Properness of Localization Maps for Algebraic Groups

Definitions and mathematical statement

Let $K$ be a field, let $K^{\mathrm{sep}}$ be a separable closure, and write $\Gamma_K=\operatorname{Gal}(K^{\mathrm{sep}}/K)$. An absolutely almost simple adjoint $K$-group is a connected semisimple algebraic $K$-group $G$ with trivial center such that, over an algebraic closure of $K$, it has no proper connected normal algebraic subgroup of positive dimension. Write $H^1(K,G)$ for the pointed set of continuous Galois $1$-cocycles with values in $G(K^{\mathrm{sep}})$, modulo the equivalence $a'_\sigma=g^{-1}a_\sigma\sigma(g)$.

For a set $V$ of discrete valuations of $K$, let $K_v$ denote the completion at $v$. For each finite subset $S\subseteq V$, extension of scalars defines \[ \omega_{S,G}:H^1(K,G)\longrightarrow\prod_{v\in V\setminus S}H^1(K_v,G),\qquad \alpha\longmapsto(\alpha_{K_v})_{v\in V\setminus S}. \] For the set-theoretic formulation below, interpret properness as the requirement that the inverse image of every finite subset is finite, equivalently that every fiber is finite. Thus the required condition is \[ \text{for every }(\xi_v)_{v\in V\setminus S}\in \prod_{v\in V\setminus S}H^1(K_v,G),\qquad \#\{\alpha\in H^1(K,G):\alpha_{K_v}=\xi_v\text{ for every }v\in V\setminus S\}<\infty. \] If $V\setminus S$ is empty, the product is a singleton, so this condition requires $H^1(K,G)$ itself to be finite. The supplied record does not define properness explicitly; the intended convention requires confirmation.

The supplied clean formulation asks: characterize the fields $K$ for which there exists a natural set $V=V(K)$ of discrete valuations such that, for every absolutely almost simple adjoint $K$-group $G$ and every finite $S\subseteq V$, the map $\omega_{S,G}$ is proper in the stated sense. In particular, does this property hold for finitely generated fields of good characteristic? Here a finitely generated field means a field finitely generated over its prime subfield.

Two qualifications remain unresolved. First, the record does not specify what makes $V$ natural or what good characteristic means, including whether the characteristic restriction depends on $G$. Second, its original statement fixes $K$ and $G$ before asking for $V$, whereas the clean statement requires one choice $V(K)$ to work for every such $G$. These quantifier orders need not be equivalent. The displayed target records the clean formulation; the original formulation may instead allow $V$ to depend on $G$.

Short English statement

Characterize fields admitting a natural family of discrete valuations for which localization of nonabelian $H^1$ has finite fibers, even after omitting any finite set of valuations, uniformly over absolutely almost simple adjoint groups. The finitely generated, good-characteristic case is singled out. The meanings of natural and good characteristic, the properness convention, and whether the valuation family may depend on the group require clarification.

Sources

• Local registry: data/unsolvedmath/problems.json, supplied record with permanent UnsolvedMath ID $30003796$, exact source problem code catalogue entry, title Properness of Localization Maps for Algebraic Groups, in the Oberwolfach Reports Open Problems collection. No individual proposer is identified; the supplied proposed year is 2018.
• Supplied report citation: Quadratic Forms and Related Structures over Fields (2019), Oberwolfach Reports, pp. 1235--1291, DOI 10.4171/owr/2018/21. https://doi.org/10.4171/owr/2018/21. The supplied background describes these reports as records of mathematical workshops and their open-problem discussions.
• Supplied literature source, labeled "spinor groups with good reduction": https://arxiv.org/abs/1707.08062.
• Supplied literature source, labeled "good-reduction survey": https://arxiv.org/abs/2005.05484.
• Supplied literature source, labeled "workshop report": https://ems.press/content/serial-article-files/46742.
• Literature assessment checked at 2026-08-21: the supplied status is partially_solved. The supplied assessment states that the properness question is settled for number fields and several group classes, with partial higher-dimensional results, but not for arbitrary fields or all finitely generated good-characteristic groups. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; its status and cited claims require independent verification.
• The supplied statement status is corrected_verified; its verification note says that the clean statement was checked against the official Oberwolfach report and preserves its hypotheses, notation, and open target. This is a supplied verification claim, not verification performed for this draft. The report citation gives 2019, while the proposed year and year tag give 2018 and the DOI contains 2018; these supplied date distinctions are retained without resolving their bibliographic meaning.