Homological dimensions of ring spectra

TL;DR

This paper introduces homological dimensions for S-algebras in algebraic topology, computes these dimensions for key examples like real K-theory, and establishes foundational properties of these dimensions.

Contribution

It defines homological dimensions for S-algebras and computes the global dimensions for real K-theory spectra KO and ko at prime 2, providing new insights.

Findings

Global dimension of KO is 1, 2, or 3.

Global dimension of ko is 4 or 5.

Established basic properties of homological dimensions for S-algebras.

Abstract

We define homological dimensions for S-algebras, the generalized rings that arise in algebraic topology. We compute the homological dimensions of a number of examples, and establish some basic properties. The most difficult computation is the global dimension of real K-theory KO and its connective version ko at the prime 2. We show that the global dimension of KO is 1, 2, or 3, and the global dimension of ko is 4 or 5.