Lemma 29.32.5. Let $f : X \to S$ be a morphism of schemes. For any pair of affine opens $\mathop{\mathrm{Spec}}(A) = U \subset X$, $\mathop{\mathrm{Spec}}(R) = V \subset S$ with $f(U) \subset V$ there is a unique isomorphism

$\Gamma (U, \Omega _{X/S}) = \Omega _{A/R}.$

compatible with $\text{d}_{X/S}$ and $\text{d} : A \to \Omega _{A/R}$.

Proof. By Lemma 29.32.3 we may replace $X$ and $S$ by $U$ and $V$. Thus we may assume $X = \mathop{\mathrm{Spec}}(A)$ and $S = \mathop{\mathrm{Spec}}(R)$ and we have to show the lemma with $U = X$ and $V = S$. Consider the $A$-module $M = \Gamma (X, \Omega _{X/S})$ together with the $R$-derivation $\text{d}_{X/S} : A \to M$. Let $N$ be another $A$-module and denote $\widetilde{N}$ the quasi-coherent $\mathcal{O}_ X$-module associated to $N$, see Schemes, Section 26.7. Precomposing by $\text{d}_{X/S} : A \to M$ we get an arrow

$\alpha : \mathop{\mathrm{Hom}}\nolimits _ A(M, N) \longrightarrow \text{Der}_ R(A, N)$

Using Lemmas 29.32.2 and 29.32.4 we get identifications

$\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(\Omega _{X/S}, \widetilde{N}) = \text{Der}_ S(\mathcal{O}_ X, \widetilde{N}) = \text{Der}_ R(A, N)$

Taking global sections determines an arrow $\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(\Omega _{X/S}, \widetilde{N}) \to \mathop{\mathrm{Hom}}\nolimits _ R(M, N)$. Combining this arrow and the identifications above we get an arrow

$\beta : \text{Der}_ R(A, N) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _ R(M, N)$

Checking what happens on global sections, we find that $\alpha$ and $\beta$ are each others inverse. Hence we see that $\text{d}_{X/S} : A \to M$ satisfies the same universal property as $\text{d} : A \to \Omega _{A/R}$, see Algebra, Lemma 10.131.3. Thus the Yoneda lemma (Categories, Lemma 4.3.5) implies there is a unique isomorphism of $A$-modules $M \cong \Omega _{A/R}$ compatible with derivations. $\square$

Comment #4957 by Rubén Muñoz--Bertrand on

In order to use Lemma 26.7.1 we need $\widetilde{M}\cong\Omega_{X/S}$, but I believe that we do not know yet that it is quasi-coherent. Actually, the remark before 29.32.7 in Section 29.32 seem to imply we don't know this at this point. Am I missing something?

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