While Auslander-Reiten theory is two-dimensional in the sense that almost split sequences can be regarded as projective resolutions of simple functors of length two, the present article exhibits a class of subcategories of classical representation categories with generalized almost split sequences of length \(n\).
Let \(R\) be a complete regular local ring of dimension \(d\), and let \(\Lambda\) be a Cohen-Macaulay \(R\)-order, i.e., an \(R\)-algebra which is finitely generated and free over \(R\). Assume that \(\Lambda\) represents an isolated singularity in the sense of Auslander. Within the category \(\Lambda\text{CM}\) of maximal Cohen-Macaulay modules over \(\Lambda\), the author considers functorially finite full subcategories \(\mathcal C\) which are maximally \(l\)-orthogonal for some \(l>0\). This means that \(\mathcal C\) consists of all objects \(X\) in \(\Lambda\text{CM}\) which satisfy \(\text{Ext}^i(X,Y)=0\) for all \(Y\) in \(\mathcal C\) and \(0<i\leq l\), and similarly, \(\mathcal C\) consists of all \(X\) in \(\Lambda\text{CM}\) with \(\text{Ext}^i(Y,X)=0\) for all \(Y\) in \(\mathcal C\) and \(0<i\leq l\). It turns out that such categories \(\mathcal C\) admit generalized almost split sequences of length \(2+l\).
Among other things, the author proves a higher Auslander-Reiten formula, he shows that long almost split sequences are found in the socle of higher Ext-groups, he generalizes Auslander’s description of the Auslander-Reiten translate (in dimension \(d\)), and characterizes the higher Auslander-Reiten translate by means of derived functors.
For representation-finite Gorenstein orders \(\Lambda\) of arbitrary dimension \(d\), he gives a complete classification of maximal 1-orthogonal subcategories of \(\Lambda\text{CM}\). This classification is related to configurations of bijectives in dimension 0 (Riedtmann) and 1 (Wiedemann). The author contributes to the ubiquity of Catalan numbers by showing that in case \(\mathbb{A}_n\), maximal 1-orthogonal subcategories can be enumerated by triangulations of polygons. In the \(\mathbb{B}\)-, \(\mathbb{C}\)-, and \(\mathbb{D}\)-cases, he gives a similar enumeration.
As an interesting example, he exhibits a maximal \((d-2)\)-orthogonal subcategory of the category \(\Lambda\text{CM}\), where the isolated singularity \(\Lambda\) is the ring of invariants of a \(d\)-dimensional power series ring under a finite group without pseudo-reflections. Last, but not least, he shows that the unpleasant fact that for \(n>1\), the group \(\text{Ext}^n(X,Y)\) does not describe \(n\)-extensions up to isomorphism, vanishes if they are restricted to a maximal \((n-1)\)-orthogonal subcategory.

Reviewer: Wolfgang Rump (Stuttgart)

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.