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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