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Global and finitistic dimension of Hopf-Galois extensions

Then in this note a spectral sequence for Ext will be constructed which yields the estimate for global dimension of A in terms of the corresponding data for H and B . As an application, we obtain the Maschke-type theorems for crossed products and twisted smash products. Finally, the relationship of finitistic dimensions between A and B will be given, if H is semisimple. | Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ Research Article Turk J Math (2013) 37: 210 – 217 ¨ ITAK ˙ c TUB doi:10.3906/mat-1107-15 Global and finitistic dimension of Hopf-Galois extensions 1 Ling LIU1,∗, Qiao-Ling GUO2 College of Mathematics, Physics & Information Engineering, Zhejiang Normal University, Jinhua 321004, P. R. China 2 College of Mathematics and Information Engineering, Jiaxing University, Jiaxing 314001s, P. R. China Received: 20.07.2011 • Accepted: 03.11.2011 • Published Online: 19.03.2013 • Printed: 22.04.2013 Abstract: Let H be a Hopf algebra over a field k and A/B a right H -Galois extension. Then in this note a spectral sequence for Ext will be constructed which yields the estimate for global dimension of A in terms of the corresponding data for H and B . As an application, we obtain the Maschke-type theorems for crossed products and twisted smash products. Finally, the relationship of finitistic dimensions between A and B will be given, if H is semisimple. Key words: Hopf-Galois extension, spectral sequence, global dimension, finitistic dimension 1. Introduction and preliminaries The definition of Hopf-Galois extension has its roots in the Chase-Harrison-Rosenberg approach to Galois theory for groups acting on commutative rings (see [4]). In 1969 Chase and Sweedler extended these ideas to coaction of a Hopf algebra H acting on a commutative k -algebra, for k a commutative ring (see [5]); the general definition appears in [7] in 1981. Hopf-Galois extensions also generalize strongly graded algebras (here H is a group algebra) and certain inseparable field extensions (here the Hopf algebra is the restricted enveloping algebra of a restricted Lie algebra), twisted group rings R ∗ G of a group G acting on a ring R and so on. Let H be a Hopf algebra over a field k and A a right H -comodule algebra, i.e., A is a k -algebra together with an H -comodule structure ρA : A → A ⊗ H (with notation a → a0 ⊗ a1 ) such that ρA