The Stacks project

99.3 Morphisms of finite presentation

This section is the analogue of Limits of Spaces, Section 67.3. There we defined what it means for a transformation of functors on $\mathit{Sch}$ to be limit preserving (we suggest looking at the characterization in Limits of Spaces, Lemma 67.3.2). In Criteria for Representability, Section 94.5 we defined the notion “limit preserving on objects”. Recall that in Artin's Axioms, Section 95.11 we have defined what it means for a category fibred in groupoids over $\mathit{Sch}$ to be limit preserving. Combining these we get the following notion.

Definition 99.3.1. Let $S$ be a scheme. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. We say $f$ is limit preserving if for every directed limit $U = \mathop{\mathrm{lim}}\nolimits U_ i$ of affine schemes over $S$ the diagram

\[ \xymatrix{ \mathop{\mathrm{colim}}\nolimits \mathcal{X}_{U_ i} \ar[r] \ar[d]_ f & \mathcal{X}_ U \ar[d]^ f \\ \mathop{\mathrm{colim}}\nolimits \mathcal{Y}_{U_ i} \ar[r] & \mathcal{Y}_ U } \]

of fibre categories is $2$-cartesian.

Lemma 99.3.2. Let $S$ be a scheme. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. If $f$ is limit preserving (Definition 99.3.1), then $f$ is limit preserving on objects (Criteria for Representability, Section 94.5).

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

\[ \mathop{\mathrm{colim}}\nolimits \mathcal{X}_{U_ i} \longrightarrow (\mathop{\mathrm{colim}}\nolimits \mathcal{Y}_{U_ i}) \times _{\mathcal{Y}_ U} \mathcal{X}_ U \]

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

Lemma 99.3.3. Let $p : \mathcal{X} \to \mathcal{Y}$ and $q : \mathcal{Z} \to \mathcal{Y}$ be $1$-morphisms of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. If $p : \mathcal{X} \to \mathcal{Y}$ is limit preserving, then so is the base change $p' : \mathcal{X} \times _\mathcal {Y} \mathcal{Z} \to \mathcal{Z}$ of $p$ by $q$.

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

\[ (\mathcal{X} \times _\mathcal {Y} \mathcal{Z})_{U_ i} = \mathcal{X}_{U_ i} \times _{\mathcal{Y}_{U_ i}} \mathcal{Z}_{U_ i} \]

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

\begin{align*} \mathop{\mathrm{colim}}\nolimits (\mathcal{X} \times _\mathcal {Y} \mathcal{Z})_{U_ i} & = \mathop{\mathrm{colim}}\nolimits \mathcal{X}_{U_ i} \times _{\mathop{\mathrm{colim}}\nolimits \mathcal{Y}_{U_ i}} \mathop{\mathrm{colim}}\nolimits \mathcal{Z}_{U_ i} \\ & = \mathcal{X}_ U \times _{\mathcal{Y}_ U} \mathop{\mathrm{colim}}\nolimits \mathcal{Y}_{U_ i} \times _{\mathop{\mathrm{colim}}\nolimits \mathcal{Y}_{U_ i}} \mathop{\mathrm{colim}}\nolimits \mathcal{Z}_{U_ i} \\ & = \mathcal{X}_ U \times _{\mathcal{Y}_ U} \mathop{\mathrm{colim}}\nolimits \mathcal{Z}_{U_ i} \\ & = \mathcal{X}_ U \times _{\mathcal{Y}_ U} \mathcal{Z}_ U \times _{\mathcal{Z}_ U} \mathop{\mathrm{colim}}\nolimits \mathcal{Z}_{U_ i} \\ & = (\mathcal{X} \times _\mathcal {Y} \mathcal{Z})_ U \times _{\mathcal{Z}_ U} \mathop{\mathrm{colim}}\nolimits \mathcal{Z}_{U_ i} \end{align*}

as desired. $\square$

Lemma 99.3.4. Let $p : \mathcal{X} \to \mathcal{Y}$ and $q : \mathcal{Y} \to \mathcal{Z}$ be $1$-morphisms of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. If $p$ and $q$ are limit preserving, then so is the composition $q \circ p$.

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

\begin{align*} \mathop{\mathrm{colim}}\nolimits \mathcal{X}_{U_ i} & = \mathcal{X}_ U \times _{\mathcal{Y}_ U} \mathop{\mathrm{colim}}\nolimits \mathcal{Y}_{U_ i} \\ & = \mathcal{X}_ U \times _{\mathcal{Y}_ U} \mathcal{Y}_ U \times _{\mathcal{Z}_ U} \mathop{\mathrm{colim}}\nolimits \mathcal{Z}_{U_ i} \\ & = \mathcal{X}_ U \times _{\mathcal{Z}_ U} \mathop{\mathrm{colim}}\nolimits \mathcal{Z}_{U_ i} \end{align*}

as desired. $\square$

Lemma 99.3.5. Let $p : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. If $p$ is representable by algebraic spaces, then the following are equivalent:

  1. $p$ is limit preserving,

  2. $p$ is limit preserving on objects, and

  3. $p$ is locally of finite presentation (see Algebraic Stacks, Definition 91.10.1).

Proof. In Criteria for Representability, Lemma 94.5.3 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 99.3.2. 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

\[ \mathop{\mathrm{colim}}\nolimits \mathcal{X}_{U_ i} \longrightarrow \mathcal{X}_ U \times _{\mathcal{Y}_ U} \mathop{\mathrm{colim}}\nolimits \mathcal{Y}_{U_ i} \]

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 91.9.2) 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

\[ (\alpha , \beta _ i) : (x_ i|_ U, p(x_ i), can) \longrightarrow (x'_ i|_ U, p(x'_ i), can) \]

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 91.9.2) the functor

\[ X_{y_ i} : (\mathit{Sch}/U_ i)^{opp} \to \textit{Sets},\quad V/U_ i \mapsto \{ (x, \phi ) \mid x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X}_ V), \phi : f(x) \to y_ i|V\} /\cong \]

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

\[ \beta _ i : X_{y_ i} \to X_{y'_ i},\quad (x/V, \phi ) \mapsto (x/V, \beta _ i|_ V \circ \phi ) \]

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

\[ \xymatrix{ U \ar[d] \ar[rr] & & U_ i \ar[d]^{(x'_ i, \text{id})} \\ U_ i \ar[r]^{(x_ i, \text{id})} & X_{y_ i} \ar[r]^{\beta _ i} & X_{y'_ i} } \]

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

\[ \xymatrix{ U_{i'} \ar[d] \ar[rr] & & U_ i \ar[d]^{(x'_ i, \text{id})} \\ U_ i \ar[r]^{(x_ i, \text{id})} & X_{y_ i} \ar[r]^{\beta _ i} & X_{y'_ i} } \]

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

\[ \Delta : X_{y'_ i} \to X_{y'_ i} \times _{U_ i} X_{y'_ i} \]

is locally of finite presentation (Morphisms of Spaces, Lemma 64.28.10) 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 67.3.8. $\square$

Lemma 99.3.6. Let $p : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. The following are equivalent

  1. the diagonal $\Delta : \mathcal{X} \to \mathcal{X} \times _\mathcal {Y} \mathcal{X}$ is limit preserving, and

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

    \[ \mathop{\mathrm{colim}}\nolimits \mathcal{X}_{U_ i} \longrightarrow \mathcal{X}_ U \times _{\mathcal{Y}_ U} \mathop{\mathrm{colim}}\nolimits \mathcal{Y}_{U_ i} \]

    is fully faithful.

In particular, if $p$ is limit preserving, then $\Delta $ is too.

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

\[ \mathop{\mathrm{colim}}\nolimits \mathcal{X}_{U_ i} \longrightarrow \mathcal{X}_ U \times _{\mathcal{Y}_ U} \mathop{\mathrm{colim}}\nolimits \mathcal{Y}_{U_ i} \]

is fully faithful if and only if the functor

\[ \mathop{\mathrm{colim}}\nolimits \mathcal{X}_{U_ i} \longrightarrow \mathcal{X}_ U \times _{(\mathcal{X} \times _\mathcal {Y} \mathcal{X})_ U} \mathop{\mathrm{colim}}\nolimits (\mathcal{X} \times _\mathcal {Y} \mathcal{X})_{U_ i} \]

