Zariski Descent for the Milnor–Witt Rost–Schmid Complex

Definitions and mathematical statement

Write \[ F_q(X)=C^*(X,K_q^{MW},q) \] for the Bloch--Levine--Rost--Schmid complex named in the supplied record, with Milnor--Witt coefficients denoted by $K_q^{MW}$. The record does not specify the base field or base scheme, the admissible schemes $X$, the range of $q$, or the construction and restriction maps of this complex. These data must be recovered from the cited sources before this becomes a fully specified assertion.

Here is a conditional formulation of the descent question. Fix the intended category of schemes and allowed indices $q$, and assume that $F_q$ is a presheaf of cochain complexes on the corresponding Zariski site. For an admissible scheme $X$ and a Zariski open cover $\mathcal U=\{U_i\}_{i\in I}$, let its cosimplicial Čech complex have degree $p$ term \[ \check C^p(\mathcal U,F_q) =\prod_{(i_0,\ldots,i_p)\in I^{p+1}} F_q(U_{i_0}\cap\cdots\cap U_{i_p}), \qquad p\geq 0, \] with maps induced by restrictions. Write $\operatorname{Tot}^{\mathrm R}$ for its derived totalization. Under the interpretation of Zariski descent as descent for open covers, the question is whether the canonical map \[ F_q(X)\longrightarrow \operatorname{Tot}^{\mathrm R}\check C^\bullet(\mathcal U,F_q) \] is a quasi-isomorphism for every admissible $q$, every admissible $X$, and every Zariski open cover $\mathcal U$ of $X$. A quasi-isomorphism means a map inducing an isomorphism on cohomology in every degree.

This expresses the supplied question conditionally; it does not supply the missing hypotheses. The record also does not specify whether the intended descent condition is open-cover descent as above or the potentially stronger requirement of descent for all Zariski hypercovers.

Short English statement

Does the Bloch--Levine--Rost--Schmid complex $C^*(-,K_q^{MW},q)$ recover its value on a scheme from its values on Zariski open sets and their intersections, up to quasi-isomorphism? The admissible schemes, base assumptions, index range, and exact descent convention remain unspecified.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004169$, source problem code catalogue entry, titled "Zariski Descent for the Milnor--Witt Rost--Schmid Complex." The record gives proposed year 2019 and names no proposer. Its supplied status is open; its statement status is corrected_verified, accompanied by a claim of manual semantic, typographic, and LaTeX review. These are registry annotations, not independent verification here.
• Supplied primary citation: Algebraic K-theory (2020), Oberwolfach Reports, DOI 10.4171/owr/2019/29, pp. 1737--1790. The background describes these reports as documenting mathematical workshops and their open-problem discussions. https://doi.org/10.4171/owr/2019/29
• Supplied literature source labeled "Bachmann--Yakerson on the BLRS framework": https://arxiv.org/abs/1811.01541.
• Literature assessment dated exactly 2026-08-22: this is supplied dated curation, not independently verified. It labels the problem open and states that the BLRS complex provides a cycle model for Milnor--Witt motivic cohomology in important cases, that an analogue of Levine localization and consequently Zariski descent is explicitly left as a question, and that later work proves restricted equivalences rather than the full extracted assertion. The supplied background calls this a dated literature triage derived from OWR curation and requests independent verification of time-sensitive status and cited claims.