A1-homotopy theory of schemes by Morel F.

By Morel F.

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Defnes a natural transformation Id ---+ Ex o Sing,. T h e functor Ex o Sing, can thus be iterated to any ordinal number power (see 2). )~o Ex together with She canonical natural transformation Id ~ Exi form an I-resolution functor. 20. 8, for any Jg" and any ordinal number ~ the canonical morphism ~ ~ Exi(,~~ is a monomorphism and an I-weak equivalence. 21. - - For any suffciently large ordinal number co then for any simplicial sheaf ~g" object Ext( g ")/s I-t0cat. Proof. - - Choose a to be a cardinal large enough to ensure: - any filtering colimit of (simplicially) fibrant objects indexed by the ordered set Seq[a] is again fibrant; - for any U E T and any functor ~ : Seq[o~] ~ A~ the map colirnves~q[~l~ J ~(U) ~ colim~q[~l ~ (U) is bijective.

For any q > 0 the sequence of pointed sets Tq+,(U Xx V) ---+ Tq(X) --+ Tq(U) x Tq(V) is exact. 17. e. isomorphic to the point sheaf pt) for all q. Then Tq =pt for all q. Proof. - - Restricting Tq to the small Zariski site of X we get a family of functors satisfying the conditions of [7, T h e o r e m 1']. e. that for any point x on X we have Tq(Spec(C>x, x)) = *. Let t E Tq(Spec(~x,,)) be an element and let U = Spec(~x, ,) - {x}. T h e n dim(U) < dim(X) and by obvious induction by dimension we may assume that Tq(U) = , for all q.

E. weakly equivalent to point and in particular non empty). ~O;'(S)is contractible. Assume first that ~ - ( S ) ~:(~ and let a E ,~'(S) be an element. Consider the family of functors Ti on Sy~ of the form u lu). -functor and the associated Nisnevich sheaves are trivial since ,~" ~ pt is a weak equivalence. 17. It remains to prove that ,~'(S) is not empty. We already know that for any V / S such that ,:~(V) is not empty it is contractible. Let s be a point of S. Let us show first that there exists an open neighborhood V of s such that ,r We may clearly assume that S is local and s is the closed point of S.

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