The Stacks project

Proof. If for every directed limit $U = \mathop{\mathrm{lim}}\nolimits U_ i$ of affine schemes over $U$, the functor

is essentially surjective, then $f$ is limit preserving on objects. $\square$

Proof. This is formal. Let $U = \mathop{\mathrm{lim}}\nolimits _{i \in I} U_ i$ be the directed limit of affine schemes $U_ i$ over $S$. For each $i$ we have

Filtered colimits commute with $2$-fibre products of categories (details omitted) hence if $p$ is limit preserving we get

as desired. $\square$

Proof. This is formal. Let $U = \mathop{\mathrm{lim}}\nolimits _{i \in I} U_ i$ be the directed limit of affine schemes $U_ i$ over $S$. If $p$ and $q$ are limit preserving we get

as desired. $\square$

Proof. In Criteria for Representability, Lemma 97.5.3 we have seen that (2) and (3) are equivalent. Thus it suffices to show that (1) and (2) are equivalent. One direction we saw in Lemma 102.3.2. For the other direction, let $U = \mathop{\mathrm{lim}}\nolimits _{i \in I} U_ i$ be the directed limit of affine schemes $U_ i$ over $S$. We have to show that

is an equivalence. Since we are assuming (2) we know that it is essentially surjective. Hence we need to prove it is fully faithful. Since $p$ is faithful on fibre categories (Algebraic Stacks, Lemma 94.9.2) we see that the functor is faithful. Let $x_ i$ and $x'_ i$ be objects in the fibre category of $\mathcal{X}$ over $U_ i$. The functor above sends $x_ i$ to $(x_ i|_ U, p(x_ i), can)$ where $can$ is the canonical isomorphism $p(x_ i|_ U) \to p(x_ i)|_ U$. Thus we assume given a morphism

in the category of the right hand side of the first displayed arrow of this proof. Our task is to produce an $i' \geq i$ and a morphism $x_ i|_{U_{i'}} \to x'_ i|_{U_{i'}}$ which maps to $(\alpha , \beta _ i|_{U_{i'}})$.

Set $y_ i = p(x_ i)$ and $y'_ i = p(x'_ i)$. By (Algebraic Stacks, Lemma 94.9.2) the functor

is an algebraic space over $U_ i$ and the same is true for the analogously defined functor $X_{y'_ i}$. Since (2) is equivalent to (3) we see that $X_{y'_ i}$ is locally of finite presentation over $U_ i$. Observe that $(x_ i, \text{id})$ and $(x'_ i, \text{id})$ define $U_ i$-valued points of $X_{y_ i}$ and $X_{y'_ i}$. There is a transformation of functors

in other words, this is a morphism of algebraic spaces over $U_ i$. We claim that

commutes. Namely, this is equivalent to the condition that the pairs $(x_ i|_ U, \beta _ i|_ U)$ and $(x'_ i|_ U, \text{id})$ as in the definition of the functor $X_{y'_ i}$ are isomorphic. And the morphism $\alpha : x_ i|_ U \to x'_ i|_ U$ exactly produces such an isomorphism. Arguing backwards the reader sees that if we can find an $i' \geq i$ such that the diagram

commutes, then we obtain an isomorphism $x_ i|_{U_{i'}} \to x'_ i|_{U_{i'}}$ which is a solution to the problem posed in the preceding paragraph. However, the diagonal morphism

is locally of finite presentation (Morphisms of Spaces, Lemma 67.28.10) hence the fact that $U \to U_ i$ equalizes the two morphisms to $X_{y'_ i}$, means that for some $i' \geq i$ the morphism $U_{i'} \to U_ i$ equalizes the two morphisms, see Limits of Spaces, Proposition 70.3.10. $\square$

Proof. Let $U = \mathop{\mathrm{lim}}\nolimits U_ i$ be a directed limit of affine schemes over $S$. We claim that the functor

is fully faithful if and only if the functor

is an equivalence. This will prove the lemma. Since $(\mathcal{X} \times _\mathcal {Y} \mathcal{X})_ U = \mathcal{X}_ U \times _{\mathcal{Y}_ U} \mathcal{X}_ U$ and $(\mathcal{X} \times _\mathcal {Y} \mathcal{X})_{U_ i} = \mathcal{X}_{U_ i} \times _{\mathcal{Y}_{U_ i}} \mathcal{X}_{U_ i}$ this is a purely category theoretic assertion which we discuss in the next paragraph.

