\input{preamble} % OK, start here. % \begin{document} \title{Morphisms of Schemes} \maketitle \phantomsection \label{section-phantom} \tableofcontents \section{Introduction} \label{section-introduction} \noindent In this chapter we introduce some types of morphisms of schemes. A basic reference is \cite{EGA}. \section{Closed immersions} \label{section-closed-immersions} \noindent In this section we elucidate some of the results obtained previously on closed immersions of schemes. Recall that a morphism of schemes $i : Z \to X$ is defined to be a closed immersion if (a) $i$ induces a homeomorphism onto a closed subset of $X$, (b) $i^\sharp : \mathcal{O}_X \to i_*\mathcal{O}_Z$ is surjective, and (c) the kernel of $i^\sharp$ is locally generated by sections, see Schemes, Definitions \ref{schemes-definition-immersion} and \ref{schemes-definition-closed-immersion-locally-ringed-spaces}. It turns out that, given that $Z$ and $X$ are schemes, there are many different ways of characterizing a closed immersion. \begin{lemma} \label{lemma-closed-immersion} Let $i : Z \to X$ be a morphism of schemes. The following are equivalent: \begin{enumerate} \item The morphism $i$ is a closed immersion. \item For every affine open $\Spec(R) = U \subset X$, there exists an ideal $I \subset R$ such that $i^{-1}(U) = \Spec(R/I)$ as schemes over $U = \Spec(R)$. \item There exists an affine open covering $X = \bigcup_{j \in J} U_j$, $U_j = \Spec(R_j)$ and for every $j \in J$ there exists an ideal $I_j \subset R_j$ such that $i^{-1}(U_j) = \Spec(R_j/I_j)$ as schemes over $U_j = \Spec(R_j)$. \item The morphism $i$ induces a homeomorphism of $Z$ with a closed subset of $X$ and $i^\sharp : \mathcal{O}_X \to i_*\mathcal{O}_Z$ is surjective. \item The morphism $i$ induces a homeomorphism of $Z$ with a closed subset of $X$, the map $i^\sharp : \mathcal{O}_X \to i_*\mathcal{O}_Z$ is surjective, and the kernel $\Ker(i^\sharp)\subset \mathcal{O}_X$ is a quasi-coherent sheaf of ideals. \item The morphism $i$ induces a homeomorphism of $Z$ with a closed subset of $X$, the map $i^\sharp : \mathcal{O}_X \to i_*\mathcal{O}_Z$ is surjective, and the kernel $\Ker(i^\sharp)\subset \mathcal{O}_X$ is a sheaf of ideals which is locally generated by sections. \end{enumerate} \end{lemma} \begin{proof} Condition (6) is our definition of a closed immersion, see Schemes, Definitions \ref{schemes-definition-closed-immersion-locally-ringed-spaces} and \ref{schemes-definition-immersion}. So (6) $\Leftrightarrow$ (1). We have (1) $\Rightarrow$ (2) by Schemes, Lemma \ref{schemes-lemma-closed-subspace-scheme}. Trivially (2) $\Rightarrow$ (3). \medskip\noindent Assume (3). Each of the morphisms $\Spec(R_j/I_j) \to \Spec(R_j)$ is a closed immersion, see Schemes, Example \ref{schemes-example-closed-immersion-affines}. Hence $i^{-1}(U_j) \to U_j$ is a homeomorphism onto its image and $i^\sharp|_{U_j}$ is surjective. Hence $i$ is a homeomorphism onto its image and $i^\sharp$ is surjective since this may be checked locally. We conclude that (3) $\Rightarrow$ (4). \medskip\noindent The implication (4) $\Rightarrow$ (1) is Schemes, Lemma \ref{schemes-lemma-characterize-closed-immersions}. The implication (5) $\Rightarrow$ (6) is trivial. And the implication (6) $\Rightarrow$ (5) follows from Schemes, Lemma \ref{schemes-lemma-closed-subspace-scheme}. \end{proof} \begin{lemma} \label{lemma-closed-immersion-ideals} Let $X$ be a scheme. Let $i : Z \to X$ and $i' : Z' \to X$ be closed immersions and consider the ideal sheaves $\mathcal{I} = \Ker(i^\sharp)$ and $\mathcal{I}' = \Ker((i')^\sharp)$ of $\mathcal{O}_X$. \begin{enumerate} \item The morphism $i : Z \to X$ factors as $Z \to Z' \to X$ for some $a : Z \to Z'$ if and only if $\mathcal{I}' \subset \mathcal{I}$. If this happens, then $a$ is a closed immersion. \item We have $Z \cong Z'$ over $X$ if and only if $\mathcal{I} = \mathcal{I}'$. \end{enumerate} \end{lemma} \begin{proof} This follows from our discussion of closed subspaces in Schemes, Section \ref{schemes-section-closed-immersion} especially Schemes, Lemmas \ref{schemes-lemma-closed-immersion} and \ref{schemes-lemma-characterize-closed-subspace}. It also follows in a straightforward way from characterization (3) in Lemma \ref{lemma-closed-immersion} above. \end{proof} \begin{lemma} \label{lemma-closed-immersion-bijection-ideals} Let $X$ be a scheme. Let $\mathcal{I} \subset \mathcal{O}_X$ be a sheaf of ideals. The following are equivalent: \begin{enumerate} \item $\mathcal{I}$ is locally generated by sections as a sheaf of $\mathcal{O}_X$-modules, \item $\mathcal{I}$ is quasi-coherent as a sheaf of $\mathcal{O}_X$-modules, and \item there exists a closed immersion $i : Z \to X$ of schemes whose corresponding sheaf of ideals $\Ker(i^\sharp)$ is equal to $\mathcal{I}$. \end{enumerate} \end{lemma} \begin{proof} The equivalence of (1) and (2) is immediate from Schemes, Lemma \ref{schemes-lemma-closed-subspace-scheme}. If (1) holds, then there is a closed subspace $i : Z \to X$ with $\mathcal{I} = \Ker(i^\sharp)$ by Schemes, Definition \ref{schemes-definition-closed-subspace} and Example \ref{schemes-example-closed-subspace}. By Schemes, Lemma \ref{schemes-lemma-closed-subspace-scheme} this is a closed immersion of schemes and (3) holds. Conversely, if (3) holds, then (2) holds by Schemes, Lemma \ref{schemes-lemma-closed-subspace-scheme} (which applies because a closed immersion of schemes is a fortiori a closed immersion of locally ringed spaces). \end{proof} \begin{lemma} \label{lemma-base-change-closed-immersion} The base change of a closed immersion is a closed immersion. \end{lemma} \begin{proof} See Schemes, Lemma \ref{schemes-lemma-base-change-immersion}. \end{proof} \begin{lemma} \label{lemma-composition-closed-immersion} A composition of closed immersions is a closed immersion. \end{lemma} \begin{proof} We have seen this in Schemes, Lemma \ref{schemes-lemma-composition-immersion}, but here is another proof. Namely, it follows from the characterization (3) of closed immersions in Lemma \ref{lemma-closed-immersion}. Since if $I \subset R$ is an ideal, and $\overline{J} \subset R/I$ is an ideal, then $\overline{J} = J/I$ for some ideal $J \subset R$ which contains $I$ and $(R/I)/\overline{J} = R/J$. \end{proof} \begin{lemma} \label{lemma-closed-immersion-quasi-compact} A closed immersion is quasi-compact. \end{lemma} \begin{proof} This lemma is a duplicate of Schemes, Lemma \ref{schemes-lemma-closed-immersion-quasi-compact}. \end{proof} \begin{lemma} \label{lemma-closed-immersion-separated} A closed immersion is separated. \end{lemma} \begin{proof} This lemma is a special case of Schemes, Lemma \ref{schemes-lemma-immersions-monomorphisms}. \end{proof} \section{Immersions} \label{section-immersions} \noindent In this section we collect some facts on immersions. \begin{lemma} \label{lemma-immersion-permanence} Let $Z \to Y \to X$ be morphisms of schemes. \begin{enumerate} \item If $Z \to X$ is an immersion, then $Z \to Y$ is an immersion. \item If $Z \to X$ is a quasi-compact immersion and $Y \to X$ is quasi-separated, then $Z \to Y$ is a quasi-compact immersion. \item If $Z \to X$ is a closed immersion and $Y \to X$ is separated, then $Z \to Y$ is a closed immersion. \end{enumerate} \end{lemma} \begin{proof} In each case the proof is to contemplate the commutative diagram $$\xymatrix{ Z \ar[r] \ar[rd] & Y \times_X Z \ar[r] \ar[d] & Z \ar[d] \\ & Y \ar[r] & X }$$ where the composition of the top horizontal arrows is the identity. Let us prove (1). The first horizontal arrow is a section of $Y \times_X Z \to Z$, whence an immersion by Schemes, Lemma \ref{schemes-lemma-section-immersion}. The arrow $Y \times_X Z \to Y$ is a base change of $Z \to X$ hence an immersion (Schemes, Lemma \ref{schemes-lemma-base-change-immersion}). Finally, a composition of immersions is an immersion (Schemes, Lemma \ref{schemes-lemma-composition-immersion}). This proves (1). The other two results are proved in exactly the same manner. \end{proof} \begin{lemma} \label{lemma-factor-quasi-compact-immersion} Let $h : Z \to X$ be an immersion. If $h$ is quasi-compact, then we can factor $h = i \circ j$ with $j : Z \to \overline{Z}$ an open immersion and $i : \overline{Z} \to X$ a closed immersion. \end{lemma} \begin{proof} Note that $h$ is quasi-compact and quasi-separated (see Schemes, Lemma \ref{schemes-lemma-immersions-monomorphisms}). Hence $h_*\mathcal{O}_Z$ is a quasi-coherent sheaf of $\mathcal{O}_X$-modules by Schemes, Lemma \ref{schemes-lemma-push-forward-quasi-coherent}. This implies that $\mathcal{I} = \Ker(\mathcal{O}_X \to h_*\mathcal{O}_Z)$ is a quasi-coherent sheaf of ideals, see Schemes, Section \ref{schemes-section-quasi-coherent}. Let $\overline{Z} \subset X$ be the closed subscheme corresponding to $\mathcal{I}$, see Lemma \ref{lemma-closed-immersion-bijection-ideals}. By Schemes, Lemma \ref{schemes-lemma-characterize-closed-subspace} the morphism $h$ factors as $h = i \circ j$ where $i : \overline{Z} \to X$ is the inclusion morphism. To see that $j$ is an open immersion, choose an open subscheme $U \subset X$ such that $h$ induces a closed immersion of $Z$ into $U$. Then it is clear that $\mathcal{I}|_U$ is the sheaf of ideals corresponding to the closed immersion $Z \to U$. Hence we see that $Z = \overline{Z} \cap U$. \end{proof} \begin{lemma} \label{lemma-factor-reduced-immersion} Let $h : Z \to X$ be an immersion. If $Z$ is reduced, then we can factor $h = i \circ j$ with $j : Z \to \overline{Z}$ an open immersion and $i : \overline{Z} \to X$ a closed immersion. \end{lemma} \begin{proof} Let $\overline{Z} \subset X$ be the closure of $h(Z)$ with the reduced induced closed subscheme structure, see Schemes, Definition \ref{schemes-definition-reduced-induced-scheme}. By Schemes, Lemma \ref{schemes-lemma-map-into-reduction} the morphism $h$ factors as $h = i \circ j$ with $i : \overline{Z} \to X$ the inclusion morphism and $j : Z \to \overline{Z}$. From the definition of an immersion we see there exists an open subscheme $U \subset X$ such that $h$ factors through a closed immersion into $U$. Hence $\overline{Z} \cap U$ and $h(Z)$ are reduced closed subschemes of $U$ with the same underlying closed set. Hence by the uniqueness in Schemes, Lemma \ref{schemes-lemma-reduced-closed-subscheme} we see that $h(Z) \cong \overline{Z} \cap U$. So $j$ induces an isomorphism of $Z$ with $\overline{Z} \cap U$. In other words $j$ is an open immersion. \end{proof} \begin{example} \label{example-thibaut} Here is an example of an immersion which is not a composition of an open immersion followed by a closed immersion. Let $k$ be a field. Let $X = \Spec(k[x_1, x_2, x_3, \ldots])$. Let $U = \bigcup_{n = 1}^{\infty} D(x_n)$. Then $U \to X$ is an open immersion. Consider the ideals $$I_n = (x_1^n, x_2^n, \ldots, x_{n - 1}^n, x_n - 1, x_{n + 1}, x_{n + 2}, \ldots) \subset k[x_1, x_2, x_3, \ldots][1/x_n].