The Stacks project

Proof. Assume (1). Consider $T = \mathop{\mathrm{lim}}\nolimits _{i \in I} T_ i$ as in (2). Let $(y, x_ T)$ be an element of the fibre product $\mathop{\mathrm{colim}}\nolimits _ i G(T_ i) \times _{G(T)} F(T)$. Then $y$ comes from $y_ i \in G(T_ i)$ for some $i$. Consider the functor $F_{y_ i}$ on $(\mathit{Sch}/T_ i)_{fppf}$ as in Definition 70.3.1. We see that $x_ T \in F_{y_ i}(T)$. Moreover $T = \mathop{\mathrm{lim}}\nolimits _{i' \geq i} T_{i'}$ is a directed system of affine schemes over $T_ i$. Hence (1) implies that $x_ T$ the image of a unique element $x$ of $\mathop{\mathrm{colim}}\nolimits _{i' \geq i} F_{y_ i}(T_{i'})$. Thus $x$ is the unique element of $\mathop{\mathrm{colim}}\nolimits F(T_ i)$ which maps to the pair $(y, x_ T)$. This proves that (2) holds.

Assume (2). Let $T$ be a scheme and $y_ T \in G(T)$. We have to show that $F_{y_ T}$ is limit preserving. Let $T' = \mathop{\mathrm{lim}}\nolimits _{i \in I} T'_ i$ be an affine scheme over $T$ which is the directed limit of affine scheme $T'_ i$ over $T$. Let $x_{T'} \in F_{y_ T}$. Pick $i \in I$ which is possible as $I$ is a directed set. Denote $y_ i \in F(T'_ i)$ the image of $y_{T'}$. Then we see that $(y_ i, x_{T'})$ is an element of the fibre product $\mathop{\mathrm{colim}}\nolimits _ i G(T'_ i) \times _{G(T')} F(T')$. Hence by (2) we get a unique element $x$ of $\mathop{\mathrm{colim}}\nolimits _ i F(T'_ i)$ mapping to $(y_ i, x_{T'})$. It is clear that $x$ defines an element of $\mathop{\mathrm{colim}}\nolimits _ i F_ y(T'_ i)$ mapping to $x_{T'}$ and we win. $\square$

Proof. Let $T = \mathop{\mathrm{lim}}\nolimits _{i \in I} T_ i$ as in characterization (2) of Lemma 70.3.2. Consider the diagram of sets

By assumption the two squares are fibre product squares. Hence the outer rectangle is a fibre product diagram too which proves the lemma. $\square$

Proof. Let $T = \mathop{\mathrm{lim}}\nolimits _{i \in I} T_ i$ as in characterization (2) of Lemma 70.3.2. Consider the diagram of sets

By assumption the lower square and the outer rectangle are fibre products of sets. Hence the upper square is a fibre product square too which proves the lemma. $\square$

Proof. Omitted. Hint: This is formal. $\square$

Proof. Let $T = \mathop{\mathrm{lim}}\nolimits _{i \in I} T_ i$ as in characterization (2) of Lemma 70.3.2. Then we have

as filtered colimits commute with finite products. Our goal is thus to show that

is a fibre product diagram. This follows from the observation that given maps of sets $E' \to E$, $F \to G$, $H \to G$, and $G \to E$ we have

Some details omitted. $\square$

Proof. Assume $F$ is limit preserving. It suffices to show that $F^+$ is limit preserving, since $F^\# = (F^+)^+$, see Sites, Theorem 7.10.10. Let $T$ be an affine scheme over $S$, and let $T = \mathop{\mathrm{lim}}\nolimits T_ i$ be written as the directed limit of an inverse system of affine $S$ schemes. Recall that $F^+(T)$ is the colimit of $\check H^0(\mathcal{V}, F)$ where the limit is over all coverings of $T$ in $(\mathit{Sch}/S)_{fppf}$. Any fppf covering of an affine scheme can be refined by a standard fppf covering, see Topologies, Lemma 34.7.4. Hence we can write

Any $\mathcal{V} = \{ T_ k \to T\} _{k = 1, \ldots , n}$ in the colimit may be written as $V_ i \times _{T_ i} T$ for some $i$ and some standard fppf covering $\mathcal{V}_ i = \{ T_{i, k} \to T_ i\} _{k = 1, \ldots , n}$ of $T_ i$. Denote $\mathcal{V}_{i'} = \{ T_{i', k} \to T_{i'}\} _{k = 1, \ldots , n}$ the base change for $i' \geq i$. Then we see that

