Lemma 96.5.1. 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 (96.5.0.1) 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

and

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

## 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Your email address will not be published.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.

## Comments (0)