The Stacks project

95.11 Limit preserving

The morphism $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ is limit preserving on objects, as defined in Criteria for Representability, Section 94.5, if the functor of the definition below is essentially surjective. However, the example in Examples, Section 107.47 shows that this isn't equivalent to being limit preserving.

Definition 95.11.1. Let $S$ be a scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. We say $\mathcal{X}$ is limit preserving if for every affine scheme $T$ over $S$ which is a limit $T = \mathop{\mathrm{lim}}\nolimits T_ i$ of a directed inverse system of affine schemes $T_ i$ over $S$, we have an equivalence

\[ \mathop{\mathrm{colim}}\nolimits \mathcal{X}_{T_ i} \longrightarrow \mathcal{X}_ T \]

of fibre categories.

We spell out what this means. First, given objects $x, y$ of $\mathcal{X}$ over $T_ i$ we should have

\[ \mathop{Mor}\nolimits _{\mathcal{X}_ T}(x|_ T, y|_ T) = \mathop{\mathrm{colim}}\nolimits _{i' \geq i} \mathop{Mor}\nolimits _{\mathcal{X}_{T_ i'}}(x|_{T_ i'}, y|_{T_ i'}) \]

and second every object of $\mathcal{X}_ T$ is isomorphic to the restriction of an object over $T_ i$ for some $i$. Note that the first condition means that the presheaves $\mathit{Isom}_\mathcal {X}(x, y)$ (see Stacks, Definition 8.2.2) are limit preserving.

Lemma 95.11.2. Let $S$ be a scheme. Let $p : \mathcal{X} \to \mathcal{Y}$ and $q : \mathcal{Z} \to \mathcal{Y}$ be $1$-morphisms of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$.

  1. If $\mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ and $\mathcal{Z} \to (\mathit{Sch}/S)_{fppf}$ are limit preserving on objects and $\mathcal{Y}$ is limit preserving, then $\mathcal{X} \times _\mathcal {Y} \mathcal{Z} \to (\mathit{Sch}/S)_{fppf}$ is limit preserving on objects.

  2. If $\mathcal{X}$, $\mathcal{Y}$, and $\mathcal{Z}$ are limit preserving, then so is $\mathcal{X} \times _\mathcal {Y} \mathcal{Z}$.

Proof. This is formal. Proof of (1). Let $T = \mathop{\mathrm{lim}}\nolimits _{i \in I} T_ i$ be the directed limit of affine schemes $T_ i$ over $S$. We will prove that the functor $\mathop{\mathrm{colim}}\nolimits \mathcal{X}_{T_ i} \to \mathcal{X}_ T$ is essentially surjective. Recall that an object of the fibre product over $T$ is a quadruple $(T, x, z, \alpha )$ where $x$ is an object of $\mathcal{X}$ lying over $T$, $z$ is an object of $\mathcal{Z}$ lying over $T$, and $\alpha : p(x) \to q(z)$ is a morphism in the fibre category of $\mathcal{Y}$ over $T$. By assumption on $\mathcal{X}$ and $\mathcal{Z}$ we can find an $i$ and objects $x_ i$ and $z_ i$ over $T_ i$ such that $x_ i|_ T \cong T$ and $z_ i|_ T \cong z$. Then $\alpha $ corresponds to an isomorphism $p(x_ i)|_ T \to q(z_ i)|_ T$ which comes from an isomorphism $\alpha _{i'} : p(x_ i)|_{T_{i'}} \to q(z_ i)|_{T_{i'}}$ by our assumption on $\mathcal{Y}$. After replacing $i$ by $i'$, $x_ i$ by $x_ i|_{T_{i'}}$, and $z_ i$ by $z_ i|_{T_{i'}}$ we see that $(T_ i, x_ i, z_ i, \alpha _ i)$ is an object of the fibre product over $T_ i$ which restricts to an object isomorphic to $(T, x, z, \alpha )$ over $T$ as desired.

We omit the arguments showing that $\mathop{\mathrm{colim}}\nolimits \mathcal{X}_{T_ i} \to \mathcal{X}_ T$ is fully faithful in (2). $\square$

Lemma 95.11.3. Let $S$ be a scheme. Let $\mathcal{X}$ be an algebraic stack over $S$. Then the following are equivalent

  1. $\mathcal{X}$ is a stack in setoids and $\mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ is limit preserving on objects,

  2. $\mathcal{X}$ is a stack in setoids and limit preserving,

  3. $\mathcal{X}$ is representable by an algebraic space locally of finite presentation.

Proof. Under each of the three assumptions $\mathcal{X}$ is representable by an algebraic space $X$ over $S$, see Algebraic Stacks, Proposition 91.13.3. It is clear that (1) and (2) are equivalent as a functor between setoids is an equivalence if and only if it is surjective on isomorphism classes. Finally, (1) and (3) are equivalent by Limits of Spaces, Proposition 67.3.8. $\square$

Lemma 95.11.4. Let $S$ be a scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Assume $\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ is representable by algebraic spaces and $\mathcal{X}$ is limit preserving. Then $\Delta $ is locally of finite type.

Proof. We apply Criteria for Representability, Lemma 94.5.6. Let $V$ be an affine scheme $V$ of finite type over $S$ and let $\theta $ be an object of $\mathcal{X} \times \mathcal{X}$ over $V$. Let $F_\theta $ be an algebraic space representing $\mathcal{X} \times _{\Delta , \mathcal{X} \times \mathcal{X}, \theta } (\mathit{Sch}/V)_{fppf}$ and let $f_\theta : F_\theta \to V$ be the canonical morphism (see Algebraic Stacks, Section 91.9). It suffices to show that $F_\theta \to V$ has the corresponding properties. By Lemmas 95.11.2 and 95.11.3 we see that $F_\theta \to S$ is locally of finite presentation. It follows that $F_\theta \to V$ is locally of finite type by Morphisms of Spaces, Lemma 64.23.6. $\square$


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