The Stacks project

Lemma 90.18.7. Let $\mathcal{C}$ be a site. Let $\mathcal{A} \to \mathcal{B}$ and $\mathcal{A} \to \mathcal{B}'$ be homomorphisms of sheaves of rings on $\mathcal{C}$. Then

\[ L_{\mathcal{B} \times \mathcal{B}'/\mathcal{A}} \longrightarrow L_{\mathcal{B}/\mathcal{A}} \oplus L_{\mathcal{B}'/\mathcal{A}} \]

is an isomorphism in $D(\mathcal{B} \times \mathcal{B}')$.

Proof. By Lemma 90.18.4 it suffices to prove this for ring maps. In the case of rings this is Lemma 90.6.4. $\square$


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