The Stacks project

64.3 Morphisms of finite presentation

In this section we generalize Limits, Proposition 31.6.1 to morphisms of algebraic spaces. The motivation for the following definition comes from the proposition just cited.

Definition 64.3.1. Let $S$ be a scheme.

  1. A functor $F : (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}$ is said to be limit preserving or locally of finite presentation 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

    \[ F(T) = \mathop{\mathrm{colim}}\nolimits F(T_ i). \]

    We sometimes say that $F$ is locally of finite presentation over $S$.

  2. Let $F, G : (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}$. A transformation of functors $a : F \to G$ is limit preserving or locally of finite presentation if for every scheme $T$ over $S$ and every $y \in G(T)$ the functor

    \[ F_ y : (\mathit{Sch}/T)_{fppf}^{opp} \longrightarrow \textit{Sets}, \quad T'/T \longmapsto \{ x \in F(T') \mid a(x) = y|_{T'}\} \]

    is locally of finite presentation over $T$1. We sometimes say that $F$ is relatively limit preserving over $G$.

The functor $F_ y$ is in some sense the fiber of $a : F \to G$ over $y$, except that it is a presheaf on the big fppf site of $T$. A formula for this functor is:

64.3.1.1
\begin{equation} \label{spaces-limits-equation-fibre-map-functors} F_ y = F|_{(\mathit{Sch}/T)_{fppf}} {\times }_{G|_{(\mathit{Sch}/T)_{fppf}}} * \end{equation}

Here $*$ is the final object in the category of (pre)sheaves on $(\mathit{Sch}/T)_{fppf}$ (see Sites, Example 7.10.2) and the map $* \to G|_{(\mathit{Sch}/T)_{fppf}}$ is given by $y$. Note that if $j : (\mathit{Sch}/T)_{fppf} \to (\mathit{Sch}/S)_{fppf}$ is the localization functor, then the formula above becomes $F_ y = j^{-1}F \times _{j^{-1}G} *$ and $j_!F_ y$ is just the fiber product $F \times _{G, y} T$. (See Sites, Section 7.25, for information on localization, and especially Sites, Remark 7.25.10 for information on $j_!$ for presheaves.)

At this point we temporarily have two definitions of what it means for a morphism $X \to Y$ of algebraic spaces over $S$ to be locally of finite presentation. Namely, one by Morphisms of Spaces, Definition 61.28.1 and one using that $X \to Y$ is a transformation of functors so that Definition 64.3.1 applies (we will use the terminology “limit preserving” for this notion as much as possible). We will show in Proposition 64.3.8 that these two definitions agree.

Lemma 64.3.2. Let $S$ be a scheme. Let $a : F \to G$ be a transformation of functors $(\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}$. The following are equivalent

  1. $a : F \to G$ is limit preserving, and

  2. 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 $S$ the diagram of sets

    \[ \xymatrix{ \mathop{\mathrm{colim}}\nolimits _ i F(T_ i) \ar[r] \ar[d]_ a & F(T) \ar[d]^ a \\ \mathop{\mathrm{colim}}\nolimits _ i G(T_ i) \ar[r] & G(T) } \]

    is a fibre product diagram.

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 64.3.1. 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. $\square$

Lemma 64.3.3. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $F, G, H : (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}$. Let $a : F \to G$, $b : G \to H$ be transformations of functors. If $a$ and $b$ are limit preserving, then

\[ b \circ a : F \longrightarrow H \]

is limit preserving.

Proof. Let $T = \mathop{\mathrm{lim}}\nolimits _{i \in I} T_ i$ as in characterization (2) of Lemma 64.3.2. Consider the diagram

\[ \xymatrix{ \mathop{\mathrm{colim}}\nolimits _ i F(T_ i) \ar[r] \ar[d]_ a & F(T) \ar[d]^ a \\ \mathop{\mathrm{colim}}\nolimits _ i G(T_ i) \ar[r] \ar[d]_ b & G(T) \ar[d]^ b \\ \mathop{\mathrm{colim}}\nolimits _ i H(T_ i) \ar[r] & H(T) } \]

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

Lemma 64.3.4. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $F, G, H : (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}$. Let $a : F \to G$, $b : H \to G$ be transformations of functors. Consider the fibre product diagram

\[ \xymatrix{ H \times _{b, G, a} F \ar[r]_-{b'} \ar[d]_{a'} & F \ar[d]^ a \\ H \ar[r]^ b & G } \]

If $a$ is limit preserving, then the base change $a'$ is limit preserving.

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

Lemma 64.3.5. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $F : (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}$ be a functor. If $F$ is limit preserving then its sheafification $F^\# $ is limit preserving.

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 33.7.4. Hence we can write

\[ F^+(T) = \mathop{\mathrm{colim}}\nolimits _{\mathcal{V}\text{ standard covering }T} \check H^0(\mathcal{V}, F). \]

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

\begin{align*} \mathop{\mathrm{colim}}\nolimits _{i' \geq i} \check H^0(\mathcal{V}_ i, F) & = \mathop{\mathrm{colim}}\nolimits _{i' \geq i} \text{Equalizer}( \xymatrix{ \prod F(T_{i', k}) \ar@<1ex>[r] \ar@<-1ex>[r] & \prod F(T_{i', k} \times _{T_{i'}} T_{i', l}) } \\ & = \text{Equalizer}( \xymatrix{ \mathop{\mathrm{colim}}\nolimits _{i' \geq i} \prod F(T_{i', k}) \ar@<1ex>[r] \ar@<-1ex>[r] & \mathop{\mathrm{colim}}\nolimits _{k' \geq k} \prod F(T_{i', k} \times _{T_{i'}} T_{i', l}) } \\ & = \text{Equalizer}( \xymatrix{ \prod F(T_ k) \ar@<1ex>[r] \ar@<-1ex>[r] & \prod F(T_ k \times _ T T_ l) } \\ & = \check H^0(\mathcal{V}, F) \end{align*}

