Dominant dimension and tilting modules

TL;DR

This paper investigates the relationship between dominant dimension and the existence of special tilting modules in algebras, revealing that such modules characterize 1-Auslander-Gorenstein algebras and exploring their implications for algebra classification and conjectures.

Contribution

It establishes that the existence of a tilting module generated and cogenerated by projective-injective modules is equivalent to having dominant dimension at least 2, and characterizes algebras with tilting-cotilting modules as 1-Auslander-Gorenstein algebras.

Findings

Algebras with such tilting modules have dominant dimension ≥ 2.

Characterization of 1-Auslander-Gorenstein algebras via tilting-cotilting modules.

Connections between endomorphism algebra global dimension and Finitistic Dimension Conjecture.

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.