Lemma 10.133.9. Let $A \to B$ be a ring map. Let $g_ i \in B$, $i \in I$ be a set of generators for $B$ as an $A$-algebra. Let $M, N$ be $B$-modules. Let $D : M \to N$ be an $A$-linear map. In order to show that $D$ is a differential operator of order $k$ it suffices to show that $D \circ g_ i - g_ i \circ D$ is a differential operator of order $k - 1$ for $i \in I$.
Proof. Namely, we claim that the set of elements $g \in B$ such that $D \circ g - g \circ D$ is a differential operator of order $k - 1$ is an $A$-subalgebra of $B$. This follows from the relations
\[ D \circ (g + g') - (g + g') \circ D = (D \circ g - g \circ D) + (D \circ g' - g' \circ D) \]
and
\[ D \circ gg' - gg' \circ D = (D \circ g - g \circ D) \circ g' + g \circ (D \circ g' - g' \circ D) \]
Strictly speaking, to conclude for products we also use Lemma 10.133.2. $\square$
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