Lemma 21.39.9 (08QB)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 21: Cohomology on Sites
Section 21.39: Homology on a category
Lemma 21.39.9 (cite)

Lemma 21.39.9. Let $\mathcal{C}_ i$ , $i = 1, 2$ be categories. Let $u_ i : \mathcal{C}_1 \times \mathcal{C}_2 \to \mathcal{C}_ i$ be the projection functors. Let $B$ be a ring. Let $g_ i : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_1 \times \mathcal{C}_2), \underline{B}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ i), \underline{B})$ be the corresponding morphisms of ringed topoi, see Example 21.39.3. For $K_ i \in D(\mathcal{C}_ i, B)$ we have

$L(\pi _1 \times \pi _2)_!( g_1^{-1}K_1 \otimes _{\underline{B}}^\mathbf {L} g_2^{-1}K_2) = L\pi _{1, !}(K_1) \otimes _ B^\mathbf {L} L\pi _{2, !}(K_2)$

in $D(B)$ with obvious notation.

Proof. As both sides commute with colimits, it suffices to prove this for $K_1 = j_{U!}\underline{B}_ U$ and $K_2 = j_{V!}\underline{B}_ V$ for $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}_1)$ and $V \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}_2)$ . See construction of $L\pi _!$ in Lemma 21.37.2. In this case

$g_1^{-1}K_1 \otimes _{\underline{B}}^\mathbf {L} g_2^{-1}K_2 = g_1^{-1}K_1 \otimes _{\underline{B}} g_2^{-1}K_2 = j_{(U, V)!}\underline{B}_{(U, V)}$

Verification omitted. Hence the result follows as both the left and the right hand side of the formula of the lemma evaluate to $B$ , see construction of $L\pi _!$ in Lemma 21.37.2. $\square$

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