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U ) is a weak equivalence. AI-HOMOTOPY THEORY OF SCHEMES 101 Proof. 15). To prove the "only if" part we need an analog of [7, T h e o r e m 1~] for Nisnevich topology. Let X be a Noetherian scheme of finite dimension. e. the category of 6tale schemes over X considered with the Nisnevich topology). -functor on XH. is a family of contravariant functors Tq, q >>,0 from Xx= to the category of pointed sets, together with pointed maps OQ : Tq+ l(U x X V) ---+ Tq(X) given for all elementary distinguished squares in X ~ , such that the following two conditions hold: 1.
Let ~ be an object of A~ ") and p : ~" ~ ~ x I 88 FABIEN MOREL, VlYkl)IMIR VOEVOI)SKY be a morphism in FI. We have to show that the upper horizontal arrow in the cartesian square x. g" x I) 1 , ld• 0 5g" , 1 ,Z'xI is an I-weak equivalence. Applying the functor Sing). to this diagram we get a cartesian square (by (1)) which is I-weak equivalent to the original one (by (4)). (Id x/0) is a simplicial weak equivalence. (~f) is a simplicial weak equivalence since the simplicial model structure is proper.
We clain that H2(S,j~(Z))~:0. (Z))=~0 (since the intersection of these two open subsets is V). O n e verifies easily that since the curves CI, C2 are irreducible this element is not zero. -property For any presheaf F on (Sm/S)~,# and any left filtering diagram Xa of smooth schemes over S with affine transition morphisms and the limit scheme X we denote by F(X) the set colimaF(Xrt). For example, for any smooth S-scheme X and any point x of X the set F(Spec ~ x'~, x ) (resp. F(Spec Gx,x) h j is the filtering colimit of the sets F(U) over the categories of Zariski and Nisnevich neighborhoods of x respectively.