The Stacks project

Remark 21.34.12. Suppose that

\[ \xymatrix{ (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}_{\mathcal{C}'}) \ar[r]_ h \ar[d]_{f'} & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \ar[d]^ f \\ (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}'), \mathcal{O}_{\mathcal{D}'}) \ar[r]^ g & (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D}) } \]

is a commutative diagram of ringed topoi. Let $K, L$ be objects of $D(\mathcal{O}_\mathcal {C})$. We claim there exists a canonical base change map

\[ Lg^*Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L) \longrightarrow R(f')_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (Lh^*K, Lh^*L) \]

in $D(\mathcal{O}_{\mathcal{D}'})$. Namely, we take the map adjoint to the composition

\begin{align*} L(f')^*Lg^*Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L) & = Lh^*Lf^*Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L) \\ & \to Lh^*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L) \\ & \to R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (Lh^*K, Lh^*L) \end{align*}

where the first arrow uses the adjunction mapping $Lf^*Rf_* \to \text{id}$ and the second arrow is the canonical map constructed in Remark 21.34.11.


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