Cluster-tilted algebras are Gorenstein and stably Calabi-Yau.(English)Zbl 1128.18007
The authors show that each cluster-tilting subcategory \(\mathcal T\) of a triangulated Calabi-Yau category \(\mathcal C\) of CY-dimension 2 is Gorenstein of dimension at most 1. That is, all finitely presented projective \(\mathcal T\)-modules are of finite injective dimension and the finitely presented injective \(\mathcal T\)-modules are of finite projective dimension, and the dimensions are at most 1. In addition they show that the stable Cohen-Macaulay category of \(\text{mod}\mathcal T\) is a Calabi-Yau category of CY-dimension 3. From these results it will be derived that cluster-tilted algebras are Gorenstein of dimension at most 1 and are hereditary if they are of finite global dimension.
Furthermore the authors prove a general result about relative 3-Calabi-Yau duality over non-stable endomorphism rings. This outcome in particular relates \(\text{Ext}_{\text{mod}}{\mathcal M}^i(X,Y)\)-groups to \(\text{Ext}_{\text{mod}}{\mathcal M}^{3-i}(Y,X)\)-groups isomorphically through the duality functor \(\operatorname{Hom}_\Bbbk(\cdot,\Bbbk)\), where \(\mathcal M\) is a subcategory in an abelian \(\Bbbk\)-linear Frobenius category \(\mathcal E\) with split idempotents, and is the pre-image of a cluster-tilting subcategory \(\mathcal T\subset\mathcal C\) of the stable category \(\mathcal C=\underline{\mathcal E}\) which is supposed to have finite-dimensional \(\operatorname{Hom}\)-spaces and to be Calabi-Yau of CY-dimension 2. The authors then generalise these results on relative Calabi-Yau duality to \(d\)-Calabi-Yau categories.
Furthermore the authors prove a general result about relative 3-Calabi-Yau duality over non-stable endomorphism rings. This outcome in particular relates \(\text{Ext}_{\text{mod}}{\mathcal M}^i(X,Y)\)-groups to \(\text{Ext}_{\text{mod}}{\mathcal M}^{3-i}(Y,X)\)-groups isomorphically through the duality functor \(\operatorname{Hom}_\Bbbk(\cdot,\Bbbk)\), where \(\mathcal M\) is a subcategory in an abelian \(\Bbbk\)-linear Frobenius category \(\mathcal E\) with split idempotents, and is the pre-image of a cluster-tilting subcategory \(\mathcal T\subset\mathcal C\) of the stable category \(\mathcal C=\underline{\mathcal E}\) which is supposed to have finite-dimensional \(\operatorname{Hom}\)-spaces and to be Calabi-Yau of CY-dimension 2. The authors then generalise these results on relative Calabi-Yau duality to \(d\)-Calabi-Yau categories.
Reviewer: Bernhard Drabant (Mühlhausen)
MSC:
| 18E30 | Derived categories, triangulated categories (MSC2010) |
| 16D90 | Module categories in associative algebras |
| 18G40 | Spectral sequences, hypercohomology |
| 18G10 | Resolutions; derived functors (category-theoretic aspects) |
| 55U35 | Abstract and axiomatic homotopy theory in algebraic topology |
Keywords:
clusterCite
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