Gluing of n -Cluster Tilting Subcategories for Representation-directed Algebras
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Gluing of n -Cluster Tilting Subcategories for Representation-directed Algebras Laertis Vaso1 Received: 4 April 2019 / Accepted: 20 April 2020 / © The Author(s) 2020
Abstract Given n ≤ d < ∞, we investigate the existence of algebras of global dimension d which admit an n-cluster tilting subcategory. We construct many such examples using representation-directed algebras. First, given two representation-directed algebras A and B, a projective A-module P and an injective B-module I satisfying certain conditions, we A in such a show how we can construct a new representation-directed algebra way that the representation theory of is completely described by the representation theories of A and B. Next we introduce n-fractured subcategories which generalize n-cluster tilting subcategories for representation-directed algebras. We then show how one can construct an n-cluster tilting subcategory for by using n-fractured subcategories of A and B. As an application of our construction, we show that if n is odd and d ≥ n then there exists an algebra admitting an n-cluster tilting subcategory and having global dimension d. We show the same result if n is even and d is odd or d ≥ 2n. Keywords n-cluster tilting subcategory · Representation-directed algebra · Global dimension · Nakayama algebra Mathematics Subject Classification (2010) Primary: 16G10. Secondary: 16E10 · 16G70 · 18E10.
1 Introduction For a representation-finite algebra , classical Auslander–Reiten theory gives a complete description of the module category mod , see for example [2]. However in general the whole module category of an algebra is very hard to study. In Osamu Iyama’s higherdimensional Auslander–Reiten theory [12, 13] one moves the focus from mod to a
Presented by: Christof Geiss Laertis Vaso
[email protected] 1
L¨agerhyddsv¨agen 1, Box 480, 75106, Uppsala, Sweden
L. Vaso
suitable subcategory C ⊆ mod satisfying certain homological properties. Such a subcategory C is called an n-cluster tilting subcategory for some positive integer n; if moreover C = add(M) for some M ∈ mod , then M is called an n-cluster tilting module. An n-cluster tilting subcategory C is the setting for a higher-dimensional analogue of the classical Auslander–Reiten theory: it admits an n-Auslander–Reiten translation, n-almost split sequences and an n-Auslander–Reiten duality generalising the classical Auslander– Reiten translation, almost split sequences and Auslander–Reiten duality when n = 1. However, in general it is not easy to find n-cluster tilting subcategories. If we set d := gl. dim , then the existence of an n-cluster tilting subcategory for n > d implies that is semisimple. Hence we may restrict to the case n ≤ d. The extreme case n = d is of special interest and has been studied extensively before, for example in [15] and [9]. If C is given by a d-cluster tilting module M, it follows that C is unique and given by
C = add{τd−i () | i ≥ 0}, where τd− denotes the d-Auslander–Reiten translation. In this case is called d-representation-finite (se
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