Lemma 76.7.3. Let S be a scheme. Let f : X \to Y be a morphism of algebraic spaces over S. Consider any commutative diagram
\xymatrix{ U \ar[d]_ a \ar[r]_\psi & V \ar[d]^ b \\ X \ar[r]^ f & Y }
where the vertical arrows are étale morphisms of algebraic spaces. Then
\Omega _{X/Y}|_{U_{\acute{e}tale}} = \Omega _{U/V}
In particular, if U, V are schemes, then this is equal to the usual sheaf of differentials of the morphism of schemes U \to V.
Proof.
By Properties of Spaces, Lemma 66.18.11 and Equation (66.18.11.1) we may think of the restriction of a sheaf on X_{\acute{e}tale} to U_{\acute{e}tale} as the pullback by a_{small}. Similarly for b. By Modules on Sites, Lemma 18.33.6 we have
\Omega _{X/Y}|_{U_{\acute{e}tale}} = \Omega _{\mathcal{O}_{U_{\acute{e}tale}}/ a_{small}^{-1}f_{small}^{-1}\mathcal{O}_{Y_{\acute{e}tale}}}
Since a_{small}^{-1}f_{small}^{-1}\mathcal{O}_{Y_{\acute{e}tale}} = \psi _{small}^{-1}b_{small}^{-1}\mathcal{O}_{Y_{\acute{e}tale}} = \psi _{small}^{-1}\mathcal{O}_{V_{\acute{e}tale}} we see that the lemma holds.
\square
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