The Stacks project

Lemma 29.32.8. Let

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

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

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

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 $\mathcal{O}_ X$.

Proof. This is a special case of Modules, Lemma 17.27.12. In the case of schemes we can also use the functoriality of the conormal sheaves (see Lemma 29.31.3) and Lemma 29.32.7 to define $c_ f$. Or we can use the characterization in the last line of the lemma to glue maps defined on affine patches (see Algebra, Equation (10.131.4.1)). $\square$


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