Lemma 92.18.4. Let \mathcal{C} be a site. Let \mathcal{A} \to \mathcal{B} be a homomorphism of sheaves of rings on \mathcal{C}. Then H^ i(L_{\mathcal{B}/\mathcal{A}}) is the sheaf associated to the presheaf U \mapsto H^ i(L_{\mathcal{B}(U)/\mathcal{A}(U)}).
Proof. Let \mathcal{C}' be the site we get by endowing \mathcal{C} with the chaotic topology (presheaves are sheaves). There is a morphism of topoi f : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}') where f_* is the inclusion of sheaves into presheaves and f^{-1} is sheafification. By Lemma 92.18.3 it suffices to prove the result for \mathcal{C}', i.e., in case \mathcal{C} has the chaotic topology.
If \mathcal{C} carries the chaotic topology, then L_{\mathcal{B}/\mathcal{A}}(U) is equal to L_{\mathcal{B}(U)/\mathcal{A}(U)} because
\xymatrix{ \mathcal{A}\textit{-Alg} \ar[d]_{\text{sections over }U} \ar[r] & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \ar@<1ex>[l] \ar[d]^{\text{sections over }U} \\ \mathcal{A}(U)\textit{-Alg} \ar[r] & \textit{Sets} \ar@<1ex>[l] }
commutes. \square
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