Definition 18.34.1. Let $\mathcal{C}$ be a site. Let $\varphi : \mathcal{O}_1 \to \mathcal{O}_2$ be a homomorphism of sheaves of rings. Let $k \geq 0$ be an integer. Let $\mathcal{F}$, $\mathcal{G}$ be sheaves of $\mathcal{O}_2$-modules. A differential operator $D : \mathcal{F} \to \mathcal{G}$ of order $k$ is an is an $\mathcal{O}_1$-linear map such that for all local sections $g$ of $\mathcal{O}_2$ the map $s \mapsto D(gs) - gD(s)$ is a differential operator of order $k - 1$. For the base case $k = 0$ we define a differential operator of order $0$ to be an $\mathcal{O}_2$-linear map.
18.34 Finite order differential operators
In this section we introduce differential operators of finite order. We suggest the reader take a look at the corresponding section in the chapter on commutative algebra (Algebra, Section 10.133).
If $D : \mathcal{F} \to \mathcal{G}$ is a differential operator of order $k$, then for all local sections $g$ of $\mathcal{O}_2$ the map $gD$ is a differential operator of order $k$. The sum of two differential operators of order $k$ is another. Hence the set of all these
\[ \text{Diff}^ k(\mathcal{F}, \mathcal{G}) = \text{Diff}^ k_{\mathcal{O}_2/\mathcal{O}_1}(\mathcal{F}, \mathcal{G}) \]
is a $\Gamma (\mathcal{C}, \mathcal{O}_2)$-module. We have
\[ \text{Diff}^0(\mathcal{F}, \mathcal{G}) \subset \text{Diff}^1(\mathcal{F}, \mathcal{G}) \subset \text{Diff}^2(\mathcal{F}, \mathcal{G}) \subset \ldots \]
The rule which maps $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ to the module of differential operators $D : \mathcal{F}|_ U \to \mathcal{G}|_ U$ of order $k$ is a sheaf of $\mathcal{O}_2$-modules on the site $\mathcal{C}$. Thus we obtain a sheaf of differential operators (if we ever need this we will add a definition here).
Lemma 18.34.2. Let $\mathcal{C}$ be a site. Let $\mathcal{O}_1 \to \mathcal{O}_2$ be a map of sheaves of rings. 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 $k + k'$. $\square$
Lemma 18.34.3. 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 functorial in the $\mathcal{O}_2$-module $\mathcal{G}$.
\[ \text{Diff}^ k_{\mathcal{O}_2/\mathcal{O}_1}(\mathcal{F}, \mathcal{G}) = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_2}( \mathcal{P}^ k_{\mathcal{O}_2/\mathcal{O}_1}(\mathcal{F}), \mathcal{G}) \]
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 18.33.2. 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. $\square$
Definition 18.34.4. 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. The module $\mathcal{P}^ k_{\mathcal{O}_2/\mathcal{O}_1}(\mathcal{F})$ constructed in Lemma 18.34.3 is called the module of principal parts of order $k$ of $\mathcal{F}$.
Note that the inclusions
\[ \text{Diff}^0(\mathcal{F}, \mathcal{G}) \subset \text{Diff}^1(\mathcal{F}, \mathcal{G}) \subset \text{Diff}^2(\mathcal{F}, \mathcal{G}) \subset \ldots \]
correspond via Yoneda's lemma (Categories, Lemma 4.3.5) to surjections
\[ \ldots \to \mathcal{P}^2_{\mathcal{O}_2/\mathcal{O}_1}(\mathcal{F}) \to \mathcal{P}^1_{\mathcal{O}_2/\mathcal{O}_1}(\mathcal{F}) \to \mathcal{P}^0_{\mathcal{O}_2/\mathcal{O}_1}(\mathcal{F}) = \mathcal{F} \]
Lemma 18.34.5. Let $\mathcal{C}$ be a site. Let $\mathcal{O}_1 \to \mathcal{O}_2$ be a homomorphism of presheaves of rings. Let $\mathcal{F}$ be a presheaf of $\mathcal{O}_2$-modules. Then $\mathcal{P}^ k_{\mathcal{O}_2^\# /\mathcal{O}_1^\# }(\mathcal{F}^\# )$ is the sheaf associated to the presheaf $U \mapsto P^ k_{\mathcal{O}_2(U)/\mathcal{O}_1(U)}(\mathcal{F}(U))$.
Proof. This can be proved in exactly the same way as is done for the sheaf of differentials in Lemma 18.33.4. Perhaps a more pleasing approach is to use the universal property of Lemma 18.34.3 directly to see the equality. We omit the details. $\square$
Lemma 18.34.6. Let $\mathcal{C}$ be a site. Let $\mathcal{O}_1 \to \mathcal{O}_2$ be a homomorphism of sheaves of rings. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_2$-modules. There is a canonical short exact sequence functorial in $\mathcal{F}$ called the sequence of principal parts.
\[ 0 \to \Omega _{\mathcal{O}_2/\mathcal{O}_1} \otimes _{\mathcal{O}_2} \mathcal{F} \to \mathcal{P}^1_{\mathcal{O}_2/\mathcal{O}_1}(\mathcal{F}) \to \mathcal{F} \to 0 \]
Proof. Follows from the commutative algebra version (Algebra, Lemma 10.133.6) and Lemmas 18.33.4 and 18.34.5. $\square$
Remark 18.34.7. Let $\mathcal{C}$ be a site. Suppose given a commutative diagram of sheaves of rings a $\mathcal{B}$-module $\mathcal{F}$, a $\mathcal{B}'$-module $\mathcal{F}'$, and a $\mathcal{B}$-linear map $\mathcal{F} \to \mathcal{F}'$. Then we get a compatible system of module maps These maps are compatible with further composition of maps of this type. The easiest way to see this is to use the description of the modules $\mathcal{P}^ k_{\mathcal{B}/\mathcal{A}}(\mathcal{M})$ in terms of (local) generators and relations in the proof of Lemma 18.34.3 but it can also be seen directly from the universal property of these modules. Moreover, these maps are compatible with the short exact sequences of Lemma 18.34.6.
\[ \xymatrix{ \mathcal{B} \ar[r] & \mathcal{B}' \\ \mathcal{A} \ar[u] \ar[r] & \mathcal{A}' \ar[u] } \]
\[ \xymatrix{ \ldots \ar[r] & \mathcal{P}^2_{\mathcal{B}'/\mathcal{A}'}(\mathcal{F}') \ar[r] & \mathcal{P}^1_{\mathcal{B}'/\mathcal{A}'}(\mathcal{F}') \ar[r] & \mathcal{P}^0_{\mathcal{B}'/\mathcal{A}'}(\mathcal{F}') \\ \ldots \ar[r] & \mathcal{P}^2_{\mathcal{B}/\mathcal{A}}(\mathcal{F}) \ar[r] \ar[u] & \mathcal{P}^1_{\mathcal{B}/\mathcal{A}}(\mathcal{F}) \ar[r] \ar[u] & \mathcal{P}^0_{\mathcal{B}/\mathcal{A}}(\mathcal{F}) \ar[u] } \]
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