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Picard Invariants of Galois Algebras Over Dual Larson Hopf Orders
Author(s) -
Byott Nigel P.
Publication year - 2000
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611500012417
Subject(s) - mathematics , hopf algebra , quasitriangular hopf algebra , pure mathematics , separable space , representation theory of hopf algebras , invariant (physics) , commutative property , algebra over a field , discrete mathematics , division algebra , filtered algebra , mathematical analysis , mathematical physics
Let A be a commutative, cocommutative Hopf algebra, finitely generated and projective over its base ring R . Waterhouse asked whether the image of the class‐invariant map, taking each A ‐Galois algebra to the class in Pic( A ) of its R ‐linear dual, is the group of primitive classes in Pic( A ). We discuss functorial aspects of this problem, and relate it to Fröhlich's Hom‐description of Pic( A ) in the case that R is a Dedekind domain with field of fractions K , and A is an R ‐Hopf order in a separable K ‐Hopf algebra. We then apply this machinery to a certain class of Hopf orders A in the Hopf algebra Map( G, K ). More precisely, we give a positive answer to Waterhouse's question for A when the dual B of A is one of the Hopf orders in KG constructed by Larson, and a compatibility condition holds between the filtrations of G determined by the various completions of B . 1991 Mathematics Subject Classification : 11R33, 16W30