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.
Read Online or Download Generalized Etale Cohomology Theories PDF
Similar algebraic geometry books
This publication provides the speculation of quadratic and hermitian types over jewelry in a truly normal atmosphere. It avoids, so far as attainable, any restrict at the attribute and takes complete benefit of the functorial houses of the idea. it isn't an encyclopedic survey. It stresses the algebraic features of the speculation and avoids - is reasonably overlapping with different books on quadratic kinds (like these of Lam, Milnor-Husemöller and Scharlau).
This quantity is a homage to the reminiscence of the Spanish mathematician Federico Gaeta (1923-2007). except a ancient presentation of his existence and interplay with the classical Italian college of algebraic geometry, the quantity provides surveys and unique examine papers at the arithmetic he studied.
The focus of this quantity is at the challenge of describing the automorphism teams of affine and projective types, a classical topic in algebraic geometry the place, in either instances, the automorphism team is frequently limitless dimensional. the gathering covers a variety of subject matters and is meant for researchers within the fields of classical algebraic geometry and birational geometry (Cremona teams) in addition to affine geometry with an emphasis on algebraic staff activities and automorphism teams.
- The elements of plane analytic geometry
- Introduction to Hodge Theory
- Algebraic Geometry, Edition: version 24 Feb 2016
- Hodge Theory and Complex Algebraic Geometry I: Volume 1 (Cambridge Studies in Advanced Mathematics)
- The Geometry of Syzygies: A Second Course in Algebraic Geometry and Commutative Algebra (Graduate Texts in Mathematics)
- Algebraic cycles and motives
Extra resources for Generalized Etale Cohomology Theories
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.