71.16 Closed subspaces of relative proj
Some auxiliary lemmas about closed subspaces of relative proj. This section is the analogue of Divisors, Section 31.31.
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
In case the closed subspace is locally cut out by finitely many equations we can define it by a finite type ideal sheaf of \mathcal{A}.
Lemma 71.16.2. Let S be a scheme. Let X be a quasi-compact and quasi-separated 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 subscheme. If \pi is quasi-compact and i of finite presentation, then there exists a d > 0 and a quasi-coherent finite type \mathcal{O}_ X-submodule \mathcal{F} \subset \mathcal{A}_ d such that Z = \underline{\text{Proj}}_ X(\mathcal{A}/\mathcal{F}\mathcal{A}).
Proof.
The reader can redo the arguments used in the case of schemes. However, we will show the lemma follows from the case of schemes by a trick. Let \mathcal{I} \subset \mathcal{A} be the quasi-coherent graded ideal cutting out Z of Lemma 71.16.1. Choose an affine scheme U and a surjective étale morphism U \to X, see Properties of Spaces, Lemma 66.6.3. By the case of schemes (Divisors, Lemma 31.31.4) there exists a d > 0 and a quasi-coherent finite type \mathcal{O}_ U-submodule \mathcal{F}' \subset \mathcal{I}_ d|_ U \subset \mathcal{A}_ d|_ U such that Z \times _ X U is equal to \underline{\text{Proj}}_ U(\mathcal{A}|_ U/\mathcal{F}'\mathcal{A}|_ U). By Limits of Spaces, Lemma 70.9.2 we can find a finite type quasi-coherent submodule \mathcal{F} \subset \mathcal{I}_ d such that \mathcal{F}' \subset \mathcal{F}|_ U. Let Z' = \underline{\text{Proj}}_ X(\mathcal{A}/\mathcal{F}\mathcal{A}). Then Z' \to P is a closed immersion (Lemma 71.12.5) and Z \subset Z' as \mathcal{F}\mathcal{A} \subset \mathcal{I}. On the other hand, Z' \times _ X U \subset Z \times _ X U by our choice of \mathcal{F}. Thus Z = Z' as desired.
\square
Lemma 71.16.3. Let S be a scheme. Let X be a quasi-compact and quasi-separated 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 X be a closed subspace. Let U \subset X be an open. Assume that
\pi is quasi-compact,
i of finite presentation,
|U| \cap |\pi |(|i|(|Z|)) = \emptyset ,
U is quasi-compact,
\mathcal{A}_ n is a finite type \mathcal{O}_ X-module for all n.
Then there exists a d > 0 and a quasi-coherent finite type \mathcal{O}_ X-submodule \mathcal{F} \subset \mathcal{A}_ d with (a) Z = \underline{\text{Proj}}_ X(\mathcal{A}/\mathcal{F}\mathcal{A}) and (b) the support of \mathcal{A}_ d/\mathcal{F} is disjoint from U.
Proof.
We use the same trick as in the proof of Lemma 71.16.2 to reduce to the case of schemes. Let \mathcal{I} \subset \mathcal{A} be the quasi-coherent graded ideal cutting out Z of Lemma 71.16.1. Choose an affine scheme W and a surjective étale morphism W \to X, see Properties of Spaces, Lemma 66.6.3. By the case of schemes (Divisors, Lemma 31.31.5) there exists a d > 0 and a quasi-coherent finite type \mathcal{O}_ W-submodule \mathcal{F}' \subset \mathcal{I}_ d|_ W \subset \mathcal{A}_ d|_ W such that (a) Z \times _ X W is equal to \underline{\text{Proj}}_ W(\mathcal{A}|_ W/\mathcal{F}'\mathcal{A}|_ W) and (b) the support of \mathcal{A}_ d|_ W/\mathcal{F}' is disjoint from U \times _ X W. By Limits of Spaces, Lemma 70.9.2 we can find a finite type quasi-coherent submodule \mathcal{F} \subset \mathcal{I}_ d such that \mathcal{F}' \subset \mathcal{F}|_ W. Let Z' = \underline{\text{Proj}}_ X(\mathcal{A}/\mathcal{F}\mathcal{A}). Then Z' \to P is a closed immersion (Lemma 71.12.5) and Z \subset Z' as \mathcal{F}\mathcal{A} \subset \mathcal{I}. On the other hand, Z' \times _ X W \subset Z \times _ X W by our choice of \mathcal{F}. Thus Z = Z'. Finally, we see that \mathcal{A}_ d/\mathcal{F} is supported on X \setminus U as \mathcal{A}_ d|_ W/\mathcal{F}|_ W is a quotient of \mathcal{A}_ d|_ W/\mathcal{F}' which is supported on W \setminus U \times _ X W. Thus the lemma follows.
\square
Lemma 71.16.4. Let S be a scheme and let X be an algebraic space over S. Let \mathcal{E} be a quasi-coherent \mathcal{O}_ X-module. There is a bijection
\left\{ \begin{matrix} \text{sections }\sigma \text{ of the }
\\ \text{morphism } \mathbf{P}(\mathcal{E}) \to X
\end{matrix} \right\} \leftrightarrow \left\{ \begin{matrix} \text{surjections }\mathcal{E} \to \mathcal{L}\text{ where}
\\ \mathcal{L}\text{ is an invertible }\mathcal{O}_ X\text{-module}
\end{matrix} \right\}
In this case \sigma is a closed immersion and there is a canonical isomorphism
\mathop{\mathrm{Ker}}(\mathcal{E} \to \mathcal{L}) \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes -1} \longrightarrow \mathcal{C}_{\sigma (X)/\mathbf{P}(\mathcal{E})}
Both the bijection and isomorphism are compatible with base change.
Proof.
Because the constructions are compatible with base change, it suffices to check the statement étale locally on X. Thus we may assume X is a scheme and the result is Divisors, Lemma 31.31.6.
\square
Comments (0)