Given an Azumaya algebra with involution (A, σ) over a commutative ring R and some auxiliary data, we construct an 8-periodic chain complex involving the Witt groups of (A, σ) 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 GLn and Sp2n, 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) → 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.
Bibliographical noteFunding Information:
We are grateful to Eva Bayer-Fluckiger for suggesting us the project at hand. We further thank Eva Bayer-Fluckiger and Raman Parimala for many useful conversations and suggestions. The research was partially conducted at the department of mathematics at University of British Columbia, where the author was supported by a post-doctoral fellowship. We thank Ori Parzanchevski for encouragement and motivation.
© 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.
ASJC Scopus subject areas
- Mathematics (all)