Independent Defects and Rough Deep Ramification

Definitions and mathematical statement

A valued field is a field $K$ equipped with a valuation $v:K\to\Gamma\cup\{\infty\}$, where $\Gamma=v(K^\times)$ is an ordered abelian group, $v(0)=\infty$, $v(1)=0$, $v(xy)=v(x)+v(y)$, and $v(x+y)\geq\min\{v(x),v(y)\}$.

Source question. Is the following implication valid for every valued field $(K,v)$? \[ \left((K,v)\text{ admits only independent defects}\right) \quad\Longrightarrow\quad \left((K,v)\text{ is roughly deeply ramified}\right). \] Here both technical conditions must be understood in the sense intended by the cited Oberwolfach report. The supplied record does not define independent defect, specify the extensions over which "only independent defects" quantifies, or define roughly deeply ramified. Consequently, this is a faithful statement of the supplied question, but not a self-contained characterization of its hypotheses and conclusion. The record states no explicit restrictions on characteristic, valuation rank, or henselianity; any hypotheses inherited from the surrounding report require review. The original passage calls this the converse of an assertion labelled (i), whose statement is not supplied.

Short English statement

Must every valued field admitting only independent defects be roughly deeply ramified? The supplied record labels this question open. Its technical definitions and any hypotheses inherited from the surrounding report must be checked before this draft can serve as a self-contained mathematical statement.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005411$, title: Independent Defects and Rough Deep Ramification; exact source problem code: catalogue entry. The record attributes the question to the 2023 Oberwolfach report and supplies no individual proposer.
• Resolutions in Local Algebra and Singularity Theory (2023), Oberwolfach Reports, pp. 327--395, DOI: https://doi.org/10.4171/owr/2023/6. The supplied original excerpt identifies p. 353 and describes the question as the converse of (i).
• Supplied literature source, labelled "OWR-2023-6, valued fields": https://oa.tib.eu/renate/server/api/core/bitstreams/11773ea5-b1e5-4e9e-b650-808e9d9dc944/content.
• Literature assessment checked at 2026-08-22: the supplied curation labels the question open, reports that the converse is explicitly posed as open, and reports that no later primary proof was located. 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 supplied registry marks the statement corrected_verified and reports a direct check of the official report. These are supplied provenance claims, not verification performed for this draft.