Situation 71.11.1. Here $S$ is a scheme, $X$ is an algebraic space over $S$, and $\mathcal{A}$ is a quasi-coherent graded $\mathcal{O}_ X$-algebra.
71.11 Relative Proj
This section revisits the construction of the relative proj in the setting of algebraic spaces. The material in this section corresponds to the material in Constructions, Section 27.16 and Divisors, Section 31.30 in the case of schemes.
In Situation 71.11.1 we are going to define a functor $F : (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}$ which will turn out to be an algebraic space. We will follow (mutatis mutandis) the procedure of Constructions, Section 27.16. First, given a scheme $T$ over $S$ we define a quadruple over $T$ to be a system $(d, f : T \to X, \mathcal{L}, \psi )$
$d \geq 1$ is an integer,
$f : T \to X$ is a morphism over $S$,
$\mathcal{L}$ is an invertible $\mathcal{O}_ T$-module, and
$\psi : f^*\mathcal{A}^{(d)} \to \bigoplus _{n \geq 0}\mathcal{L}^{\otimes n}$ is a homomorphism of graded $\mathcal{O}_ T$-algebras such that $f^*\mathcal{A}_ d \to \mathcal{L}$ is surjective.
We say two quadruples $(d, f, \mathcal{L}, \psi )$ and $(d', f', \mathcal{L}', \psi ')$ are equivalent1 if and only if we have $f = f'$ and for some positive integer $m = ad = a'd'$ there exists an isomorphism $\beta : \mathcal{L}^{\otimes a} \to (\mathcal{L}')^{\otimes a'}$ with the property that $\beta \circ \psi |_{f^*\mathcal{A}^{(m)}}$ and $\psi '|_{f^*\mathcal{A}^{(m)}}$ agree as graded ring maps $f^*\mathcal{A}^{(m)} \to \bigoplus _{n \geq 0} (\mathcal{L}')^{\otimes mn}$. Given a quadruple $(d, f, \mathcal{L}, \psi )$ and a morphism $h : T' \to T$ we have the pullback $(d, f \circ h, h^*\mathcal{L}, h^*\psi )$. Pullback preserves the equivalence relation. Finally, for a quasi-compact scheme $T$ over $S$ we set
\[ F(T) = \text{the set of equivalence classes of quadruples over }T \]
and for an arbitrary scheme $T$ over $S$ we set
\[ F(T) = \mathop{\mathrm{lim}}\nolimits _{V \subset T\text{ quasi-compact open}} F(V). \]
In other words, an element $\xi $ of $F(T)$ corresponds to a compatible system of choices of elements $\xi _ V \in F(V)$ where $V$ ranges over the quasi-compact opens of $T$. Thus we have defined our functor
71.11.1.1
\begin{equation} \label{spaces-divisors-equation-proj} F : \mathit{Sch}^{opp} \longrightarrow \textit{Sets} \end{equation}
There is a morphism $F \to X$ of functors sending the quadruple $(d, f, \mathcal{L}, \psi )$ to $f$.
Lemma 71.11.2. In Situation 71.11.1. The functor $F$ above is an algebraic space. For any morphism $g : Z \to X$ where $Z$ is a scheme there is a canonical isomorphism $\underline{\text{Proj}}_ Z(g^*\mathcal{A}) = Z \times _ X F$ compatible with further base change.
Proof. It suffices to prove the second assertion, see Spaces, Lemma 65.11.3. Let $g : Z \to X$ be a morphism where $Z$ is a scheme. Let $F'$ be the functor of quadruples associated to the graded quasi-coherent $\mathcal{O}_ Z$-algebra $g^*\mathcal{A}$. Then there is a canonical isomorphism $F' = Z \times _ X F$, sending a quadruple $(d, f : T \to Z, \mathcal{L}, \psi )$ for $F'$ to $(d, g \circ f, \mathcal{L}, \psi )$ (details omitted, see proof of Constructions, Lemma 27.16.1). By Constructions, Lemmas 27.16.4, 27.16.5, and 27.16.6 and Definition 27.16.7 we see that $F'$ is representable by $\underline{\text{Proj}}_ Z(g^*\mathcal{A})$. $\square$
The lemma above tells us the following definition makes sense.
Definition 71.11.3. 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 71.11.2. We denote it $\pi : \underline{\text{Proj}}_ X(\mathcal{A}) \to X$.
