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65.28 Morphisms of finite presentation

The property “locally of finite presentation” of morphisms of schemes is étale local on the source-and-target, see Descent, Remark 35.29.7. It is also stable under base change and fpqc local on the target, see Morphisms, Lemma 29.21.4, and Descent, Lemma 35.20.11. Hence, by Lemma 65.22.1 above, we may define what it means for a morphism of algebraic spaces to be locally of finite presentation as follows and it agrees with the already existing notion defined in Section 65.3 when the morphism is representable.

Definition 65.28.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$.

  1. We say $f$ is locally of finite presentation if the equivalent conditions of Lemma 65.22.1 hold with $\mathcal{P} =$“locally of finite presentation”.

  2. Let $x \in |X|$. We say $f$ is of finite presentation at $x$ if there exists an open neighbourhood $X' \subset X$ of $x$ such that $f|_{X'} : X' \to Y$ is locally of finite presentation1.

  3. A morphism of algebraic spaces $f : X \to Y$ is of finite presentation if it is locally of finite presentation, quasi-compact and quasi-separated.

Note that a morphism of finite presentation is not just a quasi-compact morphism which is locally of finite presentation.

Lemma 65.28.2. The composition of morphisms of finite presentation is of finite presentation. The same holds for locally of finite presentation.

Proof. See Remark 65.22.3 and Morphisms, Lemma 29.21.3. Also use the result for quasi-compact and for quasi-separated morphisms (Lemmas 65.8.5 and 65.4.8). $\square$

Lemma 65.28.3. A base change of a morphism of finite presentation is of finite presentation The same holds for locally of finite presentation.

Proof. See Remark 65.22.4 and Morphisms, Lemma 29.21.4. Also use the result for quasi-compact and for quasi-separated morphisms (Lemmas 65.8.4 and 65.4.4). $\square$

Lemma 65.28.4. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent:

  1. $f$ is locally of finite presentation,

  2. for every $x \in |X|$ the morphism $f$ is of finite presentation at $x$,

  3. for every scheme $Z$ and any morphism $Z \to Y$ the morphism $Z \times _ Y X \to Z$ is locally of finite presentation,

  4. for every affine scheme $Z$ and any morphism $Z \to Y$ the morphism $Z \times _ Y X \to Z$ is locally of finite presentation,

  5. there exists a scheme $V$ and a surjective étale morphism $V \to Y$ such that $V \times _ Y X \to V$ is locally of finite presentation,

  6. there exists a scheme $U$ and a surjective étale morphism $\varphi : U \to X$ such that the composition $f \circ \varphi $ is locally of finite presentation,

  7. for every commutative diagram

    \[ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } \]

    where $U$, $V$ are schemes and the vertical arrows are étale the top horizontal arrow is locally of finite presentation,

  8. there exists a commutative diagram

    \[ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } \]

    where $U$, $V$ are schemes, the vertical arrows are étale, and $U \to X$ is surjective such that the top horizontal arrow is locally of finite presentation, and

  9. there exist Zariski coverings $Y = \bigcup _{i \in I} Y_ i$, and $f^{-1}(Y_ i) = \bigcup X_{ij}$ such that each morphism $X_{ij} \to Y_ i$ is locally of finite presentation.

Proof. Omitted. $\square$

Lemma 65.28.5. A morphism which is locally of finite presentation is locally of finite type. A morphism of finite presentation is of finite type.

Proof. Let $f : X \to Y$ be a morphism of algebraic spaces which is locally of finite presentation. This means there exists a diagram as in Lemma 65.22.1 with $h$ locally of finite presentation and surjective vertical arrow $a$. By Morphisms, Lemma 29.21.8 $h$ is locally of finite type. Hence $X \to Y$ is locally of finite type by definition. If $f$ is of finite presentation then it is quasi-compact and it follows that $f$ is of finite type. $\square$

Lemma 65.28.6. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. If $f$ is of finite presentation and $Y$ is Noetherian, then $X$ is Noetherian.

Proof. Assume $f$ is of finite presentation and $Y$ Noetherian. By Lemmas 65.28.5 and 65.23.5 we see that $X$ is locally Noetherian. As $f$ is quasi-compact and $Y$ is quasi-compact we see that $X$ is quasi-compact. As $f$ is of finite presentation it is quasi-separated (see Definition 65.28.1) and as $Y$ is Noetherian it is quasi-separated (see Properties of Spaces, Definition 64.24.1). Hence $X$ is quasi-separated by Lemma 65.4.9. Hence we have checked all three conditions of Properties of Spaces, Definition 64.24.1 and we win. $\square$

Lemma 65.28.7. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$.

