The Stacks project

64.6 Algebraic spaces

Here is the definition.

Definition 64.6.1. 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.

There are two differences with the “usual” definition, for example the definition in Knutson's book [Kn].

The first is that we require $F$ to be a sheaf in the fppf topology. One reason for doing this is that many natural examples of algebraic spaces satisfy the sheaf condition for the fppf coverings (and even for fpqc coverings). Also, one of the reasons that algebraic spaces have been so useful is via Michael Artin's results on algebraic spaces. Built into his method is a condition which guarantees the result is locally of finite presentation over $S$. Combined it somehow seems to us that the fppf topology is the natural topology to work with. In the end the category of algebraic spaces ends up being the same. See Bootstrap, Section 79.12.

The second is that we only require the diagonal map for $F$ to be representable, whereas in [Kn] it is required that it also be quasi-compact. If $F = h_ U$ for some scheme $U$ over $S$ this corresponds to the condition that $U$ be quasi-separated. Our point of view is to try to prove a certain number of the results that follow only assuming that the diagonal of $F$ be representable, and simply add an additional hypothesis wherever this is necessary. In any case it has the pleasing consequence that the following lemma is true.

Lemma 64.6.2. A scheme is an algebraic space. More precisely, given a scheme $T \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$ the representable functor $h_ T$ is an algebraic space.

Proof. The functor $h_ T$ is a sheaf by our remarks in Section 64.2. The diagonal $h_ T \to h_ T \times h_ T = h_{T \times T}$ is representable because $(\mathit{Sch}/S)_{fppf}$ has fibre products. The identity map $h_ T \to h_ T$ is surjective étale. $\square$

Definition 64.6.3. Let $F$, $F'$ be algebraic spaces over $S$. A morphism $f : F \to F'$ of algebraic spaces over $S$ is a transformation of functors from $F$ to $F'$.

The category of algebraic spaces over $S$ contains the category $(\mathit{Sch}/S)_{fppf}$ as a full subcategory via the Yoneda embedding $T/S \mapsto h_ T$. From now on we no longer distinguish between a scheme $T/S$ and the algebraic space it represents. Thus when we say “Let $f : T \to F$ be a morphism from the scheme $T$ to the algebraic space $F$”, we mean that $T \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$, that $F$ is an algebraic space over $S$, and that $f : h_ T \to F$ is a morphism of algebraic spaces over $S$.

Comments (5)

Comment #215 by David Holmes on

Typo: in final paragraph, "From now on we no longer distinghuish" should read 'distinguish'.

Comment #585 by yogesh more on

Minor typo in line 776 of the tex file "If for some scheme over , this corresponds to the condition that [<---should be ] is quasi-separated." I also suggest linking to the definition of quasi-separated.

Comment #1724 by Keenan Kidwell on

In the third text block below Tag 025Y, second-to-last line, the word "addition" should be "additional."

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