The Stacks project

91.5 Limit preserving on objects

Let $S$ be a scheme. Let $p : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. We will say that $p$ is limit preserving on objects if the following condition holds: Given any data consisting of

  1. an affine scheme $U = \mathop{\mathrm{lim}}\nolimits _{i \in I} U_ i$ which is written as the directed limit of affine schemes $U_ i$ over $S$,

  2. an object $y_ i$ of $\mathcal{Y}$ over $U_ i$ for some $i$,

  3. an object $x$ of $\mathcal{X}$ over $U$, and

  4. an isomorphism $\gamma : p(x) \to y_ i|_ U$,

then there exists an $i' \geq i$, an object $x_{i'}$ of $\mathcal{X}$ over $U_{i'}$, an isomorphism $\beta : x_{i'}|_ U \to x$, and an isomorphism $\gamma _{i'} : p(x_{i'}) \to y_ i|_{U_{i'}}$ such that

91.5.0.1
\begin{equation} \label{criteria-equation-limit-preserving} \vcenter { \xymatrix{ p(x_{i'}|_ U) \ar[d]_{p(\beta )} \ar[rr]_{\gamma _{i'}|_ U} & & (y_ i|_{U_{i'}})|_ U \ar@{=}[d] \\ p(x) \ar[rr]^\gamma & & y_ i|_ U } } \end{equation}

commutes. In this situation we say that “$(i', x_{i'}, \beta , \gamma _{i'})$ is a solution to the problem posed by our data (1), (2), (3), (4)”. The motivation for this definition comes from Limits of Spaces, Lemma 64.3.2.

Lemma 91.5.1. 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 on objects, 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 $S$, let $z_ i$ be an object of $\mathcal{Z}$ over $U_ i$ for some $i$, let $w$ be an object of $\mathcal{X} \times _\mathcal {Y} \mathcal{Z}$ over $U$, and let $\delta : p'(w) \to z_ i|_ U$ be an isomorphism. We may write $w = (U, x, z, \alpha )$ for some object $x$ of $\mathcal{X}$ over $U$ and object $z$ of $\mathcal{Z}$ over $U$ and isomorphism $\alpha : p(x) \to q(z)$. Note that $p'(w) = z$ hence $\delta : z \to z_ i|_ U$. Set $y_ i = q(z_ i)$ and $\gamma = q(\delta ) \circ \alpha : p(x) \to y_ i|_ U$. As $p$ is limit preserving on objects there exists an $i' \geq i$ and an object $x_{i'}$ of $\mathcal{X}$ over $U_{i'}$ as well as isomorphisms $\beta : x_{i'}|_ U \to x$ and $\gamma _{i'} : p(x_{i'}) \to y_ i|_{U_{i'}}$ such that (91.5.0.1) commutes. Then we consider the object $w_{i'} = (U_{i'}, x_{i'}, z_ i|_{U_{i'}}, \gamma _{i'})$ of $\mathcal{X} \times _\mathcal {Y} \mathcal{Z}$ over $U_{i'}$ and define isomorphisms

\[ w_{i'}|_ U = (U, x_{i'}|_ U, z_ i|_ U, \gamma _{i'}|_ U) \xrightarrow {(\beta , \delta ^{-1})} (U, x, z, \alpha ) = w \]

and

\[ p'(w_{i'}) = z_ i|_{U_{i'}} \xrightarrow {\text{id}} z_ i|_{U_{i'}}. \]

