### Higher spinor classes, by J. F. Jardine

This paper has appeared in Memoirs of the AMS, No. 528 (1994), so the dvi
file has been removed.

Suppose that 2 is invertible in a scheme S. The main results of this
paper include a Künneth relation for the mod 2 étale cohomology of
for products of simplicial S-schemes where one of the factors is the
classifying object BO(n), and a cohomological induction formula for
the inclusion of BWr(n) in BO(2n), where Wr(n) is the wreath product
of the symmetric group on 2 letters with the orthogonal group scheme
O(n). This induction formula is given in terms of a total Steenrod
squaring operation for mod 2 étale cohomology. Serre's formula for
the Hasse-Witt invariant of the trace form is an easy consequence of
these results. It is shown that the Stiefel-Whitney classes of an
orthogonal Galois representation are decomposable, and that the higher
spinor classes of such a representation (as defined here) vanish in
odd degrees. An induction formula is given for the higher spinor
classes of an induced representation.

J. F. Jardine <jardine@jardine.math.uwo.ca>