On The Gersten-Witt Complex of an Azumaya Algebra with Involution

Abstract

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