Algebras of infinite dominant dimension (Q1340389)

In a sense, this paper is a continuation of [the author, Commun. Algebra 20, No. 12, 3515-3540 (1992; Zbl 0782.16007)]. It provides some insight into Nakayama's conjecture that algebras with infinite dominant dimension are selfinjective. The article is heavily based on Auslander-Reiten functors and stable categories and uses, for example, the equivalent categories \(\underline{\text{mod}}\text{mod} \Lambda\) and \(\text{mod}(\underline{\text{mod}} \Lambda)\) and the categories denoted by \(\mathcal D\) and \(\underline{\mathcal D}\). Here, \(\Lambda\) is a finite dimensional algebra over a field, of infinite dominant dimension, \({\mathcal D}\subset\text{mod }\Lambda\) is the subcategory of modules of infinite dominant dimension and \(\underline{\mathcal D}\) is the corresponding stable category. Important functors are \(\underline{\text{mod}} \Lambda\otimes_{\underline{\mathcal D}}-:\text{mod}(\underline{\mathcal D})\to\text{mod}(\underline{\text{mod}} \Lambda)\), whose image is the category of the so-called induced functors and \(\text{res}:\text{mod}(\underline{\text{mod}} \Lambda)\to\text{mod}(\underline{\mathcal D})\). One of the main results is the following Theorem: If the category of the induced functors is contravariantly finite in \(\text{mod}(\underline{\text{mod}} \Lambda)\), then \(\Lambda\) is selfinjective.