50.17 Blowing up and de Rham cohomology

Fix a base scheme $S$, a smooth morphism $X \to S$, and a closed subscheme $Z \subset X$ which is also smooth over $S$. Denote $b : X' \to X$ the blowing up of $X$ along $Z$. Denote $E \subset X'$ the exceptional divisor. Picture

50.17.0.1
$$\label{derham-equation-blowup} \vcenter { \xymatrix{ E \ar[r]_ j \ar[d]_ p & X' \ar[d]^ b \\ Z \ar[r]^ i & X } }$$

Our goal in this section is to prove that the map $b^* : H_{dR}^*(X/S) \longrightarrow H_{dR}^*(X'/S)$ is injective (although a lot more can be said).

Lemma 50.17.1. Let $S$ be a scheme. Let $Z \to X$ be a closed immersion of schemes smooth over $S$. Let $b : X' \to X$ be the blowing up of $Z$ with exceptional divisor $E \subset X'$. Then $X'$ and $E$ are smooth over $S$. The morphism $p : E \to Z$ is canonically isomorphic to the projective space bundle

$\mathbf{P}(\mathcal{I}/\mathcal{I}^2) \longrightarrow Z$

where $\mathcal{I} \subset \mathcal{O}_ X$ is the ideal sheaf of $Z$. The relative $\mathcal{O}_ E(1)$ coming from the projective space bundle structure is isomorphic to the restriction of $\mathcal{O}_{X'}(-E)$ to $E$.

Proof. By Divisors, Lemma 31.22.11 the immersion $Z \to X$ is a regular immmersion, hence the ideal sheaf $\mathcal{I}$ is of finite type, hence $b$ is a projective morphism with relatively ample invertible sheaf $\mathcal{O}_{X'}(1) = \mathcal{O}_{X'}(-E)$, see Divisors, Lemmas 31.32.4 and 31.32.13. The canonical map $\mathcal{I} \to b_*\mathcal{O}_{X'}(1)$ gives a closed immersion

$X' \longrightarrow \mathbf{P}\left(\bigoplus \nolimits _{n \geq 0} \text{Sym}^ n_{\mathcal{O}_ X}(\mathcal{I})\right)$

by the very construction of the blowup. The restriction of this morphism to $E$ gives a canonical map

$E \longrightarrow \mathbf{P}\left(\bigoplus \nolimits _{n \geq 0} \text{Sym}^ n_{\mathcal{O}_ Z}(\mathcal{I}/\mathcal{I}^2)\right)$

over $Z$. Since $\mathcal{I}/\mathcal{I}^2$ is finite locally free if this canonical map is an isomorphism, then the final part of the lemma holds. Having said all of this, now the question is étale local on $X$. Namely, blowing up commutes with flat base change by Divisors, Lemma 31.32.3 and we can check smoothness after precomposing with a surjective étale morphism. Thus by the étale local structure of a closed immersion of schemes over $S$ given in More on Morphisms, Lemma 37.37.9 this reduces to the situation discussed in Section 50.16. $\square$

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 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 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 étale 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. $\square$

Lemma 50.17.3. With notation as in Lemma 50.17.1 and denoting $f : X \to S$ the structure morphism there is a canonical distinguished triangle

$\Omega ^\bullet _{X/S} \to Rb_*(\Omega ^\bullet _{X'/S}) \oplus i_*\Omega ^\bullet _{Z/S} \to i_*Rp_*(\Omega ^\bullet _{E/S}) \to \Omega ^\bullet _{X/S}[1]$

in $D(X, f^{-1}\mathcal{O}_ S)$ where the four maps

$\begin{matrix} \Omega ^\bullet _{X/S} & \to & Rb_*(\Omega ^\bullet _{X'/S}), \\ \Omega ^\bullet _{X/S} & \to & i_*\Omega ^\bullet _{Z/S}, \\ Rb_*(\Omega ^\bullet _{X'/S}) & \to & i_*Rp_*(\Omega ^\bullet _{E/S}), \\ i_*\Omega ^\bullet _{Z/S} & \to & i_*Rp_*(\Omega ^\bullet _{E/S}) \end{matrix}$

are the canonical ones (Section 50.2), except with sign reversed for one of them.

