Auslander correspondence.
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 \(\Lambda^*=\Hom_R(\Lambda,R)\). 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 [\textit{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 \textit{A. R. Buan, R. Marsh, M. Reineke, I. Reiten}, and \textit{G. Todorov} [Adv. Math.~204, No. 2, 572-618 (2006; Zbl 1127.16011)] can be interpreted in terms of maximal 1-orthogonal subcategories.
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