# The Stacks Project

## Tag 07A7

Remark 21.20.3. The construction of unbounded derived functor $Lf^*$ and $Rf_*$ allows one to construct the base change map in full generality. Namely, suppose that $$\xymatrix{ (\mathop{\textit{Sh}}\nolimits(\mathcal{C}'), \mathcal{O}_{\mathcal{C}'}) \ar[r]_{g'} \ar[d]_{f'} & (\mathop{\textit{Sh}}\nolimits(\mathcal{C}), \mathcal{O}_\mathcal{C}) \ar[d]^f \\ (\mathop{\textit{Sh}}\nolimits(\mathcal{D}'), \mathcal{O}_{\mathcal{D}'}) \ar[r]^g & (\mathop{\textit{Sh}}\nolimits(\mathcal{D}), \mathcal{O}_\mathcal{D}) }$$ is a commutative diagram of ringed topoi. Let $K$ be an object of $D(\mathcal{O}_\mathcal{C})$. Then there exists a canonical base change map $$Lg^*Rf_*K \longrightarrow R(f')_*L(g')^*K$$ in $D(\mathcal{O}_{\mathcal{D}'})$. Namely, this map is adjoint to a map $L(f')^*Lg^*Rf_*K \to L(g')^*K$. Since $L(f')^* \circ Lg^* = L(g')^* \circ Lf^*$ we see this is the same as a map $L(g')^*Lf^*Rf_*K \to L(g')^*K$ which we can take to be $L(g')^*$ of the adjunction map $Lf^*Rf_*K \to K$.

The code snippet corresponding to this tag is a part of the file sites-cohomology.tex and is located in lines 3473–3500 (see updates for more information).

\begin{remark}
\label{remark-base-change}
The construction of unbounded derived functor $Lf^*$ and $Rf_*$
allows one to construct the base change map in full generality.
Namely, suppose that
$$\xymatrix{ (\Sh(\mathcal{C}'), \mathcal{O}_{\mathcal{C}'}) \ar[r]_{g'} \ar[d]_{f'} & (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \ar[d]^f \\ (\Sh(\mathcal{D}'), \mathcal{O}_{\mathcal{D}'}) \ar[r]^g & (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D}) }$$
is a commutative diagram of ringed topoi. Let $K$ be an object of
$D(\mathcal{O}_\mathcal{C})$.
Then there exists a canonical base change map
$$Lg^*Rf_*K \longrightarrow R(f')_*L(g')^*K$$
in $D(\mathcal{O}_{\mathcal{D}'})$. Namely, this map is adjoint to a map
$L(f')^*Lg^*Rf_*K \to L(g')^*K$.
Since $L(f')^* \circ Lg^* = L(g')^* \circ Lf^*$ we see this is the same
as a map $L(g')^*Lf^*Rf_*K \to L(g')^*K$
which we can take to be $L(g')^*$ of the adjunction map
$Lf^*Rf_*K \to K$.
\end{remark}

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