied dated curation labels the problem "partially solved" and assesses the higher-dimensional conjecture as unproved in full generality, with comparison and duality results in stable ranges and conditional forms extending known curve cases. This is supplied dated curation, not an independently verified literature assessment; the record itself requests independent verification of time-sensitive status and cited claims.
• The registry labels its statement corrected_verified and reports that it was checked against the official report. This is a supplied verification claim; no source visit or independent verification is asserted in this draft.