Hiroshima Mathematical Journal

On the chromatic $\e^0(M^1_{n-1})$ on $\Ga(m+1)$ for an odd prime

Rié Kitahama and Katsumi Shimomura

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Abstract

Ravenel introduced spectra $T(m)$ for $m\ge 0$ interpolating the Brown-Peterson spectrum $BP$ and the sphere spectrum $S$ in Complex cobordism and stable homotopy groups of spheres, AMS Chelsea Publishing, Providence, 2004. Since the homotopy groups of $BP$ are well known, it is interesting to study differences among the homotopy groups of $T(m)$'s to study the homotopy groups of spheres. He also introduced the localization functor $L_n$ on the stable homotopy category in "Localization with respect to certain periodic homology theories," Amer. J. Math. 106 (1984), 351–414. To study the difference of $L_nT(m)$'s for a fixed integer $n$, we consider the corresponding chromatic $E_1$-term $\e^0(M^1_{n-1})$ on $\Ga(m+1)$ for each $m$, and determine it for $m+1\ge (n-2)(n-1)$ in this paper. The results show that the structures depend on a integer $\Lt[\dfrac{m+1}{n-1}\Rt]$. Here $[x]$ denotes the greatest integer that does not exceed $x$.

Article information

Source
Hiroshima Math. J., Volume 41, Number 2 (2011), 211-222.

Dates
First available in Project Euclid: 24 August 2011

Permanent link to this document
https://projecteuclid.org/euclid.hmj/1314204562

Digital Object Identifier
doi:10.32917/hmj/1314204562

Mathematical Reviews number (MathSciNet)
MR2849155

Zentralblatt MATH identifier
1234.55014

Subjects
Primary: 55T15: Adams spectral sequences
Secondary: 55Q51: $v_n$-periodicity

Keywords
Adams-Novikov spectral sequence chromatic spectral sequence Ravenel spectrum

Citation

Kitahama, Rié; Shimomura, Katsumi. On the chromatic $\e^0(M^1_{n-1})$ on $\Ga(m+1)$ for an odd prime. Hiroshima Math. J. 41 (2011), no. 2, 211--222. doi:10.32917/hmj/1314204562. https://projecteuclid.org/euclid.hmj/1314204562


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References

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