The Stacks project

76.37 Extending properties from an open

In this section we collect a number of results of the form: If $f : X \to Y$ is a flat morphism of algebraic spaces and $f$ satisfies some property over a dense open of $Y$, then $f$ satisfies the same property over all of $Y$.

Lemma 76.37.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $V \subset Y$ be an open subspace. Assume

  1. $f$ is locally of finite presentation,

  2. $\mathcal{F}$ is of finite type and flat over $Y$,

  3. $V \to Y$ is quasi-compact and scheme theoretically dense,

  4. $\mathcal{F}|_{f^{-1}V}$ is of finite presentation.

Then $\mathcal{F}$ is of finite presentation.

Proof. It suffices to prove the pullback of $\mathcal{F}$ to a scheme surjective and étale over $X$ is of finite presentation. Hence we may assume $X$ is a scheme. Similarly, we can replace $Y$ by a scheme surjective and étale and over $Y$ (the inverse image of $V$ in this scheme is scheme theoretically dense, see Morphisms of Spaces, Section 67.17). Thus we reduce to the case of schemes which is More on Flatness, Lemma 38.11.1. $\square$

Lemma 76.37.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $V \subset Y$ be an open subspace. Assume

  1. $f$ is locally of finite type and flat,

  2. $V \to Y$ is quasi-compact and scheme theoretically dense,

  3. $f|_{f^{-1}V} : f^{-1}V \to V$ is locally of finite presentation.

Then $f$ is of locally of finite presentation.

Proof. The proof is identical to the proof of Lemma 76.37.1 except one uses More on Flatness, Lemma 38.11.2. $\square$

Lemma 76.37.3. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is flat and locally of finite type. Let $V \subset Y$ be an open subspace such that $|V| \subset |Y|$ is dense and such that $X_ V \to V$ has relative dimension $\leq d$. If also either

  1. $f$ is locally of finite presentation, or

  2. $V \to Y$ is quasi-compact,

then $f : X \to Y$ has relative dimension $\leq d$.

Proof. We may replace $Y$ by its reduction, hence we may assume $Y$ is reduced. Then $V$ is scheme theoretically dense in $Y$, see Morphisms of Spaces, Lemma 67.17.7. By definition the property of having relative dimension $\leq d$ can be checked on an étale covering, see Morphisms of Spaces, Sections 67.33. Thus it suffices to prove $f$ has relative dimension $\leq d$ after replacing $X$ by a scheme surjective and étale over $X$. Similarly, we can replace $Y$ by a scheme surjective and étale and over $Y$. The inverse image of $V$ in this scheme is scheme theoretically dense, see Morphisms of Spaces, Section 67.17. Since a scheme theoretically dense open of a scheme is in particular dense, we reduce to the case of schemes which is More on Flatness, Lemma 38.11.3. $\square$

Lemma 76.37.4. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is flat and proper. Let $V \to Y$ be an open subspace with $|V| \subset |Y|$ dense such that $X_ V \to V$ is finite. If also either $f$ is locally of finite presentation or $V \to Y$ is quasi-compact, then $f$ is finite.

Proof. By Lemma 76.37.3 the fibres of $f$ have dimension zero. By Morphisms of Spaces, Lemma 67.34.6 this implies that $f$ is locally quasi-finite. By Morphisms of Spaces, Lemma 67.51.1 this implies that $f$ is representable. We can check whether $f$ is finite étale locally on $Y$, hence we may assume $Y$ is a scheme. Since $f$ is representable, we reduce to the case of schemes which is More on Flatness, Lemma 38.11.4. $\square$

Lemma 76.37.5. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $V \subset Y$ be an open subspace. If

  1. $f$ is separated, locally of finite type, and flat,

  2. $f^{-1}(V) \to V$ is an isomorphism, and

  3. $V \to Y$ is quasi-compact and scheme theoretically dense,

then $f$ is an open immersion.

Proof. Applying Lemma 76.37.2 we see that $f$ is locally of finite presentation. Applying Lemma 76.37.3 we see that $f$ has relative dimension $\leq 0$. By Morphisms of Spaces, Lemma 67.34.6 this implies that $f$ is locally quasi-finite. By Morphisms of Spaces, Lemma 67.51.1 this implies that $f$ is representable. By Descent on Spaces, Lemma 74.11.14 we can check whether $f$ is an open immersion étale locally on $Y$. Hence we may assume that $Y$ is a scheme. Since $f$ is representable, we reduce to the case of schemes which is More on Flatness, Lemma 38.11.5. $\square$


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 0875. Beware of the difference between the letter 'O' and the digit '0'.