Lemma 21.15.1. Let
\xymatrix{ (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}_{\mathcal{C}'}) \ar[r]_{g'} \ar[d]_{f'} & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \ar[d]^ f \\ (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}'), \mathcal{O}_{\mathcal{D}'}) \ar[r]^ g & (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D}) }
be a commutative diagram of ringed topoi. Let \mathcal{F}^\bullet be a bounded below complex of \mathcal{O}_\mathcal {C}-modules. Assume both g and g' are flat. Then there exists a canonical base change map
g^*Rf_*\mathcal{F}^\bullet \longrightarrow R(f')_*(g')^*\mathcal{F}^\bullet
in D^{+}(\mathcal{O}_{\mathcal{D}'}).
Proof.
Choose injective resolutions \mathcal{F}^\bullet \to \mathcal{I}^\bullet and (g')^*\mathcal{F}^\bullet \to \mathcal{J}^\bullet . By Lemma 21.14.2 we see that (g')_*\mathcal{J}^\bullet is a complex of injectives representing R(g')_*(g')^*\mathcal{F}^\bullet . Hence by Derived Categories, Lemmas 13.18.6 and 13.18.7 the arrow \beta in the diagram
\xymatrix{ (g')_*(g')^*\mathcal{F}^\bullet \ar[r] & (g')_*\mathcal{J}^\bullet \\ \mathcal{F}^\bullet \ar[u]^{adjunction} \ar[r] & \mathcal{I}^\bullet \ar[u]_\beta }
exists and is unique up to homotopy. Pushing down to \mathcal{D} we get
f_*\beta : f_*\mathcal{I}^\bullet \longrightarrow f_*(g')_*\mathcal{J}^\bullet = g_*(f')_*\mathcal{J}^\bullet
By adjunction of g^* and g_* we get a map of complexes g^*f_*\mathcal{I}^\bullet \to (f')_*\mathcal{J}^\bullet . Note that this map is unique up to homotopy since the only choice in the whole process was the choice of the map \beta and everything was done on the level of complexes.
\square
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