Frame Bundles as Stack Fiber Products

Definitions and mathematical statement

The supplied record concerns a rank-$n$ vector bundle $E$ on $S$ and its classifying map $f_E:S\to BGL_n$. It does not specify the base, the category containing $S$, the topology used for the classifying stack, or the map $\mathrm{pt}\to BGL_n$. Consequently, the displayed fiber product requires additional conventions.

One precise conditional formulation is the following. Fix a base scheme $B$, a $B$-scheme $S$, and a finite locally free $\mathcal O_S$-module $E$ of constant rank $n$. Interpret $BGL_{n,B}$ as the fppf classifying stack of rank-$n$ vector bundles, and interpret $\mathrm{pt}\to BGL_n$ as the morphism $B\to BGL_{n,B}$ classifying $\mathcal O_B^{\oplus n}$. These are explicit supplementary conventions, not hypotheses specified in the source record.

Define the frame functor of $E$ on $S$-schemes by \[ \operatorname{Fr}_S(E)(T\xrightarrow{g}S) = \operatorname{Isom}_{\mathcal O_T} \left(\mathcal O_T^{\oplus n},g^*E\right). \] The recoverable clarification asks for the geometric interpretation of the stack fiber product, with the requested identification \[ S\mathbin{\times^{(2)}_{BGL_{n,B}}}B \simeq \operatorname{Fr}_S(E), \] where $\times^{(2)}$ denotes the $2$-fiber product and the frame functor is viewed as its associated fppf sheaf or stack over $S$. This states the identification requested by the supplied record; no proof is provided.

The source qualifications prevent promoting this clarification to a verified open problem. The registry labels the item open, but its supplied statement-verification note says that the official report immediately identifies the fiber product with the frame bundle and treats the question as an answered clarification. Its statement status is unrecoverable, with no clean statement supplied. The dated literature assessment instead describes the extracted text as the meta-question "How to make this geometric?" in a discussion of stacks, says that it is not a complete mathematical claim, and assigns no definite resolution. These supplied descriptions are retained without independently adjudicating them.

Short English statement

Under explicitly stated scheme and stack conventions, the question asks how pulling back the trivial-bundle point along the classifying map of $E$ describes the bundle of frames of $E$. The supplied record contains conflicting status descriptions: an open label, an assessment of an incomplete meta-question, and a verification note describing an already answered clarification. This draft therefore records a conditional formulation requiring human review.

Sources

• Local UnsolvedMath registry: supplied complete record for permanent ID $30003457$, titled Frame Bundles as Stack Fiber Products; exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The proposer is unspecified; the supplied proposed year is $2017$.
• Supplied report citation: Arbeitsgemeinschaft: Higher Gross Zagier Formulas (2018), Oberwolfach Reports, pp. 1067--1134, DOI 10.4171/owr/2017/18. Supplied source URL and literature-sources link, labeled "Original Oberwolfach report": https://doi.org/10.4171/owr/2017/18. The background attributes the collection to workshop reports and their open-problem discussions.
• Literature assessment checked on 2026-08-21: this is supplied dated curation, not independently verified. It labels the item open while describing "How to make this geometric?" as an incomplete meta-question about stacks for which no definite resolution can be assigned. The background requests independent verification of time-sensitive status and cited claims.
• The supplied statement-verification note attributes the frame-bundle identification directly to the official report and describes the extracted question as an answered clarification, not an open problem. This attribution has not been independently checked here.