The Stacks project

Lemma 18.33.3. Let $\mathcal{C}$ be a site. Let $\mathcal{O}_1 \to \mathcal{O}_2$ be a map of sheaves of rings. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_2$-modules. Let $k \geq 0$. There exists a sheaf of $\mathcal{O}_2$-modules $\mathcal{P}^ k_{\mathcal{O}_2/\mathcal{O}_1}(\mathcal{F})$ and a canonical isomorphism

\[ \text{Diff}^ k_{\mathcal{O}_2/\mathcal{O}_1}(\mathcal{F}, \mathcal{G}) = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_2}( \mathcal{P}^ k_{\mathcal{O}_2/\mathcal{O}_1}(\mathcal{F}), \mathcal{G}) \]

functorial in the $\mathcal{O}_2$-module $\mathcal{G}$.

Proof. The existence follows from general category theoretic arguments (insert future reference here), but we will also give a direct construction as this construction will be useful in the future proofs. We will freely use the notation introduced in the proof of Lemma 18.32.2. Given any differential operator $D : \mathcal{F} \to \mathcal{G}$ we obtain an $\mathcal{O}_2$-linear map $L_ D : \mathcal{O}_2[\mathcal{F}] \to \mathcal{G}$ sending $[m]$ to $D(m)$. If $D$ has order $0$ then $L_ D$ annihilates the local sections

\[ [m + m'] - [m] - [m'],\quad g_0[m] - [g_0m] \]

where $g_0$ is a local section of $\mathcal{O}_2$ and $m, m'$ are local sections of $\mathcal{F}$. If $D$ has order $1$, then $L_ D$ annihilates the local sections

\[ [m + m' - [m] - [m'],\quad f[m] - [fm], \quad g_0g_1[m] - g_0[g_1m] - g_1[g_0m] + [g_1g_0m] \]

where $f$ is a local section of $\mathcal{O}_1$, $g_0, g_1$ are local sections of $\mathcal{O}_2$, and $m, m'$ are local sections of $\mathcal{F}$. If $D$ has order $k$, then $L_ D$ annihilates the local sections $[m + m'] - [m] - [m']$, $f[m] - [fm]$, and the local sections

\[ g_0g_1\ldots g_ k[m] - \sum g_0 \ldots \hat g_ i \ldots g_ k[g_ im] + \ldots +(-1)^{k + 1}[g_0\ldots g_ km] \]

Conversely, if $L : \mathcal{O}_2[\mathcal{F}] \to \mathcal{G}$ is an $\mathcal{O}_2$-linear map annihilating all the local sections listed in the previous sentence, then $m \mapsto L([m])$ is a differential operator of order $k$. Thus we see that $\mathcal{P}^ k_{\mathcal{O}_2/\mathcal{O}_1}(\mathcal{F})$ is the quotient of $\mathcal{O}_2[\mathcal{F}]$ by the $\mathcal{O}_2$-submodule generated by these local sections. $\square$


Comments (0)


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Your email address will not be published.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 09CT. Beware of the difference between the letter 'O' and the digit '0'.