The Galois algebras and the Azumaya Galois extensions (Q1607915)

Summary: Let \(B\) be a Galois algebra over a commutative ring \(R\) with Galois group \(G\), \(C\) the center of \(B\), \(K=\{g\in G\mid g(c)=c\) for all \(c\in C\}\), \(J_g=\{b\in B\mid bx=g(x)b\) for all \(x\in B\}\) for each \(g\in K\), and \(B_K=(\bigoplus\sum_{g\in K}J_g)\). Then \(B_K\) is a central weakly Galois algebra with Galois group induced by \(K\). Moreover, an Azumaya Galois extension \(B\) with Galois group \(K\) is characterized by using \(B_K\).