Dominant Auslander-Gorenstein algebras and Koszul duality

Abstract

We introduce the class of dominant Auslander-Gorenstein algebras as a generalisation of higher Auslander algebras and minimal Auslander-Gorenstein algebras, and give their basic properties. We also introduce mixed (pre)cluster tilting modules as a generalisation of (pre)cluster tilting modules, and establish an Auslander type correspondence by showing that dominant Auslander-Gorenstein (respectively, Auslander-regular) algebras correspond bijectively with mixed precluster (respectively, cluster) tilting modules.It turns out that dominant Auslander-regular algebras have a remarkable interaction with Koszul duality, namely Koszul dominant Auslander-regular algebras are closed under Koszul duality. As an application, we use the class of dominant Auslander-regular algebras to answer a question by Green about a characterisation of the Koszul dual of an Auslander algebra. We show that the Koszul dual of the Auslander algebras of hereditary or self-injective representation-finite algebras are higher Auslander algebras, and describe the corresponding cluster tilting modules explicitly.

Homological dimensions of the Jacobson radical

“…In [21], where this notion is introduced, it is required that dom dim A ≥ 2, but this is only needed to obtain an Auslander type correspondence with precluster tilting objects and not relevant for most results, so we drop it; this is in line with [6]. Minimal Auslander-Gorenstein algebras are a subclass of Auslander-Gorenstein rings A introduced by Auslander.…”

Homological dimensions of the Jacobson radical

“…In [20], where this notion is introduced, it is required that dom dim A ≥ 2, but this is only really necessary to obtain an Auslander type correspondence with precluster tilting objects and not relevant for most theoretical results, so we drop it; this is in line with [6]. Minimal Auslander-Gorenstein algebras are a subsclass of Auslander-Gorenstein rings A introduced by Auslander.…”

Higher Morita–Tachikawa correspondence

“…Not only the Auslander's correspondence but also many known correspondences like Iyama's higher Auslander correspondence [21] and Iyama-Solberg's correspondence [20] are also specialisations of the Morita-Tachikawa correspondence. Moreover, algebras arising from these correspondences have appeared in many different areas like cluster theory, homological algebra and algebraic Lie theory to name a few (see [22] and, for instance, see also [8] and the references therein). The Morita-Tachikawa correspondence also brought interest to define and study many new classes of algebras for example: Morita algebras [23], gendo-Frobenius algebras [30] and gendo-symmetric algebras [14] as the counterparts through Morita-Tachikawa correspondence of self-injective, Frobenius and symmetric algebras, respectively.…”