On Galois Extensions of Composite Algebras

In this work we study the Galois extension of some composite commutative free R-algebras. Given \(A_1\) and \(A_2\) two commutative free R-algebras. Let \(G_1\) be a finite subgroup of \(AutA_1\) and \(G_2\) a finite subgroup of \(AutA_2\) , we prove two fundamental results on Galois extensions. The first is that \(A_1 \otimes _R A_2\) is \(1 \otimes G2\) -Galois over \(A_1\) if and only if \(A_2\) is \(G_2\) -Galois over R. The second result is that if \(A_i^{G_i} = R\) for \(i = 1, 2\) , then \(A_1 \otimes _R A_2\) is \(G_1 \otimes G_2\) -Galois over R if and only if \(A_1\) is \(G_1\) -Galois over R and \(A_2\) is \(G_2\) -Galois over R. As an application, we consider the case where R is a commutative graded ring, free as \(R_0\) -module and A is a strongly graded commutative R-algebra such that \(A_0\) is free as \(R_0\) -module. In this situation, we show that if \(G_1\) is a finite subgroup of \(\mathit {AUTR}\) (where \(\mathit {AUTR}\) is the group of all graded automorphisms of R) and R is \(G_1\) -Galois over \(R_0\) and \(A_0\) is \(G_2\) -Galois over \(R_0\) , where \(G_2\) a finite subgroup \(Aut_{R_0} A_0\) , then A is \(G_1 \otimes G_2\) -Galois over \(R_0\) .