Lemma 50.9.1. Given a commutative diagram

\[ \xymatrix{ X' \ar[r]_ f \ar[d] & X \ar[d] \\ S' \ar[r] & S } \]

of schemes the diagrams

\[ \xymatrix{ \mathop{\mathrm{Pic}}\nolimits (X') \ar[d]_{c_1^{dR}} & \mathop{\mathrm{Pic}}\nolimits (X) \ar[d]^{c_1^{dR}} \ar[l]^{f^*} \\ H^2_{dR}(X'/S') & H^2_{dR}(X/S) \ar[l]_{f^*} } \quad \xymatrix{ \mathop{\mathrm{Pic}}\nolimits (X') \ar[d]_{c_1^{Hodge}} & \mathop{\mathrm{Pic}}\nolimits (X) \ar[d]^{c_1^{Hodge}} \ar[l]^{f^*} \\ H^1(X', \Omega ^1_{X'/S'}) & H^1(X, \Omega ^1_{X/S}) \ar[l]_{f^*} } \]

commute.

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