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.