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By John F. Jardine

A generalized etale cohomology conception is a conception that is represented by way of a presheaf of spectra on an etale web site for an algebraic style, in analogy with the best way a typical spectrum represents a cohomology concept for areas. Examples contain etale cohomology and etale K-theory. This publication provides new and entire proofs of either Thomason's descent theorem for Bott periodic K-theory and the Nisnevich descent theorem. In doing so, it exposes lots of the significant principles of the homotopy thought of presheaves of spectra, and generalized etale homology theories particularly. The remedy comprises, for the aim of appropriately facing cup product buildings, a improvement of strong homotopy idea for n-fold spectra, that is then promoted to the extent of presheaves of n-fold spectra. This ebook can be of curiosity to all researchers operating in fields concerning algebraic K-theory. The ideas offered listed below are primarily combinatorial, and for that reason algebraic. an intensive historical past in conventional sturdy homotopy conception isn't assumed.

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Putting u = 0, we obtain [II, § 1J 29 MAPS OF VARIETIES INTO ABELIAN VARIETIES d . x, which shows that the restriction of

H(P g ). It is symmetric in P v ... , P g , and the point v = H(Pv ... , P g ) is rational over k(u). Hence there exists a rational map (J. U = v. Taking into account Theorem 4, and recalling that I(P') = h(P') = 0, we see that (J. is a homomorphism. This proves our theorem. Let 1 : V ~ A be a rational map of a variety into an abelian variety. Then 1 induces a homomorphism of the group of cycles on V into A as follows. We denote by Zr(V) the group of cycles of dimension r on V. Let a = ~ ni(xi ) be an element of Zo(V)· We put I(a) = ~ ni(t(x i )).

We shall say that g* is the homomorphism induced by g. By an abuse of language, we shall sometimes say that A is an Albanese variety, and that I is a canonical map. THEOREM 11. Let V be a variety. Then there exists an Albanese variety (A, f) 01 V. The abelian variety A is uniquely determined up to a birational isomorphism, and I is determined up to a translation. Proof: The uniqueness of A and I up to a translation is an immediate consequence of the uniqueness of g* in (ii). To prove the existence, we note that the theorem is birational in V.

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