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.
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机译：我们为$ S $代数定义了同构维，S $-代数是代数拓扑中出现的广义环。我们计算了许多示例的同源性维，并建立了一些基本属性。最困难的计算是实数$ K $-理论$ KO $及其在素数2时的连接版本$ ko $的全局维度。我们证明$ KO $的全局维度为2或3，而$ KO $的全局维度为$ ko $是4或5。
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