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


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)
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