The Picard group is an important invariant of the $K(n)$-local category. If the prime $p$ is relatively large compared to the height $n$, the Picard group of the $K(n)$-local category is purely algebraic. In \cref{chapter:finitetype}, we describe the necessary and sufficient numerical condition when an element $X$ in the Picard group of $K(2)$-local category at prime $p \geqslant 5$ has the property that $\pi_kX$ is a finitely generated $\Z_p$-module for all $k \in \Z$. If the prime $p$ is small, it is not necessarily purely algebraic. In \cref{chapter:grosshopkinsdual}, we show that the kernel of the map from the Picard group of the $K(n)$-local category to the algebraic Picard group is nontrivial for all heights at prime $2$. To detect nontrivial elements in the kernel, we study the Gross--Hopkins dual of $E_n^{hC_2}$, an invertible modules over $E_n^{hC_2}$, where $E_n$ is the height $n$ Lubin--Tate spectrum. In particular, we show that $E_n^{hC_2}$ is Gross--Hopkins self-dual up to a $n+4$ shift. In \cref{chapter:hurewitczimage}, we show that the Hopf elements, the Kervaire classes, and the $\bar{\kappa}$-family in the stable homotopy groups of spheres are detected by the Hurewicz map from the sphere spectrum to the $E_n^{hC_2}$.

Creator

• Li, Guchuan

DOI

• https://doi.org/10.21985/n2-ncpx-x310

Subject

• Mathematics

Language

• en

Alternate Identifier

• http://dissertations.umi.com/northwestern:14617
• etdadmin_upload_661882

Keyword

• Chromatic Homotopy Theory
• Invertible spectra

Date created

• 2019-01-01

Resource type

• Dissertation

Rights statement

• In Copyright