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. $\square$

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 09CK. Beware of the difference between the letter 'O' and the digit '0'.