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

\[ \xymatrix{ \mathop{\mathrm{colim}}\nolimits \mathcal{C}_ i \ar[d]_ p \ar[r]_ f & \mathcal{C} \ar[d]^ p \\ \mathop{\mathrm{colim}}\nolimits \mathcal{D}_ i \ar[r]^ g & \mathcal{D} } \]

Then we claim that

\[ A : \mathop{\mathrm{colim}}\nolimits \mathcal{C}_ i \longrightarrow \mathcal{C} \times _\mathcal {D} \mathop{\mathrm{colim}}\nolimits \mathcal{D}_ i \]

is fully faithful if and only if the functor

\[ B : \mathop{\mathrm{colim}}\nolimits \mathcal{C}_ i \longrightarrow \mathcal{C} \times _{\Delta , \mathcal{C} \times _\mathcal {D} \mathcal{C}, f \times _ g f} \mathop{\mathrm{colim}}\nolimits (\mathcal{C}_ i \times _{\mathcal{D}_ i} \mathcal{C}_ i) \]

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

\[ A' : \mathcal{C}' \to \mathcal{C} \times _\mathcal {D} \mathcal{D}' \quad \text{and}\quad B' : \mathcal{C}' \longrightarrow \mathcal{C} \times _{\Delta , \mathcal{C} \times _\mathcal {D} \mathcal{C}, f \times _ g f} (\mathcal{C}' \times _{\mathcal{D}'} \mathcal{C}') \]

Thus it suffices to prove that if

\[ \xymatrix{ \mathcal{C}' \ar[d]_ p \ar[r]_ f & \mathcal{C} \ar[d]^ p \\ \mathcal{D}' \ar[r]^ g & \mathcal{D} } \]

is a commutative diagram of groupoids, then $A'$ is fully faithful if and only if $B'$ is an equivalence. This follows from Categories, Lemma 4.34.9 (with trivial, i.e., punctual, base category) because

\[ \mathcal{C} \times _{\Delta , \mathcal{C} \times _\mathcal {D} \mathcal{C}, f \times _ g f} (\mathcal{C}' \times _{\mathcal{D}'} \mathcal{C}') = \mathcal{C}' \times _{A', \mathcal{C} \times _\mathcal {D} \mathcal{D}', A'} \mathcal{C}' \]

This finishes the proof. $\square$

Lemma 99.3.7. Let $S$ be a scheme. Let $\mathcal{X}$ be an algebraic stack over $S$. If $\mathcal{X} \to S$ is locally of finite presentation, then $\mathcal{X}$ is limit preserving in the sense of Artin's Axioms, Definition 95.11.1 (equivalently: the morphism $\mathcal{X} \to S$ is limit preserving).

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 98.26. We can write $\mathcal{X} = [U/R]$ for some smooth groupoid in algebraic spaces $(U, R, s, t, c)$, see Algebraic Stacks, Lemma 91.16.2. 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 67.3.8) 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

\[ \mathop{\mathrm{colim}}\nolimits [U/R]_{T_\lambda } \longrightarrow [U/R]_ T \]

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 75.23.1 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

\[ s \circ r_{\lambda , ij} = u_{\lambda , i} \circ \text{pr}_0 \quad \text{and}\quad t \circ r_{\lambda , ij} = u_{\lambda , j} \circ \text{pr}_1, \]

and

\[ c \circ (r_{\lambda , jk} \circ \text{pr}_{12}, r_{\lambda , ij} \circ \text{pr}_{01}) = r_{\lambda , ik} \circ \text{pr}_{02}. \]

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 75.23.1. $\square$

reference

Proposition 99.3.8. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. The following are equivalent

  1. $f$ is limit preserving,

  2. $f$ is limit preserving on objects, and

  3. $f$ is locally of finite presentation.

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

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

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 99.3.7. 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 98.26.7 the morphism $\Delta $ is locally of finite presentation. Thus the argument above shows that $\Delta $ is limit preserving on objects. By Lemma 99.3.5 this implies that $\Delta $ is limit preserving. By Lemma 99.3.6 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 94.5.1 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 94.5.2 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 94.5.3. This is exactly the condition that $f$ is locally of finite presentation, see Morphisms of Stacks, Definition 98.26.1. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0CMQ. Beware of the difference between the letter 'O' and the digit '0'.