The relative Picard group of a comodule algebra and Harrison cohomology

Details The relative Picard group of a comodule algebra and Harrison cohomology The relative Picard group of a comodule algebra and Harrison cohomology

2004-10-07

arxiv.org/abs/math/0410209v1

Mathematics

Rings and Algebras

12 pages

Scientific paper

Let $A$ be a commutative comodule algebra over a commutative bialgebra $H$. The group of invertible relative Hopf modules maps to the Picard group of $A$, and the kernel is described as a quotient group of the group of invertible grouplike elements of the coring $A\ot H$, or as a Harrison cohomology group. Our methods are based on elementary $K$-theory. The Hilbert 90 Theorem follows as a corollary. The part of the Picard group of the coinvariants that becomes trivial after base extension embeds in the Harrison cohomology group, and the image is contained in a well-defined subgroup $E$. It equals $E$ if $H$ is a cosemisimple Hopf algebra over a field.

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