The Stacks project

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

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

2. 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.