Here the second equality holds because filtered colimits are exact. The third equality holds because $F$ is limit preserving and because $\mathop{\mathrm{lim}}\nolimits _{i' \geq i} T_{i', k} = T_ k$ and $\mathop{\mathrm{lim}}\nolimits _{i' \geq i} T_{i', k} \times _{T_{i'}} T_{i', l} = T_ k \times _ T T_ l$ by Limits, Lemma 32.2.3. If we use this for all coverings at the same time we obtain

The switch of the order of the colimits is allowed by Categories, Lemma 4.14.10. $\square$

Proof. Let $T$ be an affine scheme over $S$. Let $I$ be a directed set, and let $T_ i$ be an inverse system of affine schemes over $S$ such that $T = \mathop{\mathrm{lim}}\nolimits T_ i$. We have to show that the canonical map $\mathop{\mathrm{colim}}\nolimits F(T_ i) \to F(T)$ is bijective.

Choose some $0 \in I$ and choose a standard fppf covering $\{ V_{0, k} \to T_{0}\} _{k = 1, \ldots , m}$ which refines the pullback $\{ U_ j \times _ S T_0 \to T_0\} $ of the given fppf covering of $S$. For each $i \geq 0$ we set $V_{i, k} = T_ i \times _{T_0} V_{0, k}$, and we set $V_ k = T \times _{T_0} V_{0, k}$. Note that $V_ k = \mathop{\mathrm{lim}}\nolimits _{i \geq 0} V_{i, k}$, see Limits, Lemma 32.2.3.

Suppose that $x, x' \in \mathop{\mathrm{colim}}\nolimits F(T_ i)$ map to the same element of $F(T)$. Say $x, x'$ are given by elements $x_ i, x'_ i \in F(T_ i)$ for some $i \in I$ (we may choose the same $i$ for both as $I$ is directed). By assumption (2) and the fact that $x_ i, x'_ i$ map to the same element of $F(T)$ this implies that

for some suitably large $i' \in I$. We can choose the same $i'$ for each $k$ as $k \in \{ 1, \ldots , m\} $ ranges over a finite set. Since $\{ V_{i', k} \to T_{i'}\} $ is an fppf covering and $F$ is a sheaf this implies that $x_ i|_{T_{i'}} = x'_ i|_{T_{i'}}$ as desired. This proves that the map $\mathop{\mathrm{colim}}\nolimits F(T_ i) \to F(T)$ is injective.

To show surjectivity we argue in a similar fashion. Let $x \in F(T)$. By assumption (2) for each $k$ we can choose a $i$ such that $x|_{V_ k}$ comes from an element $x_{i, k} \in F(V_{i, k})$. As before we may choose a single $i$ which works for all $k$. By the injectivity proved above we see that

for some large enough $i'$. Hence by the sheaf condition of $F$ the elements $x_{i, k}|_{V_{i', k}}$ glue to an element $x_{i'} \in F(T_{i'})$ as desired. $\square$

Proof. Suppose that $T$ is a scheme and $y \in G^\# (T)$. We have to show the functor $F^\# _ y : (\mathit{Sch}/T)_{fppf}^{opp} \to \textit{Sets}$ constructed from $F^\# \to G^\# $ and $y$ as in Definition 70.3.1 is limit preserving. By Equation (70.3.1.1) we see that $F^\# _ y$ is a sheaf. Choose an fppf covering $\{ V_ j \to T\} _{j \in J}$ such that $y|_{V_ j}$ comes from an element $y_ j \in F(V_ j)$. Note that the restriction of $F^\# $ to $(\mathit{Sch}/V_ j)_{fppf}$ is just $F^\# _{y_ j}$. If we can show that $F^\# _{y_ j}$ is limit preserving then Lemma 70.3.8 guarantees that $F^\# _ y$ is limit preserving and we win. This reduces us to the case $y \in G(T)$.