$$ Note that $I_n k[x_1, x_2, x_3, \ldots][1/x_nx_m] = (1)$ for any $m \not = n$. Hence the quasi-coherent ideals $\widetilde I_n$ on $D(x_n)$ agree on $D(x_nx_m)$, namely $\widetilde I_n|_{D(x_nx_m)} = \mathcal{O}_{D(x_n x_m)}$ if $n \not = m$. Hence these ideals glue to a quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_U$. Let $Z \subset U$ be the closed subscheme corresponding to $\mathcal{I}$. Thus $Z \to X$ is an immersion. \medskip\noindent We claim that we cannot factor $Z \to X$ as $Z \to \overline{Z} \to X$, where $\overline{Z} \to X$ is closed and $Z \to \overline{Z}$ is open. Namely, $\overline{Z}$ would have to be defined by an ideal $I \subset k[x_1, x_2, x_3, \ldots]$ such that $I_n = I k[x_1, x_2, x_3, \ldots][1/x_n]$. But the only element $f \in k[x_1, x_2, x_3, \ldots]$ which ends up in all $I_n$ is $0$! Hence $I$ does not exist. \end{example} \begin{lemma} \label{lemma-check-immersion} Let $f : Y \to X$ be a morphism of schemes. If for all $y \in Y$ there is an open subscheme $f(y) \in U \subset X$ such that $f|_{f^{-1}(U)} : f^{-1}(U) \to U$ is an immersion, then $f$ is an immersion. \end{lemma} \begin{proof} This statement follows readily from the discussion of closed subschemes at the end of Schemes, Section \ref{schemes-section-immersions} but we will also give a detailed proof. Let $Z \subset X$ be the closure of $f(Y)$. Since taking closures commutes with restricting to opens, we see from the assumption that $f(Y) \subset Z$ is open. Hence $Z' = Z \setminus f(Y)$ is closed. Hence $X' = X \setminus Z'$ is an open subscheme of $X$ and $f$ factors as $f : Y \to X'$ followed by the inclusion. If $y \in Y$ and $U \subset X$ is as in the statement of the lemma, then $U' = X' \cap U$ is an open neighbourhood of $f'(y)$ such that $(f')^{-1}(U') \to U'$ is an immersion (Lemma \ref{lemma-immersion-permanence}) with closed image. Hence it is a closed immersion, see Schemes, Lemma \ref{schemes-lemma-immersion-when-closed}. Since being a closed immersion is local on the target (for example by Lemma \ref{lemma-closed-immersion}) we conclude that $f'$ is a closed immersion as desired. \end{proof} \section{Closed immersions and quasi-coherent sheaves} \label{section-closed-immersions-quasi-coherent} \noindent The following lemma finally does for quasi-coherent sheaves on schemes what Modules, Lemma \ref{modules-lemma-i-star-exact} does for abelian sheaves. See also the discussion in Modules, Section \ref{modules-section-closed-immersion}. \begin{lemma} \label{lemma-i-star-equivalence} Let $i : Z \to X$ be a closed immersion of schemes. Let $\mathcal{I} \subset \mathcal{O}_X$ be the quasi-coherent sheaf of ideals cutting out $Z$. The functor $$i_* : \QCoh(\mathcal{O}_Z) \longrightarrow \QCoh(\mathcal{O}_X)$$ is exact, fully faithful, with essential image those quasi-coherent $\mathcal{O}_X$-modules $\mathcal{G}$ such that $\mathcal{I}\mathcal{G} = 0$. \end{lemma} \begin{proof} A closed immersion is quasi-compact and separated, see Lemmas \ref{lemma-closed-immersion-quasi-compact} and \ref{lemma-closed-immersion-separated}. Hence Schemes, Lemma \ref{schemes-lemma-push-forward-quasi-coherent} applies and the pushforward of a quasi-coherent sheaf on $Z$ is indeed a quasi-coherent sheaf on $X$. \medskip\noindent By Modules, Lemma \ref{modules-lemma-i-star-equivalence} the functor $i_*$ is fully faithful. \medskip\noindent Now we turn to the description of the essential image of the functor $i_*$. We have $\mathcal{I}(i_*\mathcal{F}) = 0$ for any quasi-coherent $\mathcal{O}_Z$-module, for example by Modules, Lemma \ref{modules-lemma-i-star-equivalence}. Next, suppose that $\mathcal{G}$ is any quasi-coherent $\mathcal{O}_X$-module such that $\mathcal{I}\mathcal{G} = 0$. It suffices to show that the canonical map $$\mathcal{G} \longrightarrow i_* i^*\mathcal{G}$$ is an isomorphism\footnote{This was proved in a more general situation in the proof of Modules, Lemma \ref{modules-lemma-i-star-equivalence}.}. In the case of schemes and quasi-coherent modules, working affine locally on $X$ and using Lemma \ref{lemma-closed-immersion} and Schemes, Lemma \ref{schemes-lemma-widetilde-pullback} it suffices to prove the following algebraic statement: Given a ring $R$, an ideal $I$ and an $R$-module $N$ such that $IN = 0$ the canonical map $$N \longrightarrow N \otimes_R R/I,\quad n \longmapsto n \otimes 1$$ is an isomorphism of $R$-modules. Proof of this easy algebra fact is omitted. \end{proof} \noindent Let $i : Z \to X$ be a closed immersion. Because of the lemma above we often, by abuse of notation, denote $\mathcal{F}$ the sheaf $i_*\mathcal{F}$ on $X$. \begin{lemma} \label{lemma-largest-quasi-coherent-subsheaf} Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Let $\mathcal{G} \subset \mathcal{F}$ be a $\mathcal{O}_X$-submodule. There exists a unique quasi-coherent $\mathcal{O}_X$-submodule $\mathcal{G}' \subset \mathcal{G}$ with the following property: For every quasi-coherent $\mathcal{O}_X$-module $\mathcal{H}$ the map $$\Hom_{\mathcal{O}_X}(\mathcal{H}, \mathcal{G}') \longrightarrow \Hom_{\mathcal{O}_X}(\mathcal{H}, \mathcal{G})$$ is bijective. In particular $\mathcal{G}'$ is the largest quasi-coherent $\mathcal{O}_X$-submodule of $\mathcal{F}$ contained in $\mathcal{G}$. \end{lemma} \begin{proof} Let $\mathcal{G}_a$, $a \in A$ be the set of quasi-coherent $\mathcal{O}_X$-submodules contained in $\mathcal{G}$. Then the image $\mathcal{G}'$ of $$\bigoplus\nolimits_{a \in A} \mathcal{G}_a \longrightarrow \mathcal{F}$$ is quasi-coherent as the image of a map of quasi-coherent sheaves on $X$ is quasi-coherent and since a direct sum of quasi-coherent sheaves is quasi-coherent, see Schemes, Section \ref{schemes-section-quasi-coherent}. The module $\mathcal{G}'$ is contained in $\mathcal{G}$. Hence this is the largest quasi-coherent $\mathcal{O}_X$-module contained in $\mathcal{G}$. \medskip\noindent To prove the formula, let $\mathcal{H}$ be a quasi-coherent $\mathcal{O}_X$-module and let $\alpha : \mathcal{H} \to \mathcal{G}$ be an $\mathcal{O}_X$-module map. The image of the composition $\mathcal{H} \to \mathcal{G} \to \mathcal{F}$ is quasi-coherent as the image of a map of quasi-coherent sheaves. Hence it is contained in $\mathcal{G}'$. Hence $\alpha$ factors through $\mathcal{G}'$ as desired. \end{proof} \begin{lemma} \label{lemma-i-upper-shriek} Let $i : Z \to X$ be a closed immersion of schemes. There is a functor\footnote{This is likely nonstandard notation.} $i^! : \QCoh(\mathcal{O}_X) \to \QCoh(\mathcal{O}_Z)$ which is a right adjoint to $i_*$. (Compare Modules, Lemma \ref{modules-lemma-i-star-right-adjoint}.) \end{lemma} \begin{proof} Given quasi-coherent $\mathcal{O}_X$-module $\mathcal{G}$ we consider the subsheaf $\mathcal{H}_Z(\mathcal{G})$ of $\mathcal{G}$ of local sections annihilated by $\mathcal{I}$. By Lemma \ref{lemma-largest-quasi-coherent-subsheaf} there is a canonical largest quasi-coherent $\mathcal{O}_X$-submodule $\mathcal{H}_Z(\mathcal{G})'$. By construction we have $$\Hom_{\mathcal{O}_X}(i_*\mathcal{F}, \mathcal{H}_Z(\mathcal{G})') = \Hom_{\mathcal{O}_X}(i_*\mathcal{F}, \mathcal{G})$$ for any quasi-coherent $\mathcal{O}_Z$-module $\mathcal{F}$. Hence we can set $i^!\mathcal{G} = i^*(\mathcal{H}_Z(\mathcal{G})')$. Details omitted. \end{proof} \noindent Using the $1$-to-$1$ corresponding between quasi-coherent sheaves of ideals and closed subschemes (see Lemma \ref{lemma-closed-immersion-bijection-ideals}) we can define scheme theoretic intersections and unions of closed subschemes. \begin{definition} \label{definition-scheme-theoretic-intersection-union} Let $X$ be a scheme. Let $Z, Y \subset X$ be closed subschemes corresponding to quasi-coherent ideal sheaves $\mathcal{I}, \mathcal{J} \subset \mathcal{O}_X$. The {\it scheme theoretic intersection} of $Z$ and $Y$ is the closed subscheme of $X$ cut out by $\mathcal{I} + \mathcal{J}$. The {\it scheme theoretic union} of $Z$ and $Y$ is the closed subscheme of $X$ cut out by $\mathcal{I} \cap \mathcal{J}$. \end{definition} \begin{lemma} \label{lemma-scheme-theoretic-intersection} Let $X$ be a scheme. Let $Z, Y \subset X$ be closed subschemes. Let $Z \cap Y$ be the scheme theoretic intersection of $Z$ and $Y$. Then $Z \cap Y \to Z$ and $Z \cap Y \to Y$ are closed immersions and $$\xymatrix{ Z \cap Y \ar[r] \ar[d] & Z \ar[d] \\ Y \ar[r] & X }$$ is a cartesian diagram of schemes, i.e., $Z \cap Y = Z \times_X Y$. \end{lemma} \begin{proof} The morphisms $Z \cap Y \to Z$ and $Z \cap Y \to Y$ are closed immersions by Lemma \ref{lemma-closed-immersion-ideals}. Let $U = \Spec(A)$ be an affine open of $X$ and let $Z \cap U$ and $Y \cap U$ correspond to the ideals $I \subset A$ and $J \subset A$. Then $Z \cap Y \cap U$ corresponds to $I + J \subset A$. Since $A/I \otimes_A A/J = A/(I + J)$ we see that the diagram is cartesian by our description of fibre products of schemes in Schemes, Section \ref{schemes-section-fibre-products}. \end{proof} \begin{lemma} \label{lemma-scheme-theoretic-union} Let $S$ be a scheme. Let $X, Y \subset S$ be closed subschemes. Let $X \cup Y$ be the scheme theoretic union of $X$ and $Y$. Let $X \cap Y$ be the scheme theoretic intersection of $X$ and $Y$. Then $X \to X \cup Y$ and $Y \to X \cup Y$ are closed immersions, there is a short exact sequence $$0 \to \mathcal{O}_{X \cup Y} \to \mathcal{O}_X \times \mathcal{O}_Y \to \mathcal{O}_{X \cap Y} \to 0$$ of $\mathcal{O}_S$-modules, and the diagram $$\xymatrix{ X \cap Y \ar[r] \ar[d] & X \ar[d] \\ Y \ar[r] & X \cup Y }$$ is cocartesian in the category of schemes, i.e., $X \cup Y = X \amalg_{X \cap Y} Y$. \end{lemma} \begin{proof} The morphisms $X \to X \cup Y$ and $Y \to X \cup Y$ are closed immersions by Lemma \ref{lemma-closed-immersion-ideals}. In the short exact sequence we use the equivalence of Lemma \ref{lemma-i-star-equivalence} to think of quasi-coherent modules on closed subschemes of $S$ as quasi-coherent modules on $S$. For the first map in the sequence we use the canonical maps $\mathcal{O}_{X \cup Y} \to \mathcal{O}_X$ and $\mathcal{O}_{X \cup Y} \to \mathcal{O}_Y$ and for the second map we use the canonical map $\mathcal{O}_X \to \mathcal{O}_{X \cap Y}$ and the negative of the canonical map $\mathcal{O}_Y \to \mathcal{O}_{X \cap Y}$. Then to check exactness we may work affine locally. Let $U = \Spec(A)$ be an affine open of $S$ and let $X \cap U$ and $Y \cap U$ correspond to the ideals $I \subset A$ and $J \subset A$. Then $(X \cup Y) \cap U$ corresponds to $I \cap J \subset A$ and $X \cap Y \cap U$ corresponds to $I + J \subset A$. Thus exactness follows from the exactness of $$0 \to A/I \cap J \to A/I \times A/J \to A/(I + J) \to 0$$ To show the diagram is cocartesian, suppose we are given a scheme $T$ and morphisms of schemes $f : X \to T$, $g : Y \to T$ agreeing as morphisms $X \cap Y \to T$. Goal: Show there exists a unique morphism $h : X \cup Y \to T$ agreeing with $f$ and $g$. To construct $h$ we may work affine locally on $X \cup Y$, see Schemes, Section \ref{schemes-section-glueing-schemes}. If $s \in X$, $s \not \in Y$, then $X \to X \cup Y$ is an isomorphism in a neighbourhood of $s$ and it is clear how to construct $h$. Similarly for $s \in Y$, $s \not \in X$. For $s \in X \cap Y$ we can pick an affine open $V = \Spec(B) \subset T$ containing $f(s) = g(s)$. Then we can choose an affine open $U = \Spec(A) \subset S$ containing $s$ such that $f(X \cap U)$ and $g(Y \cap U)$ are contained in $V$. The morphisms $f|_{X \cap U}$ and $g|_{Y \cap V}$ into $V$ correspond to ring maps $$B \to A/I \quad\text{and}\quad B \to A/J$$ which agree as maps into $A/(I + J)$. By the short exact sequence displayed above there is a unique lift of these ring homomorphism to a ring map $B \to A/I \cap J$ as desired. \end{proof} \section{Supports of modules} \label{section-support} \noindent In this section we collect some elementary results on supports of quasi-coherent modules on schemes. Recall that the support of a sheaf of modules has been defined in Modules, Section \ref{modules-section-support}. On the other hand, the support of a module was defined in Algebra, Section \ref{algebra-section-support}. These match. \begin{lemma} \label{lemma-support-affine-open} Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $\Spec(A) = U \subset X$ be an affine open, and set $M = \Gamma(U, \mathcal{F})$. Let $x \in U$, and let $\mathfrak p \subset A$ be the corresponding prime. The following are equivalent \begin{enumerate} \item $\mathfrak p$ is in the support of $M$, and \item $x$ is in the support of $\mathcal{F}$. \end{enumerate} \end{lemma} \begin{proof} This follows from the equality $\mathcal{F}_x = M_{\mathfrak p}$, see Schemes, Lemma \ref{schemes-lemma-spec-sheaves} and the definitions. \end{proof} \begin{lemma} \label{lemma-support-closed-specialization} Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. The support of $\mathcal{F}$ is closed under specialization. \end{lemma} \begin{proof} If $x' \leadsto x$ is a specialization and $\mathcal{F}_x = 0$ then $\mathcal{F}_{x'}$ is zero, as $\mathcal{F}_{x'}$ is a localization of the module $\mathcal{F}_x$. Hence the complement of $\text{Supp}(\mathcal{F})$ is closed under generalization. \end{proof} \noindent For finite type quasi-coherent modules the support is closed, can be checked on fibres, and commutes with base change. \begin{lemma} \label{lemma-support-finite-type} Let $\mathcal{F}$ be a finite type quasi-coherent module on a scheme $X$. Then \begin{enumerate} \item The support of $\mathcal{F}$ is closed. \item For $x \in X$ we have $$x \in \text{Supp}(\mathcal{F}) \Leftrightarrow \mathcal{F}_x \not = 0 \Leftrightarrow \mathcal{F}_x \otimes_{\mathcal{O}_{X, x}} \kappa(x) \not = 0.$$ \item For any morphism of schemes $f : Y \to X$ the pullback $f^*\mathcal{F}$ is of finite type as well and we have $\text{Supp}(f^*\mathcal{F}) = f^{-1}(\text{Supp}(\mathcal{F}))$. \end{enumerate} \end{lemma} \begin{proof} Part (1) is a reformulation of Modules, Lemma \ref{modules-lemma-support-finite-type-closed}. You can also combine Lemma \ref{lemma-support-affine-open}, Properties, Lemma \ref{properties-lemma-finite-type-module}, and Algebra, Lemma \ref{algebra-lemma-support-closed} to see this. The first equivalence in (2) is the definition of support, and the second equivalence follows from Nakayama's lemma, see Algebra, Lemma \ref{algebra-lemma-NAK}. Let $f : Y \to X$ be a morphism of schemes. Note that $f^*\mathcal{F}$ is of finite type by Modules, Lemma \ref{modules-lemma-pullback-finite-type}. For the final assertion, let $y \in Y$ with image $x \in X$. Recall that $$(f^*\mathcal{F})_y = \mathcal{F}_x \otimes_{\mathcal{O}_{X, x}} \mathcal{O}_{Y, y},$$ see Sheaves, Lemma \ref{sheaves-lemma-stalk-pullback-modules}. Hence $(f^*\mathcal{F})_y \otimes \kappa(y)$ is nonzero if and only if $\mathcal{F}_x \otimes \kappa(x)$ is nonzero. By (2) this implies $x \in \text{Supp}(\mathcal{F})$ if and only if $y \in \text{Supp}(f^*\mathcal{F})$, which is the content of assertion (3). \end{proof} \begin{lemma} \label{lemma-scheme-theoretic-support} Let $\mathcal{F}$ be a finite type quasi-coherent module on a scheme $X$. There exists a smallest closed subscheme $i : Z \to X$ such that there exists a quasi-coherent $\mathcal{O}_Z$-module $\mathcal{G}$ with $i_*\mathcal{G} \cong \mathcal{F}$. Moreover: \begin{enumerate} \item If $\Spec(A) \subset X$ is any affine open, and $\mathcal{F}|_{\Spec(A)} = \widetilde{M}$ then $Z \cap \Spec(A) = \Spec(A/I)$ where $I = \text{Ann}_A(M)$. \item The quasi-coherent sheaf $\mathcal{G}$ is unique up to unique isomorphism. \item The quasi-coherent sheaf $\mathcal{G}$ is of finite type. \item The support of $\mathcal{G}$ and of $\mathcal{F}$ is $Z$. \end{enumerate} \end{lemma} \begin{proof} Suppose that $i' : Z' \to X$ is a closed subscheme which satisfies the description on open affines from the lemma. Then by Lemma \ref{lemma-i-star-equivalence} we see that $\mathcal{F} \cong i'_*\mathcal{G}'$ for some unique quasi-coherent sheaf $\mathcal{G}'$ on $Z'$. Furthermore, it is clear that $Z'$ is the smallest closed subscheme with this property (by the same lemma). Finally, using Properties, Lemma \ref{properties-lemma-finite-type-module} and Algebra, Lemma \ref{algebra-lemma-finite-over-subring} it follows that $\mathcal{G}'$ is of finite type. We have $\text{Supp}(\mathcal{G}') = Z$ by Algebra, Lemma \ref{algebra-lemma-support-closed}. Hence, in order to prove the lemma it suffices to show that the characterization in (1) actually does define a closed subscheme. And, in order to do this it suffices to prove that the given rule produces a quasi-coherent sheaf of ideals, see Lemma \ref{lemma-closed-immersion-bijection-ideals}. This comes down to the following algebra fact: If $A$ is a ring, $f \in A$, and $M$ is a finite $A$-module, then $\text{Ann}_A(M)_f = \text{Ann}_{A_f}(M_f)$. We omit the proof. \end{proof} \begin{definition} \label{definition-scheme-theoretic-support} Let $X$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module of finite type. The {\it scheme theoretic support of $\mathcal{F}$} is the closed subscheme $Z \subset X$ constructed in Lemma \ref{lemma-scheme-theoretic-support}. \end{definition} \noindent In this situation we often think of $\mathcal{F}$ as a quasi-coherent sheaf of finite type on $Z$ (via the equivalence of categories of Lemma \ref{lemma-i-star-equivalence}). \section{Scheme theoretic image} \label{section-scheme-theoretic-image} \noindent Caution: Some of the material in this section is ultra-general and behaves differently from what you might expect. \begin{lemma} \label{lemma-scheme-theoretic-image} Let $f : X \to Y$ be a morphism of schemes. There exists a closed subscheme $Z \subset Y$ such that $f$ factors through $Z$ and such that for any other closed subscheme $Z' \subset Y$ such that $f$ factors through $Z'$ we have $Z \subset Z'$. \end{lemma} \begin{proof} Let $\mathcal{I} = \Ker(\mathcal{O}_Y \to f_*\mathcal{O}_X)$. If $\mathcal{I}$ is quasi-coherent then we just take $Z$ to be the closed subscheme determined by $\mathcal{I}$, see Lemma \ref{lemma-closed-immersion-bijection-ideals}. This works by Schemes, Lemma \ref{schemes-lemma-characterize-closed-subspace}. In general the same lemma requires us to show that there exists a largest quasi-coherent sheaf of ideals $\mathcal{I}'$ contained in $\mathcal{I}$. This follows from Lemma \ref{lemma-largest-quasi-coherent-subsheaf}. \end{proof} \begin{definition} \label{definition-scheme-theoretic-image} Let $f : X \to Y$ be a morphism of schemes. The {\it scheme theoretic image} of $f$ is the smallest closed subscheme $Z \subset Y$ through which $f$ factors, see Lemma \ref{lemma-scheme-theoretic-image} above. \end{definition} \noindent For a morphism $f : X \to Y$ of schemes with scheme theoretic image $Z$ we often denote $f : X \to Z$ the factorization of $f$ through its scheme theoretic image. If the morphism $f$ is not quasi-compact, then (in general) \begin{enumerate} \item the set theoretic inclusion $\overline{f(X)} \subset Z$ is not an equality, i.e., $f(X) \subset Z$ is not a dense subset, and \item the construction of the scheme theoretic image does not commute with restriction to open subschemes to $Y$. \end{enumerate} In Examples, Section \ref{examples-section-scheme-theoretic-image} the reader finds an example for both phenomena. These phenomena can arise even for immersions, see Examples, Section \ref{examples-section-strange-immersion}. However, the next lemma shows that both disasters are avoided when the morphism is quasi-compact. \begin{lemma} \label{lemma-quasi-compact-scheme-theoretic-image} Let $f : X \to Y$ be a morphism of schemes. Let $Z \subset Y$ be the scheme theoretic image of $f$. If $f$ is quasi-compact then \begin{enumerate} \item the sheaf of ideals $\mathcal{I} = \Ker(\mathcal{O}_Y \to f_*\mathcal{O}_X)$ is quasi-coherent, \item the scheme theoretic image $Z$ is the closed subscheme determined by $\mathcal{I}$, \item for any open $U \subset Y$ the scheme theoretic image of $f|_{f^{-1}(U)} : f^{-1}(U) \to U$ is equal to $Z \cap U$, and \item the image $f(X) \subset Z$ is a dense subset of $Z$, in other words the morphism $X \to Z$ is dominant (see Definition \ref{definition-dominant}). \end{enumerate} \end{lemma} \begin{proof} Part (4) follows from part (3). To show (3) it suffices to prove (1) since the formation of $\mathcal{I}$ commutes with restriction to open subschemes of $Y$. And if (1) holds then in the proof of Lemma \ref{lemma-scheme-theoretic-image} we showed (2). Thus it suffices to prove that $\mathcal{I}$ is quasi-coherent. Since the property of being quasi-coherent is local we may assume $Y$ is affine. As $f$ is quasi-compact, we can find a finite affine open covering $X = \bigcup_{i = 1, \ldots, n} U_i$. Denote $f'$ the composition $$X' = \coprod U_i \longrightarrow X \longrightarrow Y.$$ Then $f_*\mathcal{O}_X$ is a subsheaf of $f'_*\mathcal{O}_{X'}$, and hence $\mathcal{I} = \Ker(\mathcal{O}_Y \to f'_*\mathcal{O}_{X'})$. By Schemes, Lemma \ref{schemes-lemma-push-forward-quasi-coherent} the sheaf $f'_*\mathcal{O}_{X'}$ is quasi-coherent on $Y$. Hence we win. \end{proof} \begin{example} \label{example-scheme-theoretic-image} If $A \to B$ is a ring map with kernel $I$, then the scheme theoretic image of $\Spec(B) \to \Spec(A)$ is the closed subscheme $\Spec(A/I)$ of $\Spec(A)$. This follows from Lemma \ref{lemma-quasi-compact-scheme-theoretic-image}. \end{example} \noindent If the morphism is quasi-compact, then the scheme theoretic image only adds points which are specializations of points in the image. \begin{lemma} \label{lemma-reach-points-scheme-theoretic-image} Let $f : X \to Y$ be a quasi-compact morphism. Let $Z$ be the scheme theoretic image of $f$. Let $z \in Z$\footnote{By Lemma \ref{lemma-quasi-compact-scheme-theoretic-image} set-theoretically $Z$ agrees with the closure of $f(X)$ in $Y$.}. There exists a valuation ring $A$ with fraction field $K$ and a commutative diagram $$\xymatrix{ \Spec(K) \ar[rr] \ar[d] & & X \ar[d] \ar[ld] \\ \Spec(A) \ar[r] & Z \ar[r] & Y }$$ such that the closed point of $\Spec(A)$ maps to $z$. In particular any point of $Z$ is the specialization of a point of $f(X)$. \end{lemma} \begin{proof} Let $z \in \Spec(R) = V \subset Y$ be an affine open neighbourhood of $z$. By Lemma \ref{lemma-quasi-compact-scheme-theoretic-image} the intersection $Z \cap V$ is the scheme theoretic image of $f^{-1}(V) \to V$. Hence we may replace $Y$ by $V$ and assume $Y = \Spec(R)$ is affine. In this case $X$ is quasi-compact as $f$ is quasi-compact. Say $X = U_1 \cup \ldots \cup U_n$ is a finite affine open covering. Write $U_i = \Spec(A_i)$. Let $I = \Ker(R \to A_1 \times \ldots \times A_n)$. By Lemma \ref{lemma-quasi-compact-scheme-theoretic-image} again we see that $Z$ corresponds to the closed subscheme $\Spec(R/I)$ of $Y$. If $\mathfrak p \subset R$ is the prime corresponding to $z$, then we see that $I_{\mathfrak p} \subset R_{\mathfrak p}$ is not an equality. Hence (as localization is exact, see Algebra, Proposition \ref{algebra-proposition-localization-exact}) we see that $R_{\mathfrak p} \to (A_1)_{\mathfrak p} \times \ldots \times (A_n)_{\mathfrak p}$ is not zero. Hence one of the rings $(A_i)_{\mathfrak p}$ is not zero. Hence there exists an $i$ and a prime $\mathfrak q_i \subset A_i$ lying over a prime $\mathfrak p_i \subset \mathfrak p$. By Algebra, Lemma \ref{algebra-lemma-dominate} we can choose a valuation ring $A \subset K = \kappa(\mathfrak q_i)$ dominating the local ring $R_{\mathfrak p}/\mathfrak p_iR_{\mathfrak p} \subset \kappa(\mathfrak q_i)$. This gives the desired diagram. Some details omitted. \end{proof} \begin{lemma} \label{lemma-factor-factor} Let $$\xymatrix{ X_1 \ar[d] \ar[r]_{f_1} & Y_1 \ar[d] \\ X_2 \ar[r]^{f_2} & Y_2 }$$ be a commutative diagram of schemes. Let $Z_i \subset Y_i$, $i = 1, 2$ be the scheme theoretic image of $f_i$. Then the morphism $Y_1 \to Y_2$ induces a morphism $Z_1 \to Z_2$ and a commutative diagram $$\xymatrix{ X_1 \ar[r] \ar[d] & Z_1 \ar[d] \ar[r] & Y_1 \ar[d] \\ X_2 \ar[r] & Z_2 \ar[r] & Y_2 }$$ \end{lemma} \begin{proof} The scheme theoretic inverse image of $Z_2$ in $Y_1$ is a closed subscheme of $Y_1$ through which $f_1$ factors. Hence $Z_1$ is contained in this. This proves the lemma. \end{proof} \begin{lemma} \label{lemma-scheme-theoretic-image-reduced} Let $f : X \to Y$ be a morphism of schemes. If $X$ is reduced, then the scheme theoretic image of $f$ is the reduced induced scheme structure on $\overline{f(X)}$. \end{lemma} \begin{proof} This is true because the reduced induced scheme structure on $\overline{f(X)}$ is clearly the smallest closed subscheme of $Y$ through which $f$ factors, see Schemes, Lemma \ref{schemes-lemma-map-into-reduction}. \end{proof} \begin{lemma} \label{lemma-scheme-theoretic-image-of-partial-section} Let $f : X \to Y$ be a separated morphism of schemes. Let $V \subset Y$ be a retrocompact open. Let $s : V \to X$ be a morphism such that $f \circ s = \text{id}_V$. Let $Y'$ be the scheme theoretic image of $s$. Then $Y' \to Y$ is an isomorphism over $V$. \end{lemma} \begin{proof} The assumption that $V$ is retrocompact in $Y$ (Topology, Definition \ref{topology-definition-quasi-compact}) means that $V \to Y$ is a quasi-compact morphism. By Schemes, Lemma \ref{schemes-lemma-quasi-compact-permanence} the morphism $s : V \to X$ is quasi-compact. Hence the construction of the scheme theoretic image $Y'$ of $s$ commutes with restriction to opens by Lemma \ref{lemma-quasi-compact-scheme-theoretic-image}. In particular, we see that $Y' \cap f^{-1}(V)$ is the scheme theoretic image of a section of the separated morphism $f^{-1}(V) \to V$. Since a section of a separated morphism is a closed immersion (Schemes, Lemma \ref{schemes-lemma-section-immersion}), we conclude that $Y' \cap f^{-1}(V) \to V$ is an isomorphism as desired. \end{proof} \section{Scheme theoretic closure and density} \label{section-scheme-theoretic-closure} \noindent We take the following definition from \cite[IV, Definition 11.10.2]{EGA}. \begin{definition} \label{definition-scheme-theoretically-dense} Let $X$ be a scheme. Let $U \subset X$ be an open subscheme. \begin{enumerate} \item The scheme theoretic image of the morphism $U \to X$ is called the {\it scheme theoretic closure of $U$ in $X$}. \item We say $U$ is {\it scheme theoretically dense in $X$} if for every open $V \subset X$ the scheme theoretic closure of $U \cap V$ in $V$ is equal to $V$. \end{enumerate} \end{definition} \noindent With this definition it is {\bf not} the case that $U$ is scheme theoretically dense in $X$ if and only if the scheme theoretic closure of $U$ is $X$, see Example \ref{example-scheme-theretically-dense-not-dense}. This is somewhat inelegant; but see Lemmas \ref{lemma-scheme-theoretically-dense-quasi-compact} and \ref{lemma-reduced-scheme-theoretically-dense} below. On the other hand, with this definition $U$ is scheme theoretically dense in $X$ if and only if for every $V \subset X$ open the ring map $\mathcal{O}_X(V) \to \mathcal{O}_X(U \cap V)$ is injective, see Lemma \ref{lemma-characterize-scheme-theoretically-dense} below. In particular we see that scheme theoretically dense implies dense which is pleasing. \begin{example} \label{example-scheme-theretically-dense-not-dense} Here is an example where scheme theoretic closure being $X$ does not imply dense for the underlying topological spaces. Let $k$ be a field. Set $A = k[x, z_1, z_2, \ldots]/(x^n z_n)$ Set $I = (z_1, z_2, \ldots) \subset A$. Consider the affine scheme $X = \Spec(A)$ and the open subscheme $U = X \setminus V(I)$. Since $A \to \prod_n A_{z_n}$ is injective we see that the scheme theoretic closure of $U$ is $X$. Consider the morphism $X \to \Spec(k[x])$. This morphism is surjective (set all $z_n = 0$ to see this). But the restriction of this morphism to $U$ is not surjective because it maps to the point $x = 0$. Hence $U$ cannot be topologically dense in $X$. \end{example} \begin{lemma} \label{lemma-scheme-theoretically-dense-quasi-compact} Let $X$ be a scheme. Let $U \subset X$ be an open subscheme. If the inclusion morphism $U \to X$ is quasi-compact, then $U$ is scheme theoretically dense in $X$ if and only if the scheme theoretic closure of $U$ in $X$ is $X$. \end{lemma} \begin{proof} Follows from Lemma \ref{lemma-quasi-compact-scheme-theoretic-image} part (3). \end{proof} \begin{example} \label{example-scheme-theoretic-closure} Let $A$ be a ring and $X = \Spec(A)$. Let $f_1, \ldots, f_n \in A$ and let $U = D(f_1) \cup \ldots \cup D(f_n)$. Let $I = \Ker(A \to \prod A_{f_i})$. Then the scheme theoretic closure of $U$ in $X$ is the closed subscheme $\Spec(A/I)$ of $X$. Note that $U \to X$ is quasi-compact. Hence by Lemma \ref{lemma-scheme-theoretically-dense-quasi-compact} we see $U$ is scheme theoretically dense in $X$ if and only if $I = 0$. \end{example} \begin{lemma} \label{lemma-characterize-scheme-theoretically-dense} Let $j : U \to X$ be an open immersion of schemes. Then $U$ is scheme theoretically dense in $X$ if and only if $\mathcal{O}_X \to j_*\mathcal{O}_U$ is injective. \end{lemma} \begin{proof} If $\mathcal{O}_X \to j_*\mathcal{O}_U$ is injective, then the same is true when restricted to any open $V$ of $X$. Hence the scheme theoretic closure of $U \cap V$ in $V$ is equal to $V$, see proof of Lemma \ref{lemma-scheme-theoretic-image}. Conversely, suppose that the scheme theoretic closure of $U \cap V$ is equal to $V$ for all opens $V$. Suppose that $\mathcal{O}_X \to j_*\mathcal{O}_U$ is not injective. Then we can find an affine open, say $\Spec(A) = V \subset X$ and a nonzero element $f \in A$ such that $f$ maps to zero in $\Gamma(V \cap U, \mathcal{O}_X)$. In this case the scheme theoretic closure of $V \cap U$ in $V$ is clearly contained in $\Spec(A/(f))$ a contradiction. \end{proof} \begin{lemma} \label{lemma-intersection-scheme-theoretically-dense} Let $X$ be a scheme. If $U$, $V$ are scheme theoretically dense open subschemes of $X$, then so is $U \cap V$. \end{lemma} \begin{proof} Let $W \subset X$ be any open. Consider the map $\mathcal{O}_X(W) \to \mathcal{O}_X(W \cap V) \to \mathcal{O}_X(W \cap V \cap U)$. By Lemma \ref{lemma-characterize-scheme-theoretically-dense} both maps are injective. Hence the composite is injective. Hence by Lemma \ref{lemma-characterize-scheme-theoretically-dense} $U \cap V$ is scheme theoretically dense in $X$. \end{proof} \begin{lemma} \label{lemma-quasi-compact-immersion} Let $h : Z \to X$ be an immersion. Assume either $h$ is quasi-compact or $Z$ is reduced. Let $\overline{Z} \subset X$ be the scheme theoretic image of $h$. Then the morphism $Z \to \overline{Z}$ is an open immersion which identifies $Z$ with a scheme theoretically dense open subscheme of $\overline{Z}$. Moreover, $Z$ is topologically dense in $\overline{Z}$. \end{lemma} \begin{proof} By Lemma \ref{lemma-factor-quasi-compact-immersion} or Lemma \ref{lemma-factor-reduced-immersion} we can factor $Z \to X$ as $Z \to \overline{Z}_1 \to X$ with $Z \to \overline{Z}_1$ open and $\overline{Z}_1 \to X$ closed. On the other hand, let $Z \to \overline{Z} \subset X$ be the scheme theoretic closure of $Z \to X$. We conclude that $\overline{Z} \subset \overline{Z}_1$. Since $Z$ is an open subscheme of $\overline{Z}_1$ it follows that $Z$ is an open subscheme of $\overline{Z}$ as well. In the case that $Z$ is reduced we know that $Z \subset \overline{Z}_1$ is topologically dense by the construction of $\overline{Z}_1$ in the proof of Lemma \ref{lemma-factor-reduced-immersion}. Hence $\overline{Z}_1$ and $\overline{Z}$ have the same underlying topological spaces. Thus $\overline{Z} \subset \overline{Z}_1$ is a closed immersion into a reduced scheme which induces a bijection on underlying topological spaces, and hence it is an isomorphism. In the case that $Z \to X$ is quasi-compact we argue as follows: The assertion that $Z$ is scheme theoretically dense in $\overline{Z}$ follows from Lemma \ref{lemma-quasi-compact-scheme-theoretic-image} part (3). The last assertion follows from Lemma \ref{lemma-quasi-compact-scheme-theoretic-image} part (4). \end{proof} \begin{lemma} \label{lemma-reduced-scheme-theoretically-dense} Let $X$ be a reduced scheme and let $U \subset X$ be an open subscheme. Then the following are equivalent \begin{enumerate} \item $U$ is topologically dense in $X$, \item the scheme theoretic closure of $U$ in $X$ is $X$, and \item $U$ is scheme theoretically dense in $X$. \end{enumerate} \end{lemma} \begin{proof} This follows from Lemma \ref{lemma-quasi-compact-immersion} and the fact that a closed subscheme $Z$ of $X$ whose underlying topological space equals $X$ must be equal to $X$ as a scheme. \end{proof} \begin{lemma} \label{lemma-reduced-subscheme-closure} Let $X$ be a scheme and let $U \subset X$ be a reduced open subscheme. Then the following are equivalent \begin{enumerate} \item the scheme theoretic closure of $U$ in $X$ is $X$, and \item $U$ is scheme theoretically dense in $X$. \end{enumerate} If this holds then $X$ is a reduced scheme. \end{lemma} \begin{proof} This follows from Lemma \ref{lemma-quasi-compact-immersion} and the fact that the scheme theoretic closure of $U$ in $X$ is reduced by Lemma \ref{lemma-scheme-theoretic-image-reduced}. \end{proof} \begin{lemma} \label{lemma-equality-of-morphisms} Let $S$ be a scheme. Let $X$, $Y$ be schemes over $S$. Let $f, g : X \to Y$ be morphisms of schemes over $S$. Let $U \subset X$ be an open subscheme such that $f|_U = g|_U$. If the scheme theoretic closure of $U$ in $X$ is $X$ and $Y \to S$ is separated, then $f = g$. \end{lemma} \begin{proof} Follows from the definitions and Schemes, Lemma \ref{schemes-lemma-where-are-they-equal}. \end{proof} \section{Dominant morphisms} \label{section-dominant} \noindent The definition of a morphism of schemes being dominant is a little different from what you might expect if you are used to the notion of a dominant morphism of varieties. \begin{definition} \label{definition-dominant} A morphism $f : X \to S$ of schemes is called {\it dominant} if the image of $f$ is a dense subset of $S$. \end{definition} \noindent So for example, if $k$ is an infinite field and $\lambda_1, \lambda_2, \ldots$ is a countable collection of distinct elements of $k$, then the morphism $$\coprod\nolimits_{i = 1, 2, \ldots } \Spec(k) \longrightarrow \Spec(k[x])$$ with $i$th factor mapping to the point $x = \lambda_i$ is dominant. \begin{lemma} \label{lemma-generic-points-in-image-dominant} Let $f : X \to S$ be a morphism of schemes. If every generic point of every irreducible component of $S$ is in the image of $f$, then $f$ is dominant. \end{lemma} \begin{proof} This is a topological fact which follows directly from the fact that the topological space underlying a scheme is sober, see Schemes, Lemma \ref{schemes-lemma-scheme-sober}, and that every point of $S$ is contained in an irreducible component of $S$, see Topology, Lemma \ref{topology-lemma-irreducible}. \end{proof} \noindent The expectation that morphisms are dominant only if generic points of the target are in the image does hold if the morphism is quasi-compact. \begin{lemma} \label{lemma-quasi-compact-dominant} \begin{slogan} Morphisms whose image contains the generic points are dominant \end{slogan} Let $f : X \to S$ be a quasi-compact morphism of schemes. Then $f$ is dominant (if and) only if for every irreducible component $Z \subset S$ the generic point of $Z$ is in the image of $f$. \end{lemma} \begin{proof} Let $V \subset S$ be an affine open. Because $f$ is quasi-compact we may choose finitely many affine opens $U_i \subset f^{-1}(V)$, $i = 1, \ldots, n$ covering $f^{-1}(V)$. Consider the morphism of affines $$f' : \coprod\nolimits_{i = 1, \ldots, n} U_i \longrightarrow V.$$ A disjoint union of affines is affine, see Schemes, Lemma \ref{schemes-lemma-disjoint-union-affines}. Generic points of irreducible components of $V$ are exactly the generic points of the irreducible components of $S$ that meet $V$. Also, $f$ is dominant if and only if $f'$ is dominant no matter what choices of $V, n, U_i$ we make above. Thus we have reduced the lemma to the case of a morphism of affine schemes. The affine case is Algebra, Lemma \ref{algebra-lemma-image-dense-generic-points}. \end{proof} \noindent Here is a slightly more useful variant of the lemma above. \begin{lemma} \label{lemma-quasi-compact-generic-point-not-in-image} Let $f : X \to S$ be a quasi-compact morphism of schemes. Let $\eta \in S$ be a generic point of an irreducible component of $S$. If $\eta \not \in f(X)$ then there exists an open neighbourhood $V \subset S$ of $\eta$ such that $f^{-1}(V) = \emptyset$. \end{lemma} \begin{proof} Let $Z \subset S$ be the scheme theoretic image of $f$. We have to show that $\eta \not \in Z$. This follows from Lemma \ref{lemma-reach-points-scheme-theoretic-image} but can also be seen as follows. By Lemma \ref{lemma-quasi-compact-scheme-theoretic-image} the morphism $X \to Z$ is dominant, which by Lemma \ref{lemma-quasi-compact-dominant} means all the generic points of all irreducible components of $Z$ are in the image of $X \to Z$. By assumption we see that $\eta \not \in Z$ since $\eta$ would be the generic point of some irreducible component of $Z$ if it were in $Z$. \end{proof} \noindent There is another case where dominant is the same as having all generic points of irreducible components in the image. \begin{lemma} \label{lemma-dominant-finite-number-irreducible-components} Let $f : X \to S$ be a morphism of schemes. Suppose that $X$ has finitely many irreducible components. Then $f$ is dominant (if and) only if for every irreducible component $Z \subset S$ the generic point of $Z$ is in the image of $f$. If so, then $S$ has finitely many irreducible components as well. \end{lemma} \begin{proof} Assume $f$ is dominant. Say $X = Z_1 \cup Z_2 \cup \ldots \cup Z_n$ is the decomposition of $X$ into irreducible components. Let $\xi_i \in Z_i$ be its generic point, so $Z_i = \overline{\{\xi_i\}}$. Note that $f(Z_i)$ is an irreducible subset of $S$. Hence $$S = \overline{f(X)} = \bigcup \overline{f(Z_i)} = \bigcup \overline{\{f(\xi_i)\}}$$ is a finite union of irreducible subsets whose generic points are in the image of $f$. The lemma follows. \end{proof} \begin{lemma} \label{lemma-dominant-between-integral} Let $f : X \to Y$ be a morphism of integral schemes. The following are equivalent \begin{enumerate} \item $f$ is dominant, \item $f$ maps the generic point of $X$ to the generic point of $Y$, \item for some nonempty affine opens $U \subset X$ and $V \subset Y$ with $f(U) \subset V$ the ring map $\mathcal{O}_Y(V) \to \mathcal{O}_X(U)$ is injective, \item for all nonempty affine opens $U \subset X$ and $V \subset Y$ with $f(U) \subset V$ the ring map $\mathcal{O}_Y(V) \to \mathcal{O}_X(U)$ is injective, \item for some $x \in X$ with image $y = f(x) \in Y$ the local ring map $\mathcal{O}_{Y, y} \to \mathcal{O}_{X, x}$ is injective, and \item for all $x \in X$ with image $y = f(x) \in Y$ the local ring map $\mathcal{O}_{Y, y} \to \mathcal{O}_{X, x}$ is injective. \end{enumerate} \end{lemma} \begin{proof} The equivalence of (1) and (2) follows from Lemma \ref{lemma-dominant-finite-number-irreducible-components}. Let $U \subset X$ and $V \subset Y$ be nonempty affine opens with $f(U) \subset V$. Recall that the rings $A = \mathcal{O}_X(U)$ and $B = \mathcal{O}_Y(V)$ are integral domains. The morphism $f|_U : U \to V$ corresponds to a ring map $\varphi : B \to A$. The generic points of $X$ and $Y$ correspond to the prime ideals $(0) \subset A$ and $(0) \subset B$. Thus (2) is equivalent to the condition $(0) = \varphi^{-1}((0))$, i.e., to the condition that $\varphi$ is injective. In this way we see that (2), (3), and (4) are equivalent. Similarly, given $x$ and $y$ as in (5) the local rings $\mathcal{O}_{X, x}$ and $\mathcal{O}_{Y, y}$ are domains and the prime ideals $(0) \subset \mathcal{O}_{X, x}$ and $(0) \subset \mathcal{O}_{Y, y}$ correspond to the generic points of $X$ and $Y$ (via the identification of the spectrum of the local ring at $x$ with the set of points specializing to $x$, see Schemes, Lemma \ref{schemes-lemma-specialize-points}). Thus we can argue in the exact same manner as above to see that (2), (5), and (6) are equivalent. \end{proof} \section{Surjective morphisms} \label{section-surjective} \begin{definition} \label{definition-surjective} A morphism of schemes is said to be {\it surjective} if it is surjective on underlying topological spaces. \end{definition} \begin{lemma} \label{lemma-composition-surjective} The composition of surjective morphisms is surjective. \end{lemma} \begin{proof} Omitted. \end{proof} \begin{lemma} \label{lemma-when-point-maps-to-pair} Let $X$ and $Y$ be schemes over a base scheme $S$. Given points $x \in X$ and $y \in Y$, there is a point of $X \times_S Y$ mapping to $x$ and $y$ under the projections if and only if $x$ and $y$ lie above the same point of $S$. \end{lemma} \begin{proof} The condition is obviously necessary, and the converse follows from the proof of Schemes, Lemma \ref{schemes-lemma-points-fibre-product}. \end{proof} \begin{lemma} \label{lemma-base-change-surjective} The base change of a surjective morphism is surjective. \end{lemma} \begin{proof} Let $f: X \to Y$ be a morphism of schemes over a base scheme $S$. If $S' \to S$ is a morphism of schemes, let $p: X_{S'} \to X$ and $q: Y_{S'} \to Y$ be the canonical projections. The commutative square $$\xymatrix{ X_{S'} \ar[d]_{f_{S'}} \ar[r]_p & X \ar[d]^{f} \\ Y_{S'} \ar[r]^{q} & Y. }$$ identifies $X_{S'}$ as a fibre product of $X \to Y$ and $Y_{S'} \to Y$. Let $Z$ be a subset of the underlying topological space of $X$. Then $q^{-1}(f(Z)) = f_{S'}(p^{-1}(Z))$, because $y' \in q^{-1}(f(Z))$ if and only if $q(y') = f(x)$ for some $x \in Z$, if and only if, by Lemma \ref{lemma-when-point-maps-to-pair}, there exists $x' \in X_{S'}$ such that $f_{S'}(x') = y'$ and $p(x') = x$. In particular taking $Z = X$ we see that if $f$ is surjective so is the base change $f_{S'}: X_{S'} \to Y_{S'}$. \end{proof} \begin{example} \label{example-injective-not-preserved-base-change} Bijectivity is not stable under base change, and so neither is injectivity. For example consider the bijection $\Spec(\mathbf{C}) \to \Spec(\mathbf{R})$. The base change $\Spec(\mathbf{C} \otimes_{\mathbf{R}} \mathbf{C}) \to \Spec(\mathbf{C})$ is not injective, since there is an isomorphism $\mathbf{C} \otimes_{\mathbf{R}} \mathbf{C} \cong \mathbf{C} \times \mathbf{C}$ (the decomposition comes from the idempotent $\frac{1 \otimes 1 + i \otimes i}{2}$) and hence $\Spec(\mathbf{C} \otimes_{\mathbf{R}} \mathbf{C})$ has two points. \end{example} \begin{lemma} \label{lemma-surjection-from-quasi-compact} Let $$\xymatrix{ X \ar[rr]_f \ar[rd]_p & & Y \ar[dl]^q \\ & Z }$$ be a commutative diagram of morphisms of schemes. If $f$ is surjective and $p$ is quasi-compact, then $q$ is quasi-compact. \end{lemma} \begin{proof} Let $W \subset Z$ be a quasi-compact open. By assumption $p^{-1}(W)$ is quasi-compact. Hence by Topology, Lemma \ref{topology-lemma-image-quasi-compact} the inverse image $q^{-1}(W) = f(p^{-1}(W))$ is quasi-compact too. This proves the lemma. \end{proof} \section{Radicial and universally injective morphisms} \label{section-radicial} \noindent In this section we define what it means for a morphism of schemes to be {\it radicial} and what it means for a morphism of schemes to be {\it universally injective}. We then show that these notions agree. The reason for introducing both is that in the case of algebraic spaces there are corresponding notions which may not always agree. \begin{definition} \label{definition-universally-injective} Let $f : X \to S$ be a morphism. \begin{enumerate} \item We say that $f$ is {\it universally injective} if and only if for any morphism of schemes $S' \to S$ the base change $f' : X_{S'} \to S'$ is injective (on underlying topological spaces). \item We say $f$ is {\it radicial} if $f$ is injective as a map of topological spaces, and for every $x \in X$ the field extension $\kappa(x)/\kappa(f(x))$ is purely inseparable. \end{enumerate} \end{definition} \begin{lemma} \label{lemma-universally-injective} Let $f : X \to S$ be a morphism of schemes. The following are equivalent: \begin{enumerate} \item For every field $K$ the induced map $\Mor(\Spec(K), X) \to \Mor(\Spec(K), S)$ is injective. \item The morphism $f$ is universally injective. \item The morphism $f$ is radicial. \item The diagonal morphism $\Delta_{X/S} : X \longrightarrow X \times_S X$ is surjective. \end{enumerate} \end{lemma} \begin{proof} Let $K$ be a field, and let $s : \Spec(K) \to S$ be a morphism. Giving a morphism $x : \Spec(K) \to X$ such that $f \circ x = s$ is the same as giving a section of the projection $X_K = \Spec(K) \times_S X \to \Spec(K)$, which in turn is the same as giving a point $x \in X_K$ whose residue field is $K$. Hence we see that (2) implies (1). \medskip\noindent Conversely, suppose that (1) holds. Assume that $x, x' \in X_{S'}$ map to the same point $s' \in S'$. Choose a commutative diagram $$\xymatrix{ K & \kappa(x) \ar[l] \\ \kappa(x') \ar[u] & \kappa(s') \ar[l] \ar[u] }$$ of fields. By Schemes, Lemma \ref{schemes-lemma-characterize-points} we get two morphisms $a, a' : \Spec(K) \to X_{S'}$. One corresponding to the point $x$ and the embedding $\kappa(x) \subset K$ and the other corresponding to the point $x'$ and the embedding $\kappa(x') \subset K$. Also we have $f' \circ a = f' \circ a'$. Condition (1) now implies that the compositions of $a$ and $a'$ with $X_{S'} \to X$ are equal. Since $X_{S'}$ is the fibre product of $S'$ and $X$ over $S$ we see that $a = a'$. Hence $x = x'$. Thus (1) implies (2). \medskip\noindent If there are two different points $x, x' \in X$ mapping to the same point of $s$ then (2) is violated. If for some $s = f(x)$, $x \in X$ the field extension $\kappa(x)/\kappa(s)$ is not purely inseparable, then we may find a field extension $K/\kappa(s)$ such that $\kappa(x)$ has two $\kappa(s)$-homomorphisms into $K$. By Schemes, Lemma \ref{schemes-lemma-characterize-points} this implies that the map $\Mor(\Spec(K), X) \to \Mor(\Spec(K), S)$ is not injective, and hence (1) is violated. Thus we see that the equivalent conditions (1) and (2) imply $f$ is radicial, i.e., they imply (3). \medskip\noindent Assume (3). By Schemes, Lemma \ref{schemes-lemma-characterize-points} a morphism $\Spec(K) \to X$ is given by a pair $(x, \kappa(x) \to K)$. Property (3) says exactly that associating to the pair $(x, \kappa(x) \to K)$ the pair $(s, \kappa(s) \to \kappa(x) \to K)$ is injective. In other words (1) holds. At this point we know that (1), (2) and (3) are all equivalent. \medskip\noindent Finally, we prove the equivalence of (4) with (1), (2) and (3). A point of $X \times_S X$ is given by a quadruple $(x_1, x_2, s, \mathfrak p)$, where $x_1, x_2 \in X$, $f(x_1) = f(x_2) = s$ and $\mathfrak p \subset \kappa(x_1) \otimes_{\kappa(s)} \kappa(x_2)$ is a prime ideal, see Schemes, Lemma \ref{schemes-lemma-points-fibre-product}. If $f$ is universally injective, then by taking $S'=X$ in the definition of universally injective, $\Delta_{X/S}$ must be surjective since it is a section of the injective morphism $X \times_S X \longrightarrow X$. Conversely, if $\Delta_{X/S}$ is surjective, then always $x_1 = x_2 = x$ and there is exactly one such prime ideal $\mathfrak p$, which means that $\kappa(s) \subset \kappa(x)$ is purely inseparable. Hence $f$ is radicial. Alternatively, if $\Delta_{X/S}$ is surjective, then for any $S' \to S$ the base change $\Delta_{X_{S'}/S'}$ is surjective which implies that $f$ is universally injective. This finishes the proof of the lemma. \end{proof} \begin{lemma} \label{lemma-universally-injective-separated} A universally injective morphism is separated. \end{lemma} \begin{proof} Combine Lemma \ref{lemma-universally-injective} with the remark that $X \to S$ is separated if and only if the image of $\Delta_{X/S}$ is closed in $X \times_S X$, see Schemes, Definition \ref{schemes-definition-separated} and the discussion following it. \end{proof} \begin{lemma} \label{lemma-base-change-universally-injective} A base change of a universally injective morphism is universally injective. \end{lemma} \begin{proof} This is formal. \end{proof} \begin{lemma} \label{lemma-composition-universally-injective} A composition of radicial morphisms is radicial, and so the same holds for the equivalent condition of being universally injective. \end{lemma} \begin{proof} Omitted. \end{proof} \section{Affine morphisms} \label{section-affine} \begin{definition} \label{definition-affine} A morphism of schemes $f : X \to S$ is called {\it affine} if the inverse image of every affine open of $S$ is an affine open of $X$. \end{definition} \begin{lemma} \label{lemma-affine-separated} An affine morphism is separated and quasi-compact. \end{lemma} \begin{proof} Let $f : X \to S$ be affine. Quasi-compactness is immediate from Schemes, Lemma \ref{schemes-lemma-quasi-compact-affine}. We will show $f$ is separated using Schemes, Lemma \ref{schemes-lemma-characterize-separated}. Let $x_1, x_2 \in X$ be points of $X$ which map to the same point $s \in S$. Choose any affine open $W \subset S$ containing $s$. By assumption $f^{-1}(W)$ is affine. Apply the lemma cited with $U = V = f^{-1}(W)$. \end{proof} \begin{lemma} \label{lemma-characterize-affine} \begin{reference} \cite[II, Corollary 1.3.2]{EGA} \end{reference} Let $f : X \to S$ be a morphism of schemes. The following are equivalent \begin{enumerate} \item The morphism $f$ is affine. \item There exists an affine open covering $S = \bigcup W_j$ such that each $f^{-1}(W_j)$ is affine. \item There exists a quasi-coherent sheaf of $\mathcal{O}_S$-algebras $\mathcal{A}$ and an isomorphism $X \cong \underline{\Spec}_S(\mathcal{A})$ of schemes over $S$. See Constructions, Section \ref{constructions-section-spec} for notation. \end{enumerate} Moreover, in this case $X = \underline{\Spec}_S(f_*\mathcal{O}_X)$. \end{lemma} \begin{proof} It is obvious that (1) implies (2). \medskip\noindent Assume $S = \bigcup_{j \in J} W_j$ is an affine open covering such that each $f^{-1}(W_j)$ is affine. By Schemes, Lemma \ref{schemes-lemma-quasi-compact-affine} we see that $f$ is quasi-compact. By Schemes, Lemma \ref{schemes-lemma-characterize-quasi-separated} we see the morphism $f$ is quasi-separated. Hence by Schemes, Lemma \ref{schemes-lemma-push-forward-quasi-coherent} the sheaf $\mathcal{A} = f_*\mathcal{O}_X$ is a quasi-coherent sheaf of $\mathcal{O}_S$-algebras. Thus we have the scheme $g : Y = \underline{\Spec}_S(\mathcal{A}) \to S$ over $S$. The identity map $\text{id} : \mathcal{A} = f_*\mathcal{O}_X \to f_*\mathcal{O}_X$ provides, via the definition of the relative spectrum, a morphism $can : X \to Y$ over $S$, see Constructions, Lemma \ref{constructions-lemma-canonical-morphism}. By assumption and the lemma just cited the restriction $can|_{f^{-1}(W_j)} : f^{-1}(W_j) \to g^{-1}(W_j)$ is an isomorphism. Thus $can$ is an isomorphism. We have shown that (2) implies (3). \medskip\noindent Assume (3). By Constructions, Lemma \ref{constructions-lemma-spec-properties} we see that the inverse image of every affine open is affine, and hence the morphism is affine by definition. \end{proof} \begin{remark} \label{remark-direct-argument} We can also argue directly that (2) implies (1) in Lemma \ref{lemma-characterize-affine} above as follows. Assume $S = \bigcup W_j$ is an affine open covering such that each $f^{-1}(W_j)$ is affine. First argue that $\mathcal{A} = f_*\mathcal{O}_X$ is quasi-coherent as in the proof above. Let $\Spec(R) = V \subset S$ be affine open. We have to show that $f^{-1}(V)$ is affine. Set $A = \mathcal{A}(V) = f_*\mathcal{O}_X(V) = \mathcal{O}_X(f^{-1}(V))$. By Schemes, Lemma \ref{schemes-lemma-morphism-into-affine} there is a canonical morphism $\psi : f^{-1}(V) \to \Spec(A)$ over $\Spec(R) = V$. By Schemes, Lemma \ref{schemes-lemma-good-subcover} there exists an integer $n \geq 0$, a standard open covering $V = \bigcup_{i = 1, \ldots, n} D(h_i)$, $h_i \in R$, and a map $a : \{1, \ldots, n\} \to J$ such that each $D(h_i)$ is also a standard open of the affine scheme $W_{a(i)}$. The inverse image of a standard open under a morphism of affine schemes is standard open, see Algebra, Lemma \ref{algebra-lemma-spec-functorial}. Hence we see that $f^{-1}(D(h_i))$ is a standard open of $f^{-1}(W_{a(i)})$, in particular that $f^{-1}(D(h_i))$ is affine. Because $\mathcal{A}$ is quasi-coherent we have $A_{h_i} = \mathcal{A}(D(h_i)) = \mathcal{O}_X(f^{-1}(D(h_i)))$, so $f^{-1}(D(h_i))$ is the spectrum of $A_{h_i}$. It follows that the morphism $\psi$ induces an isomorphism of the open $f^{-1}(D(h_i))$ with the open $\Spec(A_{h_i})$ of $\Spec(A)$. Since $f^{-1}(V) = \bigcup f^{-1}(D(h_i))$ and $\Spec(A) = \bigcup \Spec(A_{h_i})$ we win. \end{remark} \begin{lemma} \label{lemma-affine-equivalence-algebras} Let $S$ be a scheme. There is an anti-equivalence of categories $$\begin{matrix} \text{Schemes affine} \\ \text{over }S \end{matrix} \longleftrightarrow \begin{matrix} \text{quasi-coherent sheaves} \\ \text{of }\mathcal{O}_S\text{-algebras} \end{matrix}$$ which associates to $f : X \to S$ the sheaf $f_*\mathcal{O}_X$. Moreover, this equivalence is compatible with arbitrary base change. \end{lemma} \begin{proof} The functor from right to left is given by $\underline{\Spec}_S$. The two functors are mutually inverse by Lemma \ref{lemma-characterize-affine} and Constructions, Lemma \ref{constructions-lemma-spec-properties} part (3). The final statement is Constructions, Lemma \ref{constructions-lemma-spec-properties} part (2). \end{proof} \begin{lemma} \label{lemma-affine-equivalence-modules} Let $f : X \to S$ be an affine morphism of schemes. Let $\mathcal{A} = f_*\mathcal{O}_X$. The functor $\mathcal{F} \mapsto f_*\mathcal{F}$ induces an equivalence of categories $$\left\{ \begin{matrix} \text{category of quasi-coherent}\\ \mathcal{O}_X\text{-modules} \end{matrix} \right\} \longrightarrow \left\{ \begin{matrix} \text{category of quasi-coherent}\\ \mathcal{A}\text{-modules} \end{matrix} \right\}$$ Moreover, an $\mathcal{A}$-module is quasi-coherent as an $\mathcal{O}_S$-module if and only if it is quasi-coherent as an $\mathcal{A}$-module. \end{lemma} \begin{proof} Omitted. \end{proof} \begin{lemma} \label{lemma-composition-affine} The composition of affine morphisms is affine. \end{lemma} \begin{proof} Let $f : X \to Y$ and $g : Y \to Z$ be affine morphisms. Let $U \subset Z$ be affine open. Then $g^{-1}(U)$ is affine by assumption on $g$. Whereupon $f^{-1}(g^{-1}(U))$ is affine by assumption on $f$. Hence $(g \circ f)^{-1}(U)$ is affine. \end{proof} \begin{lemma} \label{lemma-base-change-affine} The base change of an affine morphism is affine. \end{lemma} \begin{proof} Let $f : X \to S$ be an affine morphism. Let $S' \to S$ be any morphism. Denote $f' : X_{S'} = S' \times_S X \to S'$ the base change of $f$. For every $s' \in S'$ there exists an open affine neighbourhood $s' \in V \subset S'$ which maps into some open affine $U \subset S$. By assumption $f^{-1}(U)$ is affine. By the material in Schemes, Section \ref{schemes-section-fibre-products} we see that $f^{-1}(U)_V = V \times_U f^{-1}(U)$ is affine and equal to $(f')^{-1}(V)$. This proves that $S'$ has an open covering by affines whose inverse image under $f'$ is affine. We conclude by Lemma \ref{lemma-characterize-affine} above. \end{proof} \begin{lemma} \label{lemma-closed-immersion-affine} A closed immersion is affine. \end{lemma} \begin{proof} The first indication of this is Schemes, Lemma \ref{schemes-lemma-closed-immersion-affine-case}. See Schemes, Lemma \ref{schemes-lemma-closed-subspace-scheme} for a complete statement. \end{proof} \begin{lemma} \label{lemma-affine-s-open} Let $X$ be a scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Let $s \in \Gamma(X, \mathcal{L})$. The inclusion morphism $j : X_s \to X$ is affine. \end{lemma} \begin{proof} This follows from Properties, Lemma \ref{properties-lemma-affine-cap-s-open} and the definition. \end{proof} \begin{lemma} \label{lemma-affine-permanence} Suppose $g : X \to Y$ is a morphism of schemes over $S$. \begin{enumerate} \item If $X$ is affine over $S$ and $\Delta : Y \to Y \times_S Y$ is affine, then $g$ is affine. \item If $X$ is affine over $S$ and $Y$ is separated over $S$, then $g$ is affine. \item A morphism from an affine scheme to a scheme with affine diagonal is affine. \item A morphism from an affine scheme to a separated scheme is affine. \end{enumerate} \end{lemma} \begin{proof} Proof of (1). The base change $X \times_S Y \to Y$ is affine by Lemma \ref{lemma-base-change-affine}. The morphism $(1, g) : X \to X \times_S Y$ is the base change of $Y \to Y \times_S Y$ by the morphism $X \times_S Y \to Y \times_S Y$. Hence it is affine by Lemma \ref{lemma-base-change-affine}. The composition of affine morphisms is affine (see Lemma \ref{lemma-composition-affine}) and (1) follows. Part (2) follows from (1) as a closed immersion is affine (see Lemma \ref{lemma-closed-immersion-affine}) and $Y/S$ separated means $\Delta$ is a closed immersion. Parts (3) and (4) are special cases of (1) and (2). \end{proof} \begin{lemma} \label{lemma-morphism-affines-affine} A morphism between affine schemes is affine. \end{lemma} \begin{proof} Immediate from Lemma \ref{lemma-affine-permanence} with $S = \Spec(\mathbf{Z})$. It also follows directly from the equivalence of (1) and (2) in Lemma \ref{lemma-characterize-affine}. \end{proof} \begin{lemma} \label{lemma-Artinian-affine} Let $S$ be a scheme. Let $A$ be an Artinian ring. Any morphism $\Spec(A) \to S$ is affine. \end{lemma} \begin{proof} Omitted. \end{proof} \begin{lemma} \label{lemma-get-affine} Let $j : Y \to X$ be an immersion of schemes. Assume there exists an open $U \subset X$ with complement $Z = X \setminus U$ such that \begin{enumerate} \item $U \to X$ is affine, \item $j^{-1}(U) \to U$ is affine, and \item $j(Y) \cap Z$ is closed. \end{enumerate} Then $j$ is affine. In particular, if $X$ is affine, so is $Y$. \end{lemma} \begin{proof} By Schemes, Definition \ref{schemes-definition-immersion} there exists an open subscheme $W \subset X$ such that $j$ factors as a closed immersion $i : Y \to W$ followed by the inclusion morphism $W \to X$. Since a closed immersion is affine (Lemma \ref{lemma-closed-immersion-affine}), we see that for every $x \in W$ there is an affine open neighbourhood of $x$ in $X$ whose inverse image under $j$ is affine. If $x \in U$, then the same thing is true by assumption (2). Finally, assume $x \in Z$ and $x \not \in W$. Then $x \not \in j(Y) \cap Z$. By assumption (3) we can find an affine open neighbourhood $V \subset X$ of $x$ which does not meet $j(Y) \cap Z$. Then $j^{-1}(V) = j^{-1}(V \cap U)$ which is affine by assumptions (1) and (2). It follows that $j$ is affine by Lemma \ref{lemma-characterize-affine}. \end{proof} \section{Families of ample invertible modules} \label{section-families-ample-invertible-modules} \noindent A short section on the notion of a family of ample invertible modules. \begin{definition} \label{definition-family-ample-invertible-modules} \begin{reference} \cite[II Definition 2.2.4]{SGA6} \end{reference} Let $X$ be a scheme. Let $\{\mathcal{L}_i\}_{i \in I}$ be a family of invertible $\mathcal{O}_X$-modules. We say $\{\mathcal{L}_i\}_{i \in I}$ is an {\it ample family of invertible modules on $X$} if \begin{enumerate} \item $X$ is quasi-compact, and \item for every $x \in X$ there exists an $i \in I$, an $n \geq 1$, and $s \in \Gamma(X, \mathcal{L}_i^{\otimes n})$ such that $x \in X_s$ and $X_s$ is affine. \end{enumerate} \end{definition} \noindent If $\{\mathcal{L}_i\}_{i \in I}$ is an ample family of invertible modules on a scheme $X$, then there exists a finite subset $I' \subset I$ such that $\{\mathcal{L}_i\}_{i \in I'}$ is an ample family of invertible modules on $X$ (follows immediately from quasi-compactness). A scheme having an ample family of invertible modules has an affine diagonal by the next lemma and hence is a fortiori quasi-separated. \begin{lemma} \label{lemma-affine-diagonal} Let $X$ be a scheme such that for every point $x \in X$ there exists an invertible $\mathcal{O}_X$-module $\mathcal{L}$ and a global section $s \in \Gamma(X, \mathcal{L})$ such that $x \in X_s$ and $X_s$ is affine. Then the diagonal of $X$ is an affine morphism. \end{lemma} \begin{proof} Given invertible $\mathcal{O}_X$-modules $\mathcal{L}$, $\mathcal{M}$ and global sections $s \in \Gamma(X, \mathcal{L})$, $t \in \Gamma(X, \mathcal{M})$ such that $X_s$ and $X_t$ are affine we have to prove $X_s \cap X_t$ is affine. Namely, then Lemma \ref{lemma-characterize-affine} applied to $\Delta : X \to X \times X$ and the fact that $\Delta^{-1}(X_s \times X_t) = X_s \cap X_t$ shows that $\Delta$ is affine. The fact that $X_s \cap X_t$ is affine follows from Properties, Lemma \ref{properties-lemma-affine-cap-s-open}. \end{proof} \begin{remark} \label{remark-affine-s-opens-cover-family} In Properties, Lemma \ref{properties-lemma-affine-s-opens-cover-quasi-separated} we see that a scheme which has an ample invertible module is separated. This is wrong for schemes having an ample family of invertible modules. Namely, let $X$ be as in Schemes, Example \ref{schemes-example-affine-space-zero-doubled} with $n = 1$, i.e., the affine line with zero doubled. We use the notation of that example except that we write $x$ for $x_1$ and $y$ for $y_1$. There is, for every integer $n$, an invertible sheaf $\mathcal{L}_n$ on $X$ which is trivial on $X_1$ and $X_2$ and whose transition function $U_{12} \to U_{21}$ is $f(x) \mapsto y^n f(y)$. The global sections of $\mathcal{L}_n$ are pairs $(f(x), g(y)) \in k[x] \oplus k[y]$ such that $y^n f(y) = g(y)$. The sections $s = (1, y)$ of $\mathcal{L}_1$ and $t = (x, 1)$ of $\mathcal{L}_{-1}$ determine an open affine cover because $X_s = X_1$ and $X_t = X_2$. Therefore $X$ has an ample family of invertible modules but it is not separated. \end{remark} \section{Quasi-affine morphisms} \label{section-quasi-affine} \noindent Recall that a scheme $X$ is called {\it quasi-affine} if it is quasi-compact and isomorphic to an open subscheme of an affine scheme, see Properties, Definition \ref{properties-definition-quasi-affine}. \begin{definition} \label{definition-quasi-affine} A morphism of schemes $f : X \to S$ is called {\it quasi-affine} if the inverse image of every affine open of $S$ is a quasi-affine scheme. \end{definition} \begin{lemma} \label{lemma-quasi-affine-separated} A quasi-affine morphism is separated and quasi-compact. \end{lemma} \begin{proof} Let $f : X \to S$ be quasi-affine. Quasi-compactness is immediate from Schemes, Lemma \ref{schemes-lemma-quasi-compact-affine}. Let $U \subset S$ be an affine open. If we can show that $f^{-1}(U)$ is a separated scheme, then $f$ is separated (Schemes, Lemma \ref{schemes-lemma-characterize-separated} shows that being separated is local on the base). By assumption $f^{-1}(U)$ is isomorphic to an open subscheme of an affine scheme. An affine scheme is separated and hence every open subscheme of an affine scheme is separated as desired. \end{proof} \begin{lemma} \label{lemma-characterize-quasi-affine} Let $f : X \to S$ be a morphism of schemes. The following are equivalent \begin{enumerate} \item The morphism $f$ is quasi-affine. \item There exists an affine open covering $S = \bigcup W_j$ such that each $f^{-1}(W_j)$ is quasi-affine. \item There exists a quasi-coherent sheaf of $\mathcal{O}_S$-algebras $\mathcal{A}$ and a quasi-compact open immersion $$\xymatrix{ X \ar[rr] \ar[rd] & & \underline{\Spec}_S(\mathcal{A}) \ar[dl] \\ & S & }$$ over $S$. \item Same as in (3) but with $\mathcal{A} = f_*\mathcal{O}_X$ and the horizontal arrow the canonical morphism of Constructions, Lemma \ref{constructions-lemma-canonical-morphism}. \end{enumerate} \end{lemma} \begin{proof} It is obvious that (1) implies (2) and that (4) implies (3). \medskip\noindent Assume $S = \bigcup_{j \in J} W_j$ is an affine open covering such that each $f^{-1}(W_j)$ is quasi-affine. By Schemes, Lemma \ref{schemes-lemma-quasi-compact-affine} we see that $f$ is quasi-compact. By Schemes, Lemma \ref{schemes-lemma-characterize-quasi-separated} we see the morphism $f$ is quasi-separated. Hence by Schemes, Lemma \ref{schemes-lemma-push-forward-quasi-coherent} the sheaf $\mathcal{A} = f_*\mathcal{O}_X$ is a quasi-coherent sheaf of $\mathcal{O}_X$-algebras. Thus we have the scheme $g : Y = \underline{\Spec}_S(\mathcal{A}) \to S$ over $S$. The identity map $\text{id} : \mathcal{A} = f_*\mathcal{O}_X \to f_*\mathcal{O}_X$ provides, via the definition of the relative spectrum, a morphism $can : X \to Y$ over $S$, see Constructions, Lemma \ref{constructions-lemma-canonical-morphism}. By assumption, the lemma just cited, and Properties, Lemma \ref{properties-lemma-quasi-affine} the restriction $can|_{f^{-1}(W_j)} : f^{-1}(W_j) \to g^{-1}(W_j)$ is a quasi-compact open immersion. Thus $can$ is a quasi-compact open immersion. We have shown that (2) implies (4). \medskip\noindent Assume (3). Choose any affine open $U \subset S$. By Constructions, Lemma \ref{constructions-lemma-spec-properties} we see that the inverse image of $U$ in the relative spectrum is affine. Hence we conclude that $f^{-1}(U)$ is quasi-affine (note that quasi-compactness is encoded in (3) as well). Thus (3) implies (1). \end{proof} \begin{lemma} \label{lemma-composition-quasi-affine} The composition of quasi-affine morphisms is quasi-affine. \end{lemma} \begin{proof} Let $f : X \to Y$ and $g : Y \to Z$ be quasi-affine morphisms. Let $U \subset Z$ be affine open. Then $g^{-1}(U)$ is quasi-affine by assumption on $g$. Let $j : g^{-1}(U) \to V$ be a quasi-compact open immersion into an affine scheme $V$. By Lemma \ref{lemma-characterize-quasi-affine} above we see that $f^{-1}(g^{-1}(U))$ is a quasi-compact open subscheme of the relative spectrum $\underline{\Spec}_{g^{-1}(U)}(\mathcal{A})$ for some quasi-coherent sheaf of $\mathcal{O}_{g^{-1}(U)}$-algebras $\mathcal{A}$. By Schemes, Lemma \ref{schemes-lemma-push-forward-quasi-coherent} the sheaf $\mathcal{A}' = j_*\mathcal{A}$ is a quasi-coherent sheaf of $\mathcal{O}_V$-algebras with the property that $j^*\mathcal{A}' = \mathcal{A}$. Hence we get a commutative diagram $$\xymatrix{ f^{-1}(g^{-1}(U)) \ar[r] & \underline{\Spec}_{g^{-1}(U)}(\mathcal{A}) \ar[r] \ar[d] & \underline{\Spec}_V(\mathcal{A}') \ar[d] \\ & g^{-1}(U) \ar[r]^j & V }$$ with the square being a fibre square, see Constructions, Lemma \ref{constructions-lemma-spec-properties}. Note that the upper right corner is an affine scheme. Hence $(g \circ f)^{-1}(U)$ is quasi-affine. \end{proof} \begin{lemma} \label{lemma-base-change-quasi-affine} The base change of a quasi-affine morphism is quasi-affine. \end{lemma} \begin{proof} Let $f : X \to S$ be a quasi-affine morphism. By Lemma \ref{lemma-characterize-quasi-affine} above we can find a quasi-coherent sheaf of $\mathcal{O}_S$-algebras $\mathcal{A}$ and a quasi-compact open immersion $X \to \underline{\Spec}_S(\mathcal{A})$ over $S$. Let $g : S' \to S$ be any morphism. Denote $f' : X_{S'} = S' \times_S X \to S'$ the base change of $f$. Since the base change of a quasi-compact open immersion is a quasi-compact open immersion we see that $X_{S'} \to \underline{\Spec}_{S'}(g^*\mathcal{A})$ is a quasi-compact open immersion (we have used Schemes, Lemmas \ref{schemes-lemma-quasi-compact-preserved-base-change} and \ref{schemes-lemma-base-change-immersion} and Constructions, Lemma \ref{constructions-lemma-spec-properties}). By Lemma \ref{lemma-characterize-quasi-affine} again we conclude that $X_{S'} \to S'$ is quasi-affine. \end{proof} \begin{lemma} \label{lemma-quasi-compact-immersion-quasi-affine} A quasi-compact immersion is quasi-affine. \end{lemma} \begin{proof} Let $X \to S$ be a quasi-compact immersion. We have to show the inverse image of every affine open is quasi-affine. Hence, assuming $S$ is an affine scheme, we have to show $X$ is quasi-affine. By Lemma \ref{lemma-quasi-compact-immersion} the morphism $X \to S$ factors as $X \to Z \to S$ where $Z$ is a closed subscheme of $S$ and $X \subset Z$ is a quasi-compact open. Since $S$ is affine Lemma \ref{lemma-closed-immersion} implies $Z$ is affine. Hence we win. \end{proof} \begin{lemma} \label{lemma-affine-quasi-affine} Let $S$ be a scheme. Let $X$ be an affine scheme. A morphism $f : X \to S$ is quasi-affine if and only if it is quasi-compact. In particular any morphism from an affine scheme to a quasi-separated scheme is quasi-affine. \end{lemma} \begin{proof} Let $V \subset S$ be an affine open. Then $f^{-1}(V)$ is an open subscheme of the affine scheme $X$, hence quasi-affine if and only if it is quasi-compact. This proves the first assertion. The quasi-compactness of any $f : X \to S$ where $X$ is affine and $S$ quasi-separated follows from Schemes, Lemma \ref{schemes-lemma-quasi-compact-permanence} applied to $X \to S \to \Spec(\mathbf{Z})$. \end{proof} \begin{lemma} \label{lemma-quasi-affine-permanence} Suppose $g : X \to Y$ is a morphism of schemes over $S$. If $X$ is quasi-affine over $S$ and $Y$ is quasi-separated over $S$, then $g$ is quasi-affine. In particular, any morphism from a quasi-affine scheme to a quasi-separated scheme is quasi-affine. \end{lemma} \begin{proof} The base change $X \times_S Y \to Y$ is quasi-affine by Lemma \ref{lemma-base-change-quasi-affine}. The morphism $X \to X \times_S Y$ is a quasi-compact immersion as $Y \to S$ is quasi-separated, see Schemes, Lemma \ref{schemes-lemma-section-immersion}. A quasi-compact immersion is quasi-affine by Lemma \ref{lemma-quasi-compact-immersion-quasi-affine} and the composition of quasi-affine morphisms is quasi-affine (see Lemma \ref{lemma-composition-quasi-affine}). Thus we win. \end{proof} \section{Types of morphisms defined by properties of ring maps} \label{section-properties-ring-maps} \noindent In this section we study what properties of ring maps allow one to define local properties of morphisms of schemes. \begin{definition} \label{definition-property-local} Let $P$ be a property of ring maps. \begin{enumerate} \item We say that $P$ is {\it local} if the following hold: \begin{enumerate} \item For any ring map $R \to A$, and any $f \in R$ we have $P(R \to A) \Rightarrow P(R_f \to A_f)$. \item For any rings $R$, $A$, any $f \in R$, $a\in A$, and any ring map $R_f \to A$ we have $P(R_f \to A) \Rightarrow P(R \to A_a)$. \item For any ring map $R \to A$, and $a_i \in A$ such that $(a_1, \ldots, a_n) = A$ then $\forall i, P(R \to A_{a_i}) \Rightarrow P(R \to A)$. \end{enumerate} \item We say that $P$ is {\it stable under base change} if for any ring maps $R \to A$, $R \to R'$ we have $P(R \to A) \Rightarrow P(R' \to R' \otimes_R A)$. \item We say that $P$ is {\it stable under composition} if for any ring maps $A \to B$, $B \to C$ we have $P(A \to B) \wedge P(B \to C) \Rightarrow P(A \to C)$. \end{enumerate} \end{definition} \begin{definition} \label{definition-locally-P} Let $P$ be a property of ring maps. Let $f : X \to S$ be a morphism of schemes. We say $f$ is {\it locally of type $P$} if for any $x \in X$ there exists an affine open neighbourhood $U$ of $x$ in $X$ which maps into an affine open $V \subset S$ such that the induced ring map $\mathcal{O}_S(V) \to \mathcal{O}_X(U)$ has property $P$. \end{definition} \noindent This is not a good'' definition unless the property $P$ is a local property. Even if $P$ is a local property we will not automatically use this definition to say that a morphism is locally of type $P$'' unless we also explicitly state the definition elsewhere. \begin{lemma} \label{lemma-locally-P} Let $f : X \to S$ be a morphism of schemes. Let $P$ be a property of ring maps. Let $U$ be an affine open of $X$, and $V$ an affine open of $S$ such that $f(U) \subset V$. If $f$ is locally of type $P$ and $P$ is local, then $P(\mathcal{O}_S(V) \to \mathcal{O}_X(U))$ holds. \end{lemma} \begin{proof} As $f$ is locally of type $P$ for every $u \in U$ there exists an affine open $U_u \subset X$ mapping into an affine open $V_u \subset S$ such that $P(\mathcal{O}_S(V_u) \to \mathcal{O}_X(U_u))$ holds. Choose an open neighbourhood $U'_u \subset U \cap U_u$ of $u$ which is standard affine open in both $U$ and $U_u$, see Schemes, Lemma \ref{schemes-lemma-standard-open-two-affines}. By Definition \ref{definition-property-local} (1)(b) we see that $P(\mathcal{O}_S(V_u) \to \mathcal{O}_X(U'_u))$ holds. Hence we may assume that $U_u \subset U$ is a standard affine open. Choose an open neighbourhood $V'_u \subset V \cap V_u$ of $f(u)$ which is standard affine open in both $V$ and $V_u$, see Schemes, Lemma \ref{schemes-lemma-standard-open-two-affines}. Then $U'_u = f^{-1}(V'_u) \cap U_u$ is a standard affine open of $U_u$ (hence of $U$) and we have $P(\mathcal{O}_S(V'_u) \to \mathcal{O}_X(U'_u))$ by Definition \ref{definition-property-local} (1)(a). Hence we may assume both $U_u \subset U$ and $V_u \subset V$ are standard affine open. Applying Definition \ref{definition-property-local} (1)(b) one more time we conclude that $P(\mathcal{O}_S(V) \to \mathcal{O}_X(U_u))$ holds. Because $U$ is quasi-compact we may choose a finite number of points $u_1, \ldots, u_n \in U$ such that $$U = U_{u_1} \cup \ldots \cup U_{u_n}.$$ By Definition \ref{definition-property-local} (1)(c) we conclude that $P(\mathcal{O}_S(V) \to \mathcal{O}_X(U))$ holds. \end{proof} \begin{lemma} \label{lemma-locally-P-characterize} Let $P$ be a local property of ring maps. Let $f : X \to S$ be a morphism of schemes. The following are equivalent \begin{enumerate} \item The morphism $f$ is locally of type $P$. \item For every affine opens $U \subset X$, $V \subset S$ with $f(U) \subset V$ we have $P(\mathcal{O}_S(V) \to \mathcal{O}_X(U))$. \item There exists an open covering $S = \bigcup_{j \in J} V_j$ and open coverings $f^{-1}(V_j) = \bigcup_{i \in I_j} U_i$ such that each of the morphisms $U_i \to V_j$, $j\in J, i\in I_j$ is locally of type $P$. \item There exists an affine open covering $S = \bigcup_{j \in J} V_j$ and affine open coverings $f^{-1}(V_j) = \bigcup_{i \in I_j} U_i$ such that $P(\mathcal{O}_S(V_j) \to \mathcal{O}_X(U_i))$ holds, for all $j\in J, i\in I_j$. \end{enumerate} Moreover, if $f$ is locally of type $P$ then for any open subschemes $U \subset X$, $V \subset S$ with $f(U) \subset V$ the restriction $f|_U : U \to V$ is locally of type $P$. \end{lemma} \begin{proof} This follows from Lemma \ref{lemma-locally-P} above. \end{proof} \begin{lemma} \label{lemma-composition-type-P} Let $P$ be a property of ring maps. Assume $P$ is local and stable under composition. The composition of morphisms locally of type $P$ is locally of type $P$. \end{lemma} \begin{proof} Let $f : X \to Y$ and $g : Y \to Z$ be morphisms locally of type $P$. Let $x \in X$. Choose an affine open neighbourhood $W \subset Z$ of $g(f(x))$. Choose an affine open neighbourhood $V \subset g^{-1}(W)$ of $f(x)$. Choose an affine open neighbourhood $U \subset f^{-1}(V)$ of $x$. By Lemma \ref{lemma-locally-P-characterize} the ring maps $\mathcal{O}_Z(W) \to \mathcal{O}_Y(V)$ and $\mathcal{O}_Y(V) \to \mathcal{O}_X(U)$ satisfy $P$. Hence $\mathcal{O}_Z(W) \to \mathcal{O}_X(U)$ satisfies $P$ as $P$ is assumed stable under composition. \end{proof} \begin{lemma} \label{lemma-base-change-type-P} Let $P$ be a property of ring maps. Assume $P$ is local and stable under base change. The base change of a morphism locally of type $P$ is locally of type $P$. \end{lemma} \begin{proof} Let $f : X \to S$ be a morphism locally of type $P$. Let $S' \to S$ be any morphism. Denote $f' : X_{S'} = S' \times_S X \to S'$ the base change of $f$. For every $s' \in S'$ there exists an open affine neighbourhood $s' \in V' \subset S'$ which maps into some open affine $V \subset S$. By Lemma \ref{lemma-locally-P-characterize} the open $f^{-1}(V)$ is a union of affines $U_i$ such that the ring maps $\mathcal{O}_S(V) \to \mathcal{O}_X(U_i)$ all satisfy $P$. By the material in Schemes, Section \ref{schemes-section-fibre-products} we see that $f^{-1}(U)_{V'} = V' \times_V f^{-1}(V)$ is the union of the affine opens $V' \times_V U_i$. Since $\mathcal{O}_{X_{S'}}(V' \times_V U_i) = \mathcal{O}_{S'}(V') \otimes_{\mathcal{O}_S(V)} \mathcal{O}_X(U_i)$ we see that the ring maps $\mathcal{O}_{S'}(V') \to \mathcal{O}_{X_{S'}}(V' \times_V U_i)$ satisfy $P$ as $P$ is assumed stable under base change. \end{proof} \begin{lemma} \label{lemma-properties-local} The following properties of a ring map $R \to A$ are local. \begin{enumerate} \item (Isomorphism on local rings.) For every prime $\mathfrak q$ of $A$ lying over $\mathfrak p \subset R$ the ring map $R \to A$ induces an isomorphism $R_{\mathfrak p} \to A_{\mathfrak q}$. \item (Open immersion.) For every prime $\mathfrak q$ of $A$ there exists an $f \in R$, $\varphi(f) \not \in \mathfrak q$ such that the ring map $\varphi : R \to A$ induces an isomorphism $R_f \to A_f$. \item (Reduced fibres.) For every prime $\mathfrak p$ of $R$ the fibre ring $A \otimes_R \kappa(\mathfrak p)$ is reduced. \item (Fibres of dimension at most $n$.) For every prime $\mathfrak p$ of $R$ the fibre ring $A \otimes_R \kappa(\mathfrak p)$ has Krull dimension at most $n$. \item (Locally Noetherian on the target.) The ring map $R \to A$ has the property that $A$ is Noetherian. \item Add more here as needed\footnote{But only those properties that are not already dealt with separately elsewhere.}. \end{enumerate} \end{lemma} \begin{proof} Omitted. \end{proof} \begin{lemma} \label{lemma-properties-base-change} The following properties of ring maps are stable under base change. \begin{enumerate} \item (Isomorphism on local rings.) For every prime $\mathfrak q$ of $A$ lying over $\mathfrak p \subset R$ the ring map $R \to A$ induces an isomorphism $R_{\mathfrak p} \to A_{\mathfrak q}$. \item (Open immersion.) For every prime $\mathfrak q$ of $A$ there exists an $f \in R$, $\varphi(f) \not \in \mathfrak q$ such that the ring map $\varphi : R \to A$ induces an isomorphism $R_f \to A_f$. \item Add more here as needed\footnote{But only those properties that are not already dealt with separately elsewhere.}. \end{enumerate} \end{lemma} \begin{proof} Omitted. \end{proof} \begin{lemma} \label{lemma-properties-composition} The following properties of ring maps are stable under composition. \begin{enumerate} \item (Isomorphism on local rings.) For every prime $\mathfrak q$ of $A$ lying over $\mathfrak p \subset R$ the ring map $R \to A$ induces an isomorphism $R_{\mathfrak p} \to A_{\mathfrak q}$. \item (Open immersion.) For every prime $\mathfrak q$ of $A$ there exists an $f \in R$, $\varphi(f) \not \in \mathfrak q$ such that the ring map $\varphi : R \to A$ induces an isomorphism $R_f \to A_f$. \item (Locally Noetherian on the target.) The ring map $R \to A$ has the property that $A$ is Noetherian. \item Add more here as needed\footnote{But only those properties that are not already dealt with separately elsewhere.}. \end{enumerate} \end{lemma} \begin{proof} Omitted. \end{proof} \section{Morphisms of finite type} \label{section-finite-type} \noindent Recall that a ring map $R \to A$ is said to be of finite type if $A$ is isomorphic to a quotient of $R[x_1, \ldots, x_n]$ as an $R$-algebra, see Algebra, Definition \ref{algebra-definition-finite-type}. \begin{definition} \label{definition-finite-type} Let $f : X \to S$ be a morphism of schemes. \begin{enumerate} \item We say that $f$ is of {\it finite type at $x \in X$} if there exists an affine open neighbourhood $\Spec(A) = U \subset X$ of $x$ and an affine open $\Spec(R) = V \subset S$ with $f(U) \subset V$ such that the induced ring map $R \to A$ is of finite type. \item We say that $f$ is {\it locally of finite type} if it is of finite type at every point of $X$. \item We say that $f$ is of {\it finite type} if it is locally of finite type and quasi-compact. \end{enumerate} \end{definition} \begin{lemma} \label{lemma-locally-finite-type-characterize} Let $f : X \to S$ be a morphism of schemes. The following are equivalent \begin{enumerate} \item The morphism $f$ is locally of finite type. \item For all affine opens $U \subset X$, $V \subset S$ with $f(U) \subset V$ the ring map $\mathcal{O}_S(V) \to \mathcal{O}_X(U)$ is of finite type. \item There exist an open covering $S = \bigcup_{j \in J} V_j$ and open coverings $f^{-1}(V_j) = \bigcup_{i \in I_j} U_i$ such that each of the morphisms $U_i \to V_j$, $j\in J, i\in I_j$ is locally of finite type. \item There exist an affine open covering $S = \bigcup_{j \in J} V_j$ and affine open coverings $f^{-1}(V_j) = \bigcup_{i \in I_j} U_i$ such that the ring map $\mathcal{O}_S(V_j) \to \mathcal{O}_X(U_i)$ is of finite type, for all $j\in J, i\in I_j$. \end{enumerate} Moreover, if $f$ is locally of finite type then for any open subschemes $U \subset X$, $V \subset S$ with $f(U) \subset V$ the restriction $f|_U : U \to V$ is locally of finite type. \end{lemma} \begin{proof} This follows from Lemma \ref{lemma-locally-P} if we show that the property $R \to A$ is of finite type'' is local. We check conditions (a), (b) and (c) of Definition \ref{definition-property-local}. By Algebra, Lemma \ref{algebra-lemma-base-change-finiteness} being of finite type is stable under base change and hence we conclude (a) holds. By Algebra, Lemma \ref{algebra-lemma-compose-finite-type} being of finite type is stable under composition and trivially for any ring $R$ the ring map $R \to R_f$ is of finite type. We conclude (b) holds. Finally, property (c) is true according to Algebra, Lemma \ref{algebra-lemma-cover-upstairs}. \end{proof} \begin{lemma} \label{lemma-composition-finite-type} The composition of two morphisms which are locally of finite type is locally of finite type. The same is true for morphisms of finite type. \end{lemma} \begin{proof} In the proof of Lemma \ref{lemma-locally-finite-type-characterize} we saw that being of finite type is a local property of ring maps. Hence the first statement of the lemma follows from Lemma \ref{lemma-composition-type-P} combined with the fact that being of finite type is a property of ring maps that is stable under composition, see Algebra, Lemma \ref{algebra-lemma-compose-finite-type}. By the above and the fact that compositions of quasi-compact morphisms are quasi-compact, see Schemes, Lemma \ref{schemes-lemma-composition-quasi-compact} we see that the composition of morphisms of finite type is of finite type. \end{proof} \begin{lemma} \label{lemma-base-change-finite-type} The base change of a morphism which is locally of finite type is locally of finite type. The same is true for morphisms of finite type. \end{lemma} \begin{proof} In the proof of Lemma \ref{lemma-locally-finite-type-characterize} we saw that being of finite type is a local property of ring maps. Hence the first statement of the lemma follows from Lemma \ref{lemma-base-change-type-P} combined with the fact that being of finite type is a property of ring maps that is stable under base change, see Algebra, Lemma \ref{algebra-lemma-base-change-finiteness}. By the above and the fact that a base change of a quasi-compact morphism is quasi-compact, see Schemes, Lemma \ref{schemes-lemma-quasi-compact-preserved-base-change} we see that the base change of a morphism of finite type is a morphism of finite type. \end{proof} \begin{lemma} \label{lemma-immersion-locally-finite-type} A closed immersion is of finite type. An immersion is locally of finite type. \end{lemma} \begin{proof} This is true because an open immersion is a local isomorphism, and a closed immersion is obviously of finite type. \end{proof} \begin{lemma} \label{lemma-finite-type-noetherian} Let $f : X \to S$ be a morphism. If $S$ is (locally) Noetherian and $f$ (locally) of finite type then $X$ is (locally) Noetherian. \end{lemma} \begin{proof} This follows immediately from the fact that a ring of finite type over a Noetherian ring is Noetherian, see Algebra, Lemma \ref{algebra-lemma-Noetherian-permanence}. (Also: use the fact that the source of a quasi-compact morphism with quasi-compact target is quasi-compact.) \end{proof} \begin{lemma} \label{lemma-finite-type-Noetherian-quasi-separated} Let $f : X \to S$ be locally of finite type with $S$ locally Noetherian. Then $f$ is quasi-separated. \end{lemma} \begin{proof} In fact, it is true that $X$ is quasi-separated, see Properties, Lemma \ref{properties-lemma-locally-Noetherian-quasi-separated} and Lemma \ref{lemma-finite-type-noetherian} above. Then apply Schemes, Lemma \ref{schemes-lemma-compose-after-separated} to conclude that $f$ is quasi-separated. \end{proof} \begin{lemma} \label{lemma-permanence-finite-type} Let $X \to Y$ be a morphism of schemes over a base scheme $S$. If $X$ is locally of finite type over $S$, then $X \to Y$ is locally of finite type. \end{lemma} \begin{proof} Via Lemma \ref{lemma-locally-finite-type-characterize} this translates into the following algebra