The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Lemma 10.131.3. Let $R \to S$ be a ring map. Let $M$ be an $S$-module. Let $k \geq 0$. There exists an $S$-module $P^ k_{S/R}(M)$ and a canonical isomorphism

\[ \text{Diff}^ k_{S/R}(M, N) = \mathop{\mathrm{Hom}}\nolimits _ S(P^ k_{S/R}(M), N) \]

functorial in the $S$-module $N$.

Proof. The existence of $P^ k_{S/R}(M)$ follows from general category theoretic arguments (insert future reference here), but we will also give a construction. Set $F = \bigoplus _{m \in M} S[m]$ where $[m]$ is a symbol indicating the basis element in the summand corresponding to $m$. Given any differential operator $D : M \to N$ we obtain an $S$-linear map $L_ D : F \to N$ sending $[m]$ to $D(m)$. If $D$ has order $0$, then $L_ D$ annihilates the elements

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

where $g_0 \in S$ and $m, m' \in M$. If $D$ has order $1$, then $L_ D$ annihilates the elements

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

where $f \in R$, $g_0, g_1 \in S$, and $m \in M$. If $D$ has order $k$, then $L_ D$ annihilates the elements $[m + m'] - [m] - [m']$, $f[m] - [fm]$, and the elements

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

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


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