homological stability. Again, the spectral sequence we consider has terms involving the relative homology of general linear groups. We can hence apply Sah’s theorem again to obtain

For a division ring $D$ with infinite centre, the relative homology modules $$H_k\bigl(\operatorname{U}({\mathcal H}_{n+1}\perp E), \operatorname{U}({\mathcal H}_n\perp E);{\mathbb Z}\bigr)$$ vanish for $n\geq 2$ if $k=1$ and for $n\geq k\geq 2$. If the centre of $D$ is finite, relative homology vanishes for $n\geq 2k$.

This is an improvement over the results by Mirzaii and van der Kallen in [@MaB:HSU:02] and [@Mir:HSU:05], where homological stability for unitary groups with stability range $n\geq k+1$ has already been proved. Their result is valid for a much larger class of rings, namely local rings with infinite residue fields, but only for the case of maximal Witt index, that is for $E=\{0\}$.

Rather surprisingly, the following strong result can also be proved using this method.

For an infinite field $D$, we have $$H_k\bigl(\operatorname{SO}_{n+1,n+1}(D),\operatorname{SO}_{n,n}(D);{\mathbb Z}\bigr) =0$$ for $n\geq 2$ if $k=1$ and for $n\geq k\geq 2$. If $D$ is a finite field, then the relative homology groups vanish for $n\geq 2k$.

### The construction of a relative spectral sequence

In the following, we give an outline of the method used 