Section 67.3 (03HA): Properties of representable morphisms

67.3 Properties of representable morphisms

Let $S$ be a scheme. Let $f : X \to Y$ be a representable morphism of algebraic spaces. In Spaces, Section 65.5 we defined what it means for $f$ to have property $\mathcal{P}$ in case $\mathcal{P}$ is a property of morphisms of schemes which

is preserved under any base change, see Schemes, Definition 26.18.3, and
is fppf local on the base, see Descent, Definition 35.22.1.

Namely, in this case we say $f$ has property $\mathcal{P}$ if and only if for every scheme $U$ and any morphism $U \to Y$ the morphism of schemes $X \times _ Y U \to U$ has property $\mathcal{P}$.

According to the lists in Spaces, Section 65.4 this applies to the following properties: (1)(a) closed immersions, (1)(b) open immersions, (1)(c) quasi-compact immersions, (2) quasi-compact, (3) universally-closed, (4) (quasi-)separated, (5) monomorphism, (6) surjective, (7) universally injective, (8) affine, (9) quasi-affine, (10) (locally) of finite type, (11) (locally) quasi-finite, (12) (locally) of finite presentation, (13) locally of finite type of relative dimension $d$, (14) universally open, (15) flat, (16) syntomic, (17) smooth, (18) unramified (resp. G-unramified), (19) étale, (20) proper, (21) finite or integral, (22) finite locally free, (23) universally submersive, (24) universal homeomorphism, and (25) immersion.

In this chapter we will redefine these notions for not necessarily representable morphisms of algebraic spaces. Whenever we do this we will make sure that the new definition agrees with the old one, in order to avoid ambiguity.

Note that the definition above applies whenever $X$ is a scheme, since a morphism from a scheme to an algebraic space is representable. And in particular it applies when both $X$ and $Y$ are schemes. In Spaces, Lemma 65.5.3 we have seen that in this case the definitions match, and no ambiguity arise.

Furthermore, in Spaces, Lemma 65.5.5 we have seen that the property of representable morphisms of algebraic spaces so defined is stable under arbitrary base change by a morphism of algebraic spaces. And finally, in Spaces, Lemmas 65.5.4 and 65.5.7 we have seen that if $\mathcal{P}$ is stable under compositions, which holds for the properties (1)(a), (1)(b), (1)(c), (2) – (25), except (13) above, then taking products of representable morphisms preserves property $\mathcal{P}$ and compositions of representable morphisms preserves property $\mathcal{P}$.

We will use these facts below, and whenever we do we will simply refer to this section as a reference.

Comments (3)

Comment #4880 by Olivier de Gaay Fortman on January 12, 2020 at 12:37

Condition (2) in the third line of this section, 'fppf local on the base', seems to be hyperlinked to the wrong place only when I download the pdf. It works well on the website, but on the pdf the link sends me to Lemma 03MU, about universal injectiveness, whereas it should send me to Definition 02KO.

Comment #5160 by Johan on June 09, 2020 at 11:16

This is a limitation of using cross file references. If you download all the chapter pdfs in the same directory and then use a good pdf viewer, then the cross file links should work. Or you can download the entire book.

Comment #11604 by Elías Guisado on July 07, 2026 at 10:25

The following result might fit to this section (it is the analogous result to Properties of Algebraic Spaces, Lemma 66.7.1, but for properties of morphisms). It is stated without proof in Olsson's Algebraic Spaces and Stacks, 5.4.3.

Lemma. Let $\mathcal{P}$ be a property of morphisms of schemes satisfying:

is preserved under any base change, see Schemes, Definition 26.18.3, and

is étale local on the base, see Descent, Definition 35.22.1.

(This is almost like the hypotheses from Algebraic Spaces, Definition 65.5.1 but we require étale locality on the base, rather than fppf locality.) Let $S$ be a scheme and let $f:X\to Y$ be a representable morphism of algebraic spaces over $S$ . The following are equivalent:

For some scheme $U$ and surjective étale morphism $U\to Y$ , the morphism of schemes $X\times_YU\to U$ has property $\mathcal{P}$ .

For every scheme $U$ and every morphism $U\to Y$ , the morphism of schemes $X\times_YV\to V$ has property $\mathcal{P}$ . (That is, $f$ has $\mathcal{P}$ as in Algebraic Spaces, Definition 65.5.1.)

Proof. The implication (2) $\Rightarrow$ (1) is immediate. For the converse, choose a surjective étale morphism $U\to Y$ with $U$ a scheme such that $X\times_YU\to U$ has $\mathcal{P}$ and let $V$ be a $Y$ -scheme. Let $W=V\times_YX\times_YU$ . That is, every face in this commutative cube is cartesian. Note that every vertex of the cube other than $X$ and $Y$ is a scheme. In the cube diagram, since the top square is cartesian and $\mathcal{P}$ is preserved under base-change, $W\to U\times_YV$ has $\mathcal{P}$ . On the other hand, $U\times_YV\to V$ is étale surjective. Thus, since $\mathcal{P}$ is étale local on the base, we conclude that $X\times_YV\to V$ has $\mathcal{P}$ , as desired. $\square$

The Stacks project

67.3 Properties of representable morphisms

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