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