# zbMATH — the first resource for mathematics

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
Full Text: