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 2 or 3, and the global dimension of $ko$ is 4 or 5.
"Homological dimensions of ring spectra." Homology Homotopy Appl. 15 (2) 53 - 71, 2013.