An 8-periodic exact sequence of Witt groups of Azumaya algebras with involution

Abstract

Given an Azumaya algebra with involution $$(A,\sigma )$$ over a commutative ring R and some auxiliary data, we construct an 8-periodic chain complex involving the Witt groups of $$(A,\sigma )$$ and other algebras with involution, and prove it is exact when R is semilocal. When R is a field, this recovers an 8-periodic exact sequence of Witt groups of Grenier-Boley and Mahmoudi, which in turn generalizes exact sequences of Parimala–Sridharan–Suresh and Lewis. We apply this result in several ways: We establish the Grothendieck–Serre conjecture on principal homogeneous bundles and the local purity conjecture for certain outer forms of $${\mathbf {GL}}_n$$ and $${\mathbf {Sp}}_{2n}$$ , provided some assumptions on R. We show that a 1-hermitian form over a quadratic étale or quaternion Azumaya algebra over a semilocal ring R is isotropic if and only if its trace (a quadratic form over R) is isotropic, generalizing a result of Jacobson. We also apply it to characterize the kernel of the restriction map $$W(R)\rightarrow W(S)$$ when R is a (non-semilocal) 2-dimensional regular domain and S is a quadratic étale R-algebra, generalizing a theorem of Pfister. In the process, we establish many fundamental results concerning Azumaya algebras with involution and hermitian forms over them.