Stronger Obstructions to Zero-Cycles Beyond Étale Homotopy

Definitions and mathematical statement

For a scheme $X$ of finite type over a field $k$, a zero-cycle is a finite formal sum \[ z=\sum_{x\in X_{\mathrm{cl}}}n_x[x],\qquad n_x\in{\mathbb{Z}}, \] where $X_{\mathrm{cl}}$ denotes the set of closed points of $X$. Its degree over $k$ is \[ \deg_k(z)=\sum_x n_x[k(x):k]. \] A zero-cycle is effective if all its coefficients are nonnegative. These definitions supply terminology only: the record does not specify the base field, the class of schemes or varieties, a required degree, or whether effectiveness is required. Without any additional requirement, the zero-cycle $0$ always exists, so an obstruction to the existence of unrestricted zero-cycles would have no nontrivial target.

Source question. Can one construct an obstruction to the intended zero-cycle existence problem that is strictly stronger than the obstruction arising from stable $\text{\'etale}$ homotopy?

The intended zero-cycle problem and the stable $\text{\'etale}$ homotopy obstruction are not defined in the supplied record. Consequently, a fully specified mathematical statement cannot be recovered from this evidence alone. The following gives only a conditional comparison schema. Once the intended arithmetic and geometric setting is specified, let $T$ be its set of admissible test data, let $G\subseteq T$ consist of data realized by zero-cycles satisfying the intended requirements, and let $S_{\mathrm{st}}\subseteq T$ consist of data passing the specified stable $\text{\'etale}$ homotopy obstruction. Assuming that obstruction is sound, $G\subseteq S_{\mathrm{st}}$. A strictly stronger sound obstruction would be a defined criterion with passing set $S_{\mathrm{new}}$ such that \[ G\subseteq S_{\mathrm{new}}\subsetneq S_{\mathrm{st}}\subseteq T. \] In particular, strictness requires data in $S_{\mathrm{st}}\setminus S_{\mathrm{new}}$. This schema does not specify what the test data are, identify the baseline obstruction, or provide the requested construction. Whether the intended comparison concerns individual local data or existence of any compatible data also remains unresolved.

Short English statement

The question asks for a zero-cycle obstruction that detects a failure missed by stable $\text{\'etale}$ homotopy. The supplied record leaves the field, varieties, degree or effectiveness requirements, and comparison framework unspecified. The statement therefore records the question and a conditional meaning of strict strength, pending mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003764$, source problem code catalogue entry, titled "Stronger Obstructions to Zero-Cycles Beyond Étale Homotopy," in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied.
• Supplied report citation: Mini-Workshop: Arithmetic Geometry and Symmetries around Galois and Fundamental Groups (2019), Oberwolfach Reports, DOI 10.4171/owr/2018/17, pp. 1021--1080. https://doi.org/10.4171/owr/2018/17. The registry separately gives proposed year 2018; the relationship between these dates is unresolved.
• Supplied literature source: Harpaz--Schlank, "etale homotopy obstruction." https://arxiv.org/abs/1002.1423.
• Supplied literature source: Corwin--Schlank, "Brauer and etale obstructions." https://arxiv.org/abs/2006.11699.
• Supplied literature source: "OWR stable-homotopy report." https://ems.press/content/serial-article-files/47479?nt=1.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that étale and stable homotopy methods define and compare stronger obstructions while the general zero-cycle obstruction problem posed here remains unsettled. This is supplied dated curation, not an independently verified literature assessment. The background expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that mathematical content was preserved while notation, typography, and omitted definitions were repaired. These are supplied editorial assertions; they do not resolve the missing hypotheses or constitute independent mathematical verification here.