Arithmetic localisation and completion of spectra

TL;DR

This paper explores properties of p-local and p-complete spectra, focusing on their finiteness, dualisability, and module categories, providing clarifications and new insights into their structure.

Contribution

It establishes that every dualisable p-complete spectrum is a p-completion of a finite spectrum and shows the module category over the p-complete sphere has homological Brown representability.

Findings

Dualisable p-complete spectra are p-completions of finite spectra

Modules over the p-complete sphere have homological Brown representability

Provides homological criteria for finiteness in spectra

Abstract

This is an exposition of facts about p-local spectra, p-complete spectra and modules over the p-complete sphere spectrum, including homological criteria for finiteness. Most things are well-known to the experts, with a couple of potential exceptions: every dualisable p-complete spectrum is the p-completion of a finite spectrum, and the category of modules over the p-complete sphere has homological Brown representability.