Dominant dimension and tilting modules

Abstract

We study which algebras have tilting modules that are both generated and cogenerated by projective-injective modules. Crawley-Boevey and Sauter have shown that Auslander algebras have such tilting modules; and for algebras of global dimension 2, Auslander algebras are classified by the existence of such tilting modules.In this paper, we show that the existence of such a tilting module is equivalent to the algebra having dominant dimension at least 2, independent of its global dimension. In general such a tilting module is not necessarily cotilting. Here, we show that the algebras which have a tilting-cotilting module generated-cogenerated by projective-injective modules are precisely 1-Auslander-Gorenstein algebras.When considering such a tilting module, without the assumption that it is cotilting, we study the global dimension of its endomorphism algebra, and discuss a connection with the Finitistic Dimension Conjecture. Furthermore, as special cases, we show that triangular matrix algebras obtained from Auslander algebras and certain injective modules, have such a tilting module. We also give a description of which Nakayama algebras have such a tilting module.

Special Tilting Modules for Algebras With Positive Dominant Dimension

“…The definition of a minimal d-Auslander-Gorenstein algebra, due to Iyama and Solberg [16], may be found below (Definition 3.1). This result generalises [8,Lemma 1.1] for (1-)Auslander algebras, and also [21,Theorem 2.4.12] by Nguyen et al, who proved it independently in the case d = 1.…”

“…MATTHEW PRESSLAND AND JULIA SAUTER shifted algebras B k . Some of our results on the single tilting module T 1 , notably those of Section 3 on minimal 1-Auslander-Gorenstein algebras, were obtained independently by Nguyen et al [21].…”

1-Auslander-Gorenstein algebras which are tilted

“…Here, M d is the direct sum of d copies of M. Dually, we define Cogen M to be the class of all modules Y in mod Λ cogenerated by M, that is, the modules Y such that there exist an integer d ≥ 0 and a monomorphism Y → M d of Λ-modules. Nguyen, Reiten, Todorov, and Zhu in [9] studied the existence of such tilting and cotilting modules without any condition on the global dimension of Λ and gave a precise description. We first recall the definition of the dominant dimension of an algebra.…”

On relative Auslander algebras

“…Let Λ be an artin algebra such that both id Λ Λ and idΛ Λ are finite. It is known [Z] that in this case, id Λ Λ = idΛ Λ , say n. Then n is called the Gorenstein dimension of Λ, denoted by Gdim(Λ), and Λ is called the Iwanaga-Gorenstein algebra of Gorenstein dimension n, see [NRTZ,Remark 2.4.6]. Throughout the paper, we call such algebras Gorenstein of G-dimension n.…”

“…They therefore, provided a generalization of [CS,Lemma 1.1]. Moreover, they [NRTZ,Theorem 2.4.11] showed that an artin algebra Γ is a 1-Auslander-Gorenstein algebra if and only if C Γ contains a tiltingcotilting module. Recall that the notion of n-Auslander-Gorenstein algebras defined recently by Iyama and Solberg [IS] as artin algebras Γ with the property that idΓ Γ ≤ n + 1 ≤ domdimΓ.…”