These combine to give a solution to the problem. $\square$

Lemma 91.5.2. 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 on objects, 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 $S$, let $z_ i$ be an object of $\mathcal{Z}$ over $U_ i$ for some $i$, let $x$ be an object of $\mathcal{X}$ over $U$, and let $\gamma : q(p(x)) \to z_ i|_ U$ be an isomorphism. As $q$ is limit preserving on objects there exist an $i' \geq i$, an object $y_{i'}$ of $\mathcal{Y}$ over $U_{i'}$, an isomorphism $\beta : y_{i'}|_ U \to p(x)$, and an isomorphism $\gamma _{i'} : q(y_{i'}) \to z_ i|_{U_{i'}}$ such that (91.5.0.1) is commutative. As $p$ is limit preserving on objects there exist an $i'' \geq i'$, an object $x_{i''}$ of $\mathcal{X}$ over $U_{i''}$, an isomorphism $\beta ' : x_{i''}|_ U \to x$, and an isomorphism $\gamma '_{i''} : p(x_{i''}) \to y_{i'}|_{U_{i''}}$ such that (91.5.0.1) is commutative. The solution is to take $x_{i''}$ over $U_{i''}$ with isomorphism

\[ q(p(x_{i''})) \xrightarrow {q(\gamma '_{i''})} q(y_{i'})|_{U_{i''}} \xrightarrow {\gamma _{i'}|_{U_{i''}}} z_ i|_{U_{i''}} \]

and isomorphism $\beta ' : x_{i''}|_ U \to x$. We omit the verification that (91.5.0.1) is commutative. $\square$

Lemma 91.5.3. 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 on objects, and

  2. $p$ is locally of finite presentation (see Algebraic Stacks, Definition 88.10.1).

Proof. Assume (2). Let $U = \mathop{\mathrm{lim}}\nolimits _{i \in I} U_ i$ be the directed limit of affine schemes $U_ i$ over $S$, let $y_ i$ be an object of $\mathcal{Y}$ over $U_ i$ for some $i$, let $x$ be an object of $\mathcal{X}$ over $U$, and let $\gamma : p(x) \to y_ i|_ U$ be an isomorphism. Let $X_{y_ i}$ denote an algebraic space over $U_ i$ representing the $2$-fibre product

\[ (\mathit{Sch}/U_ i)_{fppf} \times _{y_ i, \mathcal{Y}, p} \mathcal{X}. \]

Note that $\xi = (U, U \to U_ i, x, \gamma ^{-1})$ defines an object of this $2$-fibre product over $U$. Via the $2$-Yoneda lemma $\xi $ corresponds to a morphism $f_\xi : U \to X_{y_ i}$ over $U_ i$. By Limits of Spaces, Proposition 64.3.8 there exists an $i' \geq i$ and a morphism $f_{i'} : U_{i'} \to X_{y_ i}$ such that $f_\xi $ is the composition of $f_{i'}$ and the projection morphism $U \to U_{i'}$. Also, the $2$-Yoneda lemma tells us that $f_{i'}$ corresponds to an object $\xi _{i'} = (U_{i'}, U_{i'} \to U_ i, x_{i'}, \alpha )$ of the displayed $2$-fibre product over $U_{i'}$ whose restriction to $U$ recovers $\xi $. In particular we obtain an isomorphism $\gamma : x_{i'}|U \to x$. Note that $\alpha : y_ i|_{U_{i'}} \to p(x_{i'})$. Hence we see that taking $x_{i'}$, the isomorphism $\gamma : x_{i'}|U \to x$, and the isomorphism $\beta = \alpha ^{-1} : p(x_{i'}) \to y_ i|_{U_{i'}}$ is a solution to the problem.

