Lemma 63.8.5. Let $b : Y_1 \to Y$ be a morphism of schemes. Consider a commutative diagram of schemes
\[ \vcenter { \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^ f \\ Y' \ar[r]^ g & Y } } \quad \text{and let}\quad \vcenter { \xymatrix{ X'_1 \ar[r]_{g'_1} \ar[d]_{f'_1} & X_1 \ar[d]^{f_1} \\ Y'_1 \ar[r]^{g_1} & Y_1 } } \]
be the base change by $b$. Assume $f$ and $f'$ proper and $g$ and $g'$ separated and locally quasi-finite. For a ring $\Lambda $ and $K$ in $D(X'_{\acute{e}tale}, \Lambda )$ there is commutative diagram
\[ \xymatrix{ b^{-1}g_!Rf'_*K \ar[d] \ar[r] & g_{1, !}(b')^{-1}Rf'_*K \ar[r] & g_{1, !}Rf'_{1, *}(a')^{-1}K \ar[d] \\ b^{-1}Rf_*g'_!K \ar[r] & Rf_{1, *}a^{-1}g'_!K \ar[r] & Rf_{1, *}g'_{1, !}(a')^{-1}K } \]
in $D(Y_{1, {\acute{e}tale}}, \Lambda )$ where $a : X_1 \to X$, $a' : X'_1 \to X'$, $b' : Y'_1 \to Y'$ are the projections, the vertical maps are the arrows of Lemma 63.8.1 and the horizontal arrows are the base change map (from Étale Cohomology, Section 59.86) and the base change map of Lemma 63.3.12.
Proof.
Represent $K$ by a K-injective complex $\mathcal{J}^\bullet $ of sheaves of $\Lambda $-modules on $X'_{\acute{e}tale}$. Choose a quasi-isomorphism $g'_!\mathcal{J}^\bullet \to \mathcal{I}^\bullet $ to a K-injective complex $\mathcal{I}^\bullet $ of sheaves of $\Lambda $-modules on $X_{\acute{e}tale}$. The proof of Lemma 63.8.1 constructs $g_!Rf'_*K \to Rf_*g'_!K$ as
\[ g_!f'_*\mathcal{J}^\bullet = g_!f'_!\mathcal{J}^\bullet = f_!g'_!\mathcal{J}^\bullet = f_*g'_!\mathcal{J}^\bullet \to f_*\mathcal{I}^\bullet \]
Choose a quasi-isomorphism $(a')^{-1}\mathcal{J}^\bullet \to \mathcal{J}_1^\bullet $ to a K-injective complex $\mathcal{J}_1^\bullet $ of sheaves of $\Lambda $-modules on $X'_{1, {\acute{e}tale}}$. Then we can pick a diagram of complexes
\[ \xymatrix{ g'_{1, !}\mathcal{J}_1^\bullet \ar[rr] & & \mathcal{I}_1^\bullet \\ g'_{1, !}(a')^{-1}\mathcal{J}^\bullet \ar[u] \ar@{=}[r] & a^{-1}g'_!\mathcal{J}^\bullet \ar[r] & a^{-1}\mathcal{I}^\bullet \ar[u] } \]
commuting up to homotopy where all arrows are quasi-isomorphisms, the equality comes from Lemma 63.3.4, and $\mathcal{I}_1^\bullet $ is a K-injective complex of sheaves of $\Lambda $-modules on $X_{1, {\acute{e}tale}}$. The map $g_{1, !}Rf'_{1, *}(a')^{-1}K \to Rf_{1, *}g'_{1, !}(a')^{-1}K$ is given by
\[ g_{1, !}f'_{1, *}\mathcal{J}_1^\bullet = g_{1, !}f'_{1, !}\mathcal{J}_1^\bullet = f_{1, !}g'_{1, !}\mathcal{J}_1^\bullet = f_{1, *}g'_{1, !}\mathcal{J}_1^\bullet \to f_{1, *}\mathcal{I}_1^\bullet \]
The identifications across the $3$ equal signs in both arrows are compatible with pullback maps, i.e., the diagram
\[ \xymatrix{ b^{-1}g_!f'_*\mathcal{J}^\bullet \ar@{=}[d] \ar[r] & g_{1, !}(b')^{-1}f'_*\mathcal{J}^\bullet \ar[r] & g_{1, !}f'_{1, *}(a')^{-1}\mathcal{J}^\bullet \ar@{=}[d] \\ b^{-1}f_*g'_!\mathcal{J}^\bullet \ar[r] & f_{1, *}a^{-1}g'_!\mathcal{J}^\bullet \ar[r] & f_{1, *}g'_{1, !}(a')^{-1}\mathcal{J}^\bullet } \]
of complexes of abelian sheaves commutes. To show this it is enough to show the diagram commutes with $g_!, g_{1, !}, g'_!, g'_{1, !}$ replaced by $g_*, g_{1, *}, g'_*, g'_{1, *}$ (because the shriek functors are defined as subfunctors of the $*$ functors and the base change maps are defined in a manner compatible with this, see proof of Lemma 63.3.12). For this new diagram the commutativity follows from the compatibility of pullback maps with horizontal and vertical stacking of diagrams, see Sites, Remarks 7.45.3 and 7.45.4 so that going around the diagram in either direction is the pullback map for the base change of $f \circ g' = g \circ f'$ by $b$. Since of course
\[ \xymatrix{ g_{1, !}f'_{1, *}(a')^{-1}\mathcal{J}^\bullet \ar@{=}[d] \ar[r] & g_{1, !}f'_{1, *}\mathcal{J}_1^\bullet \ar@{=}[d] \\ f_{1, *}g'_{1, !}(a')^{-1}\mathcal{J}^\bullet \ar[r] & f_{1, *}g'_{1, !}\mathcal{J}_1^\bullet } \]
commutes, to finish the proof it suffices to show that
\[ \xymatrix{ b^{-1}f_*g'_!\mathcal{J}^\bullet \ar[r] \ar[d] & f_{1, *}a^{-1}g'_!\mathcal{J}^\bullet \ar[r] \ar[d] & f_{1, *}g'_{1, !}(a')^{-1}\mathcal{J}^\bullet \ar[r] & f_{1, *}g'_{1, !}\mathcal{J}_1^\bullet \ar[d] \\ b^{-1}f_*\mathcal{I}^\bullet \ar[r] & f_{1, *}a^{-1}\mathcal{I}^\bullet \ar[rr] & & f_{1, *}\mathcal{I}_1^\bullet } \]
commutes in the derived category, which holds by our choice of maps earlier.
$\square$
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