Lemma 37.9.9 (04BX)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 37: More on Morphisms
Section 37.9: Infinitesimal deformations of maps
Lemma 37.9.9 (cite)

numberstags

$\xymatrix{ X_1 \ar[d] & X_2 \ar[l]^ f \ar[d] \\ S_1 & S_2 \ar[l] }$

be a commutative diagram of schemes with $X_2 \to X_1$ and $S_2 \to S_1$ étale. Then the map $c_ f : f^*\Omega _{X_1/S_1} \to \Omega _{X_2/S_2}$ of Morphisms, Lemma 29.32.8 is an isomorphism.

Proof. We recall that an étale morphism $U \to V$ is a smooth morphism with $\Omega _{U/V} = 0$ . Using this we see that Morphisms, Lemma 29.32.9 implies $\Omega _{X_2/S_2} = \Omega _{X_2/S_1}$ and Morphisms, Lemma 29.34.16 implies that the map $f^*\Omega _{X_1/S_1} \to \Omega _{X_2/S_1}$ (for the morphism $f$ seen as a morphism over $S_1$ ) is an isomorphism. Hence the lemma follows. $\square$

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