Lemma 20.34.11. With notation as in Remark 20.34.9 the diagram

$\xymatrix{ H^ i(X, K) \times H^ j_ Z(X, M) \ar[r] \ar[d] & H^{i + j}_ Z(X, K \otimes _{\mathcal{O}_ X}^\mathbf {L} M) \ar[d] \\ H^ i(X, K) \times H^ j(X, M) \ar[r] & H^{i + j}(X, K \otimes _{\mathcal{O}_ X}^\mathbf {L} M) }$

commutes where the top horizontal arrow is the cup product of Remark 20.34.10.

Proof. Omitted. $\square$

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