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 \input{preamble} % OK, start here. % \begin{document} \title{Divisors on Algebraic Spaces} \maketitle \phantomsection \label{section-phantom} \tableofcontents \section{Introduction} \label{section-introduction} \noindent In this chapter we study divisors on algebraic spaces and related topics. A basic reference for algebraic spaces is \cite{Kn}. \section{Associated and weakly associated points} \label{section-associated} \noindent In the case of schemes we have introduced two competing notions of associated points. Namely, the usual associated points (Divisors, Section \ref{divisors-section-associated}) and the weakly associated points (Divisors, Section \ref{divisors-section-weakly-associated}). For a general algebraic space the notion of an associated point is basically useless and we don't even bother to introduce it. If the algebraic space is locally Noetherian, then we allow ourselves to use the phrase associated point'' instead of weakly associated point'' as the notions are the same for Noetherian schemes (Divisors, Lemma \ref{divisors-lemma-ass-weakly-ass}). Before we make our definition, we need a lemma. \begin{lemma} \label{lemma-associated} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $x \in |X|$. The following are equivalent \begin{enumerate} \item for some \'etale morphism $f : U \to X$ with $U$ a scheme and $u \in U$ mapping to $x$, the point $u$ is weakly associated to $f^*\mathcal{F}$, \item for every \'etale morphism $f : U \to X$ with $U$ a scheme and $u \in U$ mapping to $x$, the point $u$ is weakly associated to $f^*\mathcal{F}$, \item the maximal ideal of $\mathcal{O}_{X, \overline{x}}$ is a weakly associated prime of the stalk $\mathcal{F}_{\overline{x}}$. \end{enumerate} If $X$ is locally Noetherian, then these are also equivalent to \begin{enumerate} \item[(4)] for some \'etale morphism $f : U \to X$ with $U$ a scheme and $u \in U$ mapping to $x$, the point $u$ is associated to $f^*\mathcal{F}$, \item[(5)] for every \'etale morphism $f : U \to X$ with $U$ a scheme and $u \in U$ mapping to $x$, the point $u$ is associated to $f^*\mathcal{F}$, \item[(6)] the maximal ideal of $\mathcal{O}_{X, \overline{x}}$ is an associated prime of the stalk $\mathcal{F}_{\overline{x}}$. \end{enumerate} \end{lemma} \begin{proof} Choose a scheme $U$ with a point $u$ and an \'etale morphism $f : U \to X$ mapping $u$ to $x$. Lift $\overline{x}$ to a geometric point of $U$ over $u$. Recall that $\mathcal{O}_{X, \overline{x}} = \mathcal{O}_{U, u}^{sh}$ where the strict henselization is with respect to our chosen lift of $\overline{x}$, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-describe-etale-local-ring}. Finally, we have $$\mathcal{F}_{\overline{x}} = (f^*\mathcal{F})_u \otimes_{\mathcal{O}_{U, u}} \mathcal{O}_{X, \overline{x}} = (f^*\mathcal{F})_u \otimes_{\mathcal{O}_{U, u}} \mathcal{O}_{U, u}^{sh}$$ by Properties of Spaces, Lemma \ref{spaces-properties-lemma-stalk-quasi-coherent}. Hence the equivalence of (1), (2), and (3) follows from More on Flatness, Lemma \ref{flat-lemma-weakly-associated-henselization}. If $X$ is locally Noetherian, then any $U$ as above is locally Noetherian, hence we see that (1), resp.\ (2) are equivalent to (4), resp.\ (5) by Divisors, Lemma \ref{divisors-lemma-ass-weakly-ass}. On the other hand, in the locally Noetherian case the local ring $\mathcal{O}_{X, \overline{x}}$ is Noetherian too (Properties of Spaces, Lemma \ref{spaces-properties-lemma-Noetherian-local-ring-Noetherian}). Hence the equivalence of (3) and (6) by the same lemma (or by Algebra, Lemma \ref{algebra-lemma-ass-weakly-ass}). \end{proof} \begin{definition} \label{definition-weakly-associated} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $x \in |X|$. \begin{enumerate} \item We say $x$ is {\it weakly associated} to $\mathcal{F}$ if the equivalent conditions (1), (2), and (3) of Lemma \ref{lemma-associated} are satisfied. \item We denote $\text{WeakAss}(\mathcal{F})$ the set of weakly associated points of $\mathcal{F}$. \item The {\it weakly associated points of $X$} are the weakly associated points of $\mathcal{O}_X$. \end{enumerate} If $X$ is locally Noetherian we will say {\it $x$ is associated to $\mathcal{F}$} if and only if $x$ is weakly associated to $\mathcal{F}$ and we set $\text{Ass}(\mathcal{F}) = \text{WeakAss}(\mathcal{F})$. Finally (still assuming $X$ is locally Noetherian), we will say {\it $x$ is an associated point of $X$} if and only if $x$ is a weakly associated point of $X$. \end{definition} \noindent At this point we can prove the obligatory lemmas. \begin{lemma} \label{lemma-weakly-ass-support} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Then $\text{WeakAss}(\mathcal{F}) \subset \text{Supp}(\mathcal{F})$. \end{lemma} \begin{proof} This is immediate from the definitions. The support of an abelian sheaf on $X$ is defined in Properties of Spaces, Definition \ref{spaces-properties-definition-support}. \end{proof} \begin{lemma} \label{lemma-ses-weakly-ass} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$ be a short exact sequence of quasi-coherent sheaves on $X$. Then $\text{WeakAss}(\mathcal{F}_2) \subset \text{WeakAss}(\mathcal{F}_1) \cup \text{WeakAss}(\mathcal{F}_3)$ and $\text{WeakAss}(\mathcal{F}_1) \subset \text{WeakAss}(\mathcal{F}_2)$. \end{lemma} \begin{proof} For every geometric point $\overline{x} \in X$ the sequence of stalks $0 \to \mathcal{F}_{1, \overline{x}} \to \mathcal{F}_{2, \overline{x}} \to \mathcal{F}_{3, \overline{x}} \to 0$ is a short exact sequence of $\mathcal{O}_{X, \overline{x}}$-modules. Hence the lemma follows from Algebra, Lemma \ref{algebra-lemma-weakly-ass}. \end{proof} \begin{lemma} \label{lemma-weakly-ass-zero} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Then $$\mathcal{F} = (0) \Leftrightarrow \text{WeakAss}(\mathcal{F}) = \emptyset$$ \end{lemma} \begin{proof} Choose a scheme $U$ and a surjective \'etale morphism $f : U \to X$. Then $\mathcal{F}$ is zero if and only if $f^*\mathcal{F}$ is zero. Hence the lemma follows from the definition and the lemma in the case of schemes, see Divisors, Lemma \ref{divisors-lemma-weakly-ass-zero}. \end{proof} \begin{lemma} \label{lemma-minimal-support-in-weakly-ass} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $x \in |X|$. If \begin{enumerate} \item $x \in \text{Supp}(\mathcal{F})$ \item $x$ is a codimension $0$ point of $X$ (Properties of Spaces, Definition \ref{spaces-properties-definition-dimension-local-ring}). \end{enumerate} Then $x \in \text{WeakAss}(\mathcal{F})$. If $\mathcal{F}$ is a finite type $\mathcal{O}_X$-module with scheme theoretic support $Z$ (Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-scheme-theoretic-support}) and $x$ is a codimension $0$ point of $Z$, then $x \in \text{WeakAss}(\mathcal{F})$. \end{lemma} \begin{proof} Since $x \in \text{Supp}(\mathcal{F})$ the stalk $\mathcal{F}_{\overline{x}}$ is not zero. Hence $\text{WeakAss}(\mathcal{F}_{\overline{x}})$ is nonempty by Algebra, Lemma \ref{algebra-lemma-weakly-ass-zero}. On the other hand, the spectrum of $\mathcal{O}_{X, \overline{x}}$ is a singleton. Hence $x$ is a weakly associated point of $\mathcal{F}$ by definition. The final statement follows as $\mathcal{O}_{X, \overline{x}} \to \mathcal{O}_{Z, \overline{z}}$ is a surjection, the spectrum of $\mathcal{O}_{Z, \overline{z}}$ is a singleton, and $\mathcal{F}_{\overline{x}}$ is a nonzero module over $\mathcal{O}_{Z, \overline{z}}$. \end{proof} \begin{lemma} \label{lemma-minimal-support-in-weakly-ass-decent} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $x \in |X|$. If \begin{enumerate} \item $X$ is decent (for example quasi-separated or locally separated), \item $x \in \text{Supp}(\mathcal{F})$ \item $x$ is not a specialization of another point in $\text{Supp}(\mathcal{F})$. \end{enumerate} Then $x \in \text{WeakAss}(\mathcal{F})$. \end{lemma} \begin{proof} (A quasi-separated algebraic space is decent, see Decent Spaces, Section \ref{decent-spaces-section-reasonable-decent}. A locally separated algebraic space is decent, see Decent Spaces, Lemma \ref{decent-spaces-lemma-locally-separated-decent}.) Choose a scheme $U$, a point $u \in U$, and an \'etale morphism $f : U \to X$ mapping $u$ to $x$. By Decent Spaces, Lemma \ref{decent-spaces-lemma-decent-no-specializations-map-to-same-point} if $u' \leadsto u$ is a nontrivial specialization, then $f(u') \not = x$. Hence we see that $u \in \text{Supp}(f^*\mathcal{F})$ is not a specialization of another point of $\text{Supp}(f^*\mathcal{F})$. Hence $u \in \text{WeakAss}(f^*\mathcal{F})$ by Divisors, Lemma \ref{lemma-minimal-support-in-weakly-ass}. \end{proof} \begin{lemma} \label{lemma-finite-ass} Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. Then $\text{Ass}(\mathcal{F}) \cap W$ is finite for every quasi-compact open $W \subset |X|$. \end{lemma} \begin{proof} Choose a quasi-compact scheme $U$ and an \'etale morphism $U \to X$ such that $W$ is the image of $|U| \to |X|$. Then $U$ is a Noetherian scheme and we may apply Divisors, Lemma \ref{divisors-lemma-finite-ass} to conclude. \end{proof} \begin{lemma} \label{lemma-restriction-injective-open-contains-weakly-ass} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. If $U \to X$ is an \'etale morphism such that $\text{WeakAss}(\mathcal{F}) \subset \Im(|U| \to |X|)$, then $\Gamma(X, \mathcal{F}) \to \Gamma(U, \mathcal{F})$ is injective. \end{lemma} \begin{proof} Let $s \in \Gamma(X, \mathcal{F})$ be a section which restricts to zero on $U$. Let $\mathcal{F}' \subset \mathcal{F}$ be the image of the map $\mathcal{O}_X \to \mathcal{F}$ defined by $s$. Then $\mathcal{F}'|_U = 0$. This implies that $\text{WeakAss}(\mathcal{F}') \cap \Im(|U| \to |X|) = \emptyset$ (by the definition of weakly associated points). On the other hand, $\text{WeakAss}(\mathcal{F}') \subset \text{WeakAss}(\mathcal{F})$ by Lemma \ref{lemma-ses-weakly-ass}. We conclude $\text{Ass}(\mathcal{F}') = \emptyset$. Hence $\mathcal{F}' = 0$ by Lemma \ref{lemma-weakly-ass-zero}. \end{proof} \begin{lemma} \label{lemma-weakass-pushforward} Let $S$ be a scheme. Let $f : X \to Y$ be a quasi-compact and quasi-separated morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $y \in |Y|$ be a point which is not in the image of $|f|$. Then $y$ is not weakly associated to $f_*\mathcal{F}$. \end{lemma} \begin{proof} By Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-pushforward} the $\mathcal{O}_Y$-module $f_*\mathcal{F}$ is quasi-coherent hence the lemma makes sense. Choose an affine scheme $V$, a point $v \in V$, and an \'etale morphism $V \to Y$ mapping $v$ to $y$. We may replace $f : X \to Y$, $\mathcal{F}$, $y$ by $X \times_Y V \to V$, $\mathcal{F}|_{X \times_Y V}$, $v$. Thus we may assume $Y$ is an affine scheme. In this case $X$ is quasi-compact, hence we can choose an affine scheme $U$ and a surjective \'etale morphism $U \to X$. Denote $g : U \to Y$ the composition. Then $f_*\mathcal{F} \subset g_*(\mathcal{F}|_U)$. By Lemma \ref{lemma-ses-weakly-ass} we reduce to the case of schemes which is Divisors, Lemma \ref{divisors-lemma-weakass-pushforward}. \end{proof} \begin{lemma} \label{lemma-check-injective-on-weakass} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a map of quasi-coherent $\mathcal{O}_X$-modules. Assume that for every $x \in |X|$ at least one of the following happens \begin{enumerate} \item $\mathcal{F}_{\overline{x}} \to \mathcal{G}_{\overline{x}}$ is injective, or \item $x \not \in \text{WeakAss}(\mathcal{F})$. \end{enumerate} Then $\varphi$ is injective. \end{lemma} \begin{proof} The assumptions imply that $\text{WeakAss}(\Ker(\varphi)) = \emptyset$ and hence $\Ker(\varphi) = 0$ by Lemma \ref{lemma-weakly-ass-zero}. \end{proof} \section{Morphisms and weakly associated points} \label{section-morphisms-weakly-associated} \begin{lemma} \label{lemma-weakly-ass-reverse-functorial} Let $S$ be a scheme. Let $f : X \to Y$ be an affine morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Then we have $$\text{WeakAss}_S(f_*\mathcal{F}) \subset f(\text{WeakAss}_X(\mathcal{F}))$$ \end{lemma} \begin{proof} Choose a scheme $V$ and a surjective \'etale morphism $V \to Y$. Set $U = X \times_Y V$. Then $U \to V$ is an affine morphism of schemes. By our definition of weakly associated points the problem is reduced to the morphism of schemes $U \to V$. This case is treated in Divisors, Lemma \ref{divisors-lemma-weakly-ass-reverse-functorial}. \end{proof} \begin{lemma} \label{lemma-ass-functorial-equal} Let $S$ be a scheme. Let $f : X \to Y$ be an affine morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. If $X$ is locally Noetherian, then we have $$\text{WeakAss}_Y(f_*\mathcal{F}) = f(\text{WeakAss}_X(\mathcal{F}))$$ \end{lemma} \begin{proof} Choose a scheme $V$ and a surjective \'etale morphism $V \to Y$. Set $U = X \times_Y V$. Then $U \to V$ is an affine morphism of schemes and $U$ is locally Noetherian. By our definition of weakly associated points the problem is reduced to the morphism of schemes $U \to V$. This case is treated in Divisors, Lemma \ref{divisors-lemma-ass-functorial-equal}. \end{proof} \begin{lemma} \label{lemma-weakly-associated-finite} Let $S$ be a scheme. Let $f : X \to Y$ be a finite morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Then $\text{WeakAss}(f_*\mathcal{F}) = f(\text{WeakAss}(\mathcal{F}))$. \end{lemma} \begin{proof} Choose a scheme $V$ and a surjective \'etale morphism $V \to Y$. Set $U = X \times_Y V$. Then $U \to V$ is a finite morphism of schemes. By our definition of weakly associated points the problem is reduced to the morphism of schemes $U \to V$. This case is treated in Divisors, Lemma \ref{divisors-lemma-weakly-associated-finite}. \end{proof} \begin{lemma} \label{lemma-weakly-ass-pullback} Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}_Y$-module. Let $x \in |X|$ and $y = f(x) \in |Y|$. If \begin{enumerate} \item $y \in \text{WeakAss}_S(\mathcal{G})$, \item $f$ is flat at $x$, and \item the dimension of the local ring of the fibre of $f$ at $x$ is zero (Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-dimension-fibre}), \end{enumerate} then $x \in \text{WeakAss}(f^*\mathcal{G})$. \end{lemma} \begin{proof} Choose a scheme $V$, a point $v \in V$, and an \'etale morphism $V \to Y$ mapping $v$ to $y$. Choose a scheme $U$, a point $u \in U$, and an \'etale morphism $U \to V \times_Y X$ mapping $v$ to a point lying over $v$ and $x$. This is possible because there is a $t \in |V \times_Y X|$ mapping to $(v, y)$ by Properties of Spaces, Lemma \ref{spaces-properties-lemma-points-cartesian}. By definition we see that the dimension of $\mathcal{O}_{U_v, u}$ is zero. Hence $u$ is a generic point of the fiber $U_v$. By our definition of weakly associated points the problem is reduced to the morphism of schemes $U \to V$. This case is treated in Divisors, Lemma \ref{divisors-lemma-weakly-ass-pullback}. \end{proof} \begin{lemma} \label{lemma-weakly-ass-change-fields} Let $K/k$ be a field extension. Let $X$ be an algebraic space over $k$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $y \in X_K$ with image $x \in X$. If $y$ is a weakly associated point of the pullback $\mathcal{F}_K$, then $x$ is a weakly associated point of $\mathcal{F}$. \end{lemma} \begin{proof} This is the translation of Divisors, Lemma \ref{divisors-lemma-weakly-ass-change-fields} into the language of algebraic spaces. We omit the details of the translation. \end{proof} \begin{lemma} \label{lemma-finite-flat-weak-assassin-up-down} Let $S$ be a scheme. Let $f : X \to Y$ be a finite flat morphism of algebraic spaces. Let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}_Y$-module. Let $x \in |X|$ be a point with image $y \in |Y|$. Then $$x \in \text{WeakAss}(g^*\mathcal{G}) \Leftrightarrow y \in \text{WeakAss}(\mathcal{G})$$ \end{lemma} \begin{proof} Follows immediately from the case of schemes (More on Flatness, Lemma \ref{flat-lemma-finite-flat-weak-assassin-up-down}) by \'etale localization. \end{proof} \begin{lemma} \label{lemma-etale-weak-assassin-up-down} Let $S$ be a scheme. Let $f : X \to Y$ be an \'etale morphism of algebraic spaces. Let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}_Y$-module. Let $x \in |X|$ be a point with image $y \in |Y|$. Then $$x \in \text{WeakAss}(f^*\mathcal{G}) \Leftrightarrow y \in \text{WeakAss}(\mathcal{G})$$ \end{lemma} \begin{proof} This is immediate from the definition of weakly associated points and in fact the corresponding lemma for the case of schemes (More on Flatness, Lemma \ref{flat-lemma-etale-weak-assassin-up-down}) is the basis for our definition. \end{proof} \section{Relative weak assassin} \label{section-relative-weak-assassin} \noindent We need a couple of lemmas to define this gadget. \begin{lemma} \label{lemma-locally-noetherian-fibre} Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $y \in |Y|$. The following are equivalent \begin{enumerate} \item for some scheme $V$, point $v \in V$, and \'etale morphism $V \to Y$ mapping $v$ to $y$, the algebraic space $X_v$ is locally Noetherian, \item for every scheme $V$, point $v \in V$, and \'etale morphism $V \to Y$ mapping $v$ to $y$, the algebraic space $X_v$ is locally Noetherian, and \item there exists a field $k$ and a morphism $\Spec(k) \to Y$ representing $y$ such that $X_k$ is locally Noetherian. \end{enumerate} If there exists a field $k_0$ and a monomorphism $\Spec(k_0) \to Y$ representing $y$, then these are also equivalent to \begin{enumerate} \item[(4)] the algebraic space $X_{k_0}$ is locally Noetherian. \end{enumerate} \end{lemma} \begin{proof} Observe that $X_v = v \times_Y X = \Spec(\kappa(v)) \times_Y X$. Hence the implications (2) $\Rightarrow$ (1) $\Rightarrow$ (3) are clear. Assume that $\Spec(k) \to Y$ is a morphism from the spectrum of a field such that $X_k$ is locally Noetherian. Let $V \to Y$ be an \'etale morphism from a scheme $V$ and let $v \in V$ a point mapping to $y$. Then the scheme $v \times_Y \Spec(k)$ is nonempty. Choose a point $w \in v \times_Y \Spec(k)$. Consider the morphisms $$X_v \longleftarrow X_w \longrightarrow X_k$$ Since $V \to Y$ is \'etale and since $w$ may be viewed as a point of $V \times_Y \Spec(k)$, we see that $\kappa(w) \supset k$ is a finite separable extension of fields (Morphisms, Lemma \ref{morphisms-lemma-etale-over-field}). Thus $X_w \to X_k$ is a finite \'etale morphism as a base change of $w \to \Spec(k)$. Hence $X_w$ is locally Noetherian (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-locally-finite-type-locally-noetherian}). The morphism $X_w \to X_v$ is a surjective, affine, flat morphism as a base change of the surjective, affine, flat morphism $w \to v$. Then the fact that $X_w$ is locally Noetherian implies that $X_v$ is locally Noetherian. This can be seen by picking a surjective \'etale morphism $U \to X$ and then using that $U_w \to U_v$ is surjective, affine, and flat. Working affine locally on the scheme $U_v$ we conclude that $U_w$ is locally Noetherian by Algebra, Lemma \ref{algebra-lemma-descent-Noetherian}. \medskip\noindent Finally, it suffices to prove that (3) implies (4) in case we have a monomorphism $\Spec(k_0) \to Y$ in the class of $y$. Then $\Spec(k) \to Y$ factors as $\Spec(k) \to \Spec(k_0) \to Y$. The argument given above then shows that $X_k$ being locally Noetherian impies that $X_{k_0}$ is locally Noetherian. \end{proof} \begin{definition} \label{definition-locally-Noetherian-fibre} Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $y \in |Y|$. We say {\it the fibre of $f$ over $y$ is locally Noetherian} if the equivalent conditions (1), (2), and (3) of Lemma \ref{lemma-locally-noetherian-fibre} are satisfied. We say {\it the fibres of $f$ are locally Noetherian} if this holds for every $y \in |Y|$. \end{definition} \noindent Of course, the usual way to guarantee locally Noetherian fibres is to assume the morphism is locally of finite type. \begin{lemma} \label{lemma-locally-finite-type-locally-Noetherian-fibres} Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. If $f$ is locally of finite type, then the fibres of $f$ are locally Noetherian. \end{lemma} \begin{proof} This follows from Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-locally-finite-type-locally-noetherian} and the fact that the spectrum of a field is Noetherian. \end{proof} \begin{lemma} \label{lemma-relative-assassin} Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $x \in |X|$ and $y = f(x) \in |Y|$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Consider commutative diagrams $$\xymatrix{ X \ar[d] & X \times_Y V \ar[d] \ar[l] & X_v \ar[d] \ar[l] \\ Y & V \ar[l] & v \ar[l] } \quad \xymatrix{ X \ar[d] & U \ar[d] \ar[l] & U_v \ar[d] \ar[l] \\ Y & V \ar[l] & \ar[l] v } \quad \xymatrix{ x \ar@{|->}[d] & x' \ar@{|->}[d] \ar@{|->}[l] & u \ar@{|->}[ld] \ar@{|->}[l] \\ y & v \ar@{|->}[l] }$$ where $V$ and $U$ are schemes, $V \to Y$ and $U \to X \times_Y V$ are \'etale, $v \in V$, $x' \in |X_v|$, $u \in U$ are points related as in the last diagram. Denote $\mathcal{F}|_{X_v}$ and $\mathcal{F}|_{U_v}$ the pullbacks of $\mathcal{F}$. The following are equivalent \begin{enumerate} \item for some $V, v, x'$ as above $x'$ is a weakly associated point of $\mathcal{F}|_{X_v}$, \item for every $V \to Y, v, x'$ as above $x'$ is a weakly associated point of $\mathcal{F}|_{X_v}$, \item for some $U, V, u, v$ as above $u$ is a weakly associated point of $\mathcal{F}|_{U_v}$, \item for every $U, V, u, v$ as above $u$ is a weakly associated point of $\mathcal{F}|_{U_v}$, \item for some field $k$ and morphism $\Spec(k) \to Y$ representing $y$ and some $t \in |X_k|$ mapping to $x$, the point $t$ is a weakly associated point of $\mathcal{F}|_{X_k}$. \end{enumerate} If there exists a field $k_0$ and a monomorphism $\Spec(k_0) \to Y$ representing $y$, then these are also equivalent to \begin{enumerate} \item[(6)] $x_0$ is a weakly associated point of $\mathcal{F}|_{X_{k_0}}$ where $x_0 \in |X_{k_0}|$ is the unique point mapping to $x$. \end{enumerate} If the fibre of $f$ over $y$ is locally Noetherian, then in conditions (1), (2), (3), (4), and (6) we may replace weakly associated'' with associated''. \end{lemma} \begin{proof} Observe that given $V, v, x'$ as in the lemma we can find $U \to X \times_Y V$ and $u \in U$ mapping to $x'$ and then the morphism $U_v \to X_v$ is \'etale. Thus it is clear that (1) and (3) are equivalent as well as (2) and (4). Each of these implies (5). We will show that (5) implies (2). Suppose given $V, v, x'$ as well as $\Spec(k) \to X$ and $t \in |X_k|$ such that the point $t$ is a weakly associated point of $\mathcal{F}|_{X_k}$. We can choose a point $w \in v \times_Y \Spec(k)$. Then we obtain the morphisms $$X_v \longleftarrow X_w \longrightarrow X_k$$ Since $V \to Y$ is \'etale and since $w$ may be viewed as a point of $V \times_Y \Spec(k)$, we see that $\kappa(w) \supset k$ is a finite separable extension of fields (Morphisms, Lemma \ref{morphisms-lemma-etale-over-field}). Thus $X_w \to X_k$ is a finite \'etale morphism as a base change of $w \to \Spec(k)$. Thus any point $x''$ of $X_w$ lying over $t$ is a weakly associated point of $\mathcal{F}|_{X_w}$ by Lemma \ref{lemma-etale-weak-assassin-up-down}. We may pick $x''$ mapping to $x'$ (Properties of Spaces, Lemma \ref{spaces-properties-lemma-points-cartesian}). Then Lemma \ref{lemma-weakly-ass-change-fields} implies that $x'$ is a weakly associated point of $\mathcal{F}|_{X_v}$. \medskip\noindent To finish the proof it suffices to show that the equivalent conditions (1) -- (5) imply (6) if we are given $\Spec(k_0) \to Y$ as in (6). In this case the morphism $\Spec(k) \to Y$ of (5) factors uniquely as $\Spec(k) \to \Spec(k_0) \to Y$. Then $x_0$ is the image of $t$ under the morphism $X_k \to X_{k_0}$. Hence the same lemma as above shows that (6) is true. \end{proof} \begin{definition} \label{definition-relative-weak-assassin} Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. The {\it relative weak assassin of $\mathcal{F}$ in $X$ over $Y$} is the set $\text{WeakAss}_{X/Y}(\mathcal{F}) \subset |X|$ consisting of those $x \in |X|$ such that the equivalent conditions of Lemma \ref{lemma-relative-assassin} are satisfied. If the fibres of $f$ are locally Noetherian (Definition \ref{definition-locally-Noetherian-fibre}) then we use the notation $\text{Ass}_{X/Y}(\mathcal{F})$. \end{definition} \noindent With this notation we can formulate some of the results already proven for schemes. \begin{lemma} \label{lemma-bourbaki} Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}_Y$-module. Assume \begin{enumerate} \item $\mathcal{F}$ is flat over $Y$, \item $X$ and $Y$ are locally Noetherian, and \item the fibres of $f$ are locally Noetherian. \end{enumerate} Then $$\text{Ass}_X(\mathcal{F} \otimes_{\mathcal{O}_X} f^*\mathcal{G}) = \{x \in \text{Ass}_{X/Y}(\mathcal{F})\text{ such that } f(x) \in \text{Ass}_Y(\mathcal{G}) \}$$ \end{lemma} \begin{proof} Via \'etale localization, this is an immediate consequence of the result for schemes, see Divisors, Lemma \ref{divisors-lemma-bourbaki}. The result for schemes is more general only because we haven't defined associated points for non-Noetherian algebraic spaces (hence we need to assume $X$ and the fibres of $X \to Y$ are locally Noetherian to even be able to formulate this result). \end{proof} \begin{lemma} \label{lemma-base-change-relative-assassin} Let $S$ be a scheme. Let $$\xymatrix{ X' \ar[d]_{f'} \ar[r]_{g'} & X \ar[d]^f \\ Y' \ar[r]^g & Y }$$ be a cartesian diagram of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module and set $\mathcal{F}' = (g')^*\mathcal{F}$. If $f$ is locally of finite type, then \begin{enumerate} \item $x' \in \text{Ass}_{X'/Y'}(\mathcal{F}') \Rightarrow g'(x') \in \text{Ass}_{X/Y}(\mathcal{F})$ \item if $x \in \text{Ass}_{X/Y}(\mathcal{F})$, then given $y' \in |Y'|$ with $f(x) = g(y')$, there exists an $x' \in \text{Ass}_{X'/Y'}(\mathcal{F}')$ with $g'(x') = x$ and $f'(x') = y'$. \end{enumerate} \end{lemma} \begin{proof} This follows from the case of schemes by \'etale localization. We write out the details completely. Choose a scheme $V$ and a surjective \'etale morphism $V \to Y$. Choose a scheme $U$ and a surjective \'etale morphism $U \to V \times_Y X$. Choose a scheme $V'$ and a surjective \'etale morphism $V' \to V \times_Y Y'$. Then $U' = V' \times_V U$ is a scheme and the morphism $U' \to X'$ is surjective and \'etale. \medskip\noindent Proof of (1). Choose $u' \in U'$ mapping to $x'$. Denote $v' \in V'$ the image of $u'$. Then $x' \in \text{Ass}_{X'/Y'}(\mathcal{F}')$ is equivalent to $u' \in \text{Ass}(\mathcal{F}|_{U'_{v'}})$ by definition (writing $\text{Ass}$ instead of $\text{WeakAss}$ makes sense as $U'_{v'}$ is locally Noetherian). Applying Divisors, Lemma \ref{divisors-lemma-base-change-relative-assassin} we see that the image $u \in U$ of $u'$ is in $\text{Ass}(\mathcal{F}|_{U_v})$ where $v \in V$ is the image of $u$. This in turn means $g'(x') \in \text{Ass}_{X/Y}(\mathcal{F})$. \medskip\noindent Proof of (2). Choose $u \in U$ mapping to $x$. Denote $v \in V$ the image of $u$. Then $x \in \text{Ass}_{X/Y}(\mathcal{F})$ is equivalent to $u \in \text{Ass}(\mathcal{F}|_{U_v})$ by definition. Choose a point $v' \in V'$ mapping to $y' \in |Y'|$ and to $v \in V$ (possible by Properties of Spaces, Lemma \ref{spaces-properties-lemma-points-cartesian}). Let $t \in \Spec(\kappa(v') \otimes_{\kappa(v)} \kappa(u))$ be a generic point of an irreducible component. Let $u' \in U'$ be the image of $t$. Applying Divisors, Lemma \ref{divisors-lemma-base-change-relative-assassin} we see that $u' \in \text{Ass}(\mathcal{F}'|_{U'_{v'}})$. This in turn means $x' \in \text{Ass}_{X'/Y'}(\mathcal{F}')$ where $x' \in |X'|$ is the image of $u'$. \end{proof} \begin{lemma} \label{lemma-base-change-relative-assassin-quasi-finite} With notation and assumptions as in Lemma \ref{lemma-base-change-relative-assassin}. Assume $g$ is locally quasi-finite, or more generally that for every $y' \in |Y'|$ the transcendence degree of $y'/g(y')$ is $0$. Then $\text{Ass}_{X'/Y'}(\mathcal{F}')$ is the inverse image of $\text{Ass}_{X/Y}(\mathcal{F})$. \end{lemma} \begin{proof} The transcendence degree of a point over its image is defined in Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-dimension-fibre}. Let $x' \in |X'|$ with image $x \in |X|$. Choose a scheme $V$ and a surjective \'etale morphism $V \to Y$. Choose a scheme $U$ and a surjective \'etale morphism $U \to V \times_Y X$. Choose a scheme $V'$ and a surjective \'etale morphism $V' \to V \times_Y Y'$. Then $U' = V' \times_V U$ is a scheme and the morphism $U' \to X'$ is surjective and \'etale. Choose $u \in U$ mapping to $x$. Denote $v \in V$ the image of $u$. Then $x \in \text{Ass}_{X/Y}(\mathcal{F})$ is equivalent to $u \in \text{Ass}(\mathcal{F}|_{U_v})$ by definition. Choose a point $u' \in U'$ mapping to $x' \in |X'|$ and to $u \in U$ (possible by Properties of Spaces, Lemma \ref{spaces-properties-lemma-points-cartesian}). Let $v' \in V'$ be the image of $u'$. Then $x' \in \text{Ass}_{X'/Y'}(\mathcal{F}')$ is equivalent to $u' \in \text{Ass}(\mathcal{F}'|_{U'_{v'}})$ by definition. Now the lemma follows from the discussion in Divisors, Remark \ref{divisors-remark-base-change-relative-assassin} applied to $u' \in \Spec(\kappa(v') \otimes_{\kappa(v)} \kappa(u))$. \end{proof} \begin{lemma} \label{lemma-relative-weak-assassin-finite} Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $i : Z \to X$ be a finite morphism. Let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}_Z$-module. Then $\text{WeakAss}_{X/Y}(i_*\mathcal{G}) = i(\text{WeakAss}_{Z/Y}(\mathcal{G}))$. \end{lemma} \begin{proof} Follows from the case of schemes (Divisors, Lemma \ref{divisors-lemma-relative-weak-assassin-finite}) by \'etale localization. Details omitted. \end{proof} \begin{lemma} \label{lemma-relative-assassin-constructible} Let $Y$ be a scheme. Let $X$ be an algebraic space of finite presentation over $Y$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module of finite presentation. Let $U \subset X$ be an open subspace such that $U \to Y$ is quasi-compact. Then the set $$E = \{y \in Y \mid \text{Ass}_{X_y}(\mathcal{F}_y) \subset |U_y|\}$$ is locally constructible in $Y$. \end{lemma} \begin{proof} Note that since $Y$ is a scheme, it makes sense to take the fibres $X_y = \Spec(\kappa(y)) \times_Y X$. (Also, by our definitions, the set $\text{Ass}_{X_y}(\mathcal{F}_y)$ is exactly the fibre of $\text{Ass}_{X/Y}(\mathcal{F}) \to Y$ over $y$, but we won't need this.) The question is local on $Y$, indeed, we have to show that $E$ is constructible if $Y$ is affine. In this case $X$ is quasi-compact. Choose an affine scheme $W$ and a surjective \'etale morphism $\varphi : W \to X$. Then $\text{Ass}_{X_y}(\mathcal{F}_y)$ is the image of $\text{Ass}_{W_y}(\varphi^*\mathcal{F}_y)$ for all $y \in Y$. Hence the lemma follows from the case of schemes for the open $\varphi^{-1}(U) \subset W$ and the morphism $W \to Y$. The case of schemes is More on Morphisms, Lemma \ref{more-morphisms-lemma-relative-assassin-constructible}. \end{proof} \section{Fitting ideals} \label{section-fitting-ideals} \noindent This section is the continuation of the discussion in Divisors, Section \ref{divisors-section-fitting-ideals}. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a finite type, quasi-coherent $\mathcal{O}_X$-module. In this situation we can construct the Fitting ideals $$0 = \text{Fit}_{-1}(\mathcal{F}) \subset \text{Fit}_0(\mathcal{F}) \subset \text{Fit}_1(\mathcal{F}) \subset \ldots \subset \mathcal{O}_X$$ as the sequence of quasi-coherent sheaves ideals characterized by the following property: for every affine $U = \Spec(A)$ \'etale over $X$ if $\mathcal{F}|_U$ corresponds to the $A$-module $M$, then $\text{Fit}_i(\mathcal{F})|_U$ corresponds to the ideal $\text{Fit}_i(M) \subset A$. This is well defined and a quasi-coherent sheaf of ideals because if $A \to B$ is an \'etale ring map, then the $i$th Fitting ideal of $M \otimes_A B$ over $B$ is equal to $\text{Fit}_i(M) B$ by More on Algebra, Lemma \ref{more-algebra-lemma-fitting-ideal-basics} part (3). More precisely (perhaps), the existence of the quasi-coherent sheaves of ideals $\text{Fit}_0(\mathcal{O}_X)$ follows (for example) from the description of quasi-coherent sheaves in Properties of Spaces, Lemma \ref{spaces-properties-lemma-characterize-quasi-coherent-small-etale} and the pullback property given in Divisors, Lemma \ref{divisors-lemma-base-change-fitting-ideal}. \medskip\noindent The advantage of constructing the Fitting ideals in this way is that we see immediately that formation of Fitting ideals commutes with \'etale localization hence many properties of the Fitting ideals immediately reduce to the corresponding properties in the case of schemes. Often we will use the discussion in Properties of Spaces, Section \ref{spaces-properties-section-properties-modules} to do the translation between properties of quasi-coherent sheaves on schemes and on algebraic spaces. \begin{lemma} \label{lemma-base-change-fitting-ideal} Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a finite type, quasi-coherent $\mathcal{O}_Y$-module. Then $f^{-1}\text{Fit}_i(\mathcal{F}) \cdot \mathcal{O}_X = \text{Fit}_i(f^*\mathcal{F})$. \end{lemma} \begin{proof} Reduces to Divisors, Lemma \ref{divisors-lemma-base-change-fitting-ideal} by \'etale localization. \end{proof} \begin{lemma} \label{lemma-fitting-ideal-of-finitely-presented} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a finitely presented $\mathcal{O}_X$-module. Then $\text{Fit}_r(\mathcal{F})$ is a quasi-coherent ideal of finite type. \end{lemma} \begin{proof} Reduces to Divisors, Lemma \ref{divisors-lemma-fitting-ideal-of-finitely-presented} by \'etale localization. \end{proof} \begin{lemma} \label{lemma-on-subscheme-cut-out-by-Fit-0} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a finite type, quasi-coherent $\mathcal{O}_X$-module. Let $Z_0 \subset X$ be the closed subspace cut out by $\text{Fit}_0(\mathcal{F})$. Let $Z \subset X$ be the scheme theoretic support of $\mathcal{F}$. Then \begin{enumerate} \item $Z \subset Z_0 \subset X$ as closed subspaces, \item $|Z| = |Z_0| = \text{Supp}(\mathcal{F})$ as closed subsets of $|X|$, \item there exists a finite type, quasi-coherent $\mathcal{O}_{Z_0}$-module $\mathcal{G}_0$ with $$(Z_0 \to X)_*\mathcal{G}_0 = \mathcal{F}.$$ \end{enumerate} \end{lemma} \begin{proof} Recall that formation of $Z$ commutes with \'etale localization, see Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-scheme-theoretic-support} (which uses Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-scheme-theoretic-support} to define $Z$). Hence (1) and (2) follow from the case of schemes, see Divisors, Lemma \ref{divisors-lemma-on-subscheme-cut-out-by-Fit-0}. To get $\mathcal{G}_0$ as in part (3) we can use that we have $\mathcal{G}$ on $Z$ as in Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-scheme-theoretic-support} and set $\mathcal{G}_0 = (Z \to Z_0)_*\mathcal{G}$. \end{proof} \begin{lemma} \label{lemma-fitting-ideal-generate-locally} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a finite type, quasi-coherent $\mathcal{O}_X$-module. Let $x \in |X|$. Then $\mathcal{F}$ can be generated by $r$ elements in an \'etale neighbourhood of $x$ if and only if $\text{Fit}_r(\mathcal{F})_{\overline{x}} = \mathcal{O}_{X, \overline{x}}$. \end{lemma} \begin{proof} Reduces to Divisors, Lemma \ref{divisors-lemma-fitting-ideal-generate-locally} by \'etale localization (as well as the description of the local ring in Properties of Spaces, Section \ref{spaces-properties-section-stalks-structure-sheaf} and the fact that the strict henselization of a local ring is faithfully flat to see that the equality over the strict henselization is equivalent to the equality over the local ring). \end{proof} \begin{lemma} \label{lemma-fitting-ideal-finite-locally-free} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a finite type, quasi-coherent $\mathcal{O}_X$-module. Let $r \geq 0$. The following are equivalent \begin{enumerate} \item $\mathcal{F}$ is finite locally free of rank $r$ \item $\text{Fit}_{r - 1}(\mathcal{F}) = 0$ and $\text{Fit}_r(\mathcal{F}) = \mathcal{O}_X$, and \item $\text{Fit}_k(\mathcal{F}) = 0$ for $k < r$ and $\text{Fit}_k(\mathcal{F}) = \mathcal{O}_X$ for $k \geq r$. \end{enumerate} \end{lemma} \begin{proof} Reduces to Divisors, Lemma \ref{divisors-lemma-fitting-ideal-finite-locally-free} by \'etale localization. \end{proof} \begin{lemma} \label{lemma-locally-free-rank-r-pullback} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a finite type, quasi-coherent $\mathcal{O}_X$-module. The closed subspaces $$X = Z_{-1} \supset Z_0 \supset Z_1 \supset Z_2 \ldots$$ defined by the Fitting ideals of $\mathcal{F}$ have the following properties \begin{enumerate} \item The intersection $\bigcap Z_r$ is empty. \item The functor $(\Sch/X)^{opp} \to \textit{Sets}$ defined by the rule $$T \longmapsto \left\{ \begin{matrix} \{*\} & \text{if }\mathcal{F}_T\text{ is locally generated by } \leq r\text{ sections} \\ \emptyset & \text{otherwise} \end{matrix} \right.$$ is representable by the open subspace $X \setminus Z_r$. \item The functor $F_r : (\Sch/X)^{opp} \to \textit{Sets}$ defined by the rule $$T \longmapsto \left\{ \begin{matrix} \{*\} & \text{if }\mathcal{F}_T\text{ locally free rank }r\\ \emptyset & \text{otherwise} \end{matrix} \right.$$ is representable by the locally closed subspace $Z_{r - 1} \setminus Z_r$ of $X$. \end{enumerate} If $\mathcal{F}$ is of finite presentation, then $Z_r \to X$, $X \setminus Z_r \to X$, and $Z_{r - 1} \setminus Z_r \to X$ are of finite presentation. \end{lemma} \begin{proof} Reduces to Divisors, Lemma \ref{divisors-lemma-locally-free-rank-r-pullback} by \'etale localization. \end{proof} \begin{lemma} \label{lemma-finite-presentation-module} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module of finite presentation. Let $X = Z_{-1} \subset Z_0 \subset Z_1 \subset \ldots$ be as in Lemma \ref{lemma-locally-free-rank-r-pullback}. Set $X_r = Z_{r - 1} \setminus Z_r$. Then $X' = \coprod_{r \geq 0} X_r$ represents the functor $$F_{flat} : \Sch/X \longrightarrow \textit{Sets},\quad\quad T \longmapsto \left\{ \begin{matrix} \{*\} & \text{if }\mathcal{F}_T\text{ flat over }T\\ \emptyset & \text{otherwise} \end{matrix} \right.$$ Moreover, $\mathcal{F}|_{X_r}$ is locally free of rank $r$ and the morphisms $X_r \to X$ and $X' \to X$ are of finite presentation. \end{lemma} \begin{proof} Reduces to Divisors, Lemma \ref{divisors-lemma-finite-presentation-module} by \'etale localization. \end{proof} \section{Effective Cartier divisors} \label{section-effective-Cartier-divisors} \noindent For some reason it seem convenient to define the notion of an effective Cartier divisor before anything else. Note that in Morphisms of Spaces, Section \ref{spaces-morphisms-section-closed-immersions} we discussed the correspondence between closed subspaces and quasi-coherent sheaves of ideals. Moreover, in Properties of Spaces, Section \ref{spaces-properties-section-properties-modules}, we discussed properties of quasi-coherent modules, in particular locally generated by $1$ element''. These references show that the following definition is compatible with the definition for schemes. \begin{definition} \label{definition-effective-Cartier-divisor} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. \begin{enumerate} \item A {\it locally principal closed subspace} of $X$ is a closed subspace whose sheaf of ideals is locally generated by $1$ element. \item An {\it effective Cartier divisor} on $X$ is a closed subspace $D \subset X$ such that the ideal sheaf $\mathcal{I}_D \subset \mathcal{O}_X$ is an invertible $\mathcal{O}_X$-module. \end{enumerate} \end{definition} \noindent Thus an effective Cartier divisor is a locally principal closed subspace, but the converse is not always true. Effective Cartier divisors are closed subspaces of pure codimension $1$ in the strongest possible sense. Namely they are locally cut out by a single element which is not a zerodivisor. In particular they are nowhere dense. \begin{lemma} \label{lemma-characterize-effective-Cartier-divisor} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $D \subset X$ be a closed subspace. The following are equivalent: \begin{enumerate} \item The subspace $D$ is an effective Cartier divisor on $X$. \item For some scheme $U$ and surjective \'etale morphism $U \to X$ the inverse image $D \times_X U$ is an effective Cartier divisor on $U$. \item For every scheme $U$ and every \'etale morphism $U \to X$ the inverse image $D \times_X U$ is an effective Cartier divisor on $U$. \item For every $x \in |D|$ there exists an \'etale morphism $(U, u) \to (X, x)$ of pointed algebraic spaces such that $U = \Spec(A)$ and $D \times_X U = \Spec(A/(f))$ with $f \in A$ not a zerodivisor. \end{enumerate} \end{lemma} \begin{proof} The equivalence of (1) -- (3) follows from Definition \ref{definition-effective-Cartier-divisor} and the references preceding it. Assume (1) and let $x \in |D|$. Choose a scheme $W$ and a surjective \'etale morphism $W \to X$. Choose $w \in D \times_X W$ mapping to $x$. By (3) $D \times_X W$ is an effective Cartier divisor on $W$. Hence we can find affine \'etale neighbourhood $U$ by choosing an affine open neighbourhood of $w$ in $W$ as in Divisors, Lemma \ref{divisors-lemma-characterize-effective-Cartier-divisor}. \medskip\noindent Assume (4). Then we see that $\mathcal{I}_D|_U$ is invertible by Divisors, Lemma \ref{divisors-lemma-characterize-effective-Cartier-divisor}. Since we can find an \'etale covering of $X$ by the collection of all such $U$ and $X \setminus D$, we conclude that $\mathcal{I}_D$ is an invertible $\mathcal{O}_X$-module. \end{proof} \begin{lemma} \label{lemma-complement-locally-principal-closed-subscheme} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $Z \subset X$ be a locally principal closed subspace. Let $U = X \setminus Z$. Then $U \to X$ is an affine morphism. \end{lemma} \begin{proof} The question is \'etale local on $X$, see Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-affine-local} and Lemma \ref{lemma-characterize-effective-Cartier-divisor}. Thus this follows from the case of schemes which is Divisors, Lemma \ref{divisors-lemma-complement-locally-principal-closed-subscheme}. \end{proof} \begin{lemma} \label{lemma-complement-effective-Cartier-divisor} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $D \subset X$ be an effective Cartier divisor. Let $U = X \setminus D$. Then $U \to X$ is an affine morphism and $U$ is scheme theoretically dense in $X$. \end{lemma} \begin{proof} Affineness is Lemma \ref{lemma-complement-locally-principal-closed-subscheme}. The density question is \'etale local on $X$ by Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-scheme-theoretically-dense}. Thus this follows from the case of schemes which is Divisors, Lemma \ref{divisors-lemma-complement-effective-Cartier-divisor}. \end{proof} \begin{lemma} \label{lemma-effective-Cartier-makes-dimension-drop} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $D \subset X$ be an effective Cartier divisor. Let $x \in |D|$. If $\dim_x(X) < \infty$, then $\dim_x(D) < \dim_x(X)$. \end{lemma} \begin{proof} Both the definition of an effective Cartier divisor and of the dimension of an an algebraic space at a point (Properties of Spaces, Definition \ref{spaces-properties-definition-dimension-at-point}) are \'etale local. Hence this lemma follows from the case of schemes which is Divisors, Lemma \ref{divisors-lemma-effective-Cartier-makes-dimension-drop}. \end{proof} \begin{definition} \label{definition-sum-effective-Cartier-divisors} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Given effective Cartier divisors $D_1$, $D_2$ on $X$ we set $D = D_1 + D_2$ equal to the closed subspace of $X$ corresponding to the quasi-coherent sheaf of ideals $\mathcal{I}_{D_1}\mathcal{I}_{D_2} \subset \mathcal{O}_S$. We call this the {\it sum of the effective Cartier divisors $D_1$ and $D_2$}. \end{definition} \noindent It is clear that we may define the sum $\sum n_iD_i$ given finitely many effective Cartier divisors $D_i$ on $X$ and nonnegative integers $n_i$. \begin{lemma} \label{lemma-sum-effective-Cartier-divisors} The sum of two effective Cartier divisors is an effective Cartier divisor. \end{lemma} \begin{proof} Omitted. \'Etale locally this reduces to the following simple algebra fact: if $f_1, f_2 \in A$ are nonzerodivisors of a ring $A$, then $f_1f_2 \in A$ is a nonzerodivisor. \end{proof} \begin{lemma} \label{lemma-sum-closed-subschemes-effective-Cartier} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $Z, Y$ be two closed subspaces of $X$ with ideal sheaves $\mathcal{I}$ and $\mathcal{J}$. If $\mathcal{I}\mathcal{J}$ defines an effective Cartier divisor $D \subset X$, then $Z$ and $Y$ are effective Cartier divisors and $D = Z + Y$. \end{lemma} \begin{proof} By Lemma \ref{lemma-characterize-effective-Cartier-divisor} this reduces to the case of schemes which is Divisors, Lemma \ref{divisors-lemma-sum-closed-subschemes-effective-Cartier}. \end{proof} \noindent Recall that we have defined the inverse image of a closed subspace under any morphism of algebraic spaces in Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-inverse-image-closed-subspace}. \begin{lemma} \label{lemma-pullback-locally-principal} Let $S$ be a scheme. Let $f : X' \to X$ be a morphism of algebraic spaces over $S$. Let $Z \subset X$ be a locally principal closed subspace. Then the inverse image $f^{-1}(Z)$ is a locally principal closed subspace of $X'$. \end{lemma} \begin{proof} Omitted. \end{proof} \begin{definition} \label{definition-pullback-effective-Cartier-divisor} Let $S$ be a scheme. Let $f : X' \to X$ be a morphism of algebraic spaces over $S$. Let $D \subset X$ be an effective Cartier divisor. We say the {\it pullback of $D$ by $f$ is defined} if the closed subspace $f^{-1}(D) \subset X'$ is an effective Cartier divisor. In this case we denote it either $f^*D$ or $f^{-1}(D)$ and we call it the {\it pullback of the effective Cartier divisor}. \end{definition} \noindent The condition that $f^{-1}(D)$ is an effective Cartier divisor is often satisfied in practice. \begin{lemma} \label{lemma-pullback-effective-Cartier-defined} Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $D \subset Y$ be an effective Cartier divisor. The pullback of $D$ by $f$ is defined in each of the following cases: \begin{enumerate} \item $f$ is flat, and \item add more here as needed. \end{enumerate} \end{lemma} \begin{proof} Omitted. \end{proof} \begin{lemma} \label{lemma-pullback-effective-Cartier-divisors-additive} Let $S$ be a scheme. Let $f : X' \to X$ be a morphism of algebraic spaces over $S$. Let $D_1$, $D_2$ be effective Cartier divisors on $X$. If the pullbacks of $D_1$ and $D_2$ are defined then the pullback of $D = D_1 + D_2$ is defined and $f^*D = f^*D_1 + f^*D_2$. \end{lemma} \begin{proof} Omitted. \end{proof} \section{Effective Cartier divisors and invertible sheaves} \label{section-effective-Cartier-invertible} \noindent Since an effective Cartier divisor has an invertible ideal sheaf (Definition \ref{definition-effective-Cartier-divisor}) the following definition makes sense. \begin{definition} \label{definition-invertible-sheaf-effective-Cartier-divisor} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ and let $D \subset X$ be an effective Cartier divisor with ideal sheaf $\mathcal{I}_D$. \begin{enumerate} \item The {\it invertible sheaf $\mathcal{O}_X(D)$ associated to $D$} is defined by $$\mathcal{O}_X(D) = \SheafHom_{\mathcal{O}_X}(\mathcal{I}_D, \mathcal{O}_X) = \mathcal{I}_D^{\otimes -1}.$$ \item The canonical section, usually denoted $1$ or $1_D$, is the global section of $\mathcal{O}_X(D)$ corresponding to the inclusion mapping $\mathcal{I}_D \to \mathcal{O}_X$. \item We write $\mathcal{O}_X(-D) = \mathcal{O}_X(D)^{\otimes -1} = \mathcal{I}_D$. \item Given a second effective Cartier divisor $D' \subset X$ we define $\mathcal{O}_X(D - D') = \mathcal{O}_X(D) \otimes_{\mathcal{O}_X} \mathcal{O}_X(-D')$. \end{enumerate} \end{definition} \noindent Some comments. We will see below that the assignment $D \mapsto \mathcal{O}_X(D)$ turns addition of effective Cartier divisors (Definition \ref{definition-sum-effective-Cartier-divisors}) into addition in the Picard group of $X$ (Lemma \ref{lemma-invertible-sheaf-sum-effective-Cartier-divisors}). However, the expression $D - D'$ in the definition above does not have any geometric meaning. More precisely, we can think of the set of effective Cartier divisors on $X$ as a commutative monoid $\text{EffCart}(X)$ whose zero element is the empty effective Cartier divisor. Then the assignment $(D, D') \mapsto \mathcal{O}_X(D - D')$ defines a group homomorphism $$\text{EffCart}(X)^{gp} \longrightarrow \text{Pic}(X)$$ where the left hand side is the group completion of $\text{EffCart}(X)$. In other words, when we write $\mathcal{O}_X(D - D')$ we may think of $D - D'$ as an element of $\text{EffCart}(X)^{gp}$. \begin{lemma} \label{lemma-conormal-effective-Cartier-divisor} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $D \subset X$ be an effective Cartier divisor. Then for the conormal sheaf we have $\mathcal{C}_{D/X} = \mathcal{I}_D|D = \mathcal{O}_X(D)^{\otimes -1}|_D$. \end{lemma} \begin{proof} Omitted. \end{proof} \begin{lemma} \label{lemma-invertible-sheaf-sum-effective-Cartier-divisors} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $D_1$, $D_2$ be effective Cartier divisors on $X$. Let $D = D_1 + D_2$. Then there is a unique isomorphism $$\mathcal{O}_X(D_1) \otimes_{\mathcal{O}_X} \mathcal{O}_X(D_2) \longrightarrow \mathcal{O}_X(D)$$ which maps $1_{D_1} \otimes 1_{D_2}$ to $1_D$. \end{lemma} \begin{proof} Omitted. \end{proof} \begin{definition} \label{definition-regular-section} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{L}$ be an invertible sheaf on $X$. A global section $s \in \Gamma(X, \mathcal{L})$ is called a {\it regular section} if the map $\mathcal{O}_X \to \mathcal{L}$, $f \mapsto fs$ is injective. \end{definition} \begin{lemma} \label{lemma-regular-section-structure-sheaf} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $f \in \Gamma(X, \mathcal{O}_X)$. The following are equivalent: \begin{enumerate} \item $f$ is a regular section, and \item for any $x \in X$ the image $f \in \mathcal{O}_{X, \overline{x}}$ is not a zerodivisor. \item for any affine $U = \Spec(A)$ \'etale over $X$ the restriction $f|_U$ is a nonzerodivisor of $A$, and \item there exists a scheme $U$ and a surjective \'etale morphism $U \to X$ such that $f|_U$ is a regular section of $\mathcal{O}_U$. \end{enumerate} \end{lemma} \begin{proof} Omitted. \end{proof} \noindent Note that a global section $s$ of an invertible $\mathcal{O}_X$-module $\mathcal{L}$ may be seen as an $\mathcal{O}_X$-module map $s : \mathcal{O}_X \to \mathcal{L}$. Its dual is therefore a map $s : \mathcal{L}^{\otimes -1} \to \mathcal{O}_X$. (See Modules on Sites, Lemma \ref{sites-modules-lemma-constructions-invertible} for the dual invertible sheaf.) \begin{definition} \label{definition-zero-scheme-s} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{L}$ be an invertible sheaf. Let $s \in \Gamma(X, \mathcal{L})$. The {\it zero scheme} of $s$ is the closed subspace $Z(s) \subset X$ defined by the quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_X$ which is the image of the map $s : \mathcal{L}^{\otimes -1} \to \mathcal{O}_X$. \end{definition} \begin{lemma} \label{lemma-zero-scheme} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Let $s \in \Gamma(X, \mathcal{L})$. \begin{enumerate} \item Consider closed immersions $i : Z \to X$ such that $i^*s \in \Gamma(Z, i^*\mathcal{L}))$ is zero ordered by inclusion. The zero scheme $Z(s)$ is the maximal element of this ordered set. \item For any morphism of algebraic spaces $f : Y \to X$ over $S$ we have $f^*s = 0$ in $\Gamma(Y, f^*\mathcal{L})$ if and only if $f$ factors through $Z(s)$. \item The zero scheme $Z(s)$ is a locally principal closed subspace of $X$. \item The zero scheme $Z(s)$ is an effective Cartier divisor on $X$ if and only if $s$ is a regular section of $\mathcal{L}$. \end{enumerate} \end{lemma} \begin{proof} Omitted. \end{proof} \begin{lemma} \label{lemma-characterize-OD} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. \begin{enumerate} \item If $D \subset X$ is an effective Cartier divisor, then the canonical section $1_D$ of $\mathcal{O}_X(D)$ is regular. \item Conversely, if $s$ is a regular section of the invertible sheaf $\mathcal{L}$, then there exists a unique effective Cartier divisor $D = Z(s) \subset X$ and a unique isomorphism $\mathcal{O}_X(D) \to \mathcal{L}$ which maps $1_D$ to $s$. \end{enumerate} The constructions $D \mapsto (\mathcal{O}_X(D), 1_D)$ and $(\mathcal{L}, s) \mapsto Z(s)$ give mutually inverse maps $$\left\{ \begin{matrix} \text{effective Cartier divisors on }X \end{matrix} \right\} \leftrightarrow \left\{ \begin{matrix} \text{pairs }(\mathcal{L}, s)\text{ consisting of an invertible}\\ \mathcal{O}_X\text{-module and a regular global section} \end{matrix} \right\}$$ \end{lemma} \begin{proof} Omitted. \end{proof} \section{Effective Cartier divisors on Noetherian spaces} \label{section-Noetherian-effective-Cartier} \noindent In the locally Noetherian setting most of the discussion of effective Cartier divisors and regular sections simplifies somewhat. \begin{lemma} \label{lemma-effective-Cartier-divisor-Sk} Let $S$ be a scheme and let $X$ be a locally Noetherian algebraic space over $S$. Let $D \subset X$ be an effective Cartier divisor. If $X$ is $(S_k)$, then $D$ is $(S_{k - 1})$. \end{lemma} \begin{proof} By our definition of the property $(S_k)$ for algebraic spaces (Properties of Spaces, Section \ref{spaces-properties-section-types-properties}) and Lemma \ref{lemma-characterize-effective-Cartier-divisor} this follows from the case of schemes (Divisors, Lemma \ref{divisors-lemma-effective-Cartier-divisor-Sk}). \end{proof} \begin{lemma} \label{lemma-normal-effective-Cartier-divisor-S1} Let $S$ be a scheme and let $X$ be a locally Noetherian normal algebraic space over $S$. Let $D \subset X$ be an effective Cartier divisor. Then $D$ is $(S_1)$. \end{lemma} \begin{proof} By our definition of normality for algebraic spaces (Properties of Spaces, Section \ref{spaces-properties-section-types-properties}) and Lemma \ref{lemma-characterize-effective-Cartier-divisor} this follows from the case of schemes (Divisors, Lemma \ref{divisors-lemma-normal-effective-Cartier-divisor-S1}). \end{proof} \section{Relative Proj} \label{section-relative-proj} \noindent 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 \ref{constructions-section-relative-proj} and Divisors, Section \ref{divisors-section-relative-proj} in the case of schemes. \begin{situation} \label{situation-relative-proj} Here $S$ is a scheme, $X$ is an algebraic space over $S$, and $\mathcal{A}$ is a quasi-coherent graded $\mathcal{O}_X$-algebra. \end{situation} \noindent In Situation \ref{situation-relative-proj} we are going to define a functor $F : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}$ which will turn out to be an algebraic space. We will follow (mutatis mutandis) the procedure of Constructions, Section \ref{constructions-section-relative-proj}. First, given a scheme $T$ over $S$ we define a {\it quadruple over $T$} to be a system $(d, f : T \to X, \mathcal{L}, \psi)$ \begin{enumerate} \item $d \geq 1$ is an integer, \item $f : T \to X$ is a morphism over $S$, \item $\mathcal{L}$ is an invertible $\mathcal{O}_T$-module, and \item $\psi : f^*\mathcal{A}^{(d)} \to \bigoplus_{n \geq 0}\mathcal{L}^{\otimes n}$ is a homomorphism of graded $\mathcal{O}_T$-algebras such that $f^*\mathcal{A}_d \to \mathcal{L}$ is surjective. \end{enumerate} We say two quadruples $(d, f, \mathcal{L}, \psi)$ and $(d', f', \mathcal{L}', \psi')$ are {\it equivalent}\footnote{This definition is motivated by Constructions, Lemma \ref{constructions-lemma-equivalent-relative}. The advantage of choosing this one is that it clearly defines an equivalence relation.} 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 {\it 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 $$F(T) = \lim_{V \subset T\text{ quasi-compact open}} F(V).