𝐸(2)-invertible spectra smashing with the Smith-Toda spectrum 𝑉(1) at the prime 3

Abstract

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.

The Brown–Comenetz dual of the K(2)-local sphere at the prime 3

“…Remark 5.5. In [15], Ichigi and Shimomura calculated π * X ∧V (1) for some X in the Picard group. These results are closely related to the calculations done in this section.…”

On Hopkins’ Picard groups for the prime 3 and chromatic level 2

“…For the sphere itself, see [21]. For a general Z, the result can be deduced after a bit of work from [16]. We include here a proof that uses our technology.…”