On the Grothendieck-Serre Conjecture for Classical Groups

Eva Bayer-Fluckiger, Uriya A. First, Raman Parimala

Published 2019-11-18Version 1

Let $(A,\sigma)$ be an Azumaya algebra with involution over a regular semilocal ring $R$ and let $\varepsilon\in \{\pm 1\}$. We prove that the Gersten-Witt complex associated to $(A,\sigma)$ and $\varepsilon$ is exact if (i) $\dim R\leq 2$, or (ii) $\dim R\leq 3$, $\mathrm{ind}\, A\leq 2$ and $\sigma$ is of the first kind, or (iii) $\dim R\leq 4$ and $\mathrm{ind}\, A$ is odd. As a corollary, we prove the Grothendieck-Serre conjecture on principal homogeneous spaces for all forms of $\mathbf{GL}_n$, $\mathbf{Sp}_{2n}$ and $\mathbf{SO}_n$ when $\dim R\leq 2$, and all outer forms of $\mathbf{GL}_{2n+1}$ when $\dim R\leq 4$. In the process, we present a new construction of the Gersten-Witt complex, defined using explicit second-residue maps.