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 \'etale 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 $W(R)\to W(S)$ when $R$ is a (non-semilocal) $2$-dimensional regular domain and $S$ is a quadratic \'etale $R$-algebra, generalizing a theorem of Pfister. In the process, we establish many fundamental results concerning Azumaya algebras with involution and hermitian forms over them.