  1. If $Y$ is locally Noetherian and $f$ locally of finite type then $f$ is locally of finite presentation.

  2. If $Y$ is locally Noetherian and $f$ of finite type and quasi-separated then $f$ is of finite presentation.

Proof. Assume $f : X \to Y$ locally of finite type and $Y$ locally Noetherian. This means there exists a diagram as in Lemma 65.22.1 with $h$ locally of finite type and surjective vertical arrow $a$. By Morphisms, Lemma 29.21.9 $h$ is locally of finite presentation. Hence $X \to Y$ is locally of finite presentation by definition. This proves (1). If $f$ is of finite type and quasi-separated then it is also quasi-compact and quasi-separated and (2) follows immediately. $\square$

Lemma 65.28.8. Let $S$ be a scheme. Let $Y$ be an algebraic space over $S$ which is quasi-compact and quasi-separated. If $X$ is of finite presentation over $Y$, then $X$ is quasi-compact and quasi-separated.

Proof. Omitted. $\square$

Lemma 65.28.9. Let $S$ be a scheme. Let $f : X \to Y$ and $Y \to Z$ be morphisms of algebraic spaces over $S$. If $X$ is locally of finite presentation over $Z$, and $Y$ is locally of finite type over $Z$, then $f$ is locally of finite presentation.

Proof. Choose a scheme $W$ and a surjective étale morphism $W \to Z$. Then choose a scheme $V$ and a surjective étale morphism $V \to W \times _ Z Y$. Finally choose a scheme $U$ and a surjective étale morphism $U \to V \times _ Y X$. By definition $U$ is locally of finite presentation over $W$ and $V$ is locally of finite type over $W$. By Morphisms, Lemma 29.21.11 the morphism $U \to V$ is locally of finite presentation. Hence $f$ is locally of finite presentation. $\square$

Lemma 65.28.10. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ with diagonal $\Delta : X \to X \times _ Y X$. If $f$ is locally of finite type then $\Delta $ is locally of finite presentation. If $f$ is quasi-separated and locally of finite type, then $\Delta $ is of finite presentation.

Proof. Note that $\Delta $ is a morphism over $X$ (via the second projection $X \times _ Y X \to X$). Assume $f$ is locally of finite type. Note that $X$ is of finite presentation over $X$ and $X \times _ Y X$ is of finite type over $X$ (by Lemma 65.23.3). Thus the first statement holds by Lemma 65.28.9. The second statement follows from the first, the definitions, and the fact that a diagonal morphism is separated (Lemma 65.4.1). $\square$

Lemma 65.28.11. An open immersion of algebraic spaces is locally of finite presentation.

Proof. An open immersion is by definition representable, hence we can use the general principle Spaces, Lemma 63.5.8 and Morphisms, Lemma 29.21.5. $\square$

Lemma 65.28.12. A closed immersion $i : Z \to X$ is of finite presentation if and only if the associated quasi-coherent sheaf of ideals $\mathcal{I} = \mathop{\mathrm{Ker}}(\mathcal{O}_ X \to i_*\mathcal{O}_ Z)$ is of finite type (as an $\mathcal{O}_ X$-module).

Proof. Let $U$ be a scheme and let $U \to X$ be a surjective étale morphism. By Lemma 65.28.4 we see that $i' : Z \times _ X U \to U$ is of finite presentation if and only if $i$ is. By Properties of Spaces, Section 64.30 we see that $\mathcal{I}$ is of finite type if and only if $\mathcal{I}|_ U = \mathop{\mathrm{Ker}}(\mathcal{O}_ U \to i'_*\mathcal{O}_{Z \times _ X U})$ is. Hence the result follows from the case of schemes, see Morphisms, Lemma 29.21.7. $\square$

[1] It seems awkward to use “locally of finite presentation at $x$”, but the current terminology may be misleading in the sense that “of finite presentation at $x$” does not mean that there is an open neighbourhood $X' \subset X$ such that $f|_{X'}$ is of finite presentation.

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