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Auslander correspondence. (English) Zbl 1115.16006
The study of maximal \(l\)-orthogonal subcategories [Adv. Math. 210, No. 1, 22-50 (2007; see the preceding review Zbl 1115.16005)] is continued. In the present article, the concept of Auslander algebra is developed in an utmost general context.
The investigation starts with an adaption of higher Auslander-Reiten theory to dualizing \(R\)-varieties. For an Abelian category \(\mathcal A\) with enough projectives over a commutative local ring \(R\), and with length-finite \(\text{Ext}^1\)-groups, let \(\mathcal B\) be a resolving subcategory with enough injectives. Then maximal \(l\)-orthogonal subcategories \(\mathcal C\) of \(\mathcal B\) are defined, and the higher Auslander-Reiten formula is proved. In case \(\mathcal C\) is a Krull-Schmidt category, existence and properties of \(n\)-almost split sequences are derived.
Now let \(R\) be a complete regular local ring of dimension \(d\), and let \(\Lambda\) be a Cohen-Macaulay \(R\)-order which represents an isolated singularity. The category \(\Lambda\text{CM}\) of maximal Cohen-Macaulay modules over \(\Lambda\) is given by the standard \(d\)-cotilting module \(\Lambda^*=\operatorname{Hom}_R(\Lambda,R)\). More generally, the author considers an arbitrary \(m\)-cotilting module \(T\) in \(\Lambda\text{CM}\). Thus \(T\) gives rise to a resolving subcategory \(\mathcal B={^\perp T}\) of \(\mathcal A=\Lambda\text{-}\mathbf{mod}\) with an injective cogenerator \(T\). Within \(\mathcal B\), the author considers maximal \((n-1)\)-orthogonal subcategories \(\mathcal C\) with an additive generator \(M\). In this general context, depending on \(d,m\), and \(n\), the Auslander algebra \(\Gamma=\text{End}_\Lambda(M)\) is characterized. For example, in case \(d=0\), this leads to the condition that \(\text{gld\,}\Gamma\leq n+1\) and \(\text{dom.dim\,}\Gamma\geq n+1\). The general conditions are, of course, more complicated. In particular, Auslander algebras of type \(d=m=n+1\) are regular of global dimension \(d\).
The author provides several applications of higher Auslander-Reiten theory. If \(\text{add\,}M\) is maximal \((d-2)\)-orthogonal, he exhibits a relationship to non-commutative crepant resolutions [M. Van den Bergh, Duke Math. J. 122, No. 3, 423-455 (2004; Zbl 1074.14013)]. Inspired by Van den Bergh’s generalization of the Bondal-Orlov conjecture, the author conjectures that the endomorphism rings of \(M\) with \(\text{add\,}M\) maximal \(l\)-orthogonal are derived equivalent, and proves this for \(l=1\). For example, let \(G\) be a finite group operating without pseudo-reflections on a \(d\)-dimensional power series ring \(\Omega\) over an algebraically closed field of characteristic zero. Assume that the ring \(\Lambda\) of invariants represents an isolated singularity. Then \(\mathcal C=\text{add\,}\Omega\) is maximal \((d-2)\)-orthogonal in \(\Lambda\text{CM}\). Generalizing Auslander’s result for \(d=2\), the author proves that the Auslander-Reiten quiver of \(\mathcal C\) coincides with the McKay quiver of \(G\).
Furthermore, he characterizes non-commutative crepant resolutions (in a slightly stronger sense) in terms of a higher representation dimension. By definition, the \(l\)-th representation dimension is restricted to those generator-cogenerators \(M\) which satisfy \(\text{Ext}^i_\Lambda(M,M)=0\) for \(i<l\). Then \(\Lambda\) admits a non-commutative crepant resolution if and only if its \((d-1)\)-th representation dimension is \(d\) (\(\geq 2\)). Finally, he shows that the recent results of Geiss, Leclerc, and Schröer on rigid modules over preprojective algebras, as well as the Ext-configurations of A. R. Buan, R. Marsh, M. Reineke, I. Reiten, and G. Todorov [Adv. Math. 204, No. 2, 572-618 (2006; Zbl 1127.16011)] can be interpreted in terms of maximal 1-orthogonal subcategories.

MSC:
16E30 Homological functors on modules (Tor, Ext, etc.) in associative algebras
16G30 Representations of orders, lattices, algebras over commutative rings
16E65 Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.)
16E10 Homological dimension in associative algebras
16D90 Module categories in associative algebras
18E10 Abelian categories, Grothendieck categories
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