$$ 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 \begin{equation} \label{equation-proj} F : \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 $f$. \begin{lemma} \label{lemma-relative-proj} In Situation \ref{situation-relative-proj}. 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. \end{lemma} \begin{proof} It suffices to prove the second assertion, see Spaces, Lemma \ref{spaces-lemma-representable-over-space}. 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 \ref{constructions-lemma-proj-base-change}). By Constructions, Lemmas \ref{constructions-lemma-equivalent-relative}, \ref{constructions-lemma-relative-proj}, and \ref{constructions-lemma-glueing-gives-functor-proj} and Definition \ref{constructions-definition-relative-proj} we see that $F'$ is representable by $\underline{\text{Proj}}_Z(g^*\mathcal{A})$. \end{proof} \noindent The lemma above tells us the following definition makes sense. \begin{definition} \label{definition-relative-proj} 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 {\it relative homogeneous spectrum of $\mathcal{A}$ over $X$}, or the {\it homogeneous spectrum of $\mathcal{A}$ over $X$}, or the {\it relative Proj of $\mathcal{A}$ over $X$} is the algebraic space $F$ over $X$ of Lemma \ref{lemma-relative-proj}. We denote it $\pi : \underline{\text{Proj}}_X(\mathcal{A}) \to X$. \end{definition} \noindent 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 \'etale morphism, where $U$ is a scheme. Set $R = U \times_X U$ with projection morphisms $s, t : R \to U$. By Lemma \ref{lemma-relative-proj} 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 \ref{spaces-properties-proposition-quasi-coherent}. 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 \ref{constructions-lemma-relative-proj-base-change}). 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 \'etale 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 \'etale localization to transfer results on the relative proj from the case of schemes to the case of algebraic spaces. \begin{lemma} \label{lemma-twists-of-structure-sheaf} In Situation \ref{situation-relative-proj}. 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 $$\psi : \pi^*\mathcal{A} \longrightarrow \bigoplus\nolimits_{n \geq 0} \mathcal{O}_{\underline{\text{Proj}}_X(\mathcal{A})}(n)$$ whose base change to any scheme over $X$ agrees with Constructions, Lemma \ref{constructions-lemma-glue-relative-proj-twists}. \end{lemma} \begin{proof} As in the discussion following Definition \ref{definition-relative-proj} choose a scheme $U$ and a surjective \'etale 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 \ref{constructions-lemma-glue-relative-proj-twists} 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 \ref{constructions-lemma-relative-proj-base-change} 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 \ref{spaces-properties-proposition-quasi-coherent} we obtain $\mathcal{O}_{P}(n)$ and $\psi$. We omit the verification of compatibility with pullback to arbitrary schemes over $X$. \end{proof} \noindent Having constructed the relative Proj we turn to some basic properties. \begin{lemma} \label{lemma-relative-proj-base-change} 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 $$r : \underline{\text{Proj}}_{X'}(g^*\mathcal{A}) \longrightarrow X' \times_X \underline{\text{Proj}}_X(\mathcal{A})$$ as well as a corresponding isomorphism $$\theta : r^*\text{pr}_2^*\left(\bigoplus\nolimits_{d \in \mathbf{Z}} \mathcal{O}_{\underline{\text{Proj}}_X(\mathcal{A})}(d)\right) \longrightarrow \bigoplus\nolimits_{d \in \mathbf{Z}} \mathcal{O}_{\underline{\text{Proj}}_{X'}(g^*\mathcal{A})}(d)$$ of $\mathbf{Z}$-graded $\mathcal{O}_{\underline{\text{Proj}}_{X'}(g^*\mathcal{A})}$-algebras. \end{lemma} \begin{proof} Let $F$ be the functor (\ref{equation-proj}) 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 \'etale localization and apply Constructions, Lemma \ref{constructions-lemma-relative-proj-base-change}). \end{proof} \begin{lemma} \label{lemma-relative-proj-separated} In Situation \ref{situation-relative-proj} the morphism $\pi : \underline{\text{Proj}}_X(\mathcal{A}) \to X$ is separated. \end{lemma} \begin{proof} By Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-separated-local} and the construction of the relative Proj this follows from the case of schemes which is Constructions, Lemma \ref{constructions-lemma-relative-proj-separated}. \end{proof} \begin{lemma} \label{lemma-relative-proj-quasi-compact} In Situation \ref{situation-relative-proj}. If one of the following holds \begin{enumerate} \item $\mathcal{A}$ is of finite type as a sheaf of $\mathcal{A}_0$-algebras, \item $\mathcal{A}$ is generated by $\mathcal{A}_1$ as an $\mathcal{A}_0$-algebra and $\mathcal{A}_1$ is a finite type $\mathcal{A}_0$-module, \item 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 $\mathcal{A}/\mathcal{F}\mathcal{A}$, \end{enumerate} then $\pi : \underline{\text{Proj}}_X(\mathcal{A}) \to X$ is quasi-compact. \end{lemma} \begin{proof} By Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-quasi-compact-local} and the construction of the relative Proj this follows from the case of schemes which is Divisors, Lemma \ref{divisors-lemma-relative-proj-quasi-compact}. \end{proof} \begin{lemma} \label{lemma-relative-proj-finite-type} In Situation \ref{situation-relative-proj}. 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. \end{lemma} \begin{proof} By Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-finite-type-local} and the construction of the relative Proj this follows from the case of schemes which is Divisors, Lemma \ref{divisors-lemma-relative-proj-finite-type}. \end{proof} \begin{lemma} \label{lemma-relative-proj-universally-closed} In Situation \ref{situation-relative-proj}. If $\mathcal{O}_X \to \mathcal{A}_0$ is an integral algebra map\footnote{In other words, the integral closure of $\mathcal{O}_X$ in $\mathcal{A}_0$, see Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-integral-closure}, equals $\mathcal{A}_0$.} 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. \end{lemma} \begin{proof} By Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-universally-closed-local} and the construction of the relative Proj this follows from the case of schemes which is Divisors, Lemma \ref{divisors-lemma-relative-proj-universally-closed}. \end{proof} \begin{lemma} \label{lemma-relative-proj-proper} In Situation \ref{situation-relative-proj}. The following conditions are equivalent \begin{enumerate} \item $\mathcal{A}_0$ is a finite type $\mathcal{O}_X$-module and $\mathcal{A}$ is of finite type as an $\mathcal{A}_0$-algebra, \item $\mathcal{A}_0$ is a finite type $\mathcal{O}_X$-module and $\mathcal{A}$ is of finite type as an $\mathcal{O}_X$-algebra. \end{enumerate} If these conditions hold, then $\pi : \underline{\text{Proj}}_X(\mathcal{A}) \to X$ is proper. \end{lemma} \begin{proof} By Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-proper-local} and the construction of the relative Proj this follows from the case of schemes which is Divisors, Lemma \ref{divisors-lemma-relative-proj-universally-closed}. \end{proof} \begin{lemma} \label{lemma-relative-proj-generated-in-degree-1} 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 \begin{enumerate} \item $P$ 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 $f^*\mathcal{A}_1 \to \mathcal{L}$, \item the canonical map $\pi^*\mathcal{A}_1 \to \mathcal{O}_P(1)$ is surjective, and \item each $\mathcal{O}_P(n)$ is invertible and the multiplication maps induce isomorphisms $\mathcal{O}_P(n) \otimes_{\mathcal{O}_P} \mathcal{O}_P(m) = \mathcal{O}_P(n + m)$. \end{enumerate} \end{lemma} \begin{proof} Omitted. See Constructions, Lemma \ref{constructions-lemma-apply-relative} for the case of schemes. \end{proof} \section{Functoriality of relative proj} \label{section-functoriality-relative-proj} \noindent This section is the analogue of Constructions, Section \ref{constructions-section-functoriality-relative-proj}. \begin{lemma} \label{lemma-morphism-relative-proj} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\psi : \mathcal{A} \to \mathcal{B}$ be a map of quasi-coherent graded $\mathcal{O}_X$-algebras. Set $P = \underline{\text{Proj}}_X(\mathcal{A}) \to X$ and $Q = \underline{\text{Proj}}_X(\mathcal{B}) \to X$. There is a canonical open subspace $U(\psi) \subset Q$ and a canonical morphism of algebraic spaces $$r_\psi : U(\psi) \longrightarrow P$$ over $X$ and a map of $\mathbf{Z}$-graded $\mathcal{O}_{U(\psi)}$-algebras $$\theta = \theta_\psi : r_\psi^*\left( \bigoplus\nolimits_{d \in \mathbf{Z}} \mathcal{O}_P(d) \right) \longrightarrow \bigoplus\nolimits_{d \in \mathbf{Z}} \mathcal{O}_{U(\psi)}(d).$$ The triple $(U(\psi), r_\psi, \theta)$ is characterized by the property that for any scheme $W$ \'etale over $X$ the triple $$(U(\psi) \times_X W,\quad r_\psi|_{U(\psi) \times_X W} : U(\psi) \times_X W \to P \times_X W,\quad \theta|_{U(\psi) \times_X W})$$ is equal to the triple associated to $\psi : \mathcal{A}|_W \to \mathcal{B}|_W$ of Constructions, Lemma \ref{constructions-lemma-morphism-relative-proj}. \end{lemma} \begin{proof} This lemma follows from \'etale localization and the case of schemes, see discussion following Definition \ref{definition-relative-proj}. Details omitted. \end{proof} \begin{lemma} \label{lemma-morphism-relative-proj-transitive} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{C}$ be quasi-coherent graded $\mathcal{O}_X$-algebras. Set $P = \underline{\text{Proj}}_X(\mathcal{A})$, $Q = \underline{\text{Proj}}_X(\mathcal{B})$ and $R = \underline{\text{Proj}}_X(\mathcal{C})$. Let $\varphi : \mathcal{A} \to \mathcal{B}$, $\psi : \mathcal{B} \to \mathcal{C}$ be graded $\mathcal{O}_X$-algebra maps. Then we have $$U(\psi \circ \varphi) = r_\varphi^{-1}(U(\psi)) \quad \text{and} \quad r_{\psi \circ \varphi} = r_\varphi \circ r_\psi|_{U(\psi \circ \varphi)}.$$ In addition we have $$\theta_\psi \circ r_\psi^*\theta_\varphi = \theta_{\psi \circ \varphi}$$ with obvious notation. \end{lemma} \begin{proof} Omitted. \end{proof} \begin{lemma} \label{lemma-surjective-graded-rings-map-relative-proj} With hypotheses and notation as in Lemma \ref{lemma-morphism-relative-proj} above. Assume $\mathcal{A}_d \to \mathcal{B}_d$ is surjective for $d \gg 0$. Then \begin{enumerate} \item $U(\psi) = Q$, \item $r_\psi : Q \to R$ is a closed immersion, and \item the maps $\theta : r_\psi^*\mathcal{O}_P(n) \to \mathcal{O}_Q(n)$ are surjective but not isomorphisms in general (even if $\mathcal{A} \to \mathcal{B}$ is surjective). \end{enumerate} \end{lemma} \begin{proof} Follows from the case of schemes (Constructions, Lemma \ref{constructions-lemma-surjective-graded-rings-map-relative-proj}) by \'etale localization. \end{proof} \begin{lemma} \label{lemma-eventual-iso-graded-rings-map-relative-proj} With hypotheses and notation as in Lemma \ref{lemma-morphism-relative-proj} above. Assume $\mathcal{A}_d \to \mathcal{B}_d$ is an isomorphism for all $d \gg 0$. Then \begin{enumerate} \item $U(\psi) = Q$, \item $r_\psi : Q \to P$ is an isomorphism, and \item the maps $\theta : r_\psi^*\mathcal{O}_P(n) \to \mathcal{O}_Q(n)$ are isomorphisms. \end{enumerate} \end{lemma} \begin{proof} Follows from the case of schemes (Constructions, Lemma \ref{constructions-lemma-eventual-iso-graded-rings-map-relative-proj}) by \'etale localization. \end{proof} \begin{lemma} \label{lemma-surjective-generated-degree-1-map-relative-proj} With hypotheses and notation as in Lemma \ref{lemma-morphism-relative-proj} above. Assume $\mathcal{A}_d \to \mathcal{B}_d$ is surjective for $d \gg 0$ and that $\mathcal{A}$ is generated by $\mathcal{A}_1$ over $\mathcal{A}_0$. Then \begin{enumerate} \item $U(\psi) = Q$, \item $r_\psi : Q \to P$ is a closed immersion, and \item the maps $\theta : r_\psi^*\mathcal{O}_P(n) \to \mathcal{O}_Q(n)$ are isomorphisms. \end{enumerate} \end{lemma} \begin{proof} Follows from the case of schemes (Constructions, Lemma \ref{constructions-lemma-surjective-generated-degree-1-map-relative-proj}) by \'etale localization. \end{proof} \section{Invertible sheaves and morphisms into relative Proj} \label{section-invertible-relative-proj} \noindent It seems that we may need the following lemma somewhere. The situation is the following: \begin{enumerate} \item Let $S$ be a scheme and $Y$ an algebraic space over $S$. \item Let $\mathcal{A}$ be a quasi-coherent graded $\mathcal{O}_Y$-algebra. \item Denote $\pi : \underline{\text{Proj}}_Y(\mathcal{A}) \to Y$ the relative Proj of $\mathcal{A}$ over $Y$. \item Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. \item Let $\mathcal{L}$ be an invertible $\mathcal{O}_Y$-module. \item Let $\psi : f^*\mathcal{A} \to \bigoplus_{d \geq 0} \mathcal{L}^{\otimes d}$ be a homomorphism of graded $\mathcal{O}_X$-algebras. \end{enumerate} Given this data let $U(\psi) \subset X$ be the open subspace with $$|U(\psi)| = \bigcup\nolimits_{d \geq 1} \{\text{locus where }f^*\mathcal{A}_d \to \mathcal{L}^{\otimes d} \text{ is surjective}\}$$ Formation of $U(\psi) \subset X$ commutes with pullback by any morphism $X' \to X$. \begin{lemma} \label{lemma-invertible-map-into-relative-proj} With assumptions and notation as above. The morphism $\psi$ induces a canonical morphism of algebraic spaces over $Y$ $$r_{\mathcal{L}, \psi} : U(\psi) \longrightarrow \underline{\text{Proj}}_Y(\mathcal{A})$$ together with a map of graded $\mathcal{O}_{U(\psi)}$-algebras $$\theta : r_{\mathcal{L}, \psi}^*\left( \bigoplus\nolimits_{d \geq 0} \mathcal{O}_{\underline{\text{Proj}}_Y(\mathcal{A})}(d) \right) \longrightarrow \bigoplus\nolimits_{d \geq 0} \mathcal{L}^{\otimes d}|_{U(\psi)}$$ characterized by the following properties: \begin{enumerate} \item For $V \to Y$ \'etale and $d \geq 0$ the diagram $$\xymatrix{ \mathcal{A}_d(V) \ar[d]_{\psi} \ar[r]_{\psi} & \Gamma(V \times_Y X, \mathcal{L}^{\otimes d}) \ar[d]^{restrict} \\ \Gamma(V \times_Y \underline{\text{Proj}}_Y(\mathcal{A}), \mathcal{O}_{\underline{\text{Proj}}_Y(\mathcal{A})}(d)) \ar[r]^-\theta & \Gamma(V \times_Y U(\psi), \mathcal{L}^{\otimes d}) }$$ is commutative. \item For any $d \geq 1$ and any morphism $W \to X$ where $W$ is a scheme such that $\psi|_W : f^*\mathcal{A}_d|_W \to \mathcal{L}^{\otimes