The Galois algebras and the Azumay Galois extensions

George Szeto,Lianyong Xue

doi:10.1155/s0161171202110386

George Szeto, Lianyong Xue

Open Access

https://doi.org/10.1155/s0161171202110386

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Abstract

Let B be a Galois algebra over a commutative ring R with Galois group G, C the center of B, K = {g ∈ G | g(c) = c for all c ∈ C}, Jg{b ∈ B | bx = g(x)b for all x ∈ B} for each g ∈ K, and BK = (⊕∑g∈K Jg). Then BK 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 BK.

Highlights

Let B be a Galois algebra over a commutative ring R with Galois group G and C the center of B
We show that BK is an Azumaya algebra over Z and a central weakly Galois algebra with Galois group K|BK
In order to show that BK is a central weakly Galois algebra with Galois group K|BK, we need two lemmas

Summary

Introduction

Let B be a Galois algebra over a commutative ring R with Galois group G and C the center of B. Throughout, let B be a Galois algebra over a commutative ring R with Galois group G, C the center of B, and K = {g ∈ G | g(c) = c for all c ∈ C}. Let B be a Galois algebra over R with Galois group G, C the center of B, BG = {b ∈ B | g(b) = b for all g ∈ G}, and K = {g ∈ G | g(c) = c for all c ∈ C}.

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Results

Conclusion