Section 65.6 (025X): Algebraic spaces

65.6 Algebraic spaces

Here is the definition.

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

The presheaf $F$ is a sheaf.
The diagonal morphism $F \to F \times F$ is representable.
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 ¹.

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 80.12.

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 65.6.2. 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 65.2. 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 65.6.3. 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 $S$.

[1] See Lemma 65.5.10, Definition 65.5.1, and Remark 65.5.2.

Comments (7)

Comment #215 by David Holmes on May 17, 2013 at 16:36

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

Comment #585 by yogesh more on May 22, 2014 at 23:24

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

Comment #598 by Johan on May 23, 2014 at 20:22

Thanks! Fixed here.

Comment #1724 by Keenan Kidwell on December 10, 2015 at 18:45

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

Comment #1763 by Johan on December 15, 2015 at 19:08

Thanks, fixed here.

Comment #11675 by Elías Guisado on July 23, 2026 at 10:05

For those who care, I want to record pointers for the full proof of the equivalence between Definition 65.6.1 and Knutson's original definition of algebraic space.

As found in [Knu71, p. 92]:

Definition 1.1. An algebraic space $A$ is a functor $A: (\text{Schemes})^{\text {opp}} \rightarrow (\text{Sets})$ such that

a) $A$ is a sheaf in the étale topology.

b) (Local Representability) There exists a scheme $U$ , and a map of sheaves $U\rightarrow A$ such that for all schemes $V$ , and maps $V \rightarrow A$ , the (sheaf) fiber product $U \times_A V$ is representable and the map $U \times_A V \rightarrow V$ is induced by an étale surjective map of schemes.

c) (Quasi-separatedness) Let $U \rightarrow A$ be as in part b. Then the map of schemes inducing $U \times_A U \rightarrow U \times U$ is quasi-compact.

Actually, we will show that a)+b) is equivalent to Definition 65.6.1. Once this is established, we will show that condition c) is equivalent to “being a quasi-separated algebraic space” as in Definition 65.13.2, (3).

On the one hand, it is clear that Definition 65.6.1 implies a)+b). Conversely, a)+b) implies Definition 65.6.1 by any of the following:

[Knu71, p. 100, II.1.8.b], which states the representability of the diagonal (up to getting rid of [Knu71, p. 92, condition c)]).

Bootstrap, Lemma 80.12.2.

The following étale version of Lemma 65.11.1:

Lemma. Let $S$ be a scheme contained in $Sch_{ét}$ . Let $F$ be a sheaf on $(Sch/S)_{ét}$ such that there exists $U \in \operatorname{Ob}((Sch/S)_{ét})$ and a map $U \to F$ which is representable, surjective, and étale. Then $F$ is an algebraic space as in Definition 65.6.1.

Proof. It goes exactly as in Lemma 65.11.1 but to argue étale-sheaf-surjectivity of $U\to F$ instead of Lemma 65.5.9 one invokes Lemma 2. $\square$

On the other hand, suppose $F$ is an algebraic space. It is clear that Definition 65.13.2, (3) for $F$ implies [Knu71, p. 92, condition c)]. The converse follows from this lemma.

References

[Knu71] D. Knutson, Algebraic Spaces, Springer, 1971

Comment #11676 by Elías Guisado on July 23, 2026 at 10:48

Alternatively, the fact that [Knu71, p. 92, condition c)] implies Definition 65.13.2, (3) is [Knu71, p. 100, II.1.8.b].

The Stacks project

65.6 Algebraic spaces

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References

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