d}|_W$ is surjective we have (a) $W \to X$ factors through $U(\psi)$ and (b) composition of $W \to U(\psi)$ with $r_{\mathcal{L}, \psi}$ agrees with the morphism $W \to \underline{\text{Proj}}_Y(\mathcal{A})$ which exists by the construction of $\underline{\text{Proj}}_Y(\mathcal{A})$, see Definition \ref{definition-relative-proj}. \item Consider a commutative diagram $$\xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]^g & Y }$$ where $X'$ and $Y'$ are schemes, set $\mathcal{A}' = g^*\mathcal{A}$ and $\mathcal{L}' = (g')^*\mathcal{L}$ and denote $\psi' : (f')^*\mathcal{A} \to \bigoplus_{d \geq 0} (\mathcal{L}')^{\otimes d}$ the pullback of $\psi$. Let $U(\psi')$, $r_{\psi', \mathcal{L}'}$, and $\theta'$ be the open, morphism, and homomorphism constructed in Constructions, Lemma \ref{lemma-invertible-map-into-relative-proj}. Then $U(\psi') = (g')^{-1}(U(\psi))$ and $r_{\psi', \mathcal{L}'}$ agrees with the base change of $r_{\psi, \mathcal{L}}$ via the isomorphism $\underline{\text{Proj}}_{Y'}(\mathcal{A}') = Y' \times_Y \underline{\text{Proj}}_Y(\mathcal{A})$ of Lemma \ref{lemma-relative-proj-base-change}. Moreover, $\theta'$ is the pullback of $\theta$. \end{enumerate} \end{lemma} \begin{proof} Omitted. Hints: First we observe that for a quasi-compact scheme $W$ over $X$ the following are equivalent \begin{enumerate} \item $W \to X$ factors through $U(\psi)$, and \item there exists a $d$ such that $\psi|_W : f^*\mathcal{A}_d|_W \to \mathcal{L}^{\otimes d}|_W$ is surjective. \end{enumerate} This gives a description of $U(\psi)$ as a subfunctor of $X$ on our base category $(\Sch/S)_{fppf}$. For such a $W$ and $d$ we consider the quadruple $(d, W \to Y, \mathcal{L}|_W, \psi^{(d)}|_W)$. By definition of $\underline{\text{Proj}}_Y(\mathcal{A})$ we obtain a morphism $W \to \underline{\text{Proj}}_Y(\mathcal{A})$. By our notion of equivalence of quadruples one sees that this morphism is independent of the choice of $d$. This clearly defines a transformation of functors $r_{\psi, \mathcal{L}} : U(\psi) \to \underline{\text{Proj}}_Y(\mathcal{A})$, i.e., a morphism of algebraic spaces. By construction this morphism satisfies (2). Since the morphism constructed in Constructions, Lemma \ref{constructions-lemma-invertible-map-into-relative-proj} satisfies the same property, we see that (3) is true. \medskip\noindent To construct $\theta$ and check the compatibility (1) of the lemma, work \'etale locally on $Y$ and $X$, arguing as in the discussion following Definition \ref{definition-relative-proj}. \end{proof} \section{Relatively ample sheaves} \label{section-relatively-ample} \noindent This section is the analogue of Morphisms, Section \ref{morphisms-section-relatively-ample} for algebraic spaces. Our definition of a relatively ample invertible sheaf is as follows. \begin{definition} \label{definition-relatively-ample} Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. We say $\mathcal{L}$ is {\it relatively ample}, or {\it $f$-relatively ample}, or {\it ample on $X/Y$}, or {\it $f$-ample} if $f : X \to Y$ is representable and for every morphism $Z \to Y$ where $Z$ is a scheme, the pullback $\mathcal{L}_T$ of $\mathcal{L}$ to $X_Z = Z \times_Y X$ is ample on $X_Z/Z$ as in Morphisms, Definition \ref{morphisms-definition-relatively-ample}. \end{definition} \noindent We will almost always reduce questions about relatively ample invertible sheaves to the case of schemes. Thus in this section we have mainly sanity checks. \begin{lemma} \label{lemma-relatively-ample-sanity-check} Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Assume $Y$ is a scheme. The following are equivalent \begin{enumerate} \item $\mathcal{L}$ is ample on $X/Y$ in the sense of Definition \ref{definition-relatively-ample}, and \item $X$ is a scheme and $\mathcal{L}$ is ample on $X/Y$ in the sense of Morphisms, Definition \ref{morphisms-definition-relatively-ample}. \end{enumerate} \end{lemma} \begin{proof} This follows from the definitions and Morphisms, Lemma \ref{morphisms-lemma-ample-base-change} (which says that being relatively ample for schemes is preserved under base change). \end{proof} \begin{lemma} \label{lemma-ample-base-change} Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Let $Y' \to Y$ be a morphism of algebraic spaces over $S$. Let $f' : X' \to Y'$ be the base change of $f$ and denote $\mathcal{L}'$ the pullback of $\mathcal{L}$ to $X'$. If $\mathcal{L}$ is $f$-ample, then $\mathcal{L}'$ is $f'$-ample. \end{lemma} \begin{proof} This follows immediately from the definition! (Hint: transitivity of base change.) \end{proof} \begin{lemma} \label{lemma-relatively-ample-properties} Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. If there exists an $f$-ample invertible sheaf, then $f$ is representable, quasi-compact, and separated. \end{lemma} \begin{proof} This is clear from the definitions and Morphisms, Lemma \ref{morphisms-lemma-relatively-ample-separated}. (If in doubt, take a look at the principle of Algebraic Spaces, Lemma \ref{spaces-lemma-representable-transformations-property-implication}.) \end{proof} \begin{lemma} \label{lemma-descend-relatively-ample} Let $V \to U$ be a surjective \'etale morphism of affine schemes. Let $X$ be an algebraic space over $U$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Let $Y = V \times_U X$ and let $\mathcal{N}$ be the pullback of $\mathcal{L}$ to $Y$. The following are equivalent \begin{enumerate} \item $\mathcal{L}$ is ample on $X/U$, and \item $\mathcal{N}$ is ample on $Y/V$. \end{enumerate} \end{lemma} \begin{proof} The implication (1) $\Rightarrow$ (2) follows from Lemma \ref{lemma-ample-base-change}. Assume (2). This implies that $Y \to V$ is quasi-compact and separated (Lemma \ref{lemma-relatively-ample-properties}) and $Y$ is a scheme. Then we conclude that $X \to U$ is quasi-compact and separated (Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-quasi-compact-local} and \ref{spaces-morphisms-lemma-separated-local}). Set $\mathcal{A} = \bigoplus_{d \geq 0} f_*\mathcal{L}^{\otimes d}$. Thus is a quasi-coherent sheaf of graded $\mathcal{O}_U$-algebras (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-pushforward}). By adjunction we have a map $\psi : f^*\mathcal{A} \to \bigoplus_{d \geq 0} \mathcal{L}^{\otimes d}$. Applying Lemma \ref{lemma-invertible-map-into-relative-proj} we obtain an open subspace $U(\psi) \subset X$ and a morphism $$r_{\mathcal{L}, \psi} : U(\psi) \to \underline{\text{Proj}}_U(\mathcal{A})$$ Since $h : V \to U$ is \'etale we have $\mathcal{A}|_V = (Y \to V)_*(\bigoplus_{d \geq 0} \mathcal{N}^{\otimes d})$, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-pushforward-etale-base-change-modules}. It follows that the pullback $\psi'$ of $\psi$ to $Y$ is the adjunction map for the situation $(Y \to V, \mathcal{N})$ as in Morphisms, Lemma \ref{morphisms-lemma-characterize-relatively-ample} part (5). Since $\mathcal{N}$ is ample on $Y/V$ we conclude from the lemma just cited that $U(\psi') = Y$ and that $r_{\mathcal{N}, \psi'}$ is an open immersion. Since Lemma \ref{lemma-invertible-map-into-relative-proj} tells us that the formation of $r_{\mathcal{L}, \psi}$ commutes with base change, we conclude that $U(\psi) = X$ and that we have a commutative diagram $$\xymatrix{ Y \ar[r]_-{r'} \ar[d] & \underline{\text{Proj}}_V(\mathcal{A}|_V) \ar[d] \ar[r] & V \ar[d] \\ X \ar[r]^-r & \underline{\text{Proj}}_U(\mathcal{A}) \ar[r] & U }$$ whose squares are fibre products. We conclude that $r$ is an open immersion by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-closed-immersion-local}. Thus $X$ is a scheme. Then we can apply Morphisms, Lemma \ref{morphisms-lemma-characterize-relatively-ample} part (5) to conclude that $\mathcal{L}$ is ample on $X/U$. \end{proof} \begin{lemma} \label{lemma-relatively-ample-local} Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. The following are equivalent \begin{enumerate} \item $\mathcal{L}$ is ample on $X/Y$, \item for every scheme $Z$ and every morphism $Z \to Y$ the algebraic space $X_Z = Z \times_Y X$ is a scheme and the pullback $\mathcal{L}_Z$ is ample on $X_Z/Z$, \item for every affine scheme $Z$ and every morphism $Z \to Y$ the algebraic space $X_Z = Z \times_Y X$ is a scheme and the pullback $\mathcal{L}_Z$ is ample on $X_Z/Z$, \item there exists a scheme $V$ and a surjective \'etale morphism $V \to Y$ such that the algebraic space $X_V = V \times_Y X$ is a scheme and the pullback $\mathcal{L}_V$ is ample on $X_V/V$. \end{enumerate} \end{lemma} \begin{proof} Parts (1) and (2) are equivalent by definition. The implication (2) $\Rightarrow$ (3) is immediate. If (3) holds and $Z \to Y$ is as in (2), then we see that $X_Z \to Z$ is affine locally on $Z$ representable. Hence $X_Z$ is a scheme for example by Properties of Spaces, Lemma \ref{spaces-properties-lemma-subscheme}. Then it follows that $\mathcal{L}_Z$ is ample on $X_Z/Z$ because it holds locally on $Z$ and we can use Morphisms, Lemma \ref{morphisms-lemma-characterize-relatively-ample}. Thus (1), (2), and (3) are equivalent. Clearly these conditions imply (4). \medskip\noindent Assume (4). Let $Z \to Y$ be a morphism with $Z$ affine. Then $U = V \times_Y Z \to Z$ is a surjective \'etale morphism such that the pullback of $\mathcal{L}_Z$ by $X_U \to X_Z$ is relatively ample on $X_U/U$. Of course we may replace $U$ by an affine open. It follows that $\mathcal{L}_Z$ is ample on $X_Z/Z$ by Lemma \ref{lemma-descend-relatively-ample}. Thus (4) $\Rightarrow$ (3) and the proof is complete. \end{proof} \section{Relative ampleness and cohomology} \label{section-ample-and-proper} \noindent This section contains some results related to the results in Cohomology of Schemes, Sections \ref{coherent-section-applications-formal-functions} and \ref{coherent-section-ample-cohomology}. \medskip\noindent The following lemma is just an example of what we can do. \begin{lemma} \label{lemma-vanshing-gives-ample} Let $R$ be a Noetherian ring. Let $X$ be an algebraic space over $R$ such that the structure morphism $f : X \to \Spec(R)$ is proper. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. The following are equivalent \begin{enumerate} \item $\mathcal{L}$ is ample on $X/R$ (Definition \ref{definition-relatively-ample}), \item for every coherent $\mathcal{O}_X$-module $\mathcal{F}$ there exists an $n_0 \geq 0$ such that $H^p(X, \mathcal{F} \otimes \mathcal{L}^{\otimes n}) = 0$ for all $n \geq n_0$ and $p > 0$. \end{enumerate} \end{lemma} \begin{proof} The implication (1) $\Rightarrow$ (2) follows from Cohomology of Schemes, Lemma \ref{coherent-lemma-coherent-proper-ample} because assumption (1) implies that $X$ is a scheme. The implication (2) $\Rightarrow$ (1) is Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-Noetherian-h1-zero-invertible}. \end{proof} \begin{lemma} \label{lemma-ample-on-fibre} Let $Y$ be a Noetherian scheme. Let $X$ be an algebraic space over $Y$ such that the structure morphism $f : X \to Y$ is proper. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. Let $y \in Y$ be a point such that $X_y$ is a scheme and $\mathcal{L}_y$ is ample on $X_y$. Then there exists a $d_0$ such that for all $d \geq d_0$ we have $$R^pf_*(\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d})_y = 0 \text{ for }p > 0$$ and the map $$f_*(\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d})_y \longrightarrow H^0(X_y, \mathcal{F}_y \otimes_{\mathcal{O}_{X_y}} \mathcal{L}_y^{\otimes d})$$ is surjective. \end{lemma} \begin{proof} Note that $\mathcal{O}_{Y, y}$ is a Noetherian local ring. Consider the canonical morphism $c : \Spec(\mathcal{O}_{Y, y}) \to Y$, see Schemes, Equation (\ref{schemes-equation-canonical-morphism}). This is a flat morphism as it identifies local rings. Denote momentarily $f' : X' \to \Spec(\mathcal{O}_{Y, y})$ the base change of $f$ to this local ring. We see that $c^*R^pf_*\mathcal{F} = R^pf'_*\mathcal{F}'$ by Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-flat-base-change-cohomology}. Moreover, the fibres $X_y$ and $X'_y$ are identified. Hence we may assume that $Y = \Spec(A)$ is the spectrum of a Noetherian local ring $(A, \mathfrak m, \kappa)$ and $y \in Y$ corresponds to $\mathfrak m$. In this case $R^pf_*(\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d})_y = H^p(X, \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d})$ for all $p \geq 0$. Denote $f_y : X_y \to \Spec(\kappa)$ the projection. \medskip\noindent Let $B = \text{Gr}_\mathfrak m(A) = \bigoplus_{n \geq 0} \mathfrak m^n/\mathfrak m^{n + 1}$. Consider the sheaf $\mathcal{B} = f_y^*\widetilde{B}$ of quasi-coherent graded $\mathcal{O}_{X_y}$-algebras. We will use notation as in Cohomology of Spaces, Section \ref{spaces-cohomology-section-theorem-formal-functions} with $I$ replaced by $\mathfrak m$. Since $X_y$ is the closed subspace of $X$ cut out by $\mathfrak m\mathcal{O}_X$ we may think of $\mathfrak m^n\mathcal{F}/\mathfrak m^{n + 1}\mathcal{F}$ as a coherent $\mathcal{O}_{X_y}$-module, see Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-i-star-equivalence}. Then $\bigoplus_{n \geq 0} \mathfrak m^n\mathcal{F}/\mathfrak m^{n + 1}\mathcal{F}$ is a quasi-coherent graded $\mathcal{B}$-module of finite type because it is generated in degree zero over $\mathcal{B}$ abd because the degree zero part is $\mathcal{F}_y = \mathcal{F}/\mathfrak m \mathcal{F}$ which is a coherent $\mathcal{O}_{X_y}$-module. Hence by Cohomology of Schemes, Lemma \ref{coherent-lemma-graded-finiteness} part (2) there exists a $d_0$ such that $$H^p(X_y, \mathfrak m^n \mathcal{F}/ \mathfrak m^{n + 1}\mathcal{F} \otimes_{\mathcal{O}_{X_y}} \mathcal{L}_y^{\otimes d}) = 0$$ for all $p > 0$, $d \geq d_0$, and $n \geq 0$. By Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-relative-affine-cohomology} this is the same as the statement that $H^p(X, \mathfrak m^n \mathcal{F}/ \mathfrak m^{n + 1}\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d}) = 0$ for all $p > 0$, $d \geq d_0$, and $n \geq 0$. \medskip\noindent Consider the short exact sequences $$0 \to \mathfrak m^n\mathcal{F}/\mathfrak m^{n + 1} \mathcal{F} \to \mathcal{F}/\mathfrak m^{n + 1} \mathcal{F} \to \mathcal{F}/\mathfrak m^n \mathcal{F} \to 0$$ of coherent $\mathcal{O}_X$-modules. Tensoring with $\mathcal{L}^{\otimes d}$ is an exact functor and we obtain short exact sequences $$0 \to \mathfrak m^n\mathcal{F}/\mathfrak m^{n + 1} \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d} \to \mathcal{F}/\mathfrak m^{n + 1} \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d} \to \mathcal{F}/\mathfrak m^n \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d} \to 0$$ Using the long exact cohomology sequence and the vanishing above we conclude (using induction) that \begin{enumerate} \item $H^p(X, \mathcal{F}/\mathfrak m^n \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d}) = 0$ for all $p > 0$, $d \geq d_0$, and $n \geq 0$, and \item $H^0(X, \mathcal{F}/\mathfrak m^n \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d}) \to H^0(X_y, \mathcal{F}_y \otimes_{\mathcal{O}_{X_y}} \mathcal{L}_y^{\otimes d})$ is surjective for all $d \geq d_0$ and $n \geq 1$. \end{enumerate} By the theorem on formal functions (Cohomology of Spaces, Theorem \ref{spaces-cohomology-theorem-formal-functions}) we find that the $\mathfrak m$-adic completion of $H^p(X, \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d})$ is zero for all $d \geq d_0$ and $p > 0$. Since $H^p(X, \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d})$ is a finite $A$-module by Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-proper-over-affine-cohomology-finite} it follows from Nakayama's lemma (Algebra, Lemma \ref{algebra-lemma-NAK}) that $H^p(X, \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d})$ is zero for all $d \geq d_0$ and $p > 0$. For $p = 0$ we deduce from Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-ML-cohomology-powers-ideal} part (3) that $H^0(X, \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d}) \to H^0(X_y, \mathcal{F}_y \otimes_{\mathcal{O}_{X_y}} \mathcal{L}_y^{\otimes d})$ is surjective, which gives the final statement of the lemma. \end{proof} \begin{lemma} \label{lemma-ample-in-neighbourhood} (For a more general version see Descent on Spaces, Lemma \ref{spaces-descent-lemma-ample-in-neighbourhood}). Let $Y$ be a Noetherian scheme. Let $X$ be an algebraic space over $Y$ such that the structure morphism $f : X \to Y$ is proper. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Let $y \in Y$ be a point such that $X_y$ is a scheme and $\mathcal{L}_y$ is ample on $X_y$. Then there is an open neighbourhood $V \subset Y$ of $y$ such that $\mathcal{L}|_{f^{-1}(V)}$ is ample on $f^{-1}(V)/V$ (as in Definition \ref{definition-relatively-ample}). \end{lemma} \begin{proof} Pick $d_0$ as in Lemma \ref{lemma-ample-on-fibre} for $\mathcal{F} = \mathcal{O}_X$. Pick $d \geq d_0$ so that we can find $r \geq 0$ and sections $s_{y, 0}, \ldots, s_{y, r} \in H^0(X_y, \mathcal{L}_y^{\otimes d})$ which define a closed immersion $$\varphi_y = \varphi_{\mathcal{L}_y^{\otimes d}, (s_{y, 0}, \ldots, s_{y, r})} : X_y \to \mathbf{P}^r_{\kappa(y)}.$$ This is possible by Morphisms, Lemma \ref{morphisms-lemma-finite-type-over-affine-ample-very-ample} but we also use Morphisms, Lemma \ref{morphisms-lemma-image-proper-scheme-closed} to see that $\varphi_y$ is a closed immersion and Constructions, Section \ref{constructions-section-projective-space} for the description of morphisms into projective space in terms of invertible sheaves and sections. By our choice of $d_0$, after replacing $Y$ by an open neighbourhood of $y$, we can choose $s_0, \ldots, s_r \in H^0(X, \mathcal{L}^{\otimes d})$ mapping to $s_{y, 0}, \ldots, s_{y, r}$. Let $X_{s_i} \subset X$ be the open subspace where $s_i$ is a generator of $\mathcal{L}^{\otimes d}$. Since the $s_{y, i}$ generate $\mathcal{L}_y^{\otimes d}$ we see that $|X_y| \subset U = \bigcup |X_{s_i}|$. Since $X \to Y$ is closed, we see that there is an open neighbourhood $y \in V \subset Y$ such that $|f|^{-1}(V) \subset U$. After replacing $Y$ by $V$ we may assume that the $s_i$ generate $\mathcal{L}^{\otimes d}$. Thus we obtain a morphism $$\varphi = \varphi_{\mathcal{L}^{\otimes d}, (s_0, \ldots, s_r)} : X \longrightarrow \mathbf{P}^r_Y$$ with $\mathcal{L}^{\otimes d} \cong \varphi^*\mathcal{O}_{\mathbf{P}^r_Y}(1)$ whose base change to $y$ gives $\varphi_y$ (strictly speaking we need to write out a proof that the construction of morphisms into projective space given in Constructions, Section \ref{constructions-section-projective-space} also works to describe morphisms of algebraic spaces into projective space; we omit the details). \medskip\noindent We will finish the proof by a sleight of hand; the correct'' proof proceeds by directly showing that $\varphi$ is a closed immersion after base changing to an open neighbourhood of $y$. Namely, by Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-proper-finite-fibre-finite-in-neighbourhood} we see that $\varphi$ is a finite over an open neighbourhood of the fibre $\mathbf{P}^r_{\kappa(y)}$ of $\mathbf{P}^r_Y \to Y$ above $y$. Using that $\mathbf{P}^r_Y \to Y$ is closed, after shrinking $Y$ we may assume that $\varphi$ is finite. In particular $X$ is a scheme. Then $\mathcal{L}^{\otimes d} \cong \varphi^*\mathcal{O}_{\mathbf{P}^r_Y}(1)$ is ample by the very general Morphisms, Lemma \ref{morphisms-lemma-pullback-ample-tensor-relatively-ample}. \end{proof} \section{Closed subspaces of relative proj} \label{section-closed-in-relative-proj} \noindent Some auxiliary lemmas about closed subspaces of relative proj. This section is the analogue of Divisors, Section \ref{divisors-section-closed-in-relative-proj}. \begin{lemma} \label{lemma-closed-subscheme-proj} 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 $Z = \underline{\text{Proj}}_X(\mathcal{A}/\mathcal{I})$. \end{lemma} \begin{proof} The morphism $\pi$ is separated by Lemma \ref{lemma-relative-proj-separated}. As $\pi$ is quasi-compact, $\pi_*$ transforms quasi-coherent modules into quasi-coherent modules, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-pushforward}. 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 \ref{lemma-surjective-graded-rings-map-relative-proj}. Hence it suffices to prove $Z = Z'$ as closed subspaces of $P$. \medskip\noindent Having said this, the question is \'etale local on the base and we reduce to the case of schemes (Divisors, Lemma \ref{divisors-lemma-closed-subscheme-proj}) by \'etale localization. \end{proof} \noindent In case the closed subspace is locally cut out by finitely many equations we can define it by a finite type ideal sheaf of $\mathcal{A}$. \begin{lemma} \label{lemma-closed-subscheme-proj-finite} 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 $Z = \underline{\text{Proj}}_X(\mathcal{A}/\mathcal{F}\mathcal{A})$. \end{lemma} \begin{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 \ref{lemma-closed-subscheme-proj}. Choose an affine scheme $U$ and a surjective \'etale morphism $U \to X$, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-quasi-compact-affine-cover}. By the case of schemes (Divisors, Lemma \ref{divisors-lemma-closed-subscheme-proj-finite}) 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 \ref{spaces-limits-lemma-directed-colimit-finite-type} 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 \ref{lemma-surjective-generated-degree-1-map-relative-proj}) 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. \end{proof} \begin{lemma} \label{lemma-closed-subscheme-proj-finite-type} 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 \begin{enumerate} \item $\pi$ is quasi-compact, \item $i$ of finite presentation, \item $|U| \cap |\pi|(|i|(|Z|)) = \emptyset$, \item $U$ is quasi-compact, \item $\mathcal{A}_n$ is a finite type $\mathcal{O}_X$-module for all $n$. \end{enumerate} 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 $U$. \end{lemma} \begin{proof} We use the same trick as in the proof of Lemma \ref{lemma-closed-subscheme-proj-finite} to reduce to the case of schemes. Let $\mathcal{I} \subset \mathcal{A}$ be the quasi-coherent graded ideal cutting out $Z$ of Lemma \ref{lemma-closed-subscheme-proj}. Choose an affine scheme $W$ and a surjective \'etale morphism $W \to X$, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-quasi-compact-affine-cover}. By the case of schemes (Divisors, Lemma \ref{divisors-lemma-closed-subscheme-proj-finite-type}) 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 \ref{spaces-limits-lemma-directed-colimit-finite-type} 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 \ref{lemma-surjective-generated-degree-1-map-relative-proj}) 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. \end{proof} \begin{lemma} \label{lemma-conormal-sheaf-section-projective-bundle} 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 $$\Ker(\mathcal{E} \to \mathcal{L}) \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes -1} \longrightarrow \mathcal{C}_{\sigma(X)/\mathbf{P}(\mathcal{E})}$$ Both the bijection and isomorphism are compatible with base change. \end{lemma} \begin{proof} Because the constructions are compatible with base change, it suffices to check the statement \'etale locally on $X$. Thus we may assume $X$ is a scheme and the result is Divisors, Lemma \ref{divisors-lemma-conormal-sheaf-section-projective-bundle}. \end{proof} \section{Blowing up} \label{section-blowing-up} \noindent Blowing up is an important tool in algebraic geometry. \begin{definition} \label{definition-blow-up} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{I} \subset \mathcal{O}_X$ be a quasi-coherent sheaf of ideals, and let $Z \subset X$ be the closed subspace corresponding to $\mathcal{I}$ (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-closed-immersion-ideals}). The {\it blowing up of $X$ along $Z$}, or the {\it blowing up of $X$ in the ideal sheaf $\mathcal{I}$} is the morphism $$b : \underline{\text{Proj}}_X \left(\bigoplus\nolimits_{n \geq 0} \mathcal{I}^n\right) \longrightarrow X$$ The {\it exceptional divisor} of the blow up is the inverse image $b^{-1}(Z)$. Sometimes $Z$ is called the {\it center} of the blowup. \end{definition} \noindent We will see later that the exceptional divisor is an effective Cartier divisor. Moreover, the blowing up is characterized as the smallest'' algebraic space over $X$ such that the inverse image of $Z$ is an effective Cartier divisor. \medskip\noindent If $b : X' \to X$ is the blow up of $X$ in $Z$, then we often denote $\mathcal{O}_{X'}(n)$ the twists of the structure sheaf. Note that these are invertible $\mathcal{O}_{X'}$-modules and that $\mathcal{O}_{X'}(n) = \mathcal{O}_{X'}(1)^{\otimes n}$ because $X'$ is the relative Proj of a quasi-coherent graded $\mathcal{O}_X$-algebra which is generated in degree $1$, see Lemma \ref{lemma-relative-proj-generated-in-degree-1}. \begin{lemma} \label{lemma-blowing-up-affine} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{I} \subset \mathcal{O}_X$ be a quasi-coherent sheaf of ideals. Let $U = \Spec(A)$ be an affine scheme \'etale over $X$ and let $I \subset A$ be the ideal corresponding to $\mathcal{I}|_U$. If $X' \to X$ is the blow up of $X$ in $\mathcal{I}$, then there is a canonical isomorphism $$U \times_X X' = \text{Proj}(\bigoplus\nolimits_{d \geq 0} I^d)$$ of schemes over $U$, where the right hand side is the homogeneous spectrum of the Rees algebra of $I$ in $A$. Moreover, $U \times_X X'$ has an affine open covering by spectra of the affine blowup algebras $A[\frac{I}{a}]$. \end{lemma} \begin{proof} Note that the restriction $\mathcal{I}|_U$ is equal to the pullback of $\mathcal{I}$ via the morphism $U \to X$, see Properties of Spaces, Section \ref{spaces-properties-section-modules}. Thus the lemma follows on combining Lemma \ref{lemma-relative-proj} with Divisors, Lemma \ref{divisors-lemma-blowing-up-affine}. \end{proof} \begin{lemma} \label{lemma-flat-base-change-blowing-up} Let $S$ be a scheme. Let $X_1 \to X_2$ be a flat morphism of algebraic spaces over $S$. Let $Z_2 \subset X_2$ be a closed subspace. Let $Z_1$ be the inverse image of $Z_2$ in $X_1$. Let $X'_i$ be the blow up of $Z_i$ in $X_i$. Then there exists a cartesian diagram $$\xymatrix{ X_1' \ar[r] \ar[d] & X_2' \ar[d] \\ X_1 \ar[r] & X_2 }$$ of algebraic spaces over $S$. \end{lemma} \begin{proof} Let $\mathcal{I}_2$ be the ideal sheaf of $Z_2$ in $X_2$. Denote $g : X_1 \to X_2$ the given morphism. Then the ideal sheaf $\mathcal{I}_1$ of $Z_1$ is the image of $g^*\mathcal{I}_2 \to \mathcal{O}_{X_1}$ (see Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-inverse-image-closed-subspace} and discussion following the definition). By Lemma \ref{lemma-relative-proj-base-change} we see that $X_1 \times_{X_2} X_2'$ is the relative Proj of $\bigoplus_{n \geq 0} g^*\mathcal{I}_2^n$. Because $g$ is flat the map $g^*\mathcal{I}_2^n \to \mathcal{O}_{X_1}$ is injective with image $\mathcal{I}_1^n$. Thus we see that $X_1 \times_{X_2} X_2' = X_1'$. \end{proof} \begin{lemma} \label{lemma-blowing-up-gives-effective-Cartier-divisor} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $Z \subset X$ be a closed subspace. The blowing up $b : X' \to X$ of $Z$ in $X$ has the following properties: \begin{enumerate} \item $b|_{b^{-1}(X \setminus Z)} : b^{-1}(X \setminus Z) \to X \setminus Z$ is an isomorphism, \item the exceptional divisor $E = b^{-1}(Z)$ is an effective Cartier divisor on $X'$, \item there is a canonical isomorphism $\mathcal{O}_{X'}(-1) = \mathcal{O}_{X'}(E)$ \end{enumerate} \end{lemma} \begin{proof} Let $U$ be a scheme and let $U \to X$ be a surjective \'etale morphism. As blowing up commutes with flat base change (Lemma \ref{lemma-flat-base-change-blowing-up}) we can prove each of these statements after base change to $U$. This reduces us to the case of schemes. In this case the result is Divisors, Lemma \ref{divisors-lemma-blowing-up-gives-effective-Cartier-divisor}. \end{proof} \begin{lemma}[Universal property blowing up] \label{lemma-universal-property-blowing-up} \begin{slogan} Blow up a closed subset to make it Cartier. \end{slogan} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $Z \subset X$ be a closed subspace. Let $\mathcal{C}$ be the full subcategory of $(\textit{Spaces}/X)$ consisting