Chromatic completion (Q2792149)

Let \(p\) be a fixed prime. The category of \(p\)-local spectra can be further localized at each chromatic level \(n\) by means of the functor \(L_n\), the Bousfield localization with respect to the Johnson-Wilson spectrum \(E(n)\). A natural problem is then to try to recover global information from these local components. One way to do this is via harmonic localization \(L_\infty\), defined as Bousfield localization at the wedge of all Morava \(K\)-theories \(K(n)\). On the other hand, the chromatic completion of a \(p\)-local spectrum \(X\), \(\mathbb{C}X\), is defined as the limit of its chromatic tower \(\cdots \to L_nX\to L_{n-1}X\to\cdots\to L_0X\). In [\textit{D. C. Ravenel}, Am. J. Math. 106, 351--414 (1984; Zbl 0586.55003)], Ravenel asked whether these functors do in fact coincide.NEWLINENEWLINEA spectrum \(X\) is called harmonic if the natural map \(X\to L_{\infty}X\) is an equivalence. Likewise, \(X\) is said to be chromatically complete if \(X\to \mathbb{C}X\) is an equivalence. For instance, it is a result of Hopkins and Ravenel [\textit{M. J. Hopkins} and \textit{D. C. Ravenel}, Bol. Soc. Mat. Mex., II. Ser. 37, No. 1--2, 271--279 (1992; Zbl 0838.55010)] that symmetric spectra are harmonic, while finite spectra are known to be chromatically complete by the chromatic convergence theorem of the same authors [\textit{D. C. Ravenel}, Nilpotence and periodicity in stable homotopy theory. Princeton, NJ: Princeton University Press (1992; Zbl 0774.55001)]. In particular, finite spectra are both harmonic and chromatically complete.NEWLINENEWLINEIn one of the main results of this paper, the author shows that chromatic convergence holds for connective spectra with finite projective \(BP\)-dimension. The latter generalizes the original convergence result of Hopkins and Ravenel, as finite spectra are known to have finite projective \(BP\)-dimension. This is achieved by showing that the corresponding tower of acyclizations is pro-trivial in homotopy via a careful analysis of the Adams-Novikov filtration for such spectra. In addition, the author answers Ravenel's question in the negative by constructing a harmonic spectrum which is not chromatically complete. This relies on a useful criterion for when a harmonic spectrum is chromatically complete, which the author deduces as a consequence of his characterization of \(L_{\infty}\) as the idempotent approximation of \(\mathbb{C}\) in the sense of \textit{C. Casacuberta} and \textit{A. Frei} [J. Pure Appl. Algebra 142, No. 1, 25--33 (1999; Zbl 0931.18005)].