A higher chromatic analogue of the image of $J$

Abstract

We prove a higher chromatic analogue of Snaith’s theorem which identifies the $K$–theory spectrum as the localisation of the suspension spectrum of $ℂ{\mathbb{P}}^{\infty }$ away from the Bott class; in this result, higher Eilenberg–MacLane spaces play the role of $ℂ{\mathbb{P}}^{\infty }=K\left(\mathbb{Z},2\right)$. Using this, we obtain a partial computation of the part of the Picard-graded homotopy of the $K\left(n\right)$–local sphere indexed by powers of a spectrum which for large primes is a shift of the Gross–Hopkins dual of the sphere. Our main technical tool is a $K\left(n\right)$–local notion generalising complex orientation to higher Eilenberg–MacLane spaces. As for complex-oriented theories, such an orientation produces a one-dimensional 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.