In particular the structure morphism of the relative Proj is representable by construction. We can also think about the relative Proj via glueing. Let $\varphi : U \to X$ be a surjective étale morphism, where $U$ is a scheme. Set $R = U \times _ X U$ with projection morphisms $s, t : R \to U$. By Lemma 71.11.2 there exists a canonical isomorphism
\[ \gamma : \underline{\text{Proj}}_ U(\varphi ^*\mathcal{A}) \longrightarrow \underline{\text{Proj}}_ X(\mathcal{A}) \times _ X U \]
over $U$. Let $\alpha : t^*\varphi ^*\mathcal{A} \to s^*\varphi ^*\mathcal{A}$ be the canonical isomorphism of Properties of Spaces, Proposition 66.32.1. Then the diagram
\[ \xymatrix{ & \underline{\text{Proj}}_ U(\varphi ^*\mathcal{A}) \times _{U, s} R \ar@{=}[r] & \underline{\text{Proj}}_ R(s^*\varphi ^*\mathcal{A}) \ar[dd]_{\text{induced by }\alpha } \\ \underline{\text{Proj}}_ X(\mathcal{A}) \times _ X R \ar[ru]_{s^*\gamma } \ar[rd]^{t^*\gamma } \\ & \underline{\text{Proj}}_ U(\varphi ^*\mathcal{A}) \times _{U, t} R \ar@{=}[r] & \underline{\text{Proj}}_ R(t^*\varphi ^*\mathcal{A}) } \]
is commutative (the equal signs come from Constructions, Lemma 27.16.10). Thus, if we denote $\mathcal{A}_ U$, $\mathcal{A}_ R$ the pullback of $\mathcal{A}$ to $U$, $R$, then $P = \underline{\text{Proj}}_ X(\mathcal{A})$ has an étale covering by the scheme $P_ U = \underline{\text{Proj}}_ U(\mathcal{A}_ U)$ and $P_ U \times _ P P_ U$ is equal to $P_ R = \underline{\text{Proj}}_ R(\mathcal{A}_ R)$. Using these remarks we can argue in the usual fashion using étale localization to transfer results on the relative proj from the case of schemes to the case of algebraic spaces.
Lemma 71.11.4. In Situation 71.11.1. The relative Proj comes equipped with a quasi-coherent sheaf of $\mathbf{Z}$-graded algebras $\bigoplus _{n \in \mathbf{Z}} \mathcal{O}_{\underline{\text{Proj}}_ X(\mathcal{A})}(n)$ and a canonical homomorphism of graded algebras whose base change to any scheme over $X$ agrees with Constructions, Lemma 27.15.5.
\[ \psi : \pi ^*\mathcal{A} \longrightarrow \bigoplus \nolimits _{n \geq 0} \mathcal{O}_{\underline{\text{Proj}}_ X(\mathcal{A})}(n) \]
Proof. As in the discussion following Definition 71.11.3 choose a scheme $U$ and a surjective étale morphism $U \to X$, set $R = U \times _ X U$ with projections $s, t : R \to U$, $\mathcal{A}_ U = \mathcal{A}|_ U$, $\mathcal{A}_ R = \mathcal{A}|_ R$, and $\pi : P = \underline{\text{Proj}}_ X(\mathcal{A}) \to X$, $\pi _ U : P_ U = \underline{\text{Proj}}_ U(\mathcal{A}_ U)$ and $\pi _ R : P_ R = \underline{\text{Proj}}_ U(\mathcal{A}_ R)$. By the Constructions, Lemma 27.15.5 we have a quasi-coherent sheaf of $\mathbf{Z}$-graded $\mathcal{O}_{P_ U}$-algebras $\bigoplus _{n \in \mathbf{Z}} \mathcal{O}_{P_ U}(n)$ and a canonical map $\psi _ U : \pi _ U^*\mathcal{A}_ U \to \bigoplus _{n \geq 0} \mathcal{O}_{P_ U}(n)$ and similarly for $P_ R$. By Constructions, Lemma 27.16.10 the pullback of $\mathcal{O}_{P_ U}(n)$ and $\psi _ U$ by either projection $P_ R \to P_ U$ is equal to $\mathcal{O}_{P_ R}(n)$ and $\psi _ R$. By Properties of Spaces, Proposition 66.32.1 we obtain $\mathcal{O}_{P}(n)$ and $\psi $. We omit the verification of compatibility with pullback to arbitrary schemes over $X$. $\square$
Having constructed the relative Proj we turn to some basic properties.
Lemma 71.11.5. Let $S$ be a scheme. Let $g : X' \to X$ be a morphism of algebraic spaces over $S$ and let $\mathcal{A}$ be a quasi-coherent sheaf of graded $\mathcal{O}_ X$-algebras. Then there is a canonical isomorphism as well as a corresponding isomorphism of $\mathbf{Z}$-graded $\mathcal{O}_{\underline{\text{Proj}}_{X'}(g^*\mathcal{A})}$-algebras.
\[ r : \underline{\text{Proj}}_{X'}(g^*\mathcal{A}) \longrightarrow X' \times _ X \underline{\text{Proj}}_ X(\mathcal{A}) \]
\[ \theta : r^*\text{pr}_2^*\left(\bigoplus \nolimits _{d \in \mathbf{Z}} \mathcal{O}_{\underline{\text{Proj}}_ X(\mathcal{A})}(d)\right) \longrightarrow \bigoplus \nolimits _{d \in \mathbf{Z}} \mathcal{O}_{\underline{\text{Proj}}_{X'}(g^*\mathcal{A})}(d) \]
Proof. Let $F$ be the functor (71.11.1.1) and let $F'$ be the corresponding functor defined using $g^*\mathcal{A}$ on $X'$. We claim there is a canonical isomorphism $r : F' \to X' \times _ X F$ of functors (and of course $r$ is the isomorphism of the lemma). It suffices to construct the bijection $r : F'(T) \to X'(T) \times _{X(T)} F(T)$ for quasi-compact schemes $T$ over $S$. First, if $\xi = (d', f', \mathcal{L}', \psi ')$ is a quadruple over $T$ for $F'$, then we can set $r(\xi ) = (f', (d', g \circ f', \mathcal{L}', \psi '))$. This makes sense as $(g \circ f')^*\mathcal{A}^{(d)} = (f')^*(g^*\mathcal{A})^{(d)}$. The inverse map sends the pair $(f', (d, f, \mathcal{L}, \psi ))$ to the quadruple $(d, f', \mathcal{L}, \psi )$. We omit the proof of the final assertion (hint: reduce to the case of schemes by étale localization and apply Constructions, Lemma 27.16.10). $\square$
Lemma 71.11.6. In Situation 71.11.1 the morphism $\pi : \underline{\text{Proj}}_ X(\mathcal{A}) \to X$ is separated.
Proof. By Morphisms of Spaces, Lemma 67.4.12 and the construction of the relative Proj this follows from the case of schemes which is Constructions, Lemma 27.16.9. $\square$
Lemma 71.11.7. In Situation 71.11.1. If one of the following holds
$\mathcal{A}$ is of finite type as a sheaf of $\mathcal{A}_0$-algebras,
$\mathcal{A}$ is generated by $\mathcal{A}_1$ as an $\mathcal{A}_0$-algebra and $\mathcal{A}_1$ is a finite type $\mathcal{A}_0$-module,
there exists a finite type quasi-coherent $\mathcal{A}_0$-submodule $\mathcal{F} \subset \mathcal{A}_{+}$ such that $\mathcal{A}_{+}/\mathcal{F}\mathcal{A}$ is a locally nilpotent sheaf of ideals of $\mathcal{A}/\mathcal{F}\mathcal{A}$,
then $\pi : \underline{\text{Proj}}_ X(\mathcal{A}) \to X$ is quasi-compact.
Proof. By Morphisms of Spaces, Lemma 67.8.8 and the construction of the relative Proj this follows from the case of schemes which is Divisors, Lemma 31.30.1. $\square$
Lemma 71.11.8. In Situation 71.11.1. If $\mathcal{A}$ is of finite type as a sheaf of $\mathcal{O}_ X$-algebras, then $\pi : \underline{\text{Proj}}_ X(\mathcal{A}) \to X$ is of finite type.
Proof. By Morphisms of Spaces, Lemma 67.23.4 and the construction of the relative Proj this follows from the case of schemes which is Divisors, Lemma 31.30.2. $\square$
Lemma 71.11.9. In Situation 71.11.1. If $\mathcal{O}_ X \to \mathcal{A}_0$ is an integral algebra map2 and $\mathcal{A}$ is of finite type as an $\mathcal{A}_0$-algebra, then $\pi : \underline{\text{Proj}}_ X(\mathcal{A}) \to X$ is universally closed.
Proof. By Morphisms of Spaces, Lemma 67.9.5 and the construction of the relative Proj this follows from the case of schemes which is Divisors, Lemma 31.30.3. $\square$
Lemma 71.11.10. In Situation 71.11.1. The following conditions are equivalent
$\mathcal{A}_0$ is a finite type $\mathcal{O}_ X$-module and $\mathcal{A}$ is of finite type as an $\mathcal{A}_0$-algebra,
$\mathcal{A}_0$ is a finite type $\mathcal{O}_ X$-module and $\mathcal{A}$ is of finite type as an $\mathcal{O}_ X$-algebra.
If these conditions hold, then $\pi : \underline{\text{Proj}}_ X(\mathcal{A}) \to X$ is proper.
Proof. By Morphisms of Spaces, Lemma 67.40.2 and the construction of the relative Proj this follows from the case of schemes which is Divisors, Lemma 31.30.3. $\square$
Lemma 71.11.11. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{A}$ be a quasi-coherent sheaf of graded $\mathcal{O}_ X$-modules generated as an $\mathcal{A}_0$-algebra by $\mathcal{A}_1$. With $P = \underline{\text{Proj}}_ X(\mathcal{A})$ we have
$P$ represents the functor $F_1$ which associates to $T$ over $S$ the set of isomorphism classes of triples $(f, \mathcal{L}, \psi )$, where $f : T \to X$ is a morphism over $S$, $\mathcal{L}$ is an invertible $\mathcal{O}_ T$-module, and $\psi : f^*\mathcal{A} \to \bigoplus _{n \geq 0} \mathcal{L}^{\otimes n}$ is a map of graded $\mathcal{O}_ T$-algebras inducing a surjection $f^*\mathcal{A}_1 \to \mathcal{L}$,
the canonical map $\pi ^*\mathcal{A}_1 \to \mathcal{O}_ P(1)$ is surjective, and
each $\mathcal{O}_ P(n)$ is invertible and the multiplication maps induce isomorphisms $\mathcal{O}_ P(n) \otimes _{\mathcal{O}_ P} \mathcal{O}_ P(m) = \mathcal{O}_ P(n + m)$.
Proof. Omitted. See Constructions, Lemma 27.16.11 for the case of schemes. $\square$
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