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Algebraic $$K$$-theory and étale cohomology. (English) Zbl 0596.14012
This is a very long paper to which a short review like this could never do justice. Therefore I should firstly point out that this paper contains one of the most important discoveries concerning the role of algebraic $$K$$-theory in algebraic geometry.
Let $$\ell$$ be a prime and $$X$$ a separated, regular, noetherian scheme in with $$\ell$$ is invertible. If $${\mathcal O}_ X$$ contains $$\ell^{\nu}$$-th roots of unity there is a 2-dimensional “Bott element” in $$K_ 2({\mathcal O}_ X;{\mathbb Z}/\ell^{\nu})$$ and one may, following the reviewer [Mem. Am. Math. Soc. 221 (1979; Zbl 0413.55004), Part IV, and ibid. 280 (1983; Zbl 0529.55015)], form the localisation $$K_*(X;{\mathbb Z}/\ell^{\nu})[\frac{1}{\beta}]$$.
The author’s main theorem states that there is an isomorphism $K_*(X;{\mathbb Z}/\ell^{\nu})[\frac{1}{\beta}]\cong K_*^{top}(X;{\mathbb Z}/\ell^{\nu})$ where the latter is topological or étale $$K$$-theory, which is constructed to be computable from étale cohomology.
When $$X$$ is a smooth complex variety $$K_*^{top}(X;{\mathbb Z}/\ell^{\nu})$$ is the usual mod $$\ell^{\nu} K$$-cohomology of $$X$$ (with the classical topology). This connection permits the author to infer many fundamental consequences. For example, in § 4, he proves a purity conjecture of Grothendieck for $$\ell$$-adic étale cohomology by exploiting the relation with algebraic $$K$$-theory.
Reviewer: V. P. Snaith

##### MSC:
 14C35 Applications of methods of algebraic $$K$$-theory in algebraic geometry 14F20 Étale and other Grothendieck topologies and (co)homologies 19E20 Relations of $$K$$-theory with cohomology theories 19F27 Étale cohomology, higher regulators, zeta and $$L$$-functions ($$K$$-theoretic aspects)
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