Relations Between Witt Rings and Brauer Groups

Abstract

Let R be a commutative ring with unit element. To R we can associate the Witt ring W(R) which classifies the nondegenerate quadratic forms Q on finitely generated projective R-modules M and the Brauer group Br(R) which classifies the Azumaya algebras A over R, that is, A is a finitely generated projective R-module and, if A’ rev denotes the reversed algebra with multiplication (x, y) ↦ yx, the algebra A ⊗R A rev is canonically isomorphic to the algebra EndR(A) Let us recall that each nondegenerate quadratic form Q on a R-module M has an image in W(R), here denoted by w(M, Q) or w(Q), and that this image fullfils the following properties: w(Q) + w(Q’) = w(Q ⊗ Q’) (orthogonal sum), −w(Q) = w(−Q) and w(Q)w(Q’) = w(Q ⊗ Q’) (tensor product), for all nondegenerate quadratic forms Q and Q’ Besides, each Azumaya algebra A has an image b(A) in the Brauer group Br(R) and b(A)b(A’) = b(A ⊗R A’), b(A)-1 = b(Arev), for all R-Azumaya algebras A and A’. Moreover, the unit element in the Brauer group Br(R) is the image of the ring R.