The Stacks project

Lemma 17.28.2. 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
\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 ( 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. $\square$

Comments (2)

Comment #8560 by on

How does one argue uniqueness of ? I guess it is because plus facts (i) is -linear and (ii) the image (pre)sheaf of generates as an -module (in the sense of Definition 17.4.5). Does this sound right?

Also, I think it might be interesting to point out that is the left adjoint of the forgetful functor , as it is easy to show (this is actually how one obtains from in the third to last sentence of the proof).

There are also:

  • 2 comment(s) on Section 17.28: Modules of differentials

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