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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