Lemma 17.27.2. Let $X$ be a topological space. Let $\varphi : \mathcal{O}_1 \to \mathcal{O}_2$ be a homomorphism of sheaves of rings. The functor

is representable.

Lemma 17.27.2. Let $X$ be a topological space. Let $\varphi : \mathcal{O}_1 \to \mathcal{O}_2$ be a homomorphism of sheaves of rings. The functor

\[ \textit{Mod}(\mathcal{O}_2) \longrightarrow \textit{Ab}, \quad \mathcal{F} \longmapsto \text{Der}_{\mathcal{O}_1}(\mathcal{O}_2, \mathcal{F}) \]

is representable.

**Proof.**
This is proved in exactly the same way as the analogous statement in algebra. During this proof, for any sheaf of sets $\mathcal{F}$ on $X$, let us denote $\mathcal{O}_2[\mathcal{F}]$ the sheafification of the presheaf $U \mapsto \mathcal{O}_2(U)[\mathcal{F}(U)]$ where this denotes the free $\mathcal{O}_2(U)$-module on the set $\mathcal{F}(U)$. For $s \in \mathcal{F}(U)$ we denote $[s]$ the corresponding section of $\mathcal{O}_2[\mathcal{F}]$ over $U$. If $\mathcal{F}$ is a sheaf of $\mathcal{O}_2$-modules, then there is a canonical map

\[ c : \mathcal{O}_2[\mathcal{F}] \longrightarrow \mathcal{F} \]

which on the presheaf level is given by the rule $\sum f_ s[s] \mapsto \sum f_ s s$. We will employ the short hand $[s] \mapsto s$ to describe this map and similarly for other maps below. Consider the map of $\mathcal{O}_2$-modules

17.27.2.1

\begin{equation} \label{modules-equation-define-module-differentials} \begin{matrix} \mathcal{O}_2[\mathcal{O}_2 \times \mathcal{O}_2] \oplus \mathcal{O}_2[\mathcal{O}_2 \times \mathcal{O}_2] \oplus \mathcal{O}_2[\mathcal{O}_1]
& \longrightarrow
& \mathcal{O}_2[\mathcal{O}_2]
\\ [(a, b)] \oplus [(f, g)] \oplus [h]
& \longmapsto
& [a + b] - [a] - [b] +
\\ & & [fg] - g[f] - f[g] +
\\ & & [\varphi (h)]
\end{matrix} \end{equation}

with short hand notation as above. Set $\Omega _{\mathcal{O}_2/\mathcal{O}_1}$ equal to the cokernel of this map. Then it is clear that there exists a map of sheaves of sets

\[ \text{d} : \mathcal{O}_2 \longrightarrow \Omega _{\mathcal{O}_2/\mathcal{O}_1} \]

mapping a local section $f$ to the image of $[f]$ in $\Omega _{\mathcal{O}_2/\mathcal{O}_1}$. By construction $\text{d}$ is a $\mathcal{O}_1$-derivation. Next, let $\mathcal{F}$ be a sheaf of $\mathcal{O}_2$-modules and let $D : \mathcal{O}_2 \to \mathcal{F}$ be a $\mathcal{O}_1$-derivation. Then we can consider the $\mathcal{O}_2$-linear map $\mathcal{O}_2[\mathcal{O}_2] \to \mathcal{F}$ which sends $[g]$ to $D(g)$. It follows from the definition of a derivation that this map annihilates sections in the image of the map (17.27.2.1) and hence defines a map

\[ \alpha _ D : \Omega _{\mathcal{O}_2/\mathcal{O}_1} \longrightarrow \mathcal{F} \]

Since it is clear that $D = \alpha _ D \circ \text{d}$ the lemma is proved. $\square$

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.

## Comments (0)

There are also: