Definition 70.11.3. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{A}$ be a quasi-coherent sheaf of graded $\mathcal{O}_ X$-algebras. The relative homogeneous spectrum of $\mathcal{A}$ over $X$, or the homogeneous spectrum of $\mathcal{A}$ over $X$, or the relative Proj of $\mathcal{A}$ over $X$ is the algebraic space $F$ over $X$ of Lemma 70.11.2. We denote it $\pi : \underline{\text{Proj}}_ X(\mathcal{A}) \to X$.

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