The Stacks project

Lemma 70.16.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{A}$ be a quasi-coherent graded $\mathcal{O}_ X$-algebra. Let $\pi : P = \underline{\text{Proj}}_ X(\mathcal{A}) \to X$ be the relative Proj of $\mathcal{A}$. Let $i : Z \to P$ be a closed subspace. Denote $\mathcal{I} \subset \mathcal{A}$ the kernel of the canonical map

\[ \mathcal{A} \longrightarrow \bigoplus \nolimits _{d \geq 0} \pi _*\left((i_*\mathcal{O}_ Z)(d)\right) \]

If $\pi $ is quasi-compact, then there is an isomorphism $Z = \underline{\text{Proj}}_ X(\mathcal{A}/\mathcal{I})$.

Proof. The morphism $\pi $ is separated by Lemma 70.11.6. As $\pi $ is quasi-compact, $\pi _*$ transforms quasi-coherent modules into quasi-coherent modules, see Morphisms of Spaces, Lemma 66.11.2. Hence $\mathcal{I}$ is a quasi-coherent $\mathcal{O}_ X$-module. In particular, $\mathcal{B} = \mathcal{A}/\mathcal{I}$ is a quasi-coherent graded $\mathcal{O}_ X$-algebra. The functoriality morphism $Z' = \underline{\text{Proj}}_ X(\mathcal{B}) \to \underline{\text{Proj}}_ X(\mathcal{A})$ is everywhere defined and a closed immersion, see Lemma 70.12.3. Hence it suffices to prove $Z = Z'$ as closed subspaces of $P$.

Having said this, the question is ├ętale local on the base and we reduce to the case of schemes (Divisors, Lemma 31.31.1) by ├ętale localization. $\square$


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