Abstract. Let L 2 denote the Bousfield localization functor with respect to the Johnson-Hovey and Sadofsky, Invertible spectra in the E(n)-local stable homotopy category, showed that every invertible spectrum is homotopy equivalent to a suspension of the E(2)-local sphere L 2 S 0 at a prime p > 3. At the prime 3, it is shown, A relation between the Picard group of the E(n)-local homotopy category and E(n)-based Adams spectral sequence, that there exists an invertible spectrum X that is not homotopy equivalent to a suspension of L 2 S 0 . In this paper, we show the homotopy equivalence v 3 2 : Σ 48 L 2 V (1) V (1) ∧ X for the Smith-Toda spectrum V (1). In the same manner as this, we also show the existence of the self-map β : Σ 144 L 2 V (1) → L 2 V (1) that induces v 9 2 on the E(2) * -homology.

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Abstract. We compute a certain Ext group related to the chromatic spectral sequence for T (m), the spectrum whose BP -homology is BP * [t 1 , · · · , tm] for each m ≥ 3. The answer we get displays a kind of periodicity not seen in the corresponding computation for the sphere spectrum.

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