Weil–Arakelov and Weil–Étale Cohomology for Arithmetic Schemes

Definitions and mathematical statement

An arithmetic scheme means a separated scheme of finite type over $\operatorname{Spec}{\mathbb{Z}}$. A complex of abelian groups is perfect if it is quasi-isomorphic to a bounded complex of finitely generated projective ${\mathbb{Z}}$-modules. A complex of real vector spaces is perfect if it is quasi-isomorphic to a bounded complex of finite-dimensional real vector spaces.

Conjectural existence problem, with unresolved source parameters. For every arithmetic scheme $X$, the source asks for compactly supported Weil--étale and Weil--Arakelov cohomology complexes \[ C_{W,c}(X,n):=R\Gamma_{W,c}(X,{\mathbb{Z}}(n)),\qquad C_{\mathrm{ar},c}(X,A,n):=R\Gamma_{\mathrm{ar},c}(X,A(n)). \] The supplied record does not specify the range or quantification of the twist $n$, the allowed coefficients $A$, or the full intended derived categories. These remain parameters requiring clarification; no additional restrictions on $X$ are supplied.

The requested complexes must satisfy at least the following conditions:

1. $C_{W,c}(X,n)$ is a perfect complex of abelian groups.
2. $C_{\mathrm{ar},c}(X,{\mathbb{R}},n)$ is a perfect complex of real vector spaces.
3. Cup product with the source's fundamental class \[ \theta\in H^1(\operatorname{Spec}{\mathbb{Z}},{\mathbb{R}}(0)) \] makes the graded cohomology of $C_{\mathrm{ar},c}(X,{\mathbb{R}},n)$ into an acyclic complex. Explicitly, writing \[ V^i=H^i\left(C_{\mathrm{ar},c}(X,{\mathbb{R}},n)\right), \] the intended cup-product action must give maps $d_\theta^i:V^i\to V^{i+1}$ such that, for every $i\in{\mathbb{Z}}$, \[ d_\theta^{i+1}\circ d_\theta^i=0, \qquad \ker(d_\theta^i)=\operatorname{im}(d_\theta^{i-1}). \] This is acyclicity of the cohomology groups equipped with the cup-product differential; it does not assert that each $V^i$ vanishes.

Unresolved qualifications. The cohomology theory containing $\theta$, its normalization, and its action on compactly supported Weil--Arakelov cohomology are not defined in the supplied record. The displayed conditions alone do not characterize the intended cohomologies: zero complexes with zero action satisfy them. Additional structural requirements are therefore needed to distinguish the intended existence problem from a bare existence assertion about perfect complexes. The background mentions zeta-value properties, but provides no zeta-value formula or compatibility condition to state.

Short English statement

The supplied conjecture asks for compactly supported Weil--étale and Weil--Arakelov cohomologies for every separated scheme of finite type over ${\mathbb{Z}}$. The integral Weil--étale complex and the real Weil--Arakelov complex should be perfect, and cup product with a fundamental class should give an exact complex on the latter's cohomology. The twist range, coefficient choices, and structural requirements identifying these cohomologies are missing. This is a statement-only draft requiring human mathematical review; the supplied open status is not independently verified.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003166$, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2016 and names no individual proposer.
• Supplied report citation: Algebraic K-theory and Motivic Cohomology (2017), Oberwolfach Reports, pp. 1753--1807, DOI 10.4171/owr/2016/31. https://doi.org/10.4171/owr/2016/31. The collection describes Oberwolfach Reports as documenting workshops and their open-problem discussions. The supplied citation's 2017 title date is retained alongside its 2016 DOI and proposed-year metadata without reconciliation.
• Flach--Morin, Weil--Etale Cohomology and Zeta-Values. Supplied literature link: https://ems.press/journals/dm/articles/8965580.
• Beshenov, Weil--etale cohomology at negative integers. Supplied literature link: https://arxiv.org/abs/2102.12114.
• Supplied literature assessment, checked exactly 2026-08-21: the record labels the problem open and describes compactly supported Weil--Arakelov and Weil--étale cohomology and the listed zeta-value properties as conjectural in the stated arithmetic-scheme generality, while proved or compatible in several restricted cases. This is supplied dated curation derived from the OWR material, not an independently verified assessment. The restricted cases and zeta-value properties are not specified in this record; time-sensitive status and cited claims require independent verification.
• The registry labels its statement corrected_verified and reports that it was checked against the official Oberwolfach report, preserving the report's hypotheses, notation, and open target. These are supplied verification claims; no source visit or independent verification is asserted here.