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Chromatic completion. (English) Zbl 1402.55005
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 [D. C. Ravenel, Am. J. Math. 106, 351–414 (1984; Zbl 0586.55003)], Ravenel asked whether these functors do in fact coincide.
A 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 [M. J. Hopkins and 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 [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.
In 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 C. Casacuberta and A. Frei [J. Pure Appl. Algebra 142, No. 1, 25–33 (1999; Zbl 0931.18005)].

##### MSC:
 55P42 Stable homotopy theory, spectra 55P60 Localization and completion in homotopy theory
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##### References:
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