Dominant dimensions, derived equivalences and tilting modules

TL;DR

This paper investigates how dominant dimensions of algebras behave under derived equivalences induced by tilting modules, providing new methods and conditions for preserving infinite dominant dimensions and exploring implications for algebra classifications.

Contribution

It introduces a new method for producing derived equivalences from exact sequences and establishes conditions under which dominant dimensions are preserved or can be infinite under tilting.

Findings

Established lower bounds for dominant dimensions under derived equivalences.

Identified conditions preserving infinite dominant dimensions during tilting.

Provided the first counterexample to the closure of generalized symmetric algebras under derived equivalences.

Abstract

The Nakayama conjecture states that an algebra of infinite dominant dimension should be self-injective. Motivated by understanding this conjecture in the context of derived categories, we study dominant dimensions of algebras under derived equivalences induced by tilting modules, specifically, the infinity of dominant dimensions under tilting procedure. We first give a new method to produce derived equivalences from relatively exact sequences, and then establish relationships and lower bounds of dominant dimensions for derived equivalences induced by tilting modules. Particularly, we show that under a sufficient condition the infinity of dominant dimensions can be preserved by tilting, and get not only a class of derived equivalences between two algebras such that one of them is a Morita algebra in the sense of Kerner-Yamagata and the other is not, but also the first counterexample to a question whether generalized symmetric algebras are closed under derived equivalences.