an:03929155
Zbl 0581.14012
Dwyer, William G.; Friedlander, Eric M.
Algebraic and ??tale \(K\)-theory
EN
Trans. Am. Math. Soc. 292, 247-280 (1985).
00149214
1985
j
14C35 18F25 14F20 19F27
??tale \(K\)-theory; Lichtenbaum-Quillen conjecture
The authors develop ??tale \(K\)-theory for a noetherian \({\mathbb Z}[1/\ell]\)- algebra \(A\) and smooth schemes over \(A\). This extends the work of \textit{E. M. Friedlander} [Invent. Math. 60, 105--134 (1980; Zbl 0519.14010) and Ann. Sci. ??c. Norm. Sup??r. (4) 15, 231--256 (1982; Zbl 0537.14011)]. The importance of ??tale \(K\)-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 \textit{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 [\textit{R. W. Thomason}, Ann. Sci. ??c. Norm. Sup??r. (4) 18, 437--552 (1985; Zbl 0596.14012)]. - The authors' main application of ??tale \(K\)-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 ??tale \(K\)-theory. Nonetheless, although thereby partially superannuated, it is important from a historical point of view that this paper has finally appeared.
V. P. Snaith
Zbl 0519.14010; Zbl 0537.14011; Zbl 0501.14013; Zbl 0596.14012