Lemma 17.29.5. Let $X$ be a topological space. Let $\mathcal{O}_1 \to \mathcal{O}_2$ be a homomorphism of presheaves of rings on $X$. Let $\mathcal{F}$ be a presheaf of $\mathcal{O}_2$-modules. Then $\mathcal{P}^ k_{\mathcal{O}_2^\# /\mathcal{O}_1^\# }(\mathcal{F}^\# )$ is the sheaf associated to the presheaf $U \mapsto P^ k_{\mathcal{O}_2(U)/\mathcal{O}_1(U)}(\mathcal{F}(U))$.

**Proof.**
This can be proved in exactly the same way as is done for the sheaf of differentials in Lemma 17.28.4. Perhaps a more pleasing approach is to use the universal property of Lemma 17.29.3 directly to see the equality. We omit the details.
$\square$

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