Lemma 50.17.1. With notation as in More on Morphisms, Lemma 37.17.3 for a \geq 0 we have
the map \Omega ^ a_{X/S} \to b_*\Omega ^ a_{X'/S} is an isomorphism,
the map \Omega ^ a_{Z/S} \to p_*\Omega ^ a_{E/S} is an isomorphism,
the map Rb_*\Omega ^ a_{X'/S} \to i_*Rp_*\Omega ^ a_{E/S} is an isomorphism on cohomology sheaves in degree \geq 1.
Proof. Let \epsilon : X_1 \to X be a surjective étale morphism. Denote i_1 : Z_1 \to X_1, b_1 : X'_1 \to X_1, E_1 \subset X'_1, and p_1 : E_1 \to Z_1 the base changes of the objects considered in More on Morphisms, Lemma 37.17.3. Observe that i_1 is a closed immersion of schemes smooth over S and that b_1 is the blowing up with center Z_1 by Divisors, Lemma 31.32.3. Suppose that we can prove (1), (2), and (3) for the morphisms b_1, p_1, and i_1. Then by Lemma 50.2.2 we obtain that the pullback by \epsilon of the maps in (1), (2), and (3) are isomorphisms. As \epsilon is a surjective flat morphism we conclude. Thus working étale locally, by More on Morphisms, Lemma 37.17.1, we may assume we are in the situation discussed in Section 50.16. In this case the lemma is the same as Lemma 50.16.1. \square
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