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

TL;DR

This paper constructs an 8-periodic exact sequence of Witt groups for Azumaya algebras with involution over rings, generalizing previous results and applying to conjectures and classification problems in algebraic geometry and quadratic form theory.

Contribution

It introduces an 8-periodic chain complex of Witt groups for Azumaya algebras with involution, proving exactness over semilocal rings and extending known sequences to broader contexts.

Findings

Established an 8-periodic exact sequence of Witt groups for Azumaya algebras with involution.

Proved the sequence is exact over semilocal rings and recovered known sequences over fields.

Applied the sequence to verify conjectures and classify hermitian forms over various rings.

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.