Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On Hopkins' Picard group

Tobias Barthel, Tomer M. Schlank, Nathaniel Stapleton, Jared Weinstein

TL;DR

The authors determine the algebraic Picard group $ ext{Pic}_{n,p}^{ ext{alg}}$ for all heights $n$ and primes $p$, and deduce the topological $K(n)$-local Picard group structure via a refined comparison map. Their strategy translates invertible $K(n)$-local spectra into Morava modules and then into cohomological data on Lubin–Tate and Drinfeld moduli spaces, ultimately reducing the problem to computing a specific $H^1_{cts}$-group and its image in pro-étale cohomology of Drinfeld’s symmetric space. Central to the argument are the determinant map on formal groups, the fundamental exact sequence linking Galois cohomology to pro-étale cohomology, and the Colmez–Dospinescu–Nizioł results identifying the pro-étale cohomology with generalized Steinberg representations; together they yield a complete algebraic description and, up to the exceptional $(n,p)=(2,2)$ case, surjectivity of the comparison map. The results give explicit generators and a precise finite-projection structure, advancing Hopkins’s predictions and providing tools for concrete computations across all heights.

Abstract

We compute the algebraic Picard group of the category of $K(n)$-local spectra, for all heights $n$ and all primes $p$. In particular, we show that it is always finitely generated over $\mathbb{Z}_p$ and, whenever $n \geq 2$, is of rank $2$, thereby confirming a prediction made by Hopkins in the early 1990s. In fact, with the exception of the anomalous case $n=p=2$, we provide a full set of topological generators for these groups. Our arguments rely on recent advances in $p$-adic geometry to translate the problem to a computation on Drinfeld's symmetric space, which can then be solved using results of Colmez--Dospinescu--Niziol.

On Hopkins' Picard group

TL;DR

The authors determine the algebraic Picard group for all heights and primes , and deduce the topological -local Picard group structure via a refined comparison map. Their strategy translates invertible -local spectra into Morava modules and then into cohomological data on Lubin–Tate and Drinfeld moduli spaces, ultimately reducing the problem to computing a specific -group and its image in pro-étale cohomology of Drinfeld’s symmetric space. Central to the argument are the determinant map on formal groups, the fundamental exact sequence linking Galois cohomology to pro-étale cohomology, and the Colmez–Dospinescu–Nizioł results identifying the pro-étale cohomology with generalized Steinberg representations; together they yield a complete algebraic description and, up to the exceptional case, surjectivity of the comparison map. The results give explicit generators and a precise finite-projection structure, advancing Hopkins’s predictions and providing tools for concrete computations across all heights.

Abstract

We compute the algebraic Picard group of the category of -local spectra, for all heights and all primes . In particular, we show that it is always finitely generated over and, whenever , is of rank , thereby confirming a prediction made by Hopkins in the early 1990s. In fact, with the exception of the anomalous case , we provide a full set of topological generators for these groups. Our arguments rely on recent advances in -adic geometry to translate the problem to a computation on Drinfeld's symmetric space, which can then be solved using results of Colmez--Dospinescu--Niziol.
Paper Structure (31 sections, 52 theorems, 171 equations, 1 table)

This paper contains 31 sections, 52 theorems, 171 equations, 1 table.

Table of Contents

  1. Introduction
  2. Results
  3. Main ideas of the proof
  4. The $K(n)$-local Picard group
  5. Motivation: invertible spectra and degree
  6. Morava modules
  7. Two distinguished spheres
  8. Conjectures and results
  9. The even algebraic local Picard group
  10. The prime-to-$p$ part and a relation
  11. A reduction of \ref{['thm:picalg']}
  12. Addenda and anomalies for $p=2$
  13. A toy case: height 1
  14. Moduli of formal groups
  15. The stack $\widehat{\mathcal{M}}_{\mathop{\mathrm{FG}}\nolimits}^n$ of height $n$ formal groups
  16. ...and 16 more sections

Key Result

Theorem A

Let $p$ be a prime, assume $n\geq 2$, and write $\mathcal{Z}_{n,p}$ for the pro-cyclic groupThis notation allows us to deal with all primes uniformly. When $p$ is odd, $\mathcal{Z}_{n,p} \cong \mathbb{Z}_p \oplus \mathbb{Z}/(2p^n-2)$, while $\mathcal{Z}_{n,2} \cong \mathbb{Z}_2 \oplus \mathbb{Z}/(2^ where In all cases except possibly $(n,p)=(2,2)$, the map $\varepsilon$ is surjective. In the case

Theorems & Definitions (105)

  • Theorem A: \ref{['thm:picalg']}
  • Theorem B: \ref{['cor:pictop']}
  • Theorem C
  • Theorem D
  • Definition 2.1.1
  • Proposition 2.1.2: Hopkins--Mahowald--Sadofsky, Hovey--Strickland
  • Theorem 2.2.1: Devinatz--Hopkins
  • Lemma 2.2.2: Hopkins--Mahowald--Sadofsky
  • Definition 2.2.3
  • Proposition 2.2.4: Hopkins--Mahowald--Sadofsky, Pstragowski, Goerss--Hopkins
  • ...and 95 more