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Tag 07A7

Chapter 21: Cohomology on Sites > Section 21.20: Cohomology of unbounded complexes

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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