of $Y \to X$ such that the inverse image of $Z$ is an effective Cartier divisor on $Y$. Then the blowing up $b : X' \to X$ of $Z$ in $X$ is a final object of $\mathcal{C}$. \end{lemma} \begin{proof} We see that $b : X' \to X$ is an object of $\mathcal{C}$ according to Lemma \ref{lemma-blowing-up-gives-effective-Cartier-divisor}. Let $f : Y \to X$ be an object of $\mathcal{C}$. We have to show there exists a unique morphism $Y \to X'$ over $X$. Let $D = f^{-1}(Z)$. Let $\mathcal{I} \subset \mathcal{O}_X$ be the ideal sheaf of $Z$ and let $\mathcal{I}_D$ be the ideal sheaf of $D$. Then $f^*\mathcal{I} \to \mathcal{I}_D$ is a surjection to an invertible $\mathcal{O}_Y$-module. This extends to a map $\psi : \bigoplus f^*\mathcal{I}^d \to \bigoplus \mathcal{I}_D^d$ of graded $\mathcal{O}_Y$-algebras. (We observe that $\mathcal{I}_D^d = \mathcal{I}_D^{\otimes d}$ as $D$ is an effective Cartier divisor.) By Lemma \ref{lemma-relative-proj-generated-in-degree-1}. the triple $(f : Y \to X, \mathcal{I}_D, \psi)$ defines a morphism $Y \to X'$ over $X$. The restriction $$Y \setminus D \longrightarrow X' \setminus b^{-1}(Z) = X \setminus Z$$ is unique. The open $Y \setminus D$ is scheme theoretically dense in $Y$ according to Lemma \ref{lemma-complement-effective-Cartier-divisor}. Thus the morphism $Y \to X'$ is unique by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-equality-of-morphisms} (also $b$ is separated by Lemma \ref{lemma-relative-proj-separated}). \end{proof} \begin{lemma} \label{lemma-blow-up-effective-Cartier-divisor} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $Z \subset X$ be an effective Cartier divisor. The blowup of $X$ in $Z$ is the identity morphism of $X$. \end{lemma} \begin{proof} Immediate from the universal property of blowups (Lemma \ref{lemma-universal-property-blowing-up}). \end{proof} \begin{lemma} \label{lemma-blow-up-reduced-space} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{I} \subset \mathcal{O}_X$ be a quasi-coherent sheaf of ideals. If $X$ is reduced, then the blow up $X'$ of $X$ in $\mathcal{I}$ is reduced. \end{lemma} \begin{proof} Let $U$ be a scheme and let $U \to X$ be a surjective \'etale morphism. As blowing up commutes with flat base change (Lemma \ref{lemma-flat-base-change-blowing-up}) we can prove each of these statements after base change to $U$. This reduces us to the case of schemes. In this case the result is Divisors, Lemma \ref{divisors-lemma-blow-up-reduced-scheme}. \end{proof} \begin{lemma} \label{lemma-blowup-finite-nr-irreducibles} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $b : X' \to X$ be the blowup of $X$ is a closed subspace. If $X$ satisfies the equivalent conditions of Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-prepare-normalization} then so does $X'$. \end{lemma} \begin{proof} Follows immediately from the lemma cited in the statement, the \'etale local description of blowing ups in Lemma \ref{lemma-blowing-up-affine}, and Divisors, Lemma \ref{divisors-lemma-blow-up-and-irreducible-components}. \end{proof} \begin{lemma} \label{lemma-blow-up-pullback-effective-Cartier} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $b : X' \to X$ be a blow up of $X$ in a closed subspace. For any effective Cartier divisor $D$ on $X$ the pullback $b^{-1}D$ is defined (see Definition \ref{definition-pullback-effective-Cartier-divisor}). \end{lemma} \begin{proof} By Lemmas \ref{lemma-blowing-up-affine} and \ref{lemma-characterize-effective-Cartier-divisor} this reduces to the following algebra fact: Let $A$ be a ring, $I \subset A$ an ideal, $a \in I$, and $x \in A$ a nonzerodivisor. Then the image of $x$ in $A[\frac{I}{a}]$ is a nonzerodivisor. Namely, suppose that $x (y/a^n) = 0$ in $A[\frac{I}{a}]$. Then $a^mxy = 0$ in $A$ for some $m$. Hence $a^my = 0$ as $x$ is a nonzerodivisor. Whence $y/a^n$ is zero in $A[\frac{I}{a}]$ as desired. \end{proof} \begin{lemma} \label{lemma-blowing-up-two-ideals} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{I} \subset \mathcal{O}_X$ and $\mathcal{J}$ be quasi-coherent sheaves of ideals. Let $b : X' \to X$ be the blowing up of $X$ in $\mathcal{I}$. Let $b' : X'' \to X'$ be the blowing up of $X'$ in $b^{-1}\mathcal{J} \mathcal{O}_{X'}$. Then $X'' \to X$ is canonically isomorphic to the blowing up of $X$ in $\mathcal{I}\mathcal{J}$. \end{lemma} \begin{proof} Let $E \subset X'$ be the exceptional divisor of $b$ which is an effective Cartier divisor by Lemma \ref{lemma-blowing-up-gives-effective-Cartier-divisor}. Then $(b')^{-1}E$ is an effective Cartier divisor on $X''$ by Lemma \ref{lemma-blow-up-pullback-effective-Cartier}. Let $E' \subset X''$ be the exceptional divisor of $b'$ (also an effective Cartier divisor). Consider the effective Cartier divisor $E'' = E' + (b')^{-1}E$. By construction the ideal of $E''$ is $(b \circ b')^{-1}\mathcal{I} (b \circ b')^{-1}\mathcal{J} \mathcal{O}_{X''}$. Hence according to Lemma \ref{lemma-universal-property-blowing-up} there is a canonical morphism from $X''$ to the blowup $c : Y \to X$ of $X$ in $\mathcal{I}\mathcal{J}$. Conversely, as $\mathcal{I}\mathcal{J}$ pulls back to an invertible ideal we see that $c^{-1}\mathcal{I}\mathcal{O}_Y$ defines an effective Cartier divisor, see Lemma \ref{lemma-sum-closed-subschemes-effective-Cartier}. Thus a morphism $c' : Y \to X'$ over $X$ by Lemma \ref{lemma-universal-property-blowing-up}. Then $(c')^{-1}b^{-1}\mathcal{J}\mathcal{O}_Y = c^{-1}\mathcal{J}\mathcal{O}_Y$ which also defines an effective Cartier divisor. Thus a morphism $c'' : Y \to X''$ over $X'$. We omit the verification that this morphism is inverse to the morphism $X'' \to Y$ constructed earlier. \end{proof} \begin{lemma} \label{lemma-blowing-up-projective} Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{I} \subset \mathcal{O}_X$ be a quasi-coherent sheaf of ideals. Let $b : X' \to X$ be the blowing up of $X$ in the ideal sheaf $\mathcal{I}$. If $\mathcal{I}$ is of finite type, then $b : X' \to X$ is a proper morphism. \end{lemma} \begin{proof} Let $U$ be a scheme and let $U \to X$ be a surjective \'etale morphism. As blowing up commutes with flat base change (Lemma \ref{lemma-flat-base-change-blowing-up}) we can prove each of these statements after base change to $U$ (see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-proper-local}). This reduces us to the case of schemes. In this case the morphism $b$ is projective by Divisors, Lemma \ref{divisors-lemma-blowing-up-projective} hence proper by Morphisms, Lemma \ref{morphisms-lemma-locally-projective-proper}. \end{proof} \begin{lemma} \label{lemma-composition-finite-type-blowups} Let $S$ be a scheme and let $X$ be an algebraic space over $S$. Assume $X$ is quasi-compact and quasi-separated. Let $Z \subset X$ be a closed subspace of finite presentation. Let $b : X' \to X$ be the blowing up with center $Z$. Let $Z' \subset X'$ be a closed subspace of finite presentation. Let $X'' \to X'$ be the blowing up with center $Z'$. There exists a closed subspace $Y \subset X$ of finite presentation, such that \begin{enumerate} \item $|Y| = |Z| \cup |b|(|Z'|)$, and \item the composition $X'' \to X$ is isomorphic to the blowing up of $X$ in $Y$. \end{enumerate} \end{lemma} \begin{proof} The condition that $Z \to X$ is of finite presentation means that $Z$ is cut out by a finite type quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_X$, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-closed-immersion-finite-presentation}. Write $\mathcal{A} = \bigoplus_{n \geq 0} \mathcal{I}^n$ so that $X' = \underline{\text{Proj}}(\mathcal{A})$. Note that $X \setminus Z$ is a quasi-compact open subspace of $X$ by Limits of Spaces, Lemma \ref{spaces-limits-lemma-quasi-coherent-finite-type-ideals}. Since $b^{-1}(X \setminus Z) \to X \setminus Z$ is an isomorphism (Lemma \ref{lemma-blowing-up-gives-effective-Cartier-divisor}) the same result shows that $b^{-1}(X \setminus Z) \setminus Z'$ is quasi-compact open subspace in $X'$. Hence $U = X \setminus (Z \cup b(Z'))$ is quasi-compact open subspace in $X$. By Lemma \ref{lemma-closed-subscheme-proj-finite-type} there exist a $d > 0$ and a finite type $\mathcal{O}_X$-submodule $\mathcal{F} \subset \mathcal{I}^d$ such that $Z' = \underline{\text{Proj}}(\mathcal{A}/\mathcal{F}\mathcal{A})$ and such that the support of $\mathcal{I}^d/\mathcal{F}$ is contained in $X \setminus U$. \medskip\noindent Since $\mathcal{F} \subset \mathcal{I}^d$ is an $\mathcal{O}_X$-submodule we may think of $\mathcal{F} \subset \mathcal{I}^d \subset \mathcal{O}_X$ as a finite type quasi-coherent sheaf of ideals on $X$. Let's denote this $\mathcal{J} \subset \mathcal{O}_X$ to prevent confusion. Since $\mathcal{I}^d / \mathcal{J}$ and $\mathcal{O}/\mathcal{I}^d$ are supported on $|X| \setminus |U|$ we see that $|V(\mathcal{J})|$ is contained in $|X| \setminus |U|$. Conversely, as $\mathcal{J} \subset \mathcal{I}^d$ we see that $|Z| \subset |V(\mathcal{J})|$. Over $X \setminus Z \cong X' \setminus b^{-1}(Z)$ the sheaf of ideals $\mathcal{J}$ cuts out $Z'$ (see displayed formula below). Hence $|V(\mathcal{J})|$ equals $|Z| \cup |b|(|Z'|)$. It follows that also $|V(\mathcal{I}\mathcal{J})| = |Z| \cup |b|(|Z'|)$. Moreover, $\mathcal{I}\mathcal{J}$ is an ideal of finite type as a product of two such. We claim that $X'' \to X$ is isomorphic to the blowing up of $X$ in $\mathcal{I}\mathcal{J}$ which finishes the proof of the lemma by setting $Y = V(\mathcal{I}\mathcal{J})$. \medskip\noindent First, recall that the blow up of $X$ in $\mathcal{I}\mathcal{J}$ is the same as the blow up of $X'$ in $b^{-1}\mathcal{J} \mathcal{O}_{X'}$, see Lemma \ref{lemma-blowing-up-two-ideals}. Hence it suffices to show that the blow up of $X'$ in $b^{-1}\mathcal{J} \mathcal{O}_{X'}$ agrees with the blow up of $X'$ in $Z'$. We will show that $$b^{-1}\mathcal{J} \mathcal{O}_{X'} = \mathcal{I}_E^d \mathcal{I}_{Z'}$$ as ideal sheaves on $X''$. This will prove what we want as $\mathcal{I}_E^d$ cuts out the effective Cartier divisor $dE$ and we can use Lemmas \ref{lemma-blow-up-effective-Cartier-divisor} and \ref{lemma-blowing-up-two-ideals}. \medskip\noindent To see the displayed equality of the ideals we may work locally. With notation $A$, $I$, $a \in I$ as in Lemma \ref{lemma-blowing-up-affine} we see that $\mathcal{F}$ corresponds to an $R$-submodule $M \subset I^d$ mapping isomorphically to an ideal $J \subset R$. The condition $Z' = \underline{\text{Proj}}(\mathcal{A}/\mathcal{F}\mathcal{A})$ means that $Z' \cap \Spec(A[\frac{I}{a}])$ is cut out by the ideal generated by the elements $m/a^d$, $m \in M$. Say the element $m \in M$ corresponds to the function $f \in J$. Then in the affine blowup algebra $A' = A[\frac{I}{a}]$ we see that $f = (a^dm)/a^d = a^d (m/a^d)$. Thus the equality holds. \end{proof} \section{Strict transform} \label{section-strict-transform} \noindent This section is the analogue of Divisors, Section \ref{divisors-section-strict-transform}. Let $S$ be a scheme, let $B$ be an algebraic space over $S$, and let $Z \subset B$ be a closed subspace. Let $b : B' \to B$ be the blowing up of $B$ in $Z$ and denote $E \subset B'$ the exceptional divisor $E = b^{-1}Z$. In the following we will often consider an algebraic space $X$ over $B$ and form the cartesian diagram $$\xymatrix{ \text{pr}_{B'}^{-1}E \ar[r] \ar[d] & X \times_B B' \ar[r]_-{\text{pr}_X} \ar[d]_{\text{pr}_{B'}} & X \ar[d]^f \\ E \ar[r] & B' \ar[r] & B }$$ Since $E$ is an effective Cartier divisor (Lemma \ref{lemma-blowing-up-gives-effective-Cartier-divisor}) we see that $\text{pr}_{B'}^{-1}E \subset X \times_B B'$ is locally principal (Lemma \ref{lemma-pullback-locally-principal}). Thus the inclusion morphism of the complement of $\text{pr}_{B'}^{-1}E$ in $X \times_B B'$ is affine and in particular quasi-compact (Lemma \ref{lemma-complement-locally-principal-closed-subscheme}). Consequently, for a quasi-coherent $\mathcal{O}_{X \times_B B'}$-module $\mathcal{G}$ the subsheaf of sections supported on $|\text{pr}_{B'}^{-1}E|$ is a quasi-coherent submodule, see Limits of Spaces, Definition \ref{spaces-limits-definition-subsheaf-sections-supported-on-closed}. If $\mathcal{G}$ is a quasi-coherent sheaf of algebras, e.g., $\mathcal{G} = \mathcal{O}_{X \times_B B'}$, then this subsheaf is an ideal of $\mathcal{G}$. \begin{definition} \label{definition-strict-transform} With $Z \subset B$ and $f : X \to B$ as above. \begin{enumerate} \item Given a quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}$ the {\it strict transform} of $\mathcal{F}$ with respect to the blowup of $B$ in $Z$ is the quotient $\mathcal{F}'$ of $\text{pr}_X^*\mathcal{F}$ by the submodule of sections supported on $|\text{pr}_{B'}^{-1}E|$. \item The {\it strict transform} of $X$ is the closed subspace $X' \subset X \times_B B'$ cut out by the quasi-coherent ideal of sections of $\mathcal{O}_{X \times_B B'}$ supported on $|\text{pr}_{B'}^{-1}E|$. \end{enumerate} \end{definition} \noindent Note that taking the strict transform along a blowup depends on the closed subspace used for the blowup (and not just on the morphism $B' \to B$). \begin{lemma}[\'Etale localization and strict transform] \label{lemma-strict-transform-local} In the situation of Definition \ref{definition-strict-transform}. Let $$\xymatrix{ U \ar[r] \ar[d] & X \ar[d] \\ V \ar[r] & B }$$ be a commutative diagram of morphisms with $U$ and $V$ schemes and \'etale horizontal arrows. Let $V' \to V$ be the blowup of $V$ in $Z \times_B V$. Then \begin{enumerate} \item $V' = V \times_B B'$ and the maps $V' \to B'$ and $U \times_V V' \to X \times_B B'$ are \'etale, \item the strict transform $U'$ of $U$ relative to $V' \to V$ is equal to $X' \times_X U$ where $X'$ is the strict transform of $X$ relative to $B' \to B$, and \item for a quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}$ the restriction of the strict transform $\mathcal{F}'$ to $U \times_V V'$ is the strict transform of $\mathcal{F}|_U$ relative to $V' \to V$. \end{enumerate} \end{lemma} \begin{proof} Part (1) follows from the fact that blowup commutes with flat base change (Lemma \ref{lemma-flat-base-change-blowing-up}), the fact that