Lemma 17.28.7. Let $X$ be a topological space. Let $\mathcal{O}_1 \to \mathcal{O}_2$ be a homomorphism of sheaves of rings on $X$. Let $x \in X$. Then we have $\Omega _{\mathcal{O}_2/\mathcal{O}_1, x} = \Omega _{\mathcal{O}_{2, x}/\mathcal{O}_{1, x}}$.

Proof. This is a special case of Lemma 17.28.6 for the inclusion map $\{ x\} \to X$. An alternative proof is to use Lemma 17.28.4, Sheaves, Lemma 6.17.2, and Algebra, Lemma 10.131.5 $\square$

Comment #3610 by Herman Rohrbach on

Typo in the second sentence of the proof: "An alternative proof is to (instead of the) use Lemma..."

Comment #8564 by on

In the statement of the theorem, I think one could write "then we have an isomorphism $\Omega_{\mathcal{O}_2/\mathcal{O}_1,x}\cong\Omega_{\mathcal{O}_{2,x}/\mathcal{O}_{1,x}}$ compatible with universal derivations." (I.e., that $\varphi\circ \mathrm{d}_{\mathcal{O}_2/\mathcal{O}_1,x}=\mathrm{d}_{\mathcal{O}_{2,x}/\mathcal{O}_{1,x}}$, where $\varphi$ is the isomorphism.) The proof doesn't change, for Lemma 17.28.6 guarantees this on its statement.

Comment #8708 by on

Also, maybe it is worth adding to the statement that “taking germ,” $\Gamma(X,\Omega_{\mathcal{O}_2/\mathcal{O}_1})\to\Omega_{\mathcal{O}_{2,x}/\mathcal{O}_{1,x}}$, is compatible with universal derivations? (Symbolically, $(dt)_x=d(t_x)$). This follows from the compatibility with universal derivations either coming from Lemma 17.28.6 ($f^{-1}d_{\mathcal{O}_2/\mathcal{O}_1}\cong d_{f^{-1}\mathcal{O}_2/f^{-1}\mathcal{O}_1})$ applied to the inclusion $\{x\}\to X$ or possibly from Lemma 10.131.5 (where I guess it holds $d_{R/S}=\operatorname{colim}_i d_{R_i/S_i}$).

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