Classification of Liquid Condensed Modules

Definitions and mathematical statement

The supplied record asks about liquid condensed $R$-modules but does not specify the admissible rings $R$, the meaning of liquid, or the liquidification construction. For the purpose of restating the question only, let $\mathcal L_R$ denote the class called liquid condensed $R$-modules in the source, and let $L_R(M)$ denote the object called the liquidification of $M$, whenever the intended construction is defined. These symbols are placeholders for unspecified source notions, not independent definitions of them.

Source question, with unresolved qualifications. Characterize $\mathcal L_R$. The source presents as an equivalent task the determination of \[ L_R\left(R[\beta S]\right) \] for the objects $R[\beta S]$ that it calls compact projective generators. The admissible objects $S$, the meaning of $\beta S$, the construction $R[\beta S]$, and the ambient category in which these objects are compact projective generators are not specified in the supplied record. Nor is a defining property or universal property of liquidification supplied. The asserted equivalence is therefore retained as a source assertion, without independent justification. Until these data and their quantifiers are supplied, this is an underspecified characterization request rather than a determinate mathematical conjecture.

Short English statement

The source asks for a characterization of liquid condensed $R$-modules, or, in its stated equivalent formulation, an identification of the liquidifications of $R[\beta S]$. The hypotheses on $R$, the range of $S$, and the definitions of liquidity and liquidification are missing, so a precise quantified problem cannot yet be extracted.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30004462$, title Classification of Liquid Condensed Modules, source problem code catalogue entry, in the collection Oberwolfach Reports Open Problems. No individual proposer is identified; the proposed year is recorded as 2020.
• Supplied report citation: Non-Commutative Geometry and Cyclic Homology (2021), Oberwolfach Reports, pp. 919--960, DOI 10.4171/owr/2020/17. Supplied source URL and literature-source link, labelled "original OWR report": https://doi.org/10.4171/owr/2020/17. No specific page, speaker, or passage within the report is supplied.
• Literature assessment dated 2026-08-22: the supplied curation labels the problem open and describes the extracted question as underspecified, with no determinate mathematical statement whose resolution can be verified. This is supplied dated curation, not an independently verified literature assessment; the report has not been consulted for this draft.
• The registry labels the statement corrected_verified and says its mathematical content was preserved while notation, typography, omitted definitions, or duplicate clauses were repaired for self-contained publication. These are supplied editorial claims, not independent verification of the formulation or its mathematical correctness.