The silting form of the Auslander–Reiten conjecture

Let kk be a field, let AA be a finite-dimensional kk-algebra, and let Db(modA)\mathsf{D}^{\rm b}(\mathsf{mod} A) be its bounded derived category. An object is presilting if it has no positive self-extensions; write addX\mathsf{add}X for the additive closure of XX, and let projA\mathsf{proj} A be the category of finitely generated projective AA-modules. The silting form of the Auslander–Reiten conjecture. Db(modA)\mathsf{D}^{\rm b}(\mathsf{mod} A) has no presilting object XX such that addX\mathsf{add}X contains projA\mathsf{proj} A as a proper subcategory. This is posed as a silting-theoretic reformulation of the Auslander–Reiten conjecture, and the source does not state whether it is resolved.

Sources & referencesView supporting material

Primary source

Osamu Iyama and Dong Yang, “Silting reduction and Calabi–Yau reduction of triangulated categories”, arXiv:1408.2678 (2018).

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