The Stacks project

Lemma 17.27.12. Let

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

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 17.27.6 and 17.27.8 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

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

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 $\mathcal{O}_ X$.

Proof. Omitted. $\square$


Comments (2)

Comment #1836 by Keenan Kidwell on

In the first line after the equation displaying , "mapsto" should be "maps to."

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  • 2 comment(s) on Section 17.27: Modules of differentials

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