The Stacks project

Lemma 17.28.4. Let $X$ be a topological space. Let $\varphi : \mathcal{O}_1 \to \mathcal{O}_2$ be a homomorphism of sheaves of rings on $X$. Then $\Omega _{\mathcal{O}_2/\mathcal{O}_1}$ is the sheaf associated to the presheaf $U \mapsto \Omega _{\mathcal{O}_2(U)/\mathcal{O}_1(U)}$.

Proof. Consider the map ( There is a similar map of presheaves whose value on the open $U$ is

\[ \mathcal{O}_2(U)[\mathcal{O}_2(U) \times \mathcal{O}_2(U)] \oplus \mathcal{O}_2(U)[\mathcal{O}_2(U) \times \mathcal{O}_2(U)] \oplus \mathcal{O}_2(U)[\mathcal{O}_1(U)] \longrightarrow \mathcal{O}_2(U)[\mathcal{O}_2(U)] \]

The cokernel of this map has value $\Omega _{\mathcal{O}_2(U)/\mathcal{O}_1(U)}$ over $U$ by the construction of the module of differentials in Algebra, Definition 10.131.2. On the other hand, the sheaves in ( are the sheafifications of the presheaves above. Thus the result follows as sheafification is exact. $\square$

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