Remark 21.19.3 (07A7)—The Stacks project

Remark 21.19.3. The construction of unbounded derived functor $Lf^*$ and $Rf_*$ allows one to construct the base change map in full generality. Namely, suppose that

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

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 $Lf^*Rf_*K \to K$.

Comments (2)

Comment #9910 by Eduardo de Lorenzo Poza on December 31, 2024 at 11:25

Here we see one natural way to construct a morphism $Lg^\* Rf\_\* K \to R(f')\_\* L(g')^\* K$ . There is another way, which is just as natural: consider the adjunction map $K \to R(g')\_\* L(g')^\*K$ , apply $Rf\_\*$ to get $Rf\_\* K \to Rf\_\* R(g')\_\* L(g')^\*K = Rg\_\* R(f')\_\* L(g')^\*$ and by adjunction we obtain the desired morphism $Lg^\* Rf\_\* K \to R(f')\_\* L(g')^\* K$ . I supposr that these two morphisms obtained in two different ways are actually the same. Is there an "easy" way to see this? Maybe as some sort of universal property?

Comment #9911 by Eduardo de Lorenzo Poza on December 31, 2024 at 11:28

Here we see one natural way to construct a morphism $Lg^* Rf_* K \to R(f')_* L(g')^* K$ . There is another way, which is just as natural: consider the adjunction map $K \to R(g')_* L(g')^*K$ , apply $Rf_*$ to get $Rf_* K \to Rf_* R(g')_* L(g')^*K = Rg_* R(f')_* L(g')^*$ and by adjunction we obtain the desired morphism $Lg^* Rf_* K \to R(f')_* L(g')^* K$ . I suppose that these two morphisms obtained in two different ways are actually the same. Is there an "easy" way to see this? Maybe as some sort of universal property?

Sorry about the duplicate comment, markdown was messing with formatting in the preview.

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