From triangulated categories to abelian categories: cluster tilting in a general framework. (English) Zbl 1133.18005
Recent results on cluster algebras and cluster categories prove that there exist natural abelian quotient categories of cluster categories modulo tilting subcategories. The main aim of this interesting paper is to show that these results can be generalized to triangulated categories: If \(\mathcal{H}\) is a triangulated category and \(\mathcal{T}\) is a tilting subcategory (i.e., it is contravariantly finite, and \(X\in \mathcal{T}\) iff \(\mathrm{Ext}^{1}(X,\mathcal{T})=0\) iff \(\mathrm{Ext}^{1}(\mathcal{T},X)=0\)), then \(\mathcal{A}=\mathcal{H}/\mathcal{T}\) is abelian (Theorem 3.3). Moreover, the authors provide explicit descriptions for monomorphisms and epimorphisms in \(\mathcal{A}\), respectively for projective objects and injective objects in \(\mathcal{A}\). All these are used to prove another important result: The category \(\mathcal{A}\) has enough projectives, has enough injectives, and it is Gorenstein of Gorenstein dimension at most one.
The quotient category \(\mathcal{A}\) is also studied for the case \(\mathcal{H}\) is a \(k\)-linear triangulated category with split idempotents and with finite-dimensional \(\mathrm{Hom}\)-spaces. It is proved that if \(\mathcal{H}\) admits a Serre duality then the canonical functor \(\pi:\mathcal{H}\to \mathcal{A}\) preserves 1-orthogonal subcategories, and if \(\mathcal{H}\) is a cluster category then the restriction of \(\pi\) to \(\mathcal{T}[-1]\) is a Galois covering of the cluster tilted algebra \(\pi(\mathcal{T}[-1])\), and \(\pi\) induces a covering functor \(\mathrm{mod}(\mathcal{T}[-1])\to \mathrm{mod}(\pi(\mathcal{T}[-1]))\).
In the last section of the paper the authors study potential converse of the main result Theorem 3.3. It is pointed out that a general converse is not valid, and it is proved that if a quotient category of a triangulated category \(\mathcal{H}\) modulo a self-orthogonal subcategory \(\mathcal{T}\) is abelian then \(\mathcal{T}\) satisfies a maximality orthogonal condition which is weaker than the condition which appears in the definition of tilting subcategories.
The quotient category \(\mathcal{A}\) is also studied for the case \(\mathcal{H}\) is a \(k\)-linear triangulated category with split idempotents and with finite-dimensional \(\mathrm{Hom}\)-spaces. It is proved that if \(\mathcal{H}\) admits a Serre duality then the canonical functor \(\pi:\mathcal{H}\to \mathcal{A}\) preserves 1-orthogonal subcategories, and if \(\mathcal{H}\) is a cluster category then the restriction of \(\pi\) to \(\mathcal{T}[-1]\) is a Galois covering of the cluster tilted algebra \(\pi(\mathcal{T}[-1])\), and \(\pi\) induces a covering functor \(\mathrm{mod}(\mathcal{T}[-1])\to \mathrm{mod}(\pi(\mathcal{T}[-1]))\).
In the last section of the paper the authors study potential converse of the main result Theorem 3.3. It is pointed out that a general converse is not valid, and it is proved that if a quotient category of a triangulated category \(\mathcal{H}\) modulo a self-orthogonal subcategory \(\mathcal{T}\) is abelian then \(\mathcal{T}\) satisfies a maximality orthogonal condition which is weaker than the condition which appears in the definition of tilting subcategories.
Reviewer: Simion Sorin Breaz (Cluj-Napoca)
MSC:
| 18E30 | Derived categories, triangulated categories (MSC2010) |
| 16G20 | Representations of quivers and partially ordered sets |
| 16G70 | Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers |
| 16S99 | Associative rings and algebras arising under various constructions |
| 17B20 | Simple, semisimple, reductive (super)algebras |
Keywords:
triangulated categories; abelian categories; 1-orthogonal categories; tilting; cluster categories; Gorenstein algebrasReferences:
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