The Stacks project

50.16 Calculations

In this section we calculate some Hodge and de Rham cohomology groups for a standard blowing up.

We fix a ring $R$ and we set $S = \mathop{\mathrm{Spec}}(R)$. Fix integers $0 \leq m$ and $1 \leq n$. Consider the closed immersion

\[ Z = \mathbf{A}^ m_ S \longrightarrow \mathbf{A}^{m + n}_ S = X,\quad (a_1, \ldots , a_ m) \mapsto (a_1, \ldots , a_ m, 0, \ldots 0). \]

We are going to consider the blowing up $L$ of $X$ along the closed subscheme $Z$. Write

\[ X = \mathbf{A}^{m + n}_ S = \mathop{\mathrm{Spec}}(A) \quad \text{with}\quad A = R[x_1, \ldots , x_ m, y_1, \ldots , y_ n] \]

We will consider $A = R[x_1, \ldots , x_ m, y_1, \ldots , y_ n]$ as a graded $R$-algebra by setting $\deg (x_ i) = 0$ and $\deg (y_ j) = 1$. With this grading we have

\[ P = \text{Proj}(A) = \mathbf{A}^ m_ S \times _ S \mathbf{P}^{n - 1}_ S = Z \times _ S \mathbf{P}^{n - 1}_ S = \mathbf{P}^{n - 1}_ Z \]

Observe that the ideal cutting out $Z$ in $X$ is the ideal $A_+$. Hence $L$ is the Proj of the Rees algebra

\[ A \oplus A_+ \oplus (A_+)^2 \oplus \ldots = \bigoplus \nolimits _{d \geq 0} A_{\geq d} \]

Hence $L$ is an example of the phenomenon studied in more generality in More on Morphisms, Section 37.51; we will use the observations we made there without further mention. In particular, we have a commutative diagram

\[ \xymatrix{ P \ar[r]_0 \ar[d]_ p & L \ar[r]_-\pi \ar[d]^ b & P \ar[d]^ p \\ Z \ar[r]^ i & X \ar[r] & Z } \]

such that $\pi : L \to P$ is a line bundle over $P = Z \times _ S \mathbf{P}^{n - 1}_ S$ with zero section $0$ whose image $E = 0(P) \subset L$ is the exceptional divisor of the blowup $b$.

Lemma 50.16.1. For $a \geq 0$ we have

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

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

  3. the map $Rb_*\Omega ^ a_{L/S} \to i_*Rp_*\Omega ^ a_{P/S}$ is an isomorphism on cohomology sheaves in degree $\geq 1$.

Proof. Let us first prove part (2). Since $P = Z \times _ S \mathbf{P}^{n - 1}_ S$ we see that

\[ \Omega ^ a_{P/S} = \bigoplus \nolimits _{a = r + s} \text{pr}_1^*\Omega ^ r_{Z/S} \otimes \text{pr}_2^*\Omega ^ s_{\mathbf{P}^{n - 1}_ S/S} \]

Recalling that $p = \text{pr}_1$ by the projection formula (Cohomology, Lemma 20.54.2) we obtain

\[ p_*\Omega ^ a_{P/S} = \bigoplus \nolimits _{a = r + s} \Omega ^ r_{Z/S} \otimes \text{pr}_{1, *}\text{pr}_2^*\Omega ^ s_{\mathbf{P}^{n - 1}_ S/S} \]

By the calculations in Section 50.11 and in particular in the proof of Lemma 50.11.3 we have $\text{pr}_{1, *}\text{pr}_2^*\Omega ^ s_{\mathbf{P}^{n - 1}_ S/S} = 0$ except if $s = 0$ in which case we get $\text{pr}_{1, *}\mathcal{O}_ P = \mathcal{O}_ Z$. This proves (2).

By the material in Section 50.10 and in particular Lemma 50.10.4 we have $\pi _*\Omega ^ a_{L/S} = \Omega ^ a_{P/S} \oplus \bigoplus _{k \geq 1} \Omega ^ a_{L/S, k}$. Since the composition $\pi \circ 0$ in the diagram above is the identity morphism on $P$ to prove part (3) it suffices to show that $\Omega ^ a_{L/S, k}$ has vanishing higher cohomology for $k > 0$. By Lemmas 50.10.2 and 50.10.4 there are short exact sequences

\[ 0 \to \Omega ^ a_{P/S} \otimes \mathcal{O}_ P(k) \to \Omega ^ a_{L/S, k} \to \Omega ^{a - 1}_{P/S} \otimes \mathcal{O}_ P(k) \to 0 \]

