Finite Galois Extensions

Corollary. An algebraic group scheme of multiplicative type is diagonalizable over a finite Galois extension. 

Theorem. Let L be a finite Galois extension ofk with group T. Then the torus corresponding to X = Z[T] is the group scheme of units of L over k. 

Suppose that S/R is a finite Galois field extension with group T. If G is any group functor satisfying G(A x B) = G(A) x G(B), we can rewrite our cohomology in terms of T-actions. 

Lemma 2.2.6. Let f X — S be finite, X normal and such that every irreducible component of X dominates an irreducible component of S (and tacitly S normal). Let X be the normalization of S in the "smallest Galois extension containing the function ring R(X) of X. 

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