The Stacks project

Lemma 97.5.3. 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 94.10.1).

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 70.3.10 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 70.3.11. 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 70.3.12 that $X_ y \to T$ is locally of finite presentation. $\square$


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 06CX. Beware of the difference between the letter 'O' and the digit '0'.