Integrals in Hopf algebras over rings

Details Integrals in Hopf algebras over rings Integrals in Hopf algebras over rings

2003-07-03

arxiv.org/abs/math/0307046v1

Communications in Algebra, 32(12), 4687-4711 (2004)

Mathematics

Rings and Algebras

26 pages

Scientific paper

10.1081/AGB-200036837

Integrals in Hopf algebras are an essential tool in studying finite dimensional Hopf algebras and their action on algebras. 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 for 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.

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