Definition 10.57.1. Let $S$ be a graded ring. We define $\text{Proj}(S)$ to be the set of homogeneous prime ideals $\mathfrak p$ of $S$ such that $S_{+} \not\subset \mathfrak p$. The set $\text{Proj}(S)$ is a subset of $\mathop{\mathrm{Spec}}(S)$ and we endow it with the induced topology. The topological space $\text{Proj}(S)$ is called the homogeneous spectrum of the graded ring $S$.

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