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 (40.3.0.2) we conclude that if s is flat, then c is flat, see Morphisms, Lemma 29.25.8. \square
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