The Stacks project

Lemma 18.32.4. Let $\mathcal{C}$ be a site. Let $\varphi : \mathcal{O}_1 \to \mathcal{O}_2$ be a homomorphism of presheaves of rings. 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 (18.32.2.1). There is a similar map of presheaves whose value on $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ 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 10.130.2. On the other hand, the sheaves in (18.32.2.1) are the sheafifications of the presheaves above. Thus the result follows as sheafification is exact. $\square$


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