Lemma 17.29.3. Let $X$ be a topological space. Let $\mathcal{O}_1 \to \mathcal{O}_2$ be a map of sheaves of rings on $X$. Let $\mathcal{F}$ 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 17.28.2. 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.
$\square$
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