The Stacks project

Lemma 21.39.10. Notation and assumptions as in Example 21.39.1. If there exists a cosimplicial object $U_\bullet $ of $\mathcal{C}$ such that Lemma 21.39.7 applies, then

\[ L\pi _!(K_1 \otimes ^\mathbf {L}_{\underline{B}} K_2) = L\pi _!(K_1) \otimes ^\mathbf {L}_ B L\pi _!(K_2) \]

for all $K_ i \in D(\underline{B})$.

Proof. Consider the diagram of categories and functors

\[ \xymatrix{ & & \mathcal{C} \\ \mathcal{C} \ar[r]^-u & \mathcal{C} \times \mathcal{C} \ar[rd]^{u_2} \ar[ru]_{u_1} \\ & & \mathcal{C} } \]

where $u$ is the diagonal functor and $u_ i$ are the projection functors. This gives morphisms of ringed topoi $g$, $g_1$, $g_2$. For any object $(U_1, U_2)$ of $\mathcal{C}$ we have

\[ \mathop{\mathrm{Mor}}\nolimits _{\mathcal{C} \times \mathcal{C}}(u(U_\bullet ), (U_1, U_2)) = \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(U_\bullet , U_1) \times \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(U_\bullet , U_2) \]

which is homotopy equivalent to a point by Simplicial, Lemma 14.26.10. Thus Lemma 21.39.8 gives $L\pi _!(g^{-1}K) = L(\pi \times \pi )_!(K)$ for any $K$ in $D(\mathcal{C} \times \mathcal{C}, B)$. Take $K = g_1^{-1}K_1 \otimes _ B^\mathbf {L} g_2^{-1}K_2$. Then $g^{-1}K = K_1 \otimes ^\mathbf {L}_{\underline{B}} K_2$ because $g^{-1} = g^* = Lg^*$ commutes with derived tensor product (Lemma 21.18.4). To finish we apply Lemma 21.39.9. $\square$

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