Assume (1). Choose a scheme $T$ and a $1$-morphism $y : (\mathit{Sch}/T)_{fppf} \to \mathcal{Y}$. Let $X_ y$ be an algebraic space over $T$ representing the $2$-fibre product $(\mathit{Sch}/T)_{fppf} \times _{y, \mathcal{Y}, p} \mathcal{X}$. We have to show that $X_ y \to T$ is locally of finite presentation. To do this we will use the criterion in Limits of Spaces, Remark 64.3.9. Consider an affine scheme $U = \mathop{\mathrm{lim}}\nolimits _{i \in I} U_ i$ written as the directed limit of affine schemes over $T$. Pick any $i \in I$ and set $y_ i = y|_{U_ i}$. Also denote $i'$ an element of $I$ which is bigger than or equal to $i$. By the $2$-Yoneda lemma morphisms $U \to X_ y$ over $T$ correspond bijectively to isomorphism classes of pairs $(x, \alpha )$ where $x$ is an object of $\mathcal{X}$ over $U$ and $\alpha : y|_ U \to p(x)$ is an isomorphism. Of course giving $\alpha $ is, up to an inverse, the same thing as giving an isomorphism $\gamma : p(x) \to y_ i|_ U$. Similarly for morphisms $U_{i'} \to X_ y$ over $T$. Hence (1) guarantees that the canonical map

\[ \mathop{\mathrm{colim}}\nolimits _{i' \geq i} X_ y(U_{i'}) \longrightarrow X_ y(U) \]

is surjective in this situation. It follows from Limits of Spaces, Lemma 64.3.10 that $X_ y \to T$ is locally of finite presentation. $\square$

Lemma 91.5.4. Let $p : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Assume $p$ is representable by algebraic spaces and an open immersion. Then $p$ is limit preserving on objects.

Proof. This follows from Lemma 91.5.3 and (via the general principle Algebraic Stacks, Lemma 88.10.9) from the fact that an open immersion of algebraic spaces is locally of finite presentation, see Morphisms of Spaces, Lemma 61.28.11. $\square$

Let $S$ be a scheme. In the following lemma we need the notion of the size of an algebraic space $X$ over $S$. Namely, given a cardinal $\kappa $ we will say $X$ has $\text{size}(X) \leq \kappa $ if and only if there exists a scheme $U$ with $\text{size}(U) \leq \kappa $ (see Sets, Section 3.9) and a surjective étale morphism $U \to X$.

Lemma 91.5.5. Let $S$ be a scheme. Let $\kappa = \text{size}(T)$ for some $T \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$ such that

  1. $\mathcal{Y} \to (\mathit{Sch}/S)_{fppf}$ is limit preserving on objects,

  2. for an affine scheme $V$ locally of finite presentation over $S$ and $y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{Y}_ V)$ the fibre product $(\mathit{Sch}/V)_{fppf} \times _{y, \mathcal{Y}} \mathcal{X}$ is representable by an algebraic space of size $\leq \kappa $1,

  3. $\mathcal{X}$ and $\mathcal{Y}$ are stacks for the Zariski topology.

Then $f$ is representable by algebraic spaces.

Proof. Let $V$ be a scheme over $S$ and $y \in \mathcal{Y}_ V$. We have to prove $(\mathit{Sch}/V)_{fppf} \times _{y, \mathcal{Y}} \mathcal{X}$ is representable by an algebraic space.

Case I: $V$ is affine and maps into an affine open $\mathop{\mathrm{Spec}}(\Lambda ) \subset S$. Then we can write $V = \mathop{\mathrm{lim}}\nolimits V_ i$ with each $V_ i$ affine and of finite presentation over $\mathop{\mathrm{Spec}}(\Lambda )$, see Algebra, Lemma 10.126.2. Then $y$ comes from an object $y_ i$ over $V_ i$ for some $i$ by assumption (1). By assumption (3) the fibre product $(\mathit{Sch}/V_ i)_{fppf} \times _{y_ i, \mathcal{Y}} \mathcal{X}$ is representable by an algebraic space $Z_ i$. Then $(\mathit{Sch}/V)_{fppf} \times _{y, \mathcal{Y}} \mathcal{X}$ is representable by $Z \times _{V_ i} V$.

