TY - JOUR

T1 - An 8-periodic exact sequence of Witt groups of Azumaya algebras with involution

AU - First, Uriya A.

N1 - Publisher Copyright: © 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2023/1

Y1 - 2023/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85122669165&partnerID=8YFLogxK

U2 - https://doi.org/10.1007/s00229-021-01352-0

DO - https://doi.org/10.1007/s00229-021-01352-0

M3 - Article

SN - 0025-2611

VL - 170

SP - 313

EP - 407

JO - Manuscripta Mathematica

JF - Manuscripta Mathematica

IS - 1-2

ER -