The ghost and weak dimensions of rings and ring spectra

TL;DR

This paper introduces the ghost dimension, a new concept for measuring weak dimension in ring spectra, and proves a conjecture relating derived categories to weak dimension of rings.

Contribution

It develops a theory of ghost dimension for ring spectra and proves a conjecture connecting derived categories to weak dimension of rings.

Findings

Established the ghost dimension as a measure of weak dimension for ring spectra.

Proved the conjecture relating derived categories to the weak dimension of rings.

Abstract

The primary object of this paper is to prove the conjecture of the authors from a previous paper, explaining how to recover the weak dimension of a ring from its derived category. In the process, we develop a theory of weak dimension, which we call ghost dimension, for the generalized rings, known as ring spectra, that arise in algebraic topology.