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By Eric M. Friedlander

This booklet provides a coherent account of the present prestige of etale homotopy conception, a topological concept brought into summary algebraic geometry by way of M. Artin and B. Mazur. Eric M. Friedlander provides lots of his personal functions of this idea to algebraic topology, finite Chevalley teams, and algebraic geometry. Of specific curiosity are the discussions about the Adams Conjecture, K-theories of finite fields, and Poincare duality. simply because those functions have required repeated transformations of the unique formula of etale homotopy idea, the writer presents a brand new remedy of the rules that is extra normal and extra designated than prior versions.

One function of this e-book is to supply the elemental thoughts and result of etale homotopy idea to topologists and algebraic geometers who might then practice the speculation of their personal paintings. that allows you to such destiny functions, the writer has brought a few new structures (function complexes, relative homology and cohomology, generalized cohomology) that have instantly proved appropriate to algebraic K-theory.

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Let a f G 3 (Un,n) represent a given cohomology class a in Hn(G 3 (U .. )), some n ~ 0, and U.. ). Because the sheaf associated to the presheaf G 3 is the zero sheaf, we may choose an etale surjective map W .... Un,n such that a restricts to 0 in GlW). (W) = V..... Un. 7). Namely, the appropriate surjectivity in dimension t ~ 0 is equivalent to the lifting of geometric points, which is equivalent to lifting maps from Speck ® ~[t]. This can be readily checked using the definition of r~·( ) and the adjointness· of coskt-1 ( ) and skt-1 ( ) .

0 for any U ..... x. v* In other words, H (X. , ) is a 8-functor. 4 which relates Cech cohomology of X. with that of each of the Xn , n 2: 0. 2. Let X. be a simplicial scheme and P an abelian presheaf on X. Then there exists a first quadrant spectral sequence v Es1,t = Ht(x ,P) s s v ;;, Hs+tcx. ,P) where Ps is the restriction of P to Et(Xs). Proof. For any covering U. -> X. )) of the form Because two maps U. ::: V. over X. )) which are related by a filtration-preserving homotopy, we conclude that two such maps ind_uce the same map Therefore, we may take the colimit of the spectral sequence indexed by the left directed category of coverings U.

Is a well-defined map of simplicial schemes. Moreover, using this explicit construction, we see that if (q 0 , ¢) and (r 0 , ifr) are elements of < Il(X 0 ,G); d. data>, then a map (}: q 0 -+ r 0 of schemes over X 0 with ifrod~(} = dr(}ocp determines a map of simplicial schemes q .... r. In particular, the G-action on q 0 determines a G-action on q, so that q: x: ... X. is an element of Il(X. ,G). The same argument implies that a map (q 0 ,¢) -+(r 0 ,ifr) in determines a map q ...

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