From CM-finite to CM-free. (English) Zbl 1352.16016
Let \(\Lambda\) be an Artin algebra and denote by \(\mathsf{Gproj}(\Lambda)\) the category of finitely generated Gorenstein-projective \(\Lambda\)-modules. Recall from [A. Beligiannis, Adv. Math. 226, No. 2, 1973–2019 (2011; Zbl 1239.16016)] that \(\Lambda\) is said to be of finite Cohen-Macaulay type (CM-finite for short), if the set of isomorphism classes of indecomposable finitely generated Gorenstein-projective modules is finite. For a CM-finite algebra \(\Lambda\) consider the relative Auslander algebra \(\mathbf{Aus}(\Lambda):=\mathsf{End}_{\Lambda}(X)^{op}\), where \(X\) is the additive generator of \(\mathsf{Gproj}(\Lambda)\).
In the paper under review, the authors study CM-finite Artin algebras for not necessarily Gorenstein algebras. In particular, they show that the relative Auslander algebra \(\mathbf{Aus}(\Lambda)\) of a CM-finite Artin algebra \(\Lambda\) is CM-free, that is, the category of Gorenstein-projective \(\mathbf{Aus}(\Lambda)\)-modules coincides with the projective \(\mathbf{Aus}(\Lambda)\)-modules. Moreover, for an abelian category \(\mathcal{A}\) with enough projective objects the authors provide a description of the defect category \(\mathsf{D}^b_{\mathsf{defect}}(\mathcal{A}):=\mathsf{D}^b_{\mathsf{sg}}(\mathcal{A})/\mathsf{Image}{F}\) introduced by P. A. Bergh, S. Oppermann and D. A. Jorgensen [Q. J. Math. 66, No. 2, 459–471 (2015; Zbl 1327.13041)], where \(F\) is the canonical triangle embedding of the stable category \(\underline{\mathsf{Gproj}}(\mathcal{A})\) to the singularity category \(\mathsf{D}^b_{\mathsf{sg}}(\mathcal{A})\) (in the sense of Buchweitz and Orlov). The description is given as the Verdier quotient of \(\mathsf{K}^{-,b}(\mathsf{Proj}(\mathcal{A}))\) by the thick subcategory \[ \begin{split}\mathsf{K}^{-,b}_{\mathcal{G}}(\mathsf{Proj}(\mathcal{A})) \\ =\big\{(P,d) \in \mathsf{K}^{-,b}(\mathsf{Proj}(\mathcal{A})) \;| \;\exists n_0\in \mathbb{Z} \;\text{with} \;H^m(P)=0, \forall m\leq n_0, \mathsf{Ker}{d^{n_0}}\in \mathsf{GProj}(\mathcal{A}) \big\}. \end{split} \] As a consequence, for a CM-finite algebra \(\Lambda\) they show that there is a triangle equivalence between the defect category \(\mathsf{D}^b_{\mathsf{defect}}(\Lambda)\) of \(\Lambda\) and the singularity category \(\mathsf{D}^b_{\mathsf{sg}}(\mathbf{Aus}(\Lambda))\) of the relative Auslander algebra \(\mathbf{Aus}(\Lambda)\).
In the paper under review, the authors study CM-finite Artin algebras for not necessarily Gorenstein algebras. In particular, they show that the relative Auslander algebra \(\mathbf{Aus}(\Lambda)\) of a CM-finite Artin algebra \(\Lambda\) is CM-free, that is, the category of Gorenstein-projective \(\mathbf{Aus}(\Lambda)\)-modules coincides with the projective \(\mathbf{Aus}(\Lambda)\)-modules. Moreover, for an abelian category \(\mathcal{A}\) with enough projective objects the authors provide a description of the defect category \(\mathsf{D}^b_{\mathsf{defect}}(\mathcal{A}):=\mathsf{D}^b_{\mathsf{sg}}(\mathcal{A})/\mathsf{Image}{F}\) introduced by P. A. Bergh, S. Oppermann and D. A. Jorgensen [Q. J. Math. 66, No. 2, 459–471 (2015; Zbl 1327.13041)], where \(F\) is the canonical triangle embedding of the stable category \(\underline{\mathsf{Gproj}}(\mathcal{A})\) to the singularity category \(\mathsf{D}^b_{\mathsf{sg}}(\mathcal{A})\) (in the sense of Buchweitz and Orlov). The description is given as the Verdier quotient of \(\mathsf{K}^{-,b}(\mathsf{Proj}(\mathcal{A}))\) by the thick subcategory \[ \begin{split}\mathsf{K}^{-,b}_{\mathcal{G}}(\mathsf{Proj}(\mathcal{A})) \\ =\big\{(P,d) \in \mathsf{K}^{-,b}(\mathsf{Proj}(\mathcal{A})) \;| \;\exists n_0\in \mathbb{Z} \;\text{with} \;H^m(P)=0, \forall m\leq n_0, \mathsf{Ker}{d^{n_0}}\in \mathsf{GProj}(\mathcal{A}) \big\}. \end{split} \] As a consequence, for a CM-finite algebra \(\Lambda\) they show that there is a triangle equivalence between the defect category \(\mathsf{D}^b_{\mathsf{defect}}(\Lambda)\) of \(\Lambda\) and the singularity category \(\mathsf{D}^b_{\mathsf{sg}}(\mathbf{Aus}(\Lambda))\) of the relative Auslander algebra \(\mathbf{Aus}(\Lambda)\).
Reviewer: Chrysostomos Psaroudakis (Trondheim)
MSC:
| 16G10 | Representations of associative Artinian rings |
| 18E30 | Derived categories, triangulated categories (MSC2010) |
| 16G50 | Cohen-Macaulay modules in associative algebras |
| 18G25 | Relative homological algebra, projective classes (category-theoretic aspects) |
Keywords:
Gorenstein-projective module; Cohen-Macaulay finite algebra; relative Auslander algebra; Gorenstein defect category; Singularity categoryReferences:
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