Mackey-functor structure on the Brauer groups of a finite Galois covering of schemes

Past studies of the Brauer group of a scheme tells us the importance of the interrelationship among Brauer groups of its finite étale coverings. In this paper, we consider these groups simultaneously, and construct an integrated object “BrauerMackey functor”. We realize this as a cohomological Mackey functor on the Galois category of finite étale coverings. For any finite étale covering of schemes, we can associate two homomorphisms for Brauer groups, namely the pull-back and the norm map. These homomorphisms make Brauer groups into a bivariant functor (= Mackey functor) on the Galois category. As a corollary, Restricting to a finite Galois covering of schemes, we obtain a cohomological Mackey functor on its Galois group. This is a generalization of the result for rings by Ford [5]. Moreover, applying Bley and Boltje’s theorem [1], we can derive certain isomorphisms for the Brauer groups of intermediate coverings.