Chromatic completion.

*(English)*Zbl 1402.55005Let \(p\) 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)].

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)].

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