Relative homology, higher cluster-tilting theory and categorified Auslander-Iyama correspondence
type: Item
instance of: scholarly article
title: Relative homology, higher cluster-tilting theory and categorified Auslander-Iyama correspondence
zbMATH Open document ID: 1329.18019
author: Apostolos Beligiannis
published in: Journal of Algebra
publication date: 2015-10-13T00:00:00Z
review text: In [Representation dimension of Artin algebras. Queen Mary College Mathematics Notes. London: Queen Mary College (1971; Zbl 0331.16026)], \textit{M. Auslander} proved that there is a one-to-one correspondence between the set of Morita equivalence classes of Artin algebras with dominant dimension at least two and global dimension at most two and the set of Morita equivalence classes of Artin algebras of finite representation type. Recently \textit{O. Iyama}, extended this correspondence to higher dimensions [Adv. Math. 210, No. 1, 51--82 (2007; Zbl 1115.16006)]. In the present article under review, the first main result is to prove a categorified version of the above correspondence. More precisely, it is proved that \(n\)-cluster tilting subcategories in abelian categories with enough injectives correspond bijectively to \(n\)-Auslander categories. Also, a bijection is established between (cluster/Morita) equivalence classes of pairs \((\mathcal{C}, M)\), and left coherent and right perfect \(n\)-Auslander rings \(\mathrm{End}_{\mathcal{C}}(M)\), where \(\mathcal{C}\) is a Grothendieck category and \(M\) is a selfcompact \(n\)-cluster tilting object in \(\mathcal{C}\). Then, by considering an \((n-k)\)-corigid \((n+1)\)-cluster tilting subcategory \(\mathfrak{X}\) of a triangulated category \(\mathcal{C}\) for some integer \(0\leq k\leq \frac{n+1}{2}\), the author proves that the cluster tilted category mod-\(\mathfrak{X}\) is \(k\)-Gorenstein. Furthermore, if \(\mathcal{C}\) is \((n+1)\)-Calabi-Yau and the objects in mod-\(\mathfrak{X}\) are all of the form \(\mathcal{C}(-,A)|_{\mathfrak{X}}\) for \(A\) lies in certain extension category, it is shown that the cluster tilted category mod-\(\mathfrak{X}\) is stably \((n+2)\)-Calabi-Yau.
reviewed by: Ming Ding
Mathematics Subject Classification ID: 18G25
Mathematics Subject Classification ID: 18G20
Mathematics Subject Classification ID: 18E30
Mathematics Subject Classification ID: 18E35
Mathematics Subject Classification ID: 18E10
Mathematics Subject Classification ID: 16G10
Mathematics Subject Classification ID: 16G50
Mathematics Subject Classification ID: 13F60
zbMATH DE Number: 6493633
zbMATH Keywords: triangulated categories
zbMATH Keywords: abelian categories
zbMATH Keywords: relative homology
zbMATH Keywords: cluster tilting objects
zbMATH Keywords: Gorenstein categories
zbMATH Keywords: Calabi-Yau categories
zbMATH Keywords: Gorenstein-projective objects
zbMATH Keywords: higher Auslander categories and rings
zbMATH Keywords: Auslander correspondence
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2015.07.024
OpenAlex ID: W1908216672
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Relative homology, higher cluster-tilting theory and categorified Auslander-Iyama correspondence (DOI)
DOI: 10.1016/J.JALGEBRA.2015.07.024

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