Semifree-Module Form of the Toral-Rank Conjecture

Definitions and mathematical statement

To interpret the question, one must specify a ground field $k$ and a differential graded algebra $(A,d_A)$ whose underlying algebra is denoted in the supplied record by $\bigwedge(x_1,\ldots,x_r)$. The record does not specify the field, the degrees of the generators, the meaning of this wedge notation, or the differential $d_A$.

For a specified differential graded algebra $(A,d_A)$, a differential graded $A$-module $(M,D)$ has $D^2=0$ and, in cohomological grading, satisfies \[ D(am)=d_A(a)m+(-1)^{|a|}aD(m) \] for homogeneous $a\in A$. Such a module is semifree if it admits an exhaustive increasing filtration by differential graded submodules \[ 0=F_{-1}M\subseteq F_0M\subseteq F_1M\subseteq\cdots, \qquad M=\bigcup_{j\geq0}F_jM, \] whose successive quotients are direct sums of degree shifts of $(A,d_A)$. In the displayed presentation $M=A\otimes_k V$, $V$ denotes the graded vector space of free generators. Dimensions below mean total dimensions over $k$, so that \[ \dim_k H^*(M,D)=\sum_n\dim_k H^n(M,D), \qquad \dim_k V=\sum_n\dim_k V^n. \] These are standard interpretive conventions; the source record does not supply the grading or filtration conventions.

The supplied question asks whether, for every number $r$ of generators and every semifree module of the displayed form, \[ (M,D)=\left(\bigwedge(x_1,\ldots,x_r)\otimes_k V,D\right), \qquad \dim_k H^*(M,D)<\infty, \] one must have \[ \dim_k V\geq 2^r. \] This is a transcription of the intended implication, subject to unresolved hypotheses. In particular, the record does not say whether the wedge denotes an exterior algebra or a free graded commutative algebra, whether the generators have prescribed parity or degree, or whether the module must satisfy additional boundedness or minimality conditions. Finite-dimensional cohomology permits zero cohomology, and no nonvanishing condition on $H^*(M,D)$ or exclusion of the zero module is stated. These omissions must be resolved before treating the displayed implication as a fully specified conjecture. No additional hypotheses are imposed here.

Short English statement

The supplied problem asks whether a semifree module over an algebra written as $\bigwedge(x_1,\ldots,x_r)$, with finite total cohomology, must have at least $2^r$ free generators. The field, algebra convention, generator degrees, and possible nonvanishing hypotheses remain unspecified. The supplied curation describes this as an algebraic form of the toral-rank conjecture and labels it open; those claims have not been independently verified.

Sources

• Local registry: the supplied UnsolvedMath record, permanent ID $30001757$, exact source problem code catalogue entry. The project identifies data/unsolvedmath/problems.json as its main registry and catalog/problems.json as an additional registry. Their contents were not independently inspected for this draft.
• Supplied report citation: Arbeitsgemeinschaft: Rational Homotopy Theory in Mathematics and Physics (2011), Oberwolfach Reports, DOI 10.4171/owr/2011/18, pp. 993--1045. https://doi.org/10.4171/owr/2011/18. The record attributes the problem to workshop open-problem discussions and identifies no individual proposer.
• Supplied literature source: Rational Homotopy Theory in Mathematics and Physics, OWR 18/2011. https://doi.org/10.4171/owr/2011/18.
• Supplied literature source, with its supplied descriptive label: Bettiol--Mendes--Töben, bounds on sectional curvature. https://arxiv.org/abs/1407.6201. Its relevance to the displayed module inequality is not explained in the record.
• Literature assessment checked at 2026-08-21: the supplied curation labels the problem open and describes the displayed inequality as an algebraic form of the toral-rank conjecture whose general case remains unresolved. This is supplied dated curation, not an independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The supplied record has statement status corrected_verified and reports that the mathematical content was checked against the official Oberwolfach report, with notation, context, and question form normalized without changing the source-level claim. This verification claim is preserved as registry attribution and was not independently checked here.