Let $y \in G(T)$. In this case we claim that $F^\# _ y = (F_ y)^\# $. This follows from Equation (70.3.1.1). Thus this case follows from Lemma 70.3.7. $\square$

Proof. Assume (1). Let $T$ be a scheme and let $y \in Y(T)$. We have to show that $T \times _ Y X$ is limit preserving over $T$ in the sense of Definition 70.3.1. Hence we are reduced to proving that if $X$ is an algebraic space which is locally of finite presentation over $S$ as an algebraic space, then it is limit preserving as a functor $X : (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}$. To see this choose a presentation $X = U/R$, see Spaces, Definition 65.9.3. It follows from Morphisms of Spaces, Definition 67.28.1 that both $U$ and $R$ are schemes which are locally of finite presentation over $S$. Hence by Limits, Proposition 32.6.1 we have

whenever $T = \mathop{\mathrm{lim}}\nolimits _ i T_ i$ in $(\mathit{Sch}/S)_{fppf}$. It follows that the presheaf

is limit preserving. Hence by Lemma 70.3.7 its sheafification $X = U/R$ is limit preserving too.

Assume (2). Choose a scheme $V$ and a surjective étale morphism $V \to Y$. Next, choose a scheme $U$ and a surjective étale morphism $U \to V \times _ Y X$. By Lemma 70.3.5 the transformation of functors $V \times _ Y X \to V$ is limit preserving. By Morphisms of Spaces, Lemma 67.39.8 the morphism of algebraic spaces $U \to V \times _ Y X$ is locally of finite presentation, hence limit preserving as a transformation of functors by the first part of the proof. By Lemma 70.3.3 the composition $U \to V \times _ Y X \to V$ is limit preserving as a transformation of functors. Hence the morphism of schemes $U \to V$ is locally of finite presentation by Limits, Proposition 32.6.1 (modulo a set theoretic remark, see last paragraph of the proof). This means, by definition, that (1) holds.

Set theoretic remark. Let $U \to V$ be a morphism of $(\mathit{Sch}/S)_{fppf}$. In the statement of Limits, Proposition 32.6.1 we characterize $U \to V$ as being locally of finite presentation if for all directed inverse systems $(T_ i, f_{ii'})$ of affine schemes over $V$ we have $U(T) = \mathop{\mathrm{colim}}\nolimits V(T_ i)$, but in the current setting we may only consider affine schemes $T_ i$ over $V$ which are (isomorphic to) an object of $(\mathit{Sch}/S)_{fppf}$. So we have to make sure that there are enough affines in $(\mathit{Sch}/S)_{fppf}$ to make the proof work. Inspecting the proof of (2) $\Rightarrow $ (1) of Limits, Proposition 32.6.1 we see that the question reduces to the case that $U$ and $V$ are affine. Say $U = \mathop{\mathrm{Spec}}(A)$ and $V = \mathop{\mathrm{Spec}}(B)$. By construction of $(\mathit{Sch}/S)_{fppf}$ the spectrum of any ring of cardinality $\leq |B|$ is isomorphic to an object of $(\mathit{Sch}/S)_{fppf}$. Hence it suffices to observe that in the "only if" part of the proof of Algebra, Lemma 10.127.3 only $A$-algebras of cardinality $\leq |B|$ are used. $\square$

Proof. Choose a scheme $V$ and a surjective étale morphism $g : V \to Y$. Next, choose a scheme $U$ and a surjective étale morphism $h : U \to V \times _ Y X$. It suffices to show for $T = \mathop{\mathrm{lim}}\nolimits T_ i$ as in the lemma that the map

is surjective, because then $U \to V$ will be locally of finite presentation by Limits, Lemma 32.6.3 (modulo a set theoretic remark exactly as in the proof of Proposition 70.3.10). Thus we take $a : T \to U$ and $b_ i : T_ i \to V$ which determine the same morphism $T \to V$. Picture

By the assumption of the lemma after increasing $i$ we can find a morphism $c_ i : T_ i \to X$ such that $h \circ a = (b_ i, c_ i) \circ p_ i : T_ i \to V \times _ Y X$ and such that $f \circ c_ i = g \circ b_ i$. Since $h$ is an étale morphism of algebraic spaces (and hence locally of finite presentation), we have the surjectivity of

by Proposition 70.3.10. Hence after increasing $i$ again we can find the desired morphism $a_ i : T_ i \to U$ with $a = a_ i \circ p_ i$ and $b_ i = (U \to V) \circ a_ i$. $\square$