Integrals in Hopf Algebras over Rings

Abstract

Integrals in Hopf algebras are an essential tool in studying finite dimensional Hopf algebras and their action on rings. Over fields it has been shown by Sweedler that the existence of integrals in a Hopf algebra is equivalent to the Hopf algebra being finite dimensional. In this paper we examine how much of this is true Hopf algebras over rings. We show that over any commutative ring R that is not a field there exists a Hopf algebra H over R containing a non-zero integral but not being finitely generated as R-module. On the contrary we show that Sweedler's equivalence is still valid for free Hopf algebras or projective Hopf algebras over integral domains. Analogously for a left H-module algebra A we study the influence of non-zero left A#H-linear maps from A to A#H on H being finitely generated as R-module. Examples and application to separability are given.

Key Words:

• Hopf algebras
• Integrals
• Invariant elements
• Hopf algebra action
• Module algebras
• Separability

Mathematics Subject Classification:

• V16W30
• 57T05

Acknowledgments

I would like to dedicate this work to my uncle Morris Morgan who passed away very suddenly. Parts of the material presented here are included in the author's doctoral thesis at the Heinrich-Heine Universität Düsseldorf. The author would like to express his gratitude to his supervisor Professor Robert Wisbauer for all his help advice and encouragement. The author would also like to thank his colleague Luís António Oliveira for stimulating and helpful discussions about integrals in semigroup rings. Last but not least the author thanks the referee for helpful suggestions and comments that improved this paper. This work is supported by Fundação para a Ciência e a Tecnologia (FCT) through the Centro de Matemática da Universidade do Porto (CMUP).

Notes

^#Communicated by R. Wisbauer.

^‡In memory of my uncle Morris Morgan.