Lemma 10.133.2. Let $R \to S$ be a ring map. Let $L, M, N$ be $S$-modules. If $D : L \to M$ and $D' : M \to N$ are differential operators of order $k$ and $k'$, then $D' \circ D$ is a differential operator of order $k + k'$.

**Proof.**
Let $g \in S$. Then the map which sends $x \in L$ to

\[ D'(D(gx)) - gD'(D(x)) = D'(D(gx)) - D'(gD(x)) + D'(gD(x)) - gD'(D(x)) \]

is a sum of two compositions of differential operators of lower order. Hence the lemma follows by induction on $k + k'$. $\square$

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