The Stacks project

94.12 Algebraic stacks

Here is the definition of an algebraic stack. We remark that condition (2) implies we can make sense out of the condition in part (3) that $(\mathit{Sch}/U)_{fppf} \to \mathcal{X}$ is smooth and surjective, see discussion following Lemma 94.10.11.

Definition 94.12.1. Let $S$ be a base scheme contained in $\mathit{Sch}_{fppf}$. An algebraic stack over $S$ is a category

\[ p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf} \]

over $(\mathit{Sch}/S)_{fppf}$ with the following properties:

  1. The category $\mathcal{X}$ is a stack in groupoids over $(\mathit{Sch}/S)_{fppf}$.

  2. The diagonal $\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ is representable by algebraic spaces.

  3. There exists a scheme $U \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$ and a $1$-morphism $(\mathit{Sch}/U)_{fppf} \to \mathcal{X}$ which is surjective and smooth1.

There are some differences with other definitions found in the literature.

The first is that we require $\mathcal{X}$ to be a stack in groupoids in the fppf topology, whereas in many references the étale topology is used. It somehow seems to us that the fppf topology is the natural topology to work with. In the end the resulting $2$-category of algebraic stacks ends up being the same. This is explained in Criteria for Representability, Section 97.19.

The second is that we only require the diagonal map of $\mathcal{X}$ to be representable by algebraic spaces, whereas in most references some other conditions are imposed. Our point of view is to try to prove a certain number of the results that follow only assuming that the diagonal of $\mathcal{X}$ be representable by algebraic spaces, and simply add an additional hypothesis wherever this is necessary. It has the added benefit that any algebraic space (as defined in Spaces, Definition 65.6.1) gives rise to an algebraic stack.

The third is that in some papers it is required that there exists a scheme $U$ and a surjective and étale morphism $U \to \mathcal{X}$. In the groundbreaking paper [DM] where algebraic stacks were first introduced Deligne and Mumford used this definition and showed that the moduli stack of stable genus $g > 1$ curves is an algebraic stack which has an étale covering by a scheme. Michael Artin, see [ArtinVersal], realized that many natural results on algebraic stacks generalize to the case where one only assume a smooth covering by a scheme. Hence our choice above. To distinguish the two cases one sees the terms “Deligne-Mumford stack” and “Artin stack” used in the literature. We will reserve the term “Artin stack” for later use (insert future reference here), and continue to use “algebraic stack”, but we will use “Deligne-Mumford stack” to indicate those algebraic stacks which have an étale covering by a scheme.

Definition 94.12.2. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $\mathcal{X}$ be an algebraic stack over $S$. We say $\mathcal{X}$ is a Deligne-Mumford stack if there exists a scheme $U$ and a surjective étale morphism $(\mathit{Sch}/U)_{fppf} \to \mathcal{X}$.

We will compare our notion of a Deligne-Mumford stack with the notion as defined in the paper by Deligne and Mumford later (see insert future reference here).

The category of algebraic stacks over $S$ forms a $2$-category. Here is the precise definition.

Definition 94.12.3. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. The $2$-category of algebraic stacks over $S$ is the sub $2$-category of the $2$-category of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$ (see Categories, Definition 4.35.6) defined as follows:

  1. Its objects are those categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$ which are algebraic stacks over $S$.

  2. Its $1$-morphisms $f : \mathcal{X} \to \mathcal{Y}$ are any functors of categories over $(\mathit{Sch}/S)_{fppf}$, as in Categories, Definition 4.32.1.

  3. Its $2$-morphisms are transformations between functors over $(\mathit{Sch}/S)_{fppf}$, as in Categories, Definition 4.32.1.

In other words this $2$-category is the full sub $2$-category of $\textit{Cat}/(\mathit{Sch}/S)_{fppf}$ whose objects are algebraic stacks. Note that every $2$-morphism is automatically an isomorphism. Hence this is actually a $(2, 1)$-category and not just a $2$-category.

We will see later (insert future reference here) that this $2$-category has $2$-fibre products.

Similar to the remark above the $2$-category of algebraic stacks over $S$ is a full sub $2$-category of the $2$-category of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. It turns out that it is closed under equivalences. Here is the precise statement.

Lemma 94.12.4. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $\mathcal{X}$, $\mathcal{Y}$ be categories over $(\mathit{Sch}/S)_{fppf}$. Assume $\mathcal{X}$, $\mathcal{Y}$ are equivalent as categories over $(\mathit{Sch}/S)_{fppf}$. Then $\mathcal{X}$ is an algebraic stack if and only if $\mathcal{Y}$ is an algebraic stack. Similarly, $\mathcal{X}$ is a Deligne-Mumford stack if and only if $\mathcal{Y}$ is a Deligne-Mumford stack.

Proof. Assume $\mathcal{X}$ is an algebraic stack (resp. a Deligne-Mumford stack). By Stacks, Lemma 8.5.4 this implies that $\mathcal{Y}$ is a stack in groupoids over $\mathit{Sch}_{fppf}$. Choose an equivalence $f : \mathcal{X} \to \mathcal{Y}$ over $\mathit{Sch}_{fppf}$. This gives a $2$-commutative diagram

\[ \xymatrix{ \mathcal{X} \ar[r]_ f \ar[d]_{\Delta _\mathcal {X}} & \mathcal{Y} \ar[d]^{\Delta _\mathcal {Y}} \\ \mathcal{X} \times \mathcal{X} \ar[r]^{f \times f} & \mathcal{Y} \times \mathcal{Y} } \]

whose horizontal arrows are equivalences. This implies that $\Delta _\mathcal {Y}$ is representable by algebraic spaces according to Lemma 94.9.3. Finally, let $U$ be a scheme over $S$, and let $x : (\mathit{Sch}/U)_{fppf} \to \mathcal{X}$ be a $1$-morphism which is surjective and smooth (resp. étale). Considering the diagram

\[ \xymatrix{ (\mathit{Sch}/U)_{fppf} \ar[r]_{\text{id}} \ar[d]_ x & (\mathit{Sch}/U)_{fppf} \ar[d]^{f \circ x} \\ \mathcal{X} \ar[r]^ f & \mathcal{Y} } \]

and applying Lemma 94.10.2 we conclude that $f \circ x$ is surjective and smooth (resp. étale) as desired. $\square$

[1] In future chapters we will denote this simply $U \to \mathcal{X}$ as is customary in the literature. Another good alternative would be to formulate this condition as the existence of a representable category fibred in groupoids $\mathcal{U}$ and a surjective smooth $1$-morphism $\mathcal{U} \to \mathcal{X}$.

Comments (6)

Comment #192 by on

Michael Artin's name is written like Micheal Artin in the discussion after the first definition.

Comment #402 by on

In the pdf version of this chapter, the sentence "This is explained in Criteria for Representability, Section 69.19" links to the wrong section 19 (on change of base, rather than change of topology). The online version seems to be correct.

Comment #403 by on

Yeah, no, that is a known problem with browsing pdfs. If you use windows (which I don't) and the acrobat reader browser plugin then I think the cross pdf links work correctly... and also in some linux setups. See this blog post.

Comment #4981 by Elyes Boughattas on

Typo in the last line of the proof of lemma 91.12.4: should be .

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View Section 94.12 as pdf