Lemma 87.15.2. Let $S$ be a scheme. Let $X, Y$ be formal algebraic spaces over $S$ and let $Z$ be a sheaf whose diagonal is representable by algebraic spaces. Let $X \to Z$ and $Y \to Z$ be maps of sheaves. Then $X \times _ Z Y$ is a formal algebraic space.
Proof. Choose $\{ X_ i \to X\} $ and $\{ Y_ j \to Y\} $ as in Definition 87.11.1. Then $\{ X_ i \times _ Z Y_ j \to X \times _ Z Y\} $ is a family of maps which are representable by algebraic spaces and étale. Thus Lemma 87.15.1 tells us it suffices to show that $X \times _ Z Y$ is a formal algebraic space when $X$ and $Y$ are affine formal algebraic spaces.
Assume $X$ and $Y$ are affine formal algebraic spaces. Write $X = \mathop{\mathrm{colim}}\nolimits X_\lambda $ and $Y = \mathop{\mathrm{colim}}\nolimits Y_\mu $ as in Definition 87.9.1. Then $X \times _ Z Y = \mathop{\mathrm{colim}}\nolimits X_\lambda \times _ Z Y_\mu $. Each $X_\lambda \times _ Z Y_\mu $ is an algebraic space. For $\lambda \leq \lambda '$ and $\mu \leq \mu '$ the morphism
\[ X_\lambda \times _ Z Y_\mu \to X_\lambda \times _ Z Y_{\mu '} \to X_{\lambda '} \times _ Z Y_{\mu '} \]
is a thickening as a composition of base changes of thickenings. Thus we conclude by applying Lemma 87.13.1. $\square$
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Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (2)
Comment #1558 by Matthew Emerton on
Comment #1576 by Johan on