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Torsion Pairs and Simple-Minded Systems in Triangulated Categories

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Abstract

Let 𝒯 be a Hom-finite triangulated Krull-Schmidt category over a field k. Inspired by a definition of Koenig and Liu (Q. J. Math 63(3), 653-674, 2012), we say that a family 𝒮 ⊆ 𝒯 of pairwise orthogonal bricks is a simple-minded system if its closure under extensions is all of 𝒯. We construct torsion pairs in 𝒯 associated to any subset 𝒳 of a simple-minded system 𝒮, and use these to define left and right mutations of 𝒮 relative to 𝒳. When 𝒯 has a Serre functor ν and 𝒮 and 𝒳 are invariant under ν ∘ [1], we show that these mutations are again simple-minded systems. We are particularly interested in the case where 𝒯 = mod-Λ for a self-injective algebra Λ. In this case, our mutation procedure parallels that introduced by Koenig and Yang for simple-minded collections in D b(mod-Λ) (Koenig and Yang, 2013). It follows that the mutation of the set of simple Λ-modules relative to 𝒳 yields the images of the simple Γ-modules under a stable equivalence mod-Γ → mod-Λ, where Γ is the tilting mutation of Λ relative to 𝒳.

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Correspondence to Alex Dugas.

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Dugas, A. Torsion Pairs and Simple-Minded Systems in Triangulated Categories. Appl Categor Struct 23, 507–526 (2015). https://doi.org/10.1007/s10485-014-9365-8

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