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
$\square$

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