The Stacks project

Lemma 85.29.3. 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 85.29.1. Let $(R/U)_\bullet $ be the simplicial scheme associated to $s : R \to U$, see Definition 85.27.3. There exists a cartesian morphism $t_\bullet : (R/U)_\bullet \to X$ of simplicial schemes with low degree morphisms given by

\[ \xymatrix{ R \times _{s, U, s} R \times _{s, U, s} R \ar@<3ex>[r]_-{\text{pr}_{12}} \ar@<0ex>[r]_-{\text{pr}_{02}} \ar@<-3ex>[r]_-{\text{pr}_{01}} \ar[dd]_{(r_0, r_1, r_2) \mapsto (r_0 \circ r_1^{-1}, r_1 \circ r_2^{-1})} & R \times _{s, U, s} R \ar@<1ex>[r]_-{\text{pr}_1} \ar@<-2ex>[r]_-{\text{pr}_0} \ar[dd]_{(r_0, r_1) \mapsto r_0 \circ r_1^{-1}} & R \ar[dd]^ t \\ \\ R \times _{s, U, t} R \ar@<3ex>[r]_{\text{pr}_1} \ar@<0ex>[r]_ c \ar@<-3ex>[r]_{\text{pr}_0} & R \ar@<1ex>[r]_ s \ar@<-2ex>[r]_ t & U } \]

Proof. For arbitrary $n$ we define $(R/U)_\bullet \to X_ n$ by the rule

\[ (r_0, \ldots , r_ n) \longrightarrow (r_0 \circ r_1^{-1}, \ldots , r_{n - 1} \circ r_ n^{-1}) \]

Compatibility with degeneracy maps is clear from the description of the degeneracies in Lemma 85.29.1. 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. $\square$

Comments (0)

There are also:

  • 2 comment(s) on Section 85.29: Groupoids and simplicial schemes

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Your email address will not be published.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 07TQ. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 85.29.3 as pdf