Frobenius bimodules and flat-dominant dimensions

Abstract

We establish relations between Frobenius parts and between flat-dominant dimensions of algebras linked by Frobenius bimodules. This is motivated by the Nakayama conjecture and an approach of Martinez-Villa to the Auslander-Reiten conjecture on stable equivalences. We show that the Frobenius parts of Frobenius extensions are again Frobenius extensions. Furthermore, let A and B be finite-dimensional algebras over a field k , and let domdim( A X ) stand for the dominant dimension of an A -module X . If B M A is a Frobenius bimodule, then domdim( A ) ⩽ domdim( B M ) and domdim( B ) ⩽ domdim( A Hom B ( M , B )). In particular, if B ⊆ A is a left-split (or right-split) Frobenius extension, then domdim( A ) = domdim( B ). These results are applied to calculate flat-dominant dimensions of a number of algebras: shew group algebras, stably equivalent algebras, trivial extensions and Markov extensions. We also prove that the universal (quantised) enveloping algebras of semisimple Lie algebras are QF -3 rings in the sense of Morita.