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Higher-dimensional Auslander-Reiten theory on maximal orthogonal subcategories. (English) Zbl 1115.16005
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.

MSC:
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.)
16G30 Representations of orders, lattices, algebras over commutative rings
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