where $\Omega ^{a - 1}_{P/S} = 0$ if $a = 0$. Since $P = Z \times _ S \mathbf{P}^{n - 1}_ S$ we have

\[ \Omega ^ a_{P/S} = \bigoplus \nolimits _{i + j = a} \Omega ^ i_{Z/S} \boxtimes \Omega ^ j_{\mathbf{P}^{n - 1}_ S/S} \]

by Lemma 50.8.1. Since $\Omega ^ i_{Z/S}$ is free of finite rank we see that it suffices to show that the higher cohomology of $\mathcal{O}_ Z \boxtimes \Omega ^ j_{\mathbf{P}^{n - 1}_ S/S}(k)$ is zero for $k > 0$. This follows from Lemma 50.11.2 applied to $P = Z \times _ S \mathbf{P}^{n - 1}_ S = \mathbf{P}^{n - 1}_ Z$ and the proof of (3) is complete.

We still have to prove (1). If $n = 1$, then we are blowing up an effective Cartier divisor and $b$ is an isomorphism and we have (1). If $n > 1$, then the composition

\[ \Gamma (X, \Omega ^ a_{X/S}) \to \Gamma (L, \Omega ^ a_{L/S}) \to \Gamma (L \setminus E, \Omega ^ a_{L/S}) = \Gamma (X \setminus Z, \Omega ^ a_{X/S}) \]

is an isomorphism as $\Omega ^ a_{X/S}$ is finite free (small detail omitted). Thus the only way (1) can fail is if there are nonzero elements of $\Gamma (L, \Omega ^ a_{L/S})$ which vanish outside of $E = 0(P)$. Since $L$ is a line bundle over $P$ with zero section $0 : P \to L$, it suffices to show that on a line bundle there are no nonzero sections of a sheaf of differentials which vanish identically outside the zero section. The reader sees this is true either (preferably) by a local caculation or by using that $\Omega _{L/S, k} \subset \Omega _{L^\star /S, k}$ (see references above). $\square$

We suggest the reader skip to the next section at this point.

Lemma 50.16.2. For $a \geq 0$ there are canonical maps

\[ b^*\Omega ^ a_{X/S} \longrightarrow \Omega ^ a_{L/S} \longrightarrow b^*\Omega ^ a_{X/S} \otimes _{\mathcal{O}_ L} \mathcal{O}_ L((n - 1)E) \]

whose composition is induced by the inclusion $\mathcal{O}_ L \subset \mathcal{O}_ L((n - 1)E)$.

Proof. The first arrow in the displayed formula is discussed in Section 50.2. To get the second arrow we have to show that if we view a local section of $\Omega ^ a_{L/S}$ as a “meromorphic section” of $b^*\Omega ^ a_{X/S}$, then it has a pole of order at most $n - 1$ along $E$. To see this we work on affine local charts on $L$. Namely, recall that $L$ is covered by the spectra of the affine blowup algebras $A[\frac{I}{y_ i}]$ where $I = A_{+}$ is the ideal generated by $y_1, \ldots , y_ n$. See Algebra, Section 10.70 and Divisors, Lemma 31.32.2. By symmetry it is enough to work on the chart corresponding to $i = 1$. Then

\[ A[\frac{I}{y_1}] = R[x_1, \ldots , x_ m, y_1, t_2, \ldots , t_ n] \]

where $t_ i = y_ i/y_1$, see More on Algebra, Lemma 15.31.2. Thus the module $\Omega ^1_{L/S}$ is over the corresponding affine open freely generated by $\text{d}x_1, \ldots , \text{d}x_ m$, $\text{d}y_1$, and $\text{d}t_1, \ldots , \text{d}t_ n$. Of course, the first $m + 1$ of these generators come from $b^*\Omega ^1_{X/S}$ and for the remaining $n - 1$ we have

\[ \text{d}t_ j = \text{d}\frac{y_ j}{y_1} = \frac{1}{y_1}\text{d}y_ j - \frac{y_ j}{y_1^2}\text{d}y_1 = \frac{\text{d}y_ j - t_ j \text{d}y_1}{y_1} \]

which has a pole of order $1$ along $E$ since $E$ is cut out by $y_1$ on this chart. Since the wedges of $a$ of these elements give a basis of $\Omega ^ a_{L/S}$ over this chart, and since there are at most $n