The Stacks project

Lemma 18.33.2. Let $\mathcal{C}$ be a site. Let $\mathcal{O}_1 \to \mathcal{O}_2$ be a map of sheaves of rings. Let $\mathcal{E}, \mathcal{F}, \mathcal{G}$ be sheaves of $\mathcal{O}_2$-modules. If $D : \mathcal{E} \to \mathcal{F}$ and $D' : \mathcal{F} \to \mathcal{G}$ are differential operators of order $k$ and $k'$, then $D' \circ D$ is a differential operator of order $k + k'$.

Proof. Let $g$ be a local section of $\mathcal{O}_2$. Then the map which sends a local section $x$ of $\mathcal{E}$ 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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