An $8$-Periodic Exact Sequence of Witt Groups of Azumaya Algebras with Involution

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 ${\mathbf{G}\mathbf{L}}_n$ and ${\mathbf{S}\mathbf{p}}_{2n}$, 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.