Proposition 50.17.3. With notation as in More on Morphisms, Lemma 37.17.3 the map $\Omega ^\bullet _{X/S} \to Rb_*\Omega ^\bullet _{X'/S}$ has a splitting in $D(X, (X \to S)^{-1}\mathcal{O}_ S)$.

**Proof.**
Consider the triangle constructed in Lemma 50.17.2. We claim that the map

has a splitting whose image contains the summand $i_*\Omega ^\bullet _{Z/S}$. By Derived Categories, Lemma 13.4.11 this will show that the first arrow of the triangle has a splitting which vanishes on the summand $i_*\Omega ^\bullet _{Z/S}$ which proves the lemma. We will prove the claim by decomposing $Rp_*\Omega ^\bullet _{E/S}$ into a direct sum where the first piece corresponds to $\Omega ^\bullet _{Z/S}$ and the second piece can be lifted through $Rb_*\Omega ^\bullet _{X'/S}$.

Proof of the claim. 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 $E_ m = p^{-1}(Z_ m)$. If $p_ m : E_ m \to Z_ m$ denotes the restriction of $p$ to $E_ m$, then we have a canonical isomorphism

in $D(Z_ m, (Z_ m \to S)^{-1}\mathcal{O}_ S)$ where in degree $0$ we have the canonical map $\Omega ^\bullet _{Z_ m/S} \to Rp_{m, *}\Omega ^\bullet _{E_ m/S}$. See Remark 50.14.2. Thus we have an isomorphism

in $D(Z, (Z \to S)^{-1}\mathcal{O}_ S)$ whose restriction to the summand $\Omega ^\bullet _{Z/S} = \bigoplus \Omega ^\bullet _{Z_ m/S}$ of the source is the canonical map $\Omega ^\bullet _{Z/S} \to Rp_*(\Omega ^\bullet _{E/S})$. Consider the subcomplexes $M_ m$ and $K_ m$ of the complex $\bigoplus \nolimits _{t = 0, \ldots , N - m - 1} \Omega ^\bullet _{Z_ m/S}[-2t]$ introduced in Remark 50.14.2. We set

We have $M = K[-2]$ and by construction the map

is an isomorphism (see remark referenced above).

Consider the map

in $D(X', (X' \to S)^{-1}\mathcal{O}_ S)$ of Lemma 50.15.5 with the property that the composition

is the map $\theta '$ of Remark 50.4.3 for $c_1^{dR}(\mathcal{O}_{X'}(-E))|_ E) = c_1^{dR}(\mathcal{O}_ E(1))$. The final assertion of Remark 50.14.2 tells us that the diagram

commutes. Thus we see that we can obtain the desired splitting of the claim as the map

The relationship between $\theta '$ and $\delta $ stated above together with the commutative diagram involving $\theta '$, $\tilde\xi |_ K$, and $\tilde\xi |_ M$ above are exactly what's needed to show that this is a section to the canonical map $\Omega ^\bullet _{Z/S} \oplus Rb_*(\Omega ^\bullet _{X'/S}) \to Rp_*(\Omega ^\bullet _{E/S})$ and the proof of the claim is complete. $\square$

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