Dualizing Complexes and p-Torsion on the Fargues–Fontaine Curve

Definitions and mathematical statement

Let $p$ denote the prime occurring in the supplied record, and write $X=X_{E,F}$ for its Fargues--Fontaine curve. The record does not specify the hypotheses on the parameters $E$ and $F$. Its extracted theorem distinguishes an algebraic curve $X^{\mathrm{alg}}$ and an adic space $X^{\mathrm{ad}}$, but does not supply their precise constructions or comparison conventions. We write $\mathcal G$ for a coefficient sheaf, to distinguish it from the parameter $F$.

The requested problem is to describe a dualizing complex for $X$, and to determine the associated cohomological behavior with $p$-torsion coefficients. This is an unresolved formulation rather than a uniquely specified conjecture: the record does not identify the duality theory, coefficient ring, site, structural morphism, or normalization of the proposed dualizing complex. In particular, it does not specify whether coherent duality or a duality theory for torsion sheaves is intended. It also does not specify whether the general phrase $p$-torsion means sheaves annihilated by $p$, sheaves annihilated by a power of $p$, or sheaves whose local sections are each annihilated by some power of $p$.

The following supplied background qualifies the problem and must be retained.

1. The extracted report states that, for an $\acute{e}$tale abelian torsion sheaf $\mathcal G$ on $X^{\mathrm{alg}}$ whose torsion is prime to $p$, its Conjectures $(1)$ and $(2)$ hold. Those conjectures are not included in the record, so their mathematical content cannot be restated. The extract additionally reports \[ \operatorname{cd}_{\ell}\!\left(E(X^{\mathrm{alg}})\right)\leq 1, \qquad \ell\neq p\text{ a prime}. \] Here $\operatorname{cd}_{\ell}$ denotes $\ell$-cohomological dimension; the field notation $E(X^{\mathrm{alg}})$ is retained from the extract, whose precise conventions are missing. These are reported background assertions, not conclusions established by this draft.
2. For a constructible $\mathbf F_p$-module sheaf $\mathcal G$, the extract reports the conditional assertion \[ \text{(Stein)}\quad\Longrightarrow\quad H^i\!\left(X^{\mathrm{ad}},\mathcal G^{\mathrm{ad}}\right)=0 \quad\text{for every integer }i\geq 3. \] A $\mathbf F_p$-module sheaf is, in particular, annihilated by $p$. The notation $\mathcal G^{\mathrm{ad}}$ represents the associated sheaf appearing in the extract; its construction and the precise cohomological site are not specified there.
3. The hypothesis labelled $\text{(Stein)}$ concerns the tilt of a perfectoid space obtained by cyclotomic base change of an adification of an open subset. The extract describes $E_\infty/E$ as the cyclotomic $\mathbf Z_p$-extension and asks that the resulting tilt be a perfectoid Stein space. However, its description of the open subset mixes algebraic and adic notation, its completed-field notation is corrupted, and the purported definition ends with an incomplete union of spaces $V_n$. The conditions on $V_n$, the exhaustion, and the required transition maps are absent. Consequently, $\text{(Stein)}$ is only a label for a missing full hypothesis here, and the displayed implication is a record of the supplied conditional assertion rather than a self-contained theorem.

Thus the source supports the research questions above and the reported conditional vanishing in degrees at least $3$. It does not specify an unconditional vanishing conjecture for all $p$-torsion coefficients, a complete duality statement, or a precise relation between those two targets. These missing qualifications require human mathematical review.

Short English statement

Describe a dualizing complex for the Fargues--Fontaine curve and clarify its cohomology with $p$-torsion coefficients. The supplied extract reports vanishing in degrees $i\geq 3$ for constructible $\mathbf F_p$-coefficients under a Stein hypothesis, but that hypothesis is truncated. The curve parameters, duality framework, and scope of $p$-torsion coefficients remain unspecified.

Sources

• Supplied local registry record: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30005399$, source problem code catalogue entry, title Dualizing Complexes and p-Torsion on the Fargues--Fontaine Curve. No individual proposer is identified; the supplied proposed year is $2023$.
• Supplied report citation: Arithmetic of Shimura Varieties (2023), Oberwolfach Reports, pp. 261--326, DOI 10.4171/owr/2023/5. https://doi.org/10.4171/owr/2023/5. The record attributes the problem to workshop open-problem discussions documented in this report; it gives no precise page or speaker for this item.
• Supplied literature source, labelled "OWR-2023-5, open questions": https://oa.tib.eu/renate/bitstreams/ae468b44-47d2-4568-9d88-ffcb4023f8d4/download.
• Supplied literature source, labelled "Bartling, partial cohomology results": https://arxiv.org/abs/2206.14253. The extract refers to a talk's main result as reference [1], but does not provide the corresponding bibliography entry; its identification with this link is not established by the supplied text.
• Literature assessment checked at 2026-08-22: the supplied curation labels the problem open and describes the dualizing-complex and $p$-torsion questions as speculative, stating that existing Banach--Colmez and $\acute{e}$tale-cohomology results do not provide a general answer. 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 the statement corrected_verified and says it was reconstructed and checked against the official Oberwolfach report. That is a supplied provenance claim; no source visit or independent verification is asserted in this draft.