The Stacks project

Lemma 17.28.12. Let

be a commutative diagram of ringed spaces.

1. The canonical map $\mathcal{O}_ X \to f_*\mathcal{O}_{X'}$ composed with $f_*\text{d}_{X'/S'} : f_*\mathcal{O}_{X'} \to f_*\Omega _{X'/S'}$ is a $S$-derivation and we obtain a canonical map of $\mathcal{O}_ X$-modules $\Omega _{X/S} \to f_*\Omega _{X'/S'}$.

2. The commutative diagram

   \[ \xymatrix{ f^{-1}\mathcal{O}_ X \ar[r] & \mathcal{O}_{X'} \\ f^{-1}h^{-1}\mathcal{O}_ S \ar[u] \ar[r] & (h')^{-1}\mathcal{O}_{S'} \ar[u] } \]

   induces by Lemmas 17.28.6 and 17.28.8 a canonical map $f^{-1}\Omega _{X/S} \to \Omega _{X'/S'}$.

These two maps correspond (via adjointness of $f_*$ and $f^*$ and via $f^*\Omega _{X/S} = f^{-1}\Omega _{X/S} \otimes _{f^{-1}\mathcal{O}_ X} \mathcal{O}_{X'}$ and Sheaves, Lemma 6.20.2) to the same $\mathcal{O}_{X'}$-module homomorphism

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