Lemma 10.133.8. Let $A \to B$ be a ring map. The differentials $\text{d} : \Omega ^ i_{B/A} \to \Omega ^{i + 1}_{B/A}$ are differential operators of order $1$.

**Proof.**
Given $b \in B$ we have to show that $\text{d} \circ b - b \circ \text{d}$ is a linear operator. Thus we have to show that

To see this it suffices to check this on additive generators for $\Omega ^ i_{B/A}$. Thus it suffices to show that

is zero. This is a pleasant calculation using the Leibniz rule which is left to the reader. $\square$

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