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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