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A1-homotopy theory of schemes by Morel F.

By Morel F.

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Gersten who considered it in the context of Zariski topology in [7]. In Section 2 we first recall the most important definitions and results of Section 3 in the context of the site with interval ((Sm/S)~vi,, Al). We then discuss briefly the functoriality of our constructions with respect to S. In Section 2 we prove three theorems which play major role in further applications of our constructions. In the final section we discuss some examples of topological realizations functors. 1. Simplicial sheaves in the Nisnevich topology on smooth sites Nisnevich topology Let S be a Noetherian scheme of finite dimension.

H~,(S, F), where the left hand side refers to the Cech cohomology canonical morphisrn H~v,(S, v, , groups, is an isomorphism. Proof. 17] for the ~tale topology with the reference to [1, Th. 4(iii)] replaced by the reference to [1, Th. 4(i)]. 10. 9 is false for Zariski topology. 4,2 over a field k. 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.

V+l(,~)(U x I), x). 582" I-local9 Then the canonical morphism of sheaves a~_o(~,SZ") ~ a ~ ( , ~ ' ) is an epimorphism. In particular, if ~ f is connected (a~_o(,~ ") =pt) then so is ~ " . 3. T h e A l - h o m o t o p y c a t e g o r y o f s c h e m e s o v e r a b a s e In this section we study the basic properties of At-homotopy category of smooth schemes over a base. Modulo the conventions of the previous section the definition of the AX-homotopy category ,~;~ (S) of smooth schemes over a base scheme S takes one line - , ~ q (S) is the homotopy category of the site with interval ((Sm/S)~a, AI), where Sm/S is the category of smooth schemes (of finite type) over S and N/s refers to the Nisnevich topolog3z Nisnevich topology was introduced by Y.

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