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 (17.28.2.1). 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 (17.28.2.1) are the sheafifications of the presheaves above. Thus the result follows as sheafification is exact. $\square$

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