Lemma 76.7.7 (05Z7)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 76: More on Morphisms of Spaces
Section 76.7: Sheaf of differentials of a morphism
Lemma 76.7.7 (cite)

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Lemma 76.7.7. Let $S$ be a scheme. Let

$\xymatrix{ X'' \ar[d] \ar[r]_ g & X' \ar[d] \ar[r]_ f & X \ar[d] \\ Y'' \ar[r] & Y' \ar[r] & Y }$

be a commutative diagram of algebraic spaces over $S$ . Then we have

$c_{f \circ g} = c_ g \circ g^* c_ f$

as maps $(f \circ g)^*\Omega _{X/Y} \to \Omega _{X''/Y''}$ .

Proof. Omitted. Hint: Use the characterization of $c_ f, c_ g, c_{f \circ g}$ in terms of the effect these maps have on local sections. $\square$

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