## 87.25 Surjective morphisms

By Lemma 87.12.4 the following definition does not clash with the already existing definitions for morphisms of algebraic spaces or morphisms of formal algebraic spaces which are representable by algebraic spaces.

Definition 87.25.1. Let $S$ be a scheme. A morphism $f : X \to Y$ of formal algebraic spaces over $S$ is said to be surjective if it induces a surjective morphism $X_{red} \to Y_{red}$ on underlying reduced algebraic spaces.

Lemma 87.25.2. The composition of two surjective morphisms is a surjective morphism.

Proof. Omitted. $\square$

Lemma 87.25.3. A base change of a surjective morphism is a surjective morphism.

Proof. Omitted. $\square$

Lemma 87.25.4. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of formal algebraic spaces over $S$. The following are equivalent

1. $f$ is surjective,

2. for every scheme $T$ and morphism $T \to Y$ the projection $X \times _ Y T \to T$ is a surjective morphism of formal algebraic spaces,

3. for every affine scheme $T$ and morphism $T \to Y$ the projection $X \times _ Y T \to T$ is a surjective morphism of formal algebraic spaces,

4. there exists a covering $\{ Y_ j \to Y\}$ as in Definition 87.11.1 such that each $X \times _ Y Y_ j \to Y_ j$ is a surjective morphism of formal algebraic spaces,

5. there exists a surjective morphism $Z \to Y$ of formal algebraic spaces such that $X \times _ Y Z \to Z$ is surjective, and

6. add more here.

Proof. Omitted. $\square$

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