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June 2005 Hopf algebroids and Galois extensions
Lars Kadison
Bull. Belg. Math. Soc. Simon Stevin 12(2): 275-293 (June 2005). DOI: 10.36045/bbms/1117805089


To a finite Hopf-Galois extension $A | B$ we associate dual bialgebroids $S := End\,_BA_B$ and $T := (A \o_B A)^B$ over the centralizer $R$ using the depth two theory in [18,Kadison-Szlachányi]. First we extend results on the equivalence of certain properties of Hopf-Galois extensions with corresponding properties of the coacting Hopf algebra [21,8] to depth two extensions using coring theory [3]. Next we show that $T^{\rm op}$ is a Hopf algebroid over the centralizer $R$ via Lu's theorem \cite[23,5.1] for smash products with special modules over the Drinfel'd double, the Miyashita-Ulbrich action, the fact that $R$ is a commutative algebra in the pre-braided category of Yetter-Drinfel'd modules [28] and the equivalence of Yetter-Drinfel'd modules with modules over Drinfel'd double [24]. In our last section, an exposition of results of Sugano [29,30] leads us to a Galois correspondence between sub-Hopf algebroids of $S$ over simple subalgebras of the centralizer with finite projective intermediate simple subrings of a finite projective H-separable extension of simple rings $A \supseteq B$.


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Lars Kadison. "Hopf algebroids and Galois extensions." Bull. Belg. Math. Soc. Simon Stevin 12 (2) 275 - 293, June 2005.


Published: June 2005
First available in Project Euclid: 3 June 2005

zbMATH: 1105.16037
MathSciNet: MR2179969
Digital Object Identifier: 10.36045/bbms/1117805089

Primary: 06A15, 12F10, 13B02, 16W30

Rights: Copyright © 2005 The Belgian Mathematical Society


Vol.12 • No. 2 • June 2005
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