The Stacks project

Proposition 50.17.4. With notation as in Lemma 50.17.1 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 50.17.3. We claim that the map

\[ Rb_*(\Omega ^\bullet _{X'/S}) \oplus i_*\Omega ^\bullet _{Z/S} \to i_*Rp_*(\Omega ^\bullet _{E/S}) \]

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

\[ \tilde\xi _ m : \bigoplus \nolimits _{t = 0, \ldots , N - m - 1} \Omega ^\bullet _{Z_ m/S}[-2t] \longrightarrow Rp_{m, *}\Omega ^\bullet _{E_ m/S} \]

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

\[ \tilde\xi : \bigoplus \nolimits _ m \bigoplus \nolimits _{t = 0, \ldots , N - m - 1} \Omega ^\bullet _{Z_ m/S}[-2t] \longrightarrow Rp_*(\Omega ^\bullet _{E/S}) \]

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

\[ M = \bigoplus M_ m \quad \text{and}\quad K = \bigoplus K_ m \]

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

\[ c_{E/Z} \oplus \tilde\xi |_ M : \Omega ^\bullet _{Z/S} \oplus M \longrightarrow Rp_*(\Omega ^\bullet _{E/S}) \]

is an isomorphism (see remark referenced above).

Consider the map

\[ \delta : \Omega ^\bullet _{E/S}[-2] \longrightarrow \Omega ^\bullet _{X'/S} \]

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

\[ \Omega ^\bullet _{E/S}[-2] \longrightarrow \Omega ^\bullet _{X'/S} \longrightarrow \Omega ^\bullet _{E/S} \]

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

\[ \xymatrix{ K[-2] \ar[d]_{(\tilde\xi |_ K)[-2]} \ar[r]_{\text{id}} & M \ar[d]^{\tilde x|_ M} \\ Rp_*\Omega ^\bullet _{E/S}[-2] \ar[r]^-{Rp_*\theta '} & Rp_*\Omega ^\bullet _{E/S} } \]

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

\begin{align*} Rp_*(\Omega ^\bullet _{E/S}) & \xrightarrow {(c_{E/Z} \oplus \tilde\xi |_ M)^{-1}} \Omega ^\bullet _{Z/S} \oplus M \\ & \xrightarrow {\text{id} \oplus \text{id}^{-1}} \Omega ^\bullet _{Z/S} \oplus K[-2] \\ & \xrightarrow {\text{id} \oplus (\tilde\xi |_ K)[-2]} \Omega ^\bullet _{Z/S} \oplus Rp_*\Omega ^\bullet _{E/S}[-2] \\ & \xrightarrow {\text{id} \oplus Rb_*\delta } \Omega ^\bullet _{Z/S} \oplus Rb_*\Omega ^\bullet _{X'/S} \end{align*}

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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