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...ON PAIRS AND SIMPLE-MINDED SYSTEMS IN TRIANGULATED CATEGORIES ALEX DUGAS Abstract. Let T be a Hom-finite triangulated Krull-Schmidt category over a field k. Inspired by a definition of Koenig and Liu =-=[14]-=-, 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 simpl...

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...ver, they still generate the stable category and there is no cohomology between them in negative degrees. Such systems of objects in a stable module category have been called simple-minded systems in =-=[15]-=-. Analogous systems of objects in a derived module category (defined in a slightly different way) have been called cohomologically Schurian collections in [3] and simple-minded collections in [16]. An...

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...esult by Rickard-Rouquier [30]. As another application, we establish a bijection between families of ‘simple-minded objects’ (a piece of terminology due to J. Rickard and, independently, to König-Liu =-=[23]-=-) and hearts of t-structures in suitable triangulated categories (section 9). Further applications will be given in the forthcoming paper [20]. In establishing our main theorem, we need foundational r...

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... discuss a construction by Rickard [16]. The same construction is studied by Keller–Nicolás [9] in the context of positive dg algebras. Let r be the rank of the Grothendieck group of modΛ. Following =-=[11]-=- we say that a set of objects X1, . . . ,Xr in the bounded derived category Db(modΛ) are simple-minded if the following conditions hold (1) Hom(Xi,Σ mXj) = 0, ∀ m < 0, (2) Hom(Xi,Xj) = K if i = j...

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...We denote by sms(A) the collection of all simple-minded systems of A. 3 The definition presented above is taken from [Dug2]. This is different from the original definition of simpleminded system from =-=[KL]-=-. In [KL], simple-minded system is defined for stable module category of any artinian algebra, which does not necessarily possess any triangulated structure. Nevertheless, we are only interested in st...

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... result by Rickard-Rouquier [30]. As another application, we establish a bijection between families of ‘simple-minded objects’ (a piece of terminology due to J. Rickard and, indepently, to König-Liu =-=[23]-=-) and hearts of t-structures in suitable triangulated categories (section 9). Further applications will be given in the forthcoming paper [20]. In establishing our main theorem, we need foundational r...

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... of the derived category of a symmetric algebra, which he called periodic twists [11]. While 1This problem can be viewed as a special case of the more general question: if S is a simple-minded system =-=[14, 9]-=- in mod-A is there necessarily an algebra B and an equivalence α : mod-B → mod-A sending the simple B-modules to S? 1 there are ample similarities, we do not see any obvious way in which these constru...