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be a two-dimensional global field of characteristic
be a divisorial set of places of
. We show that for a given
, the set of
-isomorphism classes of spinor groups
-dimensional quadratic forms over
that have good reduction at all
is finite. This result yields some other finiteness properties, such as the finiteness of the genus
and the properness of the global-to-local map in Galois cohomology. The proof relies on the finiteness of the unramified cohomology groups
established in the paper. The results for spinor groups are then extended to some unitary groups and to groups of type
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