A Characterization of Central Galois Algebras

Let A be an Azumaya R-algebra over a commutative ring R of a constant rank n for some integer n, G an automorphism group of A of order n, and J sub g /sub = {a ∈ A|ax = g(x)a for all x ∈ A} for g ∈ G. Then A is a central Galois algebra over R with Galois group G if and only if ∑ sub g∈ /sub Gg is a separable R-algebra of rank n. In particular, when G is inner induced by {U sub g /sub for g ∈ G}, A is a central Galois R-algebra if and only if ∑ sub g /sub RU sub g /sub is a separable R-algebra of rank n. Thus all inner Galois groups can be computed from the multiplicative group of units of A.