The Stacks project

Lemma 76.7.6. Let $S$ be a scheme. Let

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

be a commutative diagram of algebraic spaces. The map $f^\sharp : \mathcal{O}_ X \to f_*\mathcal{O}_{X'}$ composed with the map $f_*\text{d}_{X'/Y'} : f_*\mathcal{O}_{X'} \to f_*\Omega _{X'/Y'}$ is a $Y$-derivation. Hence we obtain a canonical map of $\mathcal{O}_ X$-modules $\Omega _{X/Y} \to f_*\Omega _{X'/Y'}$, and by adjointness of $f_*$ and $f^*$ a canonical $\mathcal{O}_{X'}$-module homomorphism

\[ c_ f : f^*\Omega _{X/Y} \longrightarrow \Omega _{X'/Y'}. \]

It is uniquely characterized by the property that $f^*\text{d}_{X/Y}(t)$ mapsto $\text{d}_{X'/Y'}(f^* t)$ for any local section $t$ of $\mathcal{O}_ X$.

Proof. This is a special case of Modules on Sites, Lemma 18.33.11. $\square$

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