Canonical Decomposition Cones and TF Equivalence

Definitions and mathematical statement

Let $k$ be an algebraically closed field and let $A$ be a finite-dimensional $k$-algebra, as required by the supplied original passage. Write $\operatorname{proj}A$ for the category of finitely generated projective right $A$-modules and put \[ V_A=K_0(\operatorname{proj}A)\otimes_{{\mathbb{Z}}}{\mathbb{R}}. \] The use of right modules is an explicit convention for this draft; the supplied extract does not specify handedness, which must be checked against the references.

For $\eta\in V_A$ and a finite-dimensional right $A$-module $M$, define $\eta(M)$ by real-linear extension of \[ [P](M)=\dim_k\operatorname{Hom}_A(P,M). \] Define the two weak numerical torsion classes by \[ \begin{aligned} \overline{\mathcal T}_{\eta} &=\{M:\eta(Q)\geq 0\text{ for every quotient module }Q\text{ of }M\},\\ \overline{\mathcal F}_{\eta} &=\{M:\eta(L)\leq 0\text{ for every submodule }L\subseteq M\}. \end{aligned} \] Two weights are TF-equivalent when both of these classes agree. Thus \[ [\theta]_{\mathrm{TF}} =\{\eta\in V_A: \overline{\mathcal T}_{\eta}=\overline{\mathcal T}_{\theta} \text{ and } \overline{\mathcal F}_{\eta}=\overline{\mathcal F}_{\theta}\}. \]

For an integral class $\gamma\in K_0(\operatorname{proj}A)$, choose projectives $P_\gamma^-$ and $P_\gamma^+$ with no common indecomposable summands such that \[ \gamma=[P_\gamma^+]-[P_\gamma^-]. \] The generic canonical decomposition of $\gamma$ is the decomposition into generically indecomposable projective-presentation classes obtained from a Zariski-generic map in $\operatorname{Hom}_A(P_\gamma^-,P_\gamma^+)$. Write $\operatorname{ind}(\gamma)$ for the set of its summand classes, ignoring multiplicities, and set \[ \operatorname{ind}({\mathbb{N}}\theta) :=\bigcup_{n\geq 1}\operatorname{ind}(n\theta). \] Here $\operatorname{ind}(0)=\varnothing$; including $n=0$ consequently has no effect. These are the standard projective-presentation interpretations of notation left undefined in the supplied extract and require source confirmation.

For $S\subseteq V_A$, let \[ \operatorname{cone}(S) =\left\{\sum_{i=1}^{r}a_i s_i: r\geq 0,\ a_i\in{\mathbb{R}}_{\geq0},\ s_i\in S\right\}. \] The empty sum is zero. Let $\operatorname{relint}$ denote interior relative to the real linear span of this cone; no closure is taken before forming the relative interior.

The requested conjecture, identified as Conjecture 2 in the supplied passage, is: for every algebraically closed field $k$, every finite-dimensional $k$-algebra $A$, and every $\theta\in K_0(\operatorname{proj}A)$, does one have \[ [\theta]_{\mathrm{TF}} =\operatorname{relint}\!\left( \operatorname{cone}\left(\operatorname{ind}({\mathbb{N}}\theta)\right) \right)? \] This includes $\theta=0$, with $\operatorname{cone}(\varnothing)=\{0\}$ and $\operatorname{relint}\{0\}=\{0\}$.

The supplied passage attributes to its Theorem 1 the inclusion of the canonical-decomposition cone interior in $[\theta]_{\mathrm{TF}}$, and also asserts the inclusion for $\operatorname{ind}({\mathbb{N}}\theta)$. It states that $\operatorname{cone}(\operatorname{ind}\theta)\subseteq\operatorname{cone}(\operatorname{ind}({\mathbb{N}}\theta))$ need not be equality, referring to its Theorem 5. The extracted question interprets the report's notation $\operatorname{cone}^{\circ}$ as relative interior; its agreement with the report's exact convention, especially for the potentially infinite generating set, remains to be checked.

Short English statement

For a finite-dimensional algebra over an algebraically closed field, ask whether every real weight TF-equivalent to an integral weight $\theta$ lies precisely in the relative interior of the cone generated by the generic canonical summands of all positive multiples of $\theta$. The supplied curation labels the problem partially solved, with hereditary and $E$-tame cases proved; this is a dated supplied assessment, not an independently verified conclusion.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005755$, title "Canonical Decomposition Cones and TF Equivalence," exact source problem code catalogue entry. The record attributes the problem to the Oberwolfach Reports Open Problems collection and gives proposed year 2024; no individual proposer is supplied.
• Cluster Algebras and Its Applications (2024), Oberwolfach Reports, pp. 69--136, DOI 10.4171/owr/2024/2: https://doi.org/10.4171/owr/2024/2. This is both the supplied source report and the literature-source link labelled "original OWR report." The supplied extract identifies the target as Conjecture 2 and includes context attributed to Theorems 1 and 5; it supplies no precise page for that passage.
• Supplied literature source labelled "published primary paper": https://doi.org/10.1112/plms.12639. No paper title, authors, or theorem locator is supplied.
• Literature assessment checked at 2026-08-22: the supplied curation states that the equality is proved for hereditary and $E$-tame algebras, while the general finite-dimensional-algebra case remains unresolved. This is supplied dated curation, not independently verified here. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that the official report was checked directly. This records the supplied verification claim; it does not claim that the present draft independently visited or verified either reference.