On the Grothendieck–Serre conjecture for classical groups

We prove some new cases of the Grothendieck–Serre conjecture for classical groups. This is based on a new construction of the Gersten–Witt complex for Witt groups of Azumaya algebras with involution on regular semilocal rings, with explicit second residue maps; the complex is shown to be exact when the ring is of dimension ⩽2$\leqslant 2$ (or ⩽4$\leqslant 4$ , with additional hypotheses on the algebra with involution). Note that we do not assume that the ring contains a field.

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