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 β-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 $K\left(n\right)$-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 $K\left(n\right)$-local modules over the homotopy fixed points spectrum $E_n^{hN}$, where N is the normalizer in $G_n$ of a finite cyclic subgroup of order p, as a subquotient of a single continuous cohomology group $H^{2n+1}(N,{\pi }_{2n}{E}_n)$.