K-theory hypercohomology spectra of number rings at the prime 2
نویسنده
چکیده
Here K̂ is the `-completed periodic complex K-theory spectrum, Λ is the ring of operations [K̂, K̂], and Λ′F is the Iwasawa algebra associated to the `-adic cyclotomic extension F∞ obtained by adjoining all `-power roots of unity. The action of Λ′F on these roots of unity gives an embedding Λ′F ⊂ Λ. The Λ′F -module M∞ is the “basic Iwasawa module”. It can be defined as the étale homology group H1(R∞; Z`), where R∞ is the integral closure of R in F∞. With the hypotheses of Theorem 1.1, a theorem of Iwasawa implies that M∞ has projective dimension at most one as Λ′F -module. Combining this fact with the theorem above led to an explicit description of the Bousfield localization L ˆ K as the fibre of a certain map between wedges of suspensions of K̂, and to a complete calculation of the mod ` homology of the zero-th space ΩL ˆ K If the Lichtenbaum-Quillen conjectures hold, these results translate immediately to theorems about KR itself. Since various forms of the 2-primary Lichtenbaum-Quillen conjectures have recently been proved by Rognes and Weibel [8] [9], we digress to explain this point. In their weakest form, the 2-primary Lichtenbaum-Quillen conjectures for KR assert that there is a descent spectral sequence with E2-term H p ét(R; Z2(m)) converging to π∗KR for ∗ > 0. The strong form, conjectured by Dwyer and Friedlander, asserts that the natural map to étale K-theory KR−→KétR induces a weak equivalence on 0-connected covers. An equivalent formulation
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