Projective Modules Over Frobenius Algebras and Hopf Comodule Algebras

Abstract

This note presents some results on projective modules and the Grothendieck groups K ₀ and G ₀ for Frobenius algebras and for certain Hopf Galois extensions. Our principal technical tools are the Higman trace for Frobenius algebras and a product formula for Hattori-Stallings ranks of projectives over Hopf Galois extensions.

Key Words:

• Cartan map
• Character
• Comodule algebra
• Frobenius algebra
• Grothendieck group
• Higman trace
• Hopf algebra
• Hopf Galois extension
• Projective module
• (Hattori–Stallings) rank

2000 Mathematics Subject Classification:

• 16W30
• 16W35
• 16G10
• 20G42

ACKNOWLEDGMENTS

The first author thanks the organizers of the conference on the occasion of Mia Cohen's retirement, held at the Sde-Boker campus of Ben-Gurion University May during 24–27, 2010, for giving him the opportunity to present some related results on ring theoretic methods in the representation theory of Hopf algebras to an expert audience. Thanks are also due to the referee for a careful reading of this article.

Research of the first author supported in part by NSA Grant H98230-09-1-0026.

Notes

Called Gaschütz–Ikeda operator in [4].

Communicated by S. Montgomery.

Dedicated to Mia Cohen on the occasion of her retirement.