Dominant dimensions, derived equivalences and tilting modules

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.

28 pages. The only change in this version is that the original section 3 is divided into two parts, in order to make the structure of the paper more reasonable

References in corpus (2)

• Derived equivalences and stable equivalences of Morita type, II
• Derived equivalences from cohomological approximations, and mutations of $\Phi$-Yoneda algebras

Cited by in corpus (11)

• Special tilting modules for algebras with positive dominant dimension
• Tilting modules over Auslander-Gorenstein Algebras
• Derived equivalences, restriction to self-injective subalgebras and invariance of homological dimensions
• Dominant dimension and tilting modules
• Tilting modules and dominant dimension with respect to injective modules
• Fabric idempotents and homological dimensions
• Milnor squares of algebras, I: derived equivalences
• Ortho-symmetric modules, Gorenstein algebras and derived equivalences
• On representation-finite gendo-symmetric biserial algebras
• Frobenius bimodules and flat-dominant dimensions
• On a conjecture about Morita algebras