Remark 37.9.11. Lemma 37.9.10 can be improved in the following way. Suppose that we have commutative diagrams as in Lemma 37.9.10 but we do not assume that $X_2 \to X_1$ and $S_2 \to S_1$ are étale. Next, suppose we have $\theta _1 : a_1^*\Omega _{X_1/S_1} \to \mathcal{C}_{T_1/T'_1}$ and $\theta _2 : a_2^*\Omega _{X_2/S_2} \to \mathcal{C}_{T_2/T'_2}$ such that

$\xymatrix{ f_*\mathcal{O}_{X_2} \ar[rr]_{f_*D_2} & & f_*a_{2, *}\mathcal{C}_{T_2/T_2'} \\ \mathcal{O}_{X_1} \ar[rr]^{D_1} \ar[u]^{f^\sharp } & & a_{1, *}\mathcal{C}_{T_1/T_1'} \ar[u]_{\text{induced by }(h')^\sharp } }$

is commutative where $D_ i$ corresponds to $\theta _ i$ as in Equation (37.9.1.1). Then we have the conclusion of Lemma 37.9.10. The importance of the condition that both $X_2 \to X_1$ and $S_2 \to S_1$ are étale is that it allows us to construct a $\theta _2$ from $\theta _1$.

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