On the Gersten–Witt complex of an Azumaya algebra with involution

Abstract

Let (A,σ) be an Azumaya algebra with involution over a regular ring R. We prove that the Gersten–Witt complex of (A,σ) defined by Gille is isomorphic to the Gersten–Witt complex of (A,σ) defined by Bayer-Fluckiger, Parimala and the author. Advantages of both constructions are used to show that the Gersten–Witt complex is exact when dim⁡R≤3, indA≤2 and σ is orthogonal or symplectic. This means that the Grothendieck–Serre conjecture holds for the group R-scheme of σ-unitary elements in A under the same hypotheses; R is not required to contain a field.