Definition 27.16.7. Let $S$ be a scheme. Let $\mathcal{A}$ be a quasi-coherent sheaf of graded $\mathcal{O}_ S$-algebras. The relative homogeneous spectrum of $\mathcal{A}$ over $S$, or the homogeneous spectrum of $\mathcal{A}$ over $S$, or the relative Proj of $\mathcal{A}$ over $S$ is the scheme constructed in Lemma 27.15.4 which represents the functor $F$ (27.16.4.1), see Lemma 27.16.6. We denote it $\pi : \underline{\text{Proj}}_ S(\mathcal{A}) \to S$.

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