Let $\mathcal{I}$ be a filtered index category. Let $(\mathcal{C}_ i)$ and $(\mathcal{D}_ i)$ be systems of groupoids over $\mathcal{I}$. Let $p : (\mathcal{C}_ i) \to (\mathcal{D}_ i)$ be a map of systems of groupoids over $\mathcal{I}$. Suppose we have a functor $p : \mathcal{C} \to \mathcal{D}$ of groupoids and functors $f : \mathop{\mathrm{colim}}\nolimits \mathcal{C}_ i \to \mathcal{C}$ and $g : \mathop{\mathrm{colim}}\nolimits \mathcal{D}_ i \to \mathcal{D}$ fitting into a commutative diagram

Then we claim that

is fully faithful if and only if the functor

is an equivalence. Set $\mathcal{C}' = \mathop{\mathrm{colim}}\nolimits \mathcal{C}_ i$ and $\mathcal{D}' = \mathop{\mathrm{colim}}\nolimits \mathcal{D}_ i$. Since $2$-fibre products commute with filtered colimits we see that $A$ and $B$ become the functors

Thus it suffices to prove that if

is a commutative diagram of groupoids, then $A'$ is fully faithful if and only if $B'$ is an equivalence. This follows from Categories, Lemma 4.35.10 (with trivial, i.e., punctual, base category) because

This finishes the proof. $\square$

Proof. Choose a surjective smooth morphism $U \to \mathcal{X}$ for some scheme $U$. Then $U \to S$ is locally of finite presentation, see Morphisms of Stacks, Section 101.27. We can write $\mathcal{X} = [U/R]$ for some smooth groupoid in algebraic spaces $(U, R, s, t, c)$, see Algebraic Stacks, Lemma 94.16.2. Since $U$ is locally of finite presentation over $S$ it follows that the algebraic space $R$ is locally of finite presentation over $S$. Recall that $[U/R]$ is the stack in groupoids over $(\mathit{Sch}/S)_{fppf}$ obtained by stackyfying the category fibred in groupoids whose fibre category over $T$ is the groupoid $(U(T), R(T), s, t, c)$. Since $U$ and $R$ are limit preserving as functors (Limits of Spaces, Proposition 70.3.10) this category fibred in groupoids is limit preserving. Thus it suffices to show that fppf stackyfication preserves the property of being limit preserving. This is true (hint: use Topologies, Lemma 34.13.2). However, we give a direct proof below using that in this case we know what the stackyfication amounts to.

Let $T = \mathop{\mathrm{lim}}\nolimits T_\lambda $ be a directed limit of affine schemes over $S$. We have to show that the functor

