Spectra of derived module homomorphisms.

*(English)*Zbl 0635.55011The author has worked extensively in stable homotopy, especially the theory of spectra [see, e.g., Topology 22, 1-18 (1983; Zbl 0519.55004)]. The purpose of the present note is to introduce a homomorphism spectrum between module spectra over a given ring spectrum. This is carried out in the case where the ring and module spectra have \(A_{\infty}\) structures. The resulting homomorphism spectrum has many pleasant properties, e.g., it is a homotopy invariant, exact in each variable, and reduces to the more traditional mapping spectrum when the ring spectrum is that of spheres. The author also constructs a spectral sequence.

These results specialize to known work in several cases, including work of J. Klippenstein on Brown-Gitler spectra [Thesis, Univ. Warwick (1985)] and the Adams-like spectral sequence of Brayton Gray (unpublished) and T. Y. Lin [Indiana Univ. Math. J. 25, 135-158 (1976; Zbl 0333.55013)].

These results specialize to known work in several cases, including work of J. Klippenstein on Brown-Gitler spectra [Thesis, Univ. Warwick (1985)] and the Adams-like spectral sequence of Brayton Gray (unpublished) and T. Y. Lin [Indiana Univ. Math. J. 25, 135-158 (1976; Zbl 0333.55013)].

Reviewer: D.W.Kahn

##### MSC:

55P42 | Stable homotopy theory, spectra |

55P65 | Homotopy functors in algebraic topology |

55T25 | Generalized cohomology and spectral sequences in algebraic topology |

##### Keywords:

homomorphism spectrum between module spectra over a given ring spectrum; \(A_{\infty }\) structures; spectral sequence; Brown-Gitler spectra; Adams-like spectral sequence
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\textit{A. Robinson}, Math. Proc. Camb. Philos. Soc. 101, 249--257 (1986; Zbl 0635.55011)

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##### References:

[1] | DOI: 10.2307/1993608 · Zbl 0114.39402 |

[2] | DOI: 10.1512/iumj.1976.25.25011 · Zbl 0333.55013 |

[3] | DOI: 10.1016/0040-9383(83)90042-3 · Zbl 0519.55004 |

[4] | Robinson, J. Pure Appl. Algebra |

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