Case II: $V$ is general. Choose an affine open covering $V = \bigcup _{i \in I} V_ i$ such that each $V_ i$ maps into an affine open of $S$. We first claim that $\mathcal{Z} = (\mathit{Sch}/V)_{fppf} \times _{y, \mathcal{Y}} \mathcal{X}$ is a stack in setoids for the Zariski topology. Namely, it is a stack in groupoids for the Zariski topology by Stacks, Lemma 8.5.6. Then suppose that $z$ is an object of $\mathcal{Z}$ over a scheme $T$. Denote $g : T \to V$ the morphism corresponding to the projection of $z$ in $(\mathit{Sch}/V)_{fppf}$. Consider the Zariski sheaf $\mathit{I} = \mathit{Isom}_{\mathcal{Z}}(z, z)$. By Case I we see that $\mathit{I}|_{g^{-1}(V_ i)} = *$ (the singleton sheaf). Hence $\mathcal{I} = *$. Thus $\mathcal{Z}$ is fibred in setoids. To finish the proof we have to show that the Zariski sheaf $Z : T \mapsto \mathop{\mathrm{Ob}}\nolimits (\mathcal{Z}_ T)/\cong $ is an algebraic space, see Algebraic Stacks, Lemma 88.8.2. There is a map $p : Z \to V$ (transformation of functors) and by Case I we know that $Z_ i = p^{-1}(V_ i)$ is an algebraic space. The morphisms $Z_ i \to Z$ are representable by open immersions and $\coprod Z_ i \to Z$ is surjective (in the Zariski topology). Hence $Z$ is a sheaf for the fppf topology by Bootstrap, Lemma 74.3.11. Thus Spaces, Lemma 59.8.5 applies and we conclude that $Z$ is an algebraic space2. $\square$

Lemma 91.5.6. Let $S$ be a scheme. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Let $\mathcal{P}$ be a property of morphisms of algebraic spaces as in Algebraic Stacks, Definition 88.10.1. If

  1. $f$ is representable by algebraic spaces,

  2. $\mathcal{Y} \to (\mathit{Sch}/S)_{fppf}$ is limit preserving on objects,

  3. for an affine scheme $V$ locally of finite presentation over $S$ and $y \in \mathcal{Y}_ V$ the resulting morphism of algebraic spaces $f_ y : F_ y \to V$, see Algebraic Stacks, Equation (88.9.1.1), has property $\mathcal{P}$.

Then $f$ has property $\mathcal{P}$.

Proof. Let $V$ be a scheme over $S$ and $y \in \mathcal{Y}_ V$. We have to show that $F_ y \to V$ has property $\mathcal{P}$. Since $\mathcal{P}$ is fppf local on the base we may assume that $V$ is an affine scheme which maps into an affine open $\mathop{\mathrm{Spec}}(\Lambda ) \subset S$. Thus we can write $V = \mathop{\mathrm{lim}}\nolimits V_ i$ with each $V_ i$ affine and of finite presentation over $\mathop{\mathrm{Spec}}(\Lambda )$, see Algebra, Lemma 10.126.2. Then $y$ comes from an object $y_ i$ over $V_ i$ for some $i$ by assumption (2). By assumption (3) the morphism $F_{y_ i} \to V_ i$ has property $\mathcal{P}$. As $\mathcal{P}$ is stable under arbitrary base change and since $F_ y = F_{y_ i} \times _{V_ i} V$ we conclude that $F_ y \to V$ has property $\mathcal{P}$ as desired. $\square$

[1] The condition on size can be dropped by those ignoring set theoretic issues.
[2] To see that the set theoretic condition of that lemma is satisfied we argue as follows: First choose the open covering such that $|I| \leq \text{size}(V)$. Next, choose schemes $U_ i$ of size $\leq \max (\kappa , \text{size}(V))$ and surjective étale morphisms $U_ i \to Z_ i$; we can do this by assumption (2) and Sets, Lemma 3.9.6 (details omitted). Then Sets, Lemma 3.9.9 implies that $\coprod U_ i$ is an object of $(\mathit{Sch}/S)_{fppf}$. Hence $\coprod Z_ i$ is an algebraic space by Spaces, Lemma 59.8.4.

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