The Stacks project

Lemma 17.29.2. Let $X$ be a topological space. Let $\mathcal{O}_1 \to \mathcal{O}_2$ be a map of sheaves of rings on $X$. 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 $k + k'$. $\square$

Comments (0)

There are also:

  • 1 comment(s) on Section 17.29: Finite order differential operators

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0G3R. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 17.29.2 as pdf