The Stacks project

Lemma 50.17.5. With notation as in More on Morphisms, Lemma 37.17.3 for $a \geq 0$ there is a unique arrow $Rb_*\Omega ^ a_{X'/S} \to \Omega ^ a_{X/S}$ in $D(\mathcal{O}_ X)$ whose composition with $\Omega ^ a_{X/S} \to Rb_*\Omega ^ a_{X'/S}$ is the identity on $\Omega ^ a_{X/S}$.

Proof. We may decompose $X$ into open and closed subschemes having fixed relative dimension to $S$, see Morphisms, Lemma 29.34.12. Since the derived category $D(X, f^{-1}\mathcal{O})_ S)$ correspondingly decomposes as a product of categories, we may assume $X$ has fixed relative dimension $N$ over $S$. We may decompose $Z = \coprod Z_ m$ into open and closed subschemes of relative dimension $m \geq 0$ over $S$. The restriction $i_ m : Z_ m \to X$ of $i$ to $Z_ m$ is a regular immersion of codimension $N - m$, see Divisors, Lemma 31.22.11. Let $E = \coprod E_ m$ be the corresponding decomposition, i.e., we set $E_ m = p^{-1}(Z_ m)$. We claim that there are natural maps

\[ b^*\Omega ^ a_{X/S} \to \Omega ^ a_{X'/S} \to b^*\Omega ^ a_{X/S} \otimes _{\mathcal{O}_{X'}} \mathcal{O}_{X'}(\sum (N - m - 1)E_ m) \]

whose composition is induced by the inclusion $\mathcal{O}_{X'} \to \mathcal{O}_{X'}(\sum (N - m - 1)E_ m)$. Namely, in order to prove this, it suffices to show that the cokernel of the first arrow is locally on $X'$ annihilated by a local equation of the effective Cartier divisor $\sum (N - m - 1)E_ m$. To see this in turn we can work ├ętale locally on $X$ as in the proof of Lemma 50.17.1 and apply Lemma 50.16.2. Computing ├ętale locally using Lemma 50.16.3 we see that the induced composition

\[ \Omega ^ a_{X/S} \to Rb_*\Omega ^ a_{X'/S} \to Rb_*\left(b^*\Omega ^ a_{X/S} \otimes _{\mathcal{O}_{X'}} \mathcal{O}_{X'}(\sum (N - m - 1)E_ m)\right) \]

is an isomorphism in $D(\mathcal{O}_ X)$ which is how we obtain the existence of the map in the lemma.

For uniqueness, it suffices to show that there are no nonzero maps from $\tau _{\geq 1}Rb_*\Omega _{X'/S}$ to $\Omega ^ a_{X/S}$ in $D(\mathcal{O}_ X)$. For this it suffices in turn to show that there are no nonzero maps from $R^ qb_*\Omega _{X'/s}[-q]$ to $\Omega ^ a_{X/S}$ in $D(\mathcal{O}_ X)$ for $q \geq 1$ (details omitted). By Lemma 50.17.1 we see that $R^ qb_*\Omega _{X'/s} \cong i_*R^ qp_*\Omega ^ a_{E/S}$ is the pushforward of a module on $Z = \coprod Z_ m$. Moreover, observe that the restriction of $R^ qp_*\Omega ^ a_{E/S}$ to $Z_ m$ is nonzero only for $q < N - m$. Namely, the fibres of $E_ m \to Z_ m$ have dimension $N - m - 1$ and we can apply Limits, Lemma 32.19.2. Thus the desired vanishing follows from Lemma 50.17.4. $\square$

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