The Stacks project

65.23 Morphisms of finite type

The property “locally of finite type” 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 29.15.4, and Descent, Lemmas 35.20.10. Hence, by Lemma 65.22.1 above, we may define what it means for a morphism of algebraic spaces to be locally of finite type as follows and it agrees with the already existing notion defined in Section 65.3 when the morphism is representable.

Definition 65.23.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$.

  1. We say $f$ locally of finite type if the equivalent conditions of Lemma 65.22.1 hold with $\mathcal{P} = \text{locally of finite type}$.

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

  3. We say $f$ is of finite type if it is locally of finite type and quasi-compact.

Consider the algebraic space $\mathbf{A}^1_ k/\mathbf{Z}$ of Spaces, Example 63.14.8. The morphism $\mathbf{A}^1_ k/\mathbf{Z} \to \mathop{\mathrm{Spec}}(k)$ is of finite type.

Lemma 65.23.2. The composition of finite type morphisms is of finite type. The same holds for locally of finite type.

Lemma 65.23.3. A base change of a finite type morphism is finite type. The same holds for locally of finite type.

Lemma 65.23.4. 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 type,

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

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

  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 type,

  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 type,

  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 type,

  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, $U \to X$ is surjective, and the top horizontal arrow is locally of finite type, 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 type.

Proof. Each of the conditions (2), (3), (4), (5), (6), (7), and (9) imply condition (8) in a straightforward manner. For example, if (5) holds, then we can choose a scheme $V$ which is a disjoint union of affines and a surjective morphism $V \to Y$ (see Properties of Spaces, Lemma 64.6.1). Then $V \times _ Y X \to V$ is locally of finite type by (5). Choose a scheme $U$ and a surjective étale morphism $U \to V \times _ Y X$. Then $U \to V$ is locally of finite type by Lemma 65.23.2. Hence (8) is true.

The conditions (1), (7), and (8) are equivalent by definition.

To finish the proof, we show that (1) implies all of the conditions (2), (3), (4), (5), (6), and (9). For (2) this is immediate. For (3), (4), (5), and (9) this follows from the fact that being locally of finite type is preserved under base change, see Lemma 65.23.3. For (6) we can take $U = X$ and we're done. $\square$

Lemma 65.23.5. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. If $f$ is locally of finite type and $Y$ is locally Noetherian, then $X$ is locally Noetherian.

Proof. Let

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

be a commutative diagram where $U$, $V$ are schemes and the vertical arrows are surjective étale. If $f$ is locally of finite type, then $U \to V$ is locally of finite type. If $Y$ is locally Noetherian, then $V$ is locally Noetherian. By Morphisms, Lemma 29.15.6 we see that $U$ is locally Noetherian, which means that $X$ is locally Noetherian. $\square$

Lemma 65.23.6. Let $S$ be a scheme. Let $f : X \to Y$, $g : Y \to Z$ be morphisms of algebraic spaces over $S$. If $g \circ f : X \to Z$ is locally of finite type, then $f : X \to Y$ is locally of finite type.

Proof. We can find a diagram

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

where $U$, $V$, $W$ are schemes, the vertical arrows are étale and surjective, see Spaces, Lemma 63.11.6. At this point we can use Lemma 65.23.4 and Morphisms, Lemma 29.15.8 to conclude. $\square$

Proof. Follows from the general principle Spaces, Lemma 63.5.8 and Morphisms, Lemmas 29.15.5. $\square$


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