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The Stacks project

Lemma 71.16.1. 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 71.11.6. As \pi is quasi-compact, \pi _* transforms quasi-coherent modules into quasi-coherent modules, see Morphisms of Spaces, Lemma 67.11.2. 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 71.12.3. Hence it suffices to prove Z = Z' as closed subspaces of P.

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


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