The Stacks project

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. $\square$


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