Remark 20.34.10. With notation as in Remark 20.34.9 we obtain a canonical cup product
\begin{align*} H^ a(X, K) \times H^ b_ Z(X, M) & = H^ a(X, K) \times H^ b(Z, R\mathcal{H}_ Z(M)) \\ & \to H^ a(Z, K|_ Z) \times H^ b(Z, R\mathcal{H}_ Z(M)) \\ & \to H^{a + b}(Z, K|_ Z \otimes _{\mathcal{O}_ X|_ Z}^\mathbf {L} R\mathcal{H}_ Z(M)) \\ & \to H^{a + b}(Z, R\mathcal{H}_ Z(K \otimes _{\mathcal{O}_ X}^\mathbf {L} M)) \\ & = H^{a + b}_ Z(X, K \otimes _{\mathcal{O}_ X}^\mathbf {L} M) \end{align*}
Here the equal signs are given by Lemma 20.34.4, the first arrow is restriction to Z, the second arrow is the cup product (Section 20.31), and the third arrow is the map from Remark 20.34.9.
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