Lemma 29.32.8. Let

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

It is uniquely characterized by the property that $f^*\text{d}_{X/S}(h)$ maps to $\text{d}_{X'/S'}(f^* h)$ for any local section $h$ of $\mathcal{O}_ X$.

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