Lemma 59.86.1. Consider a cartesian diagram of schemes
\[ \xymatrix{ X \ar[d]_ f & Y \ar[l]^ h \ar[d]^ e \\ S & T \ar[l]_ g } \]
Let $\{ U_ i \to X\} $ be an étale covering such that $U_ i \to S$ factors as $U_ i \to V_ i \to S$ with $V_ i \to S$ étale and consider the cartesian diagrams
\[ \xymatrix{ U_ i \ar[d]_{f_ i} & U_ i \times _ X Y \ar[l]^{h_ i} \ar[d]^{e_ i} \\ V_ i & V_ i \times _ S T \ar[l]_{g_ i} } \]
Let $\mathcal{F}$ be a sheaf on $T_{\acute{e}tale}$. Let $K$ in $D(T_{\acute{e}tale})$. Set $K_ i = K|_{V_ i \times _ S T}$ and $\mathcal{F}_ i = \mathcal{F}|_{V_ i \times _ S T}$.
If $f_ i^{-1}g_{i, *}\mathcal{F}_ i = h_{i, *}e_ i^{-1}\mathcal{F}_ i$ for all $i$, then $f^{-1}g_*\mathcal{F} = h_*e^{-1}\mathcal{F}$.
If $f_ i^{-1}Rg_{i, *}K_ i = Rh_{i, *}e_ i^{-1}K_ i$ for all $i$, then $f^{-1}Rg_*K = Rh_*e^{-1}K$.
If $\mathcal{F}$ is an abelian sheaf and $f_ i^{-1}R^ qg_{i, *}\mathcal{F}_ i = R^ qh_{i, *}e_ i^{-1}\mathcal{F}_ i$ for all $i$, then $f^{-1}R^ qg_*\mathcal{F} = R^ qh_*e^{-1}\mathcal{F}$.
Proof.
Proof of (1). First we observe that
\[ (f^{-1}g_*\mathcal{F})|_{U_ i} = f_ i^{-1}(g_*\mathcal{F}|_{V_ i}) = f_ i^{-1}g_{i, *}\mathcal{F}_ i \]
The first equality because $U_ i \to X \to S$ is equal to $U_ i \to V_ i \to S$ and the second equality because $g_*\mathcal{F}|_{V_ i} = g_{i, *}\mathcal{F}_ i$ by Sites, Lemma 7.28.2. Similarly we have
\[ (h_*e^{-1}\mathcal{F})|_{U_ i} = h_{i, *}(e^{-1}\mathcal{F}|_{U_ i \times _ X Y}) = h_{i, *}e_ i^{-1}\mathcal{F}_ i \]
Thus if the base change maps $f_ i^{-1}g_{i, *}\mathcal{F}_ i \to h_{i, *}e_ i^{-1}\mathcal{F}_ i$ are isomorphisms for all $i$, then the base change map $f^{-1}g_*\mathcal{F} \to h_*e^{-1}\mathcal{F}$ restricts to an isomorphism over $U_ i$ for all $i$ and we conclude it is an isomorphism as $\{ U_ i \to X\} $ is an étale covering.
For the other two statements we replace the appeal to Sites, Lemma 7.28.2 by an appeal to Cohomology on Sites, Lemma 21.20.4.
$\square$
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