  We prove that every connected affine scheme of positive characteristic is a
K(pi, 1) space for the etale topology. The main ingredient is the special case
of the affine space over a field k. This is dealt with by induction on n, using
a key "Bertini-type"' statement regarding the wild ramification of l-adic local
systems on affine spaces, which might be of independent interest. Its proof
uses in an essential way recent advances in higher ramification theory due to
T. Saito. We also give rigid analytic and mixed characteristic versions of the
main result.
