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

TL;DR

This paper proves the isomorphism and exactness of the Gersten-Witt complex for Azumaya algebras with involution over regular rings under certain conditions, confirming the Grothendieck-Serre conjecture in this context.

Contribution

It establishes the equivalence of two definitions of the Gersten-Witt complex and proves its exactness under specific conditions, extending the conjecture to rings not containing a field.

Findings

Gersten-Witt complex is isomorphic under different constructions.

Exactness of the complex when dimension ≤ 3, index ≤ 2, and involution is orthogonal or symplectic.

Grothendieck-Serre conjecture holds for the group scheme of σ-unitary elements under these conditions.

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.