Proof. Choose a distinguished triangle

$C \to Rb_*\Omega ^\bullet _{X'/S} \oplus i_*\Omega ^\bullet _{Z/S} \to i_*Rp_*\Omega ^\bullet _{E/S} \to C[1]$

in $D(X, f^{-1}\mathcal{O}_ S)$. It suffices to show that $\Omega ^\bullet _{X/S}$ is isomorphic to $C$ in a manner compatible with the canonical maps. By the axioms of triangulated categories there exists a map of distinguished triangles

$\xymatrix{ C' \ar[r] \ar[d] & b_*\Omega ^\bullet _{X'/S} \oplus i_*\Omega ^\bullet _{Z/S} \ar[r] \ar[d] & i_*p_*\Omega ^\bullet _{E/S} \ar[r] \ar[d] & C'[1] \ar[d] \\ C \ar[r] & Rb_*\Omega ^\bullet _{X'/S} \oplus i_*\Omega ^\bullet _{Z/S} \ar[r] & i_*Rp_*\Omega ^\bullet _{E/S} \ar[r] & C[1] }$

By Lemma 50.17.2 part (3) and Derived Categories, Proposition 13.4.23 we conclude that $C' \to C$ is an isomorphism. By Lemma 50.17.2 part (2) the map $i_*\Omega ^\bullet _{Z/S} \to i_*p_*\Omega ^\bullet _{E/S}$ is an isomorphism. Thus $C' = b_*\Omega ^\bullet _{X'/S}$ in the derived category. Finally we use Lemma 50.17.2 part (1) tells us this is equal to $\Omega ^\bullet _{X/S}$. We omit the verification this is compatible with the canonical maps. $\square$

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. $\square$

Lemma 50.17.6 shows that producing the splitting on Hodge cohomology is a good deal easier than the result of Proposition 50.17.4. We urge the reader to skip ahead to the next section.

Lemma 50.17.5. Let $i : Z \to X$ be a closed immersion of schemes which is regular of codimension $c$. Then $\mathop{\mathrm{Ext}}\nolimits ^ q_{\mathcal{O}_ X}(i_*\mathcal{F}, \mathcal{E}) = 0$ for $q < c$ for $\mathcal{E}$ locally free on $X$ and $\mathcal{F}$ any $\mathcal{O}_ Z$-module.

Proof. By the local to global spectral sequence of $\mathop{\mathrm{Ext}}\nolimits$ it suffices to prove this affine locally on $X$. See Cohomology, Section 20.40. Thus we may assume $X = \mathop{\mathrm{Spec}}(A)$ and there exists a regular sequence $f_1, \ldots , f_ c$ in $A$ such that $Z = \mathop{\mathrm{Spec}}(A/(f_1, \ldots , f_ c))$. We may assume $c \geq 1$. Then we see that $f_1 : \mathcal{E} \to \mathcal{E}$ is injective. Since $i_*\mathcal{F}$ is annihilated by $f_1$ this shows that the lemma holds for $i = 0$ and that we have a surjection

$\mathop{\mathrm{Ext}}\nolimits ^{q - 1}_{\mathcal{O}_ X}(i_*\mathcal{F}, \mathcal{E}/f_1\mathcal{E}) \longrightarrow \mathop{\mathrm{Ext}}\nolimits ^ q_{\mathcal{O}_ X}(i_*\mathcal{F}, \mathcal{E})$

Thus it suffices to show that the source of this arrow is zero. Next we repeat this argument: if $c \geq 2$ the map $f_2 : \mathcal{E}/f_1\mathcal{E} \to \mathcal{E}/f_1\mathcal{E}$ is injective. Since $i_*\mathcal{F}$ is annihilated by $f_2$ this shows that the lemma holds for $q = 1$ and that we have a surjection

$\mathop{\mathrm{Ext}}\nolimits ^{q - 2}_{\mathcal{O}_ X}(i_*\mathcal{F}, \mathcal{E}/f_1\mathcal{E} + f_2\mathcal{E}) \longrightarrow \mathop{\mathrm{Ext}}\nolimits ^{q - 1}_{\mathcal{O}_ X}(i_*\mathcal{F}, \mathcal{E}/f_1\mathcal{E})$

Continuing in this fashion the lemma is proved. $\square$

Lemma 50.17.6. With notation as in Lemma 50.17.1 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 étale locally on $X$ as in the proof of Lemma 50.17.2 and apply Lemma 50.16.2. Computing étale 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 50.17.2 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 32.18.2. Thus the desired vanishing follows from Lemma 50.17.5. $\square$

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