The Stacks project

Lemma 40.4.1. Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. The sheaf of differentials of $R$ seen as a scheme over $U$ via $t$ is a quotient of the pullback via $t$ of the conormal sheaf of the immersion $e : U \to R$. In a formula: there is a canonical surjection $t^*\mathcal{C}_{U/R} \to \Omega _{R/U}$. If $s$ is flat, then this map is an isomorphism.

Proof. Note that $e : U \to R$ is an immersion as it is a section of the morphism $s$, see Schemes, Lemma 26.21.11. Consider the following diagram

\[ \xymatrix{ R \ar[r]_-{(1, i)} \ar[d]_ t & R \times _{s, U, t} R \ar[d]^ c \ar[rr]_{(\text{pr}_0, i \circ \text{pr}_1)} & & R \times _{t, U, t} R \\ U \ar[r]^ e & R } \]

The square on the left is cartesian, because if $a \circ b = e$, then $b = i(a)$. The composition of the horizontal maps is the diagonal morphism of $t : R \to U$. The right top horizontal arrow is an isomorphism. Hence since $\Omega _{R/U}$ is the conormal sheaf of the composition it is isomorphic to the conormal sheaf of $(1, i)$. By Morphisms, Lemma 29.31.4 we get the surjection $t^*\mathcal{C}_{U/R} \to \Omega _{R/U}$ and if $c$ is flat, then this is an isomorphism. Since $c$ is a base change of $s$ by the properties of Diagram ( we conclude that if $s$ is flat, then $c$ is flat, see Morphisms, Lemma 29.25.8. $\square$

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