2012/07/31 by Alex Dugas
Mathematics · #Algebraic structures and combinatorial models #Axiom #Equivalence (formal languages) #Functor #Homotopy and Cohomology in Algebraic Topology #Invariant (physics) #Pairwise comparison #Rings, Modules, and Algebras #Torsion (gastropod) #Triangulated category #math.RT
paper · pdf · doi:10.1007/s10485-014-9365-8
published as Appl. Cat. Structures 23 (2015), 23, 507-526 · Minor corrections. To appear in Applied Categorical Structures. The final publication is available at springerlink.com: http://link.springer.com/article/10.1007%2Fs10485-014-9365-8
openalex publication_date 2014/02/10 · arxiv created 2014/03/31 · arxiv updated 2016/06/07 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
Let T be a Hom-finite triangulated Krull-Schmidt category over a field k. Inspired by a definition of Koenig and Liu, we say that a family S of pairwise orthogonal objects in T with trivial endomorphism rings 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 is the stable module category of a self-injective algebra Λ. In this case, our mutation procedure parallels that introduced by Koenig and Yang for simple-minded collections in the derived category of Λ. 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 between Γ and Λ, where Γ is the tilting mutation of Λ relative to X.