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On Hopkins’ Picard groups for the primes 3 and chromatic level 2
, 2012
"... We give a calculation of Picard groups P ic2 of K(2)local invertible spectra and P ic(L2) of E(2)local invertible spectra, both at the prime 3. The main contribution of this paper is to calculation the subgroup κ2 of invertible spectra X with (E2)∗X ∼ = (E2)∗S 0 as continuous modules over the Mor ..."
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We give a calculation of Picard groups P ic2 of K(2)local invertible spectra and P ic(L2) of E(2)local invertible spectra, both at the prime 3. The main contribution of this paper is to calculation the subgroup κ2 of invertible spectra X with (E2)∗X ∼ = (E2)∗S 0 as continuous modules over the Morava stabilizer group G2.
A higher chromatic analogue of the image of J
, 2014
"... We prove a higher chromatic analogue of Snaith’s theorem which identifies the Ktheory spectrum as the localisation of the suspension spectrum of CP ∞ away from the Bott class; in this result, higher EilenbergMacLane spaces play the role of CP ∞ = K(Z, 2). Using this, we obtain a partial computati ..."
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We prove a higher chromatic analogue of Snaith’s theorem which identifies the Ktheory spectrum as the localisation of the suspension spectrum of CP ∞ away from the Bott class; in this result, higher EilenbergMacLane spaces play the role of CP ∞ = K(Z, 2). Using this, we obtain a partial computation of the part of the Picardgraded homotopy of the K(n)local sphere indexed by powers of a spectrum which for large primes is a shift of the GrossHopkins dual of the sphere. Our main technical tool is a K(n)local notion generalising complex orientation to higher EilenbergMacLane spaces. As for complexoriented theories, such an orientation produces a onedimensional formal group law as an invariant of the cohomology theory. As an application, we prove a theorem that gives evidence for the chromatic redshift conjecture. 1