The Stacks project

Lemma 74.7.3. 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 ├ętale 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 64.18.10 and Equation (64.18.10.1) 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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