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. $\square$

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