Relative homology, higher cluster-tilting theory and categorified Auslander-Iyama correspondence
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.
scientific article; zbMATH DE number 6493633
scientific article