is an equivalence of categories. Let us show this functor is essentially surjective. Let $x \in \mathop{\mathrm{Ob}}\nolimits ([U/R]_ T)$. In Groupoids in Spaces, Lemma 78.24.1 the reader finds a description of the category $[U/R]_ T$. In particular $x$ corresponds to an fppf covering $\{ T_ i \to T\} _{i \in I}$ and a $[U/R]$-descent datum $(u_ i, r_{ij})$ relative to this covering. After refining this covering we may assume it is a standard fppf covering of the affine scheme $T$. By Topologies, Lemma 34.13.2 we may choose a $\lambda $ and a standard fppf covering $\{ T_{\lambda , i} \to T_\lambda \} _{i \in I}$ whose base change to $T$ is equal to $\{ T_ i \to T\} _{i \in I}$. For each $i$, after increasing $\lambda $, we can find a $u_{\lambda , i} : T_{\lambda , i} \to U$ whose composition with $T_ i \to T_{\lambda , i}$ is the given morphism $u_ i$ (this is where we use that $U$ is limit preserving). Similarly, for each $i, j$, after increasing $\lambda $, we can find a $r_{\lambda , ij} : T_{\lambda , i} \times _{T_\lambda } T_{\lambda , j} \to R$ whose composition with $T_{ij} \to T_{\lambda , ij}$ is the given morphism $r_{ij}$ (this is where we use that $R$ is limit preserving). After increasing $\lambda $ we can further assume that

and

In other words, we may assume that $(u_{\lambda , i}, r_{\lambda , ij})$ is a $[U/R]$-descent datum relative to the covering $\{ T_{\lambda , i} \to T_\lambda \} _{i \in I}$. Then we obtain a corresponding object of $[U/R]$ over $T_\lambda $ whose pullback to $T$ is isomorphic to $x$ as desired. The proof of fully faithfulness works in exactly the same way using the description of morphisms in the fibre categories of $[U/T]$ given in Groupoids in Spaces, Lemma 78.24.1. $\square$

Proof. Assume (3). Let $T = \mathop{\mathrm{lim}}\nolimits T_ i$ be a directed limit of affine schemes. Consider the functor

Let $(x, y_ i, \beta )$ be an object on the right hand side, i.e., $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X}_ T)$, $y_ i \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{Y}_{T_ i})$, and $\beta : f(x) \to y_ i|_ T$ in $\mathcal{Y}_ T$. Then we can consider $(x, y_ i, \beta )$ as an object of the algebraic stack $\mathcal{X}_{y_ i} = \mathcal{X} \times _{\mathcal{Y}, y_ i} T_ i$ over $T$. Since $\mathcal{X}_{y_ i} \to T_ i$ is locally of finite presentation (as a base change of $f$) we see that it is limit preserving by Lemma 102.3.7. This means that $(x, y_ i, \beta )$ comes from an object over $T_{i'}$ for some $i' \geq i$ and unwinding the definitions we find that $(x, y_ i, \beta )$ is in the essential image of the displayed functor. In other words, the displayed functor is essentially surjective. Another formulation is that this means $f$ is limit preserving on objects. Now we apply this to the diagonal $\Delta $ of $f$. Namely, by Morphisms of Stacks, Lemma 101.27.7 the morphism $\Delta $ is locally of finite presentation. Thus the argument above shows that $\Delta $ is limit preserving on objects. By Lemma 102.3.5 this implies that $\Delta $ is limit preserving. By Lemma 102.3.6 we conclude that the displayed functor above is fully faithful. Thus it is an equivalence (as we already proved essential surjectivity) and we conclude that (1) holds.

The implication (1) $\Rightarrow $ (2) is trivial. Assume (2). Choose a scheme $V$ and a surjective smooth morphism $V \to \mathcal{Y}$. By Criteria for Representability, Lemma 97.5.1 the base change $\mathcal{X} \times _\mathcal {Y} V \to V$ is limit preserving on objects. Choose a scheme $U$ and a surjective smooth morphism $U \to \mathcal{X} \times _\mathcal {Y} V$. Since a smooth morphism is locally of finite presentation, we see that $U \to \mathcal{X} \times _\mathcal {Y} V$ is limit preserving (first part of the proof). By Criteria for Representability, Lemma 97.5.2 we find that the composition $U \to V$ is limit preserving on objects. We conclude that $U \to V$ is locally of finite presentation, see Criteria for Representability, Lemma 97.5.3. This is exactly the condition that $f$ is locally of finite presentation, see Morphisms of Stacks, Definition 101.27.1. $\square$