Lemma 84.29.3. Let $(U, R, s, t, c)$ be a groupoid scheme over a scheme $S$. Let $X$ be the simplicial scheme over $S$ constructed in Lemma 84.29.1. Let $(R/U)_\bullet $ be the simplicial scheme associated to $s : R \to U$, see Definition 84.27.3. There exists a cartesian morphism $t_\bullet : (R/U)_\bullet \to X$ of simplicial schemes with low degree morphisms given by
Proof. For arbitrary $n$ we define $(R/U)_\bullet \to X_ n$ by the rule
Compatibility with degeneracy maps is clear from the description of the degeneracies in Lemma 84.29.1. We omit the verification that the maps respect the morphisms $s^ n_ j$. Groupoids, Lemma 39.13.5 (with the roles of $s$ and $t$ reversed) shows that the two right squares are cartesian. In exactly the same manner one shows all the other squares are cartesian too. Hence the morphism is cartesian. $\square$
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Your email address will not be published.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.
There are also: