A Note on Hopkins' Picard Groups of the Stable Homotopy Categories of -Local Spectra

  • Katsumi Shimomura

    Kochi University, Japan
A Note on Hopkins' Picard Groups of the Stable Homotopy Categories of $L_n$-Local Spectra cover

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Abstract

For a stable homotopy category, M. Hopkins introduced a Picard group as a category consisting of isomorphism classes of invertible objects. For the stable homotopy category of -local spectra, M. Hovey and H. Sadofsky showed that the Picard group is actually a group containing the group of integers as a direct summand. Kamiya and the author constructed an injection from the other summand of the Picard group to the direct sum of the -terms over of the Adams–Novikov spectral sequence converging to the homotopy groups of the -localized sphere spectrum. In this paper, we show in a classical way that the injection is a bijection under a condition.

Cite this article

Katsumi Shimomura, A Note on Hopkins' Picard Groups of the Stable Homotopy Categories of -Local Spectra. Publ. Res. Inst. Math. Sci. 56 (2020), no. 1, pp. 195–205

DOI 10.4171/PRIMS/56-1-8