The Stacks project

Lemma 87.26.4. 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 a monomorphism,

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

  5. there exists a family of morphisms $\{ Y_ j \to Y\} $ such that $\coprod Y_ j \to Y$ is a surjection of sheaves on $(\mathit{Sch}/S)_{fppf}$ such that each $X \times _ Y Y_ j \to Y_ j$ is a monomorphism for all $j$,

  6. there exists a morphism $Z \to Y$ of formal algebraic spaces which is representable by algebraic spaces, surjective, flat, and locally of finite presentation such that $X \times _ Y Z \to X$ is a monomorphism, and

  7. add more here.

Proof. Omitted. $\square$

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