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For any ordinal n u m b e r o let's define as usual the iteration (oo)c0 of the previous functor; ha fact one defines a functor from the ordered set of ordinal numbers y ~< co to the "category" of functors. One proceeds by transfinite induction, requiring that if 7 = 7 ' + 1 then (~~ "t= ~B((O~) o o,/, ) and if 7 is a limit ordinal then (~off = cotm~ ,- , <~Iq, ,,,,o, R)T . 13). Let o~ be a cardinal number and , ~ an ordered set ; we shall write , ~ /> o~ if any subset of ~,~ of cardinal ~< r has an upper bound.

Z,, Zn+~ =0) be a splitting sequence of minimal length tbr ~//'. Let us choose a splitting s for the morphism p-l(Z,) ---+ Z,. Since p is 6tale we have a decomposition p - l ( Z , ) = Im(s)I_I Y where Y is a closed subset of LI Wi. Let U = X - Z, and let V = (H W i ) - Y. Clearly U and V form an elementary distinguished square over X and family of morphisms 98 FABIEN M O R E L , V I , A I ) I M I R V O E V O D S K Y ~ ' / x x U ~ U is a Nisnevich covering of U with a splitting sequence of length n - 1.

Let S be the spectrum of the semilocal ring of x0, Xl. Any Zariski open covering for S has a v i refinement which consists of exactly two open subsets and therefore t-IZ~r(S, F)= 0 for any F and any i > 1. 2 Let us show that there exists a sheaf F such that HZ~T(S, F)~:0. Choose two irreducible curves Cl, C2 on S such that C~ fh C2 = {x0, Xl} and let U = S - (C~ U C2), V = S - {x0, xl }. Denote the open embedding U ---+ S by j and the open embedding U ~ V by j ' . We clain that H2(S,j~(Z))~:0. (Z))=~0 (since the intersection of these two open subsets is V).