The Morava K-theory of algebraic K-theory spectra
The key results of the paper are: Theorem A: If \(n\geq 2\), the Morava K- theory \(K(n)_*\) of the algebraic K-theory spectrum K\({\mathbb{Z}}\) vanishes. Theorem B: The unit map f: \(Q_ 0S^ 0\to BGL{\mathbb{Z}}^+\) factors through the space Im(J). Here K(n) and Im(J) are of course defined with respect to the fixed prime p, the upper \(+\) denotes Quillen's plus-construction, and \(Q_ 0S^ 0\) is the zero-component of the infinite loop space colim \(\Omega\) \({}^ nS^ n\). Theorem A implies the vanishing of \(K(n)_*\) on any algebraic K-theory spectrum KX, where X is a ring or a scheme. Let R be the ring \({\mathbb{Z}}[1/p]\). Section 1 contains an outline of the representation theory of a p-group G over R, which allows to conclude (Theorem 1.12) that the image of the Burnside ring AG in the Grothendieck group \(G_ 0^ RRG\) of finitely generated R-free RG-modules maps injectively to the Grothendieck group \(G_ 0{\mathbb{Q}}G\) of \({\mathbb{Q}}G\). In Section 2 Theorem B is proved. Using the known factorization of \(Q_ 0S^ 0\to BU\) through Im(J), the splitting \(Q_ 0S^ 0\simeq Im(J)\times F\) and the equivalence \(Q_ 0S^ 0\simeq B\Sigma^+_{\infty}\), a completed version (Theorem 1.14) of 1.12 yields the result. The proof of Theorem A uses the work of Hopkins and Smith on finite complexes X with \(v_ n\)-selfmaps \(\tilde v_ n: S^ kX\to X\). It is easy to see (Proposition 3.3) that it sufficies to show that the \(v_ n\)-periodic homotopy groups \(\tilde v_ n^{-1}\pi^ s_*(K{\mathbb{Z}};X)\) are zero for \(n\geq 2\). The latter groups coincide with the unstable groups \(\tilde v^ 1_ n\pi_*(BGL{\mathbb{Z}}^+;X)\). The vanishing of these now follows from Theorem B and the vanishing of the \(v_ n\)-periodic homotopy groups of Im(J). The paper is very densely written and gives essentially all necessary details for the complete story. A version of the Lichtenbaum-Quillen conjectures and its relation to harmonic localization are discussed. Moreover, Section 4 and 5 contain short proofs of some known results in homotopy theory and group theory which had been used in the proofs of the main theorems.
scientific article; zbMATH DE number 4166063
scientific article