Brauer groups and Amitsur cohomology for general commutative ring extensions

Abstract

Exact couples are interconnected families of long exact sequences extending the short exact sequences usually derived from spectral sequences. This is exploited to give a long exact sequence connecting Amitsur cohomology groups H> n( S R , U) (where U means the multiplicative group) and H n( S R , Pic) and a third sequence of groups H n ( J), for every faithfully flat commutative R-algebra S. This same sequence is derived in another way without assuming faithful flatness and H n ( J) is identified explicitly as a certain subquotient of a group of isomorphism classes of pairs ( P, α) with P a rank one, projective S n-module and α an isomorphism from the coboundary of P ( in Pic S n + 1 ) to S n + 1 . (Here S n denotes repeated tensor product of S over R.) This last formulation allows us to construct a homomorphism of the relative Brauer group B( S R ) to H 2( J) which is a monomorphism when S is faithfully flat over R, and an isomorphism when some S-module is faithfully projective over R. The first approach also identifies H 2( J) with Ker[ H 2( R, U)→ H 2( S, U)], where H 2( R, U) denotes the ordinary, Grothendieck cohomology (in the étale topology, for example).