Lemma 20.34.13. With notation and assumptions as in Remark 20.34.12 the diagram

$\xymatrix{ H^ p_ Z(X, K) \ar[r] \ar[d] & H^ p_{Z'}(X, Lf^*K) \ar[d] \\ H^ p(X, K) \ar[r] & H^ p(X', Lf^*K) }$

commutes. Here the top horizontal arrow comes from the identifications $H^ p_ Z(X, K) = H^ p(Z, R\mathcal{H}_ Z(K))$ and $H^ p_{Z'}(X', Lf^*K) = H^ p(Z', R\mathcal{H}_{Z'}(K'))$, the pullback map $H^ p(Z, R\mathcal{H}_ Z(K)) \to H^ p(Z', L(f|_{Z'})^*R\mathcal{H}_ Z(K))$, and the map constructed in Remark 20.34.12.

Proof. Omitted. Hints: Using that $H^ p(Z, R\mathcal{H}_ Z(K)) = H^ p(X, i_*R\mathcal{H}_ Z(K))$ and similarly for $R\mathcal{H}_{Z'}(Lf^*K)$ this follows from the functoriality of the pullback maps and the commutative diagram used to define the map of Remark 20.34.12. $\square$

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