The semibrick-pair completion conjecture for tau-tilting finite algebras

Let Λ\Lambda be a τ\tau-tilting finite algebra, and let DU[1]\mathcal{D} \sqcup \mathcal{U}[1] be a semibrick pair, meaning a collection of bricks in the indicated degrees satisfying the relevant hom-orthogonality conditions. Let the smallest wide subcategory containing DU[1]\mathcal{D}\sqcup \mathcal{U}[1] be the wide subcategory generated by this pair. Semibrick-pair completion conjecture. The pair DU[1]\mathcal{D} \sqcup \mathcal{U}[1] is completable if and only if the smallest wide subcategory containing it has D+U|\mathcal{D}| + |\mathcal{U}| simple objects. This concerns when semibrick pairs extend to 2-term simple minded collections in the τ\tau-tilting finite setting; the supplied text does not state whether the conjecture has been resolved.

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Primary source

Emily Barnard and Eric J. Hanson, “Pairwise Compatibility for 2-Simple Minded Collections II: Preprojective Algebras and Semibrick Pairs of Full Rank”, arXiv:2010.08645 (2023).

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