The Stacks project

Definition 65.6.1. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. An algebraic space over $S$ is a presheaf

\[ F : (\mathit{Sch}/S)^{opp}_{fppf} \longrightarrow \textit{Sets} \]

with the following properties

  1. The presheaf $F$ is a sheaf.

  2. The diagonal morphism $F \to F \times F$ is representable.

  3. There exists a scheme $U \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$ and a map $h_ U \to F$ which is surjective, and étale.

Comments (4)

Comment #6511 by on

It might be better to refer to https://stacks.math.columbia.edu/tag/025W and the following remark after the third property.

Comment #6567 by on

If other people agree with #6511 then I will make this change.

Comment #6834 by DatPham on

It seems to me that the definition above does not change if we replace part (3) by the condition that is étale and surjective as a morphism of sheaves. However, surjective morphisms of algebraic spaces (as defined in {https://stacks.math.columbia.edu/tag/03MC}) are not the same as morphisms of algebraic spaces which are surjective as morphisms of sheaves (otherwise it would imply that any surjective morphism of schemes admits a section fppf locally). Is this correct or am I missing something?

There are also:

  • 5 comment(s) on Section 65.6: Algebraic spaces

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View Definition 65.6.1 as pdf