The Stacks project

Lemma 50.17.2. With notation as in Lemma 50.17.1 for $a \geq 0$ we have

  1. the map $\Omega ^ a_{X/S} \to b_*\Omega ^ a_{X'/S}$ is an isomorphism,

  2. the map $\Omega ^ a_{Z/S} \to p_*\Omega ^ a_{E/S}$ is an isomorphism,

  3. 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 ├ętale 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 Lemma 50.17.1. 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 ├ętale locally, by More on Morphisms, Lemma 37.37.9, 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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