Hopf algebroids and Galois extensions
Kadison, Lars
Bull. Belg. Math. Soc. Simon Stevin, Tome 11 (2005) no. 5, p. 275-293 / Harvested from Project Euclid
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 $A \supseteq B$.
Publié le : 2005-04-14
Classification:  Hopf-Galois extension,  Hopf algebroid,  H-separable extension,  depth two extension,  bialgebroid,  coring,  06A15,  12F10,  13B02,  16W30
@article{1117805089,
     author = {Kadison, Lars},
     title = {Hopf algebroids and Galois extensions},
     journal = {Bull. Belg. Math. Soc. Simon Stevin},
     volume = {11},
     number = {5},
     year = {2005},
     pages = { 275-293},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1117805089}
}
Kadison, Lars. Hopf algebroids and Galois extensions. Bull. Belg. Math. Soc. Simon Stevin, Tome 11 (2005) no. 5, pp.  275-293. http://gdmltest.u-ga.fr/item/1117805089/