The Stacks project

Lemma 50.23.10. Let $c \geq 0$ and

\[ \xymatrix{ Z' \ar[d]_ h \ar[r] & X' \ar[d]_ g \ar[r] & S' \ar[d] \\ Z \ar[r] & X \ar[r] & S } \]

satisfy the assumptions of Lemma 50.23.6 and assume in addition that $X \to S$ and $X' \to S'$ are smooth and that $Z \to X$ and $Z' \to X'$ are Koszul regular immersions. Then the diagram

\[ \xymatrix{ H^ q(Z, \Omega ^ p_{Z/S}) \ar[rr]_-{\gamma ^{p, q}} \ar[d] & & H^{q + c}(X, \Omega ^{p + c}_{X/S}) \ar[d] \\ H^ q(Z', \Omega ^ p_{Z'/S'}) \ar[rr]^{\gamma ^{p, q}} & & H^{q + c}(X', \Omega ^{p + c}_{X'/S'}) } \]

is commutative where $\gamma ^{p, q}$ is as in Remark 50.23.7.

Proof. This follows on combining Lemma 50.23.6 and Cohomology, Lemma 20.34.13. $\square$

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