I tried to figure out why Galois comodules is a generalization of several Galois aspects, but I could not?
I am really interested in Comodule theory, and I am very curious to know the answer for this question. I know Galois theory is a nice application of adunction, but I do not know how Galois comodules is considered a generalization of classical Galois theory of field extensions?
So could you guys help me?
Thanks,
As for your question, Galois comodules in that sense provide a useful framework to study Galois corings, which in turn are a framework to study Holf-Galois extensions, which in turn are a generalization of the theory of Galois extensions of rings to the situation in which the group is replaced by a Hopf algebra, which itself is a generalization of the good ol' galois theory for fields. Are you familiar with any of these theories Galois comodule theory is intended to generalize? If not, you should probably become familiar.
A great introduction is Susan Montgomery's book Hopf Algebras and Their Actions on Rings; there you'll get to Hopf-Galois theory and see what the relationship with classical Galois theory of fields is. Once there, you can read Brzinzki's book on corings, for example, to see how one goes from Hopf-Galois theory to the more general coring theory (and why!) and then you can easily jump with confidence into Galois comodules