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 31.2.3. If we use this for all coverings at the same time we obtain

\begin{align*} F^+(T) & = \mathop{\mathrm{colim}}\nolimits _{\mathcal{V}\text{ standard covering }T} \check H^0(\mathcal{V}, F) \\ & = \mathop{\mathrm{colim}}\nolimits _{i \in I} \mathop{\mathrm{colim}}\nolimits _{\mathcal{V}_ i\text{ standard covering }T_ i} \check H^0(T \times _{T_ i}\mathcal{V}_ i, F) \\ & = \mathop{\mathrm{colim}}\nolimits _{i \in I} F^+(T_ i) \end{align*}

The switch of the order of the colimits is allowed by Categories, Lemma 4.14.9. $\square$

Lemma 64.3.6. Let $S$ be a scheme. Let $F : (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}$ be a functor. Assume that

  1. $F$ is a sheaf, and

  2. there exists an fppf covering $\{ U_ j \to S\} _{j \in J}$ such that $F|_{(\mathit{Sch}/U_ j)_{fppf}}$ is limit preserving.

Then $F$ is limit preserving.

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 31.2.3.

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

\[ x_ i|_{V_{i', k}} = x'_ i|_{V_{i', k}} \]

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

\[ x_{i, k}|_{V_{i', k} \times _{T_{i'}} V_{i', l}} = x_{i, l}|_{V_{i', k} \times _{T_{i'}} V_{i', l}} \]

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. $\square$

Lemma 64.3.7. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $F, G : (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}$ be functors. If $a : F \to G$ is a transformation which is limit preserving, then the induced transformation of sheaves $F^\# \to G^\# $ is limit preserving.

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 64.3.1 is limit preserving. By Equation (64.3.1.1) 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 64.3.6 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 (64.3.1.1). Thus this case follows from Lemma 64.3.5. $\square$

Proposition 64.3.8. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent:

  1. The morphism $f$ is a morphism of algebraic spaces which is locally of finite presentation, see Morphisms of Spaces, Definition 61.28.1.

  2. The morphism $f : X \to Y$ is limit preserving as a transformation of functors, see Definition 64.3.1.

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 64.3.1. 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 59.9.3. It follows from Morphisms of Spaces, Definition 61.28.1 that both $U$ and $R$ are schemes which are locally of finite presentation over $S$. Hence by Limits, Proposition 31.6.1 we have

\[ U(T) = \mathop{\mathrm{colim}}\nolimits U(T_ i), \quad R(T) = \mathop{\mathrm{colim}}\nolimits R(T_ i) \]

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

\[ (\mathit{Sch}/S)_{fppf}^{opp} \longrightarrow \textit{Sets}, \quad W \longmapsto U(W)/R(W) \]

is limit preserving. Hence by Lemma 64.3.5 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 64.3.4 the transformation of functors $V \times _ Y X \to V$ is limit preserving. By Morphisms of Spaces, Lemma 61.39.8 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 64.3.3 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 31.6.1 (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 31.6.1 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 31.6.1 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 10.126.3 only $A$-algebras of cardinality $\leq |B|$ are used. $\square$

Remark 64.3.9. Here is an important special case of Proposition 64.3.8. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Then $X$ is locally of finite presentation over $S$ if and only if $X$, as a functor $(\mathit{Sch}/S)^{opp} \to \textit{Sets}$, is limit preserving. Compare with Limits, Remark 31.6.2. In fact, we will see in Lemma 64.3.10 below that it suffices if the map

\[ \mathop{\mathrm{colim}}\nolimits X(T_ i) \longrightarrow X(T) \]

is surjective whenever $T = \mathop{\mathrm{lim}}\nolimits T_ i$ is a directed limit of affine schemes over $S$.

Lemma 64.3.10. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. If for every directed limit $T = \mathop{\mathrm{lim}}\nolimits _{i \in I} T_ i$ of affine schemes over $S$ the map

\[ \mathop{\mathrm{colim}}\nolimits X(T_ i) \longrightarrow X(T) \times _{Y(T)} \mathop{\mathrm{colim}}\nolimits Y(T_ i) \]

is surjective, then $f$ is locally of finite presentation. In other words, in Proposition 64.3.8 part (2) it suffices to check surjectivity in the criterion of Lemma 64.3.2.

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

\[ \mathop{\mathrm{colim}}\nolimits U(T_ i) \longrightarrow U(T) \times _{V(T)} \mathop{\mathrm{colim}}\nolimits V(T_ i) \]

is surjective, because then $U \to V$ will be locally of finite presentation by Limits, Lemma 31.6.3 (modulo a set theoretic remark exactly as in the proof of Proposition 64.3.8). Thus we take $a : T \to U$ and $b_ i : T_ i \to V$ which determine the same morphism $T \to V$. Picture

\[ \xymatrix{ T \ar[d]_ a \ar[rr]_{p_ i} & & T_ i \ar[d]^{b_ i} \ar@{..>}[ld] \\ U \ar[r]^-h & X \times _ Y V \ar[d] \ar[r] & V \ar[d]^ g \\ & X \ar[r]^ f & Y } \]

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

\[ \mathop{\mathrm{colim}}\nolimits U(T_ i) \longrightarrow U(T) \times _{(X \times _ Y V)(T)} \mathop{\mathrm{colim}}\nolimits (X \times _ Y V)(T_ i) \]

by Proposition 64.3.8. 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 $b_ i = (U \to V) \circ a_ i$. $\square$

[1] The characterization (2) in Lemma 64.3.2 may be easier to parse.

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