The Stacks project

65.16 Scheme theoretic image

Caution: Some of the material in this section is ultra-general and behaves differently from what you might expect.

slogan

Lemma 65.16.1. 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 $Z \subset Z'$.

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 65.13.1. 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 65.14.2. $\square$

Suppose that in the situation of Lemma 65.16.1 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 64.29) we conclude. Thus the following definition agrees with the earlier definition for morphisms of schemes.

Definition 65.16.2. 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 65.16.1 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 65.16.3. 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

  1. the sheaf of ideals $\mathcal{I} = \mathop{\mathrm{Ker}}(\mathcal{O}_ Y \to f_*\mathcal{O}_ X)$ is quasi-coherent,

  2. the scheme theoretic image $Z$ is the closed subspace corresponding to $\mathcal{I}$,

  3. 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

  4. 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 65.16.1 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 65.11.2 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 65.16.4. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $X$ is reduced. Then

  1. the scheme theoretic image $Z$ of $f$ is the reduced induced algebraic space structure on $\overline{|f|(|X|)}$, and

  2. 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 64.12.4. 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 65.16.5. 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

\[ \xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[rr] \ar[d] & & X \ar[d] \ar[ld] \\ \mathop{\mathrm{Spec}}(A) \ar[r] & Z \ar[r] & Y } \]

such that the closed point of $\mathop{\mathrm{Spec}}(A)$ maps to $z$.

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 65.16.3 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 65.16.6. Let $S$ be a scheme. 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 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

\[ \xymatrix{ X_1 \ar[r] \ar[d] & Z_1 \ar[d] \ar[r] & Y_1 \ar[d] \\ X_2 \ar[r] & Z_2 \ar[r] & Y_2 } \]

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 65.16.7. 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 65.8.9 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 65.16.3. 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 65.4.7), we conclude that $Y' \cap f^{-1}(V) \to V$ is an isomorphism as desired. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 082W. Beware of the difference between the letter 'O' and the digit '0'.