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Picard groups of quotient ring spectra

Ishan Levy, Guchuan Li, Ningchuan Zhang

Abstract

We develop tools to study Picard groups of quotients of ring spectra by a finitely generated ideal, which we use to show that $\mathrm{Pic}(\mathrm{E}_n/I) = \mathbb{Z}/2$, where $\mathrm{E}_n$ is a Lubin--Tate theory and $I$ is an ideal generated by suitable powers of a regular sequence. We apply this to obtain spectral sequences computing Picard groups of $\mathrm{K}(n)$-local generalized Moore algebras, and make some preliminary computations including the height $1$ case.

Picard groups of quotient ring spectra

Abstract

We develop tools to study Picard groups of quotients of ring spectra by a finitely generated ideal, which we use to show that , where is a Lubin--Tate theory and is an ideal generated by suitable powers of a regular sequence. We apply this to obtain spectral sequences computing Picard groups of -local generalized Moore algebras, and make some preliminary computations including the height case.

Paper Structure

This paper contains 9 sections, 28 theorems, 46 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Tools for studying Picard groups
  3. Relative module spaces and pullbacks
  4. The descent tower
  5. Picard groups of regular quotients of complete local rings
  6. Computations of Picard groups of quotient ring spectra
  7. Picard groups of quotients of $\mathrm{KO}$ and Tate $\mathrm{K}$-theories
  8. Picard groups of generalized Moore spectra of type $n$
  9. Computations at height $1$

Key Result

Theorem 1.1

Let $R$ be an $\mathbb{E}_3$-ring. Suppose $v\in \pi_*(R)$ is an element such that $R/v$ admits a left unital multiplication in $R$-modules. If $n\geq 3$ and $\pi_{-1-n|v|}(R/v)=0$, then the base change map $\mathrm{Pic}(R/v^{n+1})\to \mathrm{Pic}(R/v^n)$ is injective where $R/v^{n+1}\to R/v^n$ is m

Figures (1)

  • Figure 1: The HFPSS and the Picard DSS for $\mathrm{KO}/2^{k}$, Adams grading, $\bullet=\mathbb{Z}/2$, $=\mathbb{Z}/2^{k-1}$, $=\mathbb{Z}/2^k$, $\bigtimes=(\mathbb{Z}/2^k)^\times$.

Theorems & Definitions (73)

  • Theorem 1.1: \ref{['cor:pic_inj']}
  • Theorem 1.2: \ref{['iteratedquotient']}
  • Theorem 1.3: \ref{['thm:Moore_Pic']}
  • Theorem 1.4: \ref{['thm:Pic_S0pk']}, \ref{['thm:Pic_S02k']}
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Definition 2.4: Tamme_excision2018
  • Remark 2.5
  • ...and 63 more