Bounding the $K(p-1)$-local exotic Picard group at $p>3$

Abstract

In this paper, we bound the descent filtration of the exotic Picard group $\kappa_n$, for a prime number p>3 and n=p-1. Our method involves a detailed comparison of the Picard spectral sequence, the homotopy fixed point spectral sequence, and an auxiliary $\beta$-inverted homotopy fixed point spectral sequence whose input is the Farrell-Tate cohomology of the Morava stabilizer group. Along the way, we deduce that the K(n)-local Adams-Novikov spectral sequence for the sphere has a horizontal vanishing line at $3n^2+1$ on the $E_{2n^2+2}$-page. The same analysis also allows us to express the exotic Picard group of $K(n)$-local modules over the homotopy fixed points spectrum $\mathrm{E}_n^{hN}$, where N is the normalizer in $\mathbb{G}_n$ of a finite cyclic subgroup of order p, as a subquotient of a single continuous cohomology group $H^{2n+1}(N,\pi_{2n}\mathrm{E}_n)$.