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Algebraic and étale $$K$$-theory. (English) Zbl 0581.14012
The authors develop étale $$K$$-theory for a noetherian $${\mathbb Z}[1/\ell]$$- algebra $$A$$ and smooth schemes over $$A$$. This extends the work of E. M. Friedlander [Invent. Math. 60, 105–134 (1980; Zbl 0519.14010) and Ann. Sci. Éc. Norm. Supér. (4) 15, 231–256 (1982; Zbl 0537.14011)]. The importance of étale $$K$$-theory is that it provides a computable target for algebraic $$K$$-theory. In fact there is a natural map $\phi: K_ i(A;{\mathbb Z}/\ell^{\nu})\to K_ i^{et}(A;{\mathbb Z}/\ell^{\nu})$ which is expected (the Lichtenbaum-Quillen conjecture) to be an isomorphism for “nice” $$A$$ when $$i$$ is large. In fact $$\phi$$ is onto in these circumstances [the authors, the reviewer and R. W. Thomason, Invent. Math. 66, 481–491 (1982; Zbl 0501.14013)].
The main theorem in this subject is that $$\phi$$ made “Bolt periodic” is an isomorphism [R. W. Thomason, Ann. Sci. Éc. Norm. Supér. (4) 18, 437–552 (1985; Zbl 0596.14012)]. - The authors’ main application of étale $$K$$-theory is to show that if $$A$$ is the ring of $$S$$-integers in a number field then $$\phi$$ is surjective if $$i\geq 1.$$
During the gestation period of this paper other authors – for example, J. F. Jardine, A. A. Suslin, R. W. Thomason – have increased our knowledge of algebraic $$K$$-theory and our understanding of étale $$K$$-theory. Nonetheless, although thereby partially superannuated, it is important from a historical point of view that this paper has finally appeared.
Reviewer: V. P. Snaith

##### MSC:
 14C35 Applications of methods of algebraic $$K$$-theory in algebraic geometry 18F25 Algebraic $$K$$-theory and $$L$$-theory (category-theoretic aspects) 14F20 Étale and other Grothendieck topologies and (co)homologies 19F27 Étale cohomology, higher regulators, zeta and $$L$$-functions ($$K$$-theoretic aspects)
##### Keywords:
étale $$K$$-theory; Lichtenbaum-Quillen conjecture
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