Topology Seminar

Date: February 15, 2023

Time: 3:00PM - 4:00PM

Location: BLOC 628

Speaker: Ningchuan Zhang, University of Pennsylvania

Title: Profinite Picard groups in K(n)-local homotopy theory

Abstract: The study of Picard groups in homotopy theory was initiated by Hopkins-Mahowald-Sadofsky. They gave a general framework to compute Picard groups of the categories of $K(n)$-local spectra. In the past decade, significant progress has been made in our understandings of Picard groups with new tools from higher category theory and equivariant homotopy theory. In this talk, we first review the basics of Picard groups in homotopy theory. Then we establish a descent spectral sequence to compute Picard groups of Galois extensions of the $K(n)$-local sphere. One key point in the proof is the profinite topology on the Picard group of a $K(n)$-local $E_\infty$ ring spectra. As an application, we compute Picard groups of all Galois extensions of the $K(1)$-local spheres at all primes. This is joint work in progress with Guchuan Li.