Picard groups of higher real -theory spectra at height

Abstract

Using the descent spectral sequence for a Galois extension of ring spectra, we compute the Picard group of the higher real K-theory spectra of Hopkins and Miller at height n = p − 1, for p an odd prime. More generally, we determine the Picard groups of the homotopy fixed points spectra E hG n , where En is Lubin-Tate E-theory at the prime p and height n = p − 1, and G is any finite subgroup of the extended Morava stabilizer group. We find that these Picard groups are always cyclic, generated by the suspension. 1 arXiv:1511.08064v1 [math.AT]

Examples of Descent up to Nilpotence

“…Remark 3.23. We refer to [MS16,HMS15] for a description of how these results can be used to calculate Picard groups of ring spectra. For instance, it is possible to calculate the Picard group of KU -modules relatively directly using the homotopy groups of KU (compare [BR05]), while invertible modules over KO can then be determined by descent.…”

“…There are many applications of these ideas that we shall not touch on here. For example, we refer to [MS16,HMS15] for the use of these techniques to calculate Picard groups of certain ring spectra; [Bal16,Mat15a] for applications to the classification of thick subcategories; and [CMNN16] for applications to Galois descent in algebraic K-theory.…”

The Galois group of a stable homotopy theory

“…Remark 3.2. For some applications of these ideas to computations, see the paper [Mat15a] (for descriptions of thick subcategories) and [GL,MS,HMS15] (for calculations of certain Picard groups).…”

Gross–Hopkins duals of higher real K–theory spectra

“…One may be able to test the non-triviality of X ∈ κ n by showing that X ∧ E hH n is non-trivial in Pic(E hH n ). In contrast with Pic n , the Picard group of E hH n is known and simple; Heard, Mathew, and Stojanoska [HMS15] have shown it is cyclic, so that X ∧ E hH n Σ k E hH n for some integer k. The spectra E hH n are periodic, so the problem reduces to determining whether or not k ≡ 0 modulo the periodicity.…”