**MSC:**- 20E42 Groups with a $BN$-pair; buildings, See also {51E24}
- 20G25 Linear algebraic groups over local fields and their integers

cohomology of $S$-arithmetic spin groups over number fields.

The results of the first and the second part over the Bruhat-Tits-building

and its relations to the classification of quadratic forms over $\mathbb{Z}$

and to group cohomology are used to compute the cohomology of the

spin group over the ring $\mathbb{Z}[\frac{1}{2}]$, which is associated

to an even unimodular quadratic lattice of dimension 8 over $\mathbb{Z}$,

with coefficients in the ring $\mathbb{Z}[\frac{1}{6}]$ (resp. with

coefficients in the prime fields of characteristic 5 and 7).