Lemma 67.16.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. There exists a closed subspace $Z \subset Y$ such that $f$ factors through $Z$ and such that for any other closed subspace $Z' \subset Y$ such that $f$ factors through $Z'$ we have $Z \subset Z'$.
67.16 Scheme theoretic image
Caution: Some of the material in this section is ultra-general and behaves differently from what you might expect.
Proof. Let $\mathcal{I} = \mathop{\mathrm{Ker}}(\mathcal{O}_ Y \to f_*\mathcal{O}_ X)$. If $\mathcal{I}$ is quasi-coherent then we just take $Z$ to be the closed subscheme determined by $\mathcal{I}$, see Lemma 67.13.1. In general the 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 67.14.2. $\square$
Suppose that in the situation of Lemma 67.16.1 above $X$ and $Y$ are representable. Then the closed subspace $Z \subset Y$ found in the lemma agrees with the closed subscheme $Z \subset Y$ found in Morphisms, Lemma 29.6.1. The reason is that closed subspaces (or subschemes) are in a inclusion reversing correspondence with quasi-coherent ideal sheaves on $X_{\acute{e}tale}$ and $X$. As the category of quasi-coherent modules on $X_{\acute{e}tale}$ and $X$ are the same (Properties of Spaces, Section 66.29) we conclude. Thus the following definition agrees with the earlier definition for morphisms of schemes.
Definition 67.16.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The scheme theoretic image of $f$ is the smallest closed subspace $Z \subset Y$ through which $f$ factors, see Lemma 67.16.1 above.
We often just denote $f : X \to Z$ the factorization of $f$. If the morphism $f$ is not quasi-compact, then (in general) the construction of the scheme theoretic image does not commute with restriction to open subspaces of $Y$.
Lemma 67.16.3. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $Z \subset Y$ be the scheme theoretic image of $f$. If $f$ is quasi-compact then
the sheaf of ideals $\mathcal{I} = \mathop{\mathrm{Ker}}(\mathcal{O}_ Y \to f_*\mathcal{O}_ X)$ is quasi-coherent,
the scheme theoretic image $Z$ is the closed subspace corresponding to $\mathcal{I}$,
for any étale morphism $V \to Y$ the scheme theoretic image of $X \times _ Y V \to V$ is equal to $Z \times _ Y V$, and
the image $|f|(|X|) \subset |Z|$ is a dense subset of $|Z|$.
Proof. To prove (3) it suffices to prove (1) and (2) since the formation of $\mathcal{I}$ commutes with étale localization. If (1) holds then in the proof of Lemma 67.16.1 we showed (2). Let us prove that $\mathcal{I}$ is quasi-coherent. Since the property of being quasi-coherent is étale local we may assume $Y$ is an affine scheme. As $f$ is quasi-compact, we can find an affine scheme $U$ and a surjective étale morphism $U \to X$. Denote $f'$ the composition $U \to X \to Y$. Then $f_*\mathcal{O}_ X$ is a subsheaf of $f'_*\mathcal{O}_ U$, and hence $\mathcal{I} = \mathop{\mathrm{Ker}}(\mathcal{O}_ Y \to \mathcal{O}_{X'})$. By Lemma 67.11.2 the sheaf $f'_*\mathcal{O}_ U$ is quasi-coherent on $Y$. Hence $\mathcal{I}$ is quasi-coherent as a kernel of a map between coherent modules. Finally, part (4) follows from parts (1), (2), and (3) as the ideal $\mathcal{I}$ will be the unit ideal in any point of $|Y|$ which is not contained in the closure of $|f|(|X|)$. $\square$
Lemma 67.16.4. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $X$ is reduced. Then
the scheme theoretic image $Z$ of $f$ is the reduced induced algebraic space structure on $\overline{|f|(|X|)}$, and
for any étale morphism $V \to Y$ the scheme theoretic image of $X \times _ Y V \to V$ is equal to $Z \times _ Y V$.
Proof. Part (1) is true because the reduced induced algebraic space structure on $\overline{|f|(|X|)}$ is the smallest closed subspace of $Y$ through which $f$ factors, see Properties of Spaces, Lemma 66.12.4. Part (2) follows from (1), the fact that $|V| \to |Y|$ is open, and the fact that being reduced is preserved under étale localization. $\square$
Lemma 67.16.5. Let $S$ be a scheme. Let $f : X \to Y$ be a quasi-compact morphism of algebraic spaces over $S$. Let $Z$ be the scheme theoretic image of $f$. Let $z \in |Z|$. There exists a valuation ring $A$ with fraction field $K$ and a commutative diagram such that the closed point of $\mathop{\mathrm{Spec}}(A)$ maps to $z$.
\[ \xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[rr] \ar[d] & & X \ar[d] \ar[ld] \\ \mathop{\mathrm{Spec}}(A) \ar[r] & Z \ar[r] & Y } \]
Proof. Choose an affine scheme $V$ with a point $z' \in V$ and an étale morphism $V \to Y$ mapping $z'$ to $z$. Let $Z' \subset V$ be the scheme theoretic image of $X \times _ Y V \to V$. By Lemma 67.16.3 we have $Z' = Z \times _ Y V$. Thus $z' \in Z'$. Since $f$ is quasi-compact and $V$ is affine we see that $X \times _ Y V$ is quasi-compact. Hence there exists an affine scheme $W$ and a surjective étale morphism $W \to X \times _ Y V$. Then $Z' \subset V$ is also the scheme theoretic image of $W \to V$. By Morphisms, Lemma 29.6.5 we can choose a diagram
\[ \xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r] \ar[d] & W \ar[r] \ar[d] & X \times _ Y V \ar[d] \ar[r] & X \ar[d] \\ \mathop{\mathrm{Spec}}(A) \ar[r] & Z' \ar[r] & V \ar[r] & Y } \]
such that the closed point of $\mathop{\mathrm{Spec}}(A)$ maps to $z'$. Composing with $Z' \to Z$ and $W \to X \times _ Y V \to X$ we obtain a solution. $\square$
Lemma 67.16.6. Let $S$ be a scheme. Let be a commutative diagram of algebraic spaces over $S$. 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
\[ \xymatrix{ X_1 \ar[d] \ar[r]_{f_1} & Y_1 \ar[d] \\ X_2 \ar[r]^{f_2} & Y_2 } \]
\[ \xymatrix{ X_1 \ar[r] \ar[d] & Z_1 \ar[d] \ar[r] & Y_1 \ar[d] \\ X_2 \ar[r] & Z_2 \ar[r] & Y_2 } \]
Proof. The scheme theoretic inverse image of $Z_2$ in $Y_1$ is a closed subspace of $Y_1$ through which $f_1$ factors. Hence $Z_1$ is contained in this. This proves the lemma. $\square$
Lemma 67.16.7. Let $S$ be a scheme. Let $f : X \to Y$ be a separated morphism of algebraic spaces over $S$. Let $V \subset Y$ be an open subspace such that $V \to Y$ is quasi-compact. 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 $V$.
Proof. By Lemma 67.8.9 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 67.16.3. 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 (Lemma 67.4.7), we conclude that $Y' \cap f^{-1}(V) \to V$ is an isomorphism as desired. $\square$
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