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Cluster tilting for higher Auslander algebras. (English) Zbl 1233.16014
Let \(\Lambda\) be a finite dimensional algebra. Then, by O. Iyama [Adv. Math. 210, No. 1, 22-50 (2007; Zbl 1115.16005)], the category of finite dimensional modules over \(\Lambda\) admits an \(n\)-Auslander-Reiten translate, \(\tau_n\) (and dual translate \(\tau^-_n\)). Let \(\mathcal M\) denote \(\tau_n\)-closure of \(D\Lambda\), where \(D=\operatorname{Hom}_k(-,k)\), and set \(\mathcal P(\mathcal M)\) to be the subcategory of objects in \(\mathcal M\) with projective dimension less than \(n\). Let \(\mathcal M_{\mathcal P}\) be the subcatgory of \(\mathcal M\) consisting of objects with no non-zero summands in \(\mathcal P(\mathcal M)\). Then \(\Lambda\) is said to be ‘\(n\)-complete’ if its global dimension is at most \(n\), there exists a tilting \(\Lambda\)-module \(T\) such that \(\mathcal P(\mathcal M)=\text{add\,}T\), \(\mathcal M\) is an \(n\)-cluster tilting subcategory of \(T^\perp\) (the subcategory of mod-\(\Lambda\) consisting of objects \(X\) such that \(\text{Ext}^i(T,X)=0\) for \(i>0\)), and \(\text{Ext}^i(\mathcal M_{\mathcal P},\Lambda)=0\) for \(0<i<n\).
Then it is shown that \(\Lambda\) is necessarily \(\tau_n\)-finite, i.e. global dimension at most \(n\) and \(\tau_n^l\) vanishing on \(D\Lambda\) for some \(l\). Let \(M\) be an additive generator for \(\mathcal M\). Then the ‘cone’ of \(\Lambda\) is the endomorphism algebra of \(M\). The main result of the paper is to show that, if \(\Lambda\) is \(n\)-complete, its cone is \((n+1)\)-complete.
This is a very interesting paper for several reasons. For example, the theory developed, including the above result, is a natural generalisation of some of the theory of Auslander algebras. It gives a nice way to construct \(n\)-cluster-tilting subcategories, inductively (by repeatedly applying the above result). There are close connections with the theory of Cohen-Macaulay modules over a singularity and some interesting higher dimensional Auslander-Reiten quivers appear.

MSC:
16G10 Representations of associative Artinian rings
13F60 Cluster algebras
16G20 Representations of quivers and partially ordered sets
16E30 Homological functors on modules (Tor, Ext, etc.) in associative algebras
16G70 Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers
16E65 Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.)
16G50 Cohen-Macaulay modules in associative algebras
16G60 Representation type (finite, tame, wild, etc.) of associative algebras
16E10 Homological dimension in associative algebras
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