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Bott stability in algebraic K-theory. (English) Zbl 0594.18012
Applications of algebraic K-theory to algebraic geometry and number theory, Proc. AMS-IMS-SIAM Joint Summer Res. Conf., Boulder/Colo. 1983, Part I, Contemp. Math. 55, 389-406 (1986).
[For the entire collection see Zbl 0588.00014.]
let \(\ell^{\nu}\) be a fixed prime power and X a regular scheme. The author’s fundamental comparision between algebraic and topological K- theory \(K/\ell^{\nu}(X)[\beta^{-1}]=K/\ell^{\nu Top}(X)\) [cf. the author, ”Algebraic K-theory and etale cohomology”, 2nd edition, preprint] allows some control over the Bott element \(\beta\).
In this note, the author refines the proof of his result (loc. cit.) that for reasonable regular schemes X the natural map \[ \rho: K/\ell_ n^{\nu}(X)\to K/\ell_ n^{\nu Top}(X) \] is surjective for n sufficiently large. Moreover, a bounded power of \(\beta\) annihilates all elements in the kernel of \(\rho\).
Reviewer: M.Golasinski

18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
55R50 Stable classes of vector space bundles in algebraic topology and relations to \(K\)-theory
14A15 Schemes and morphisms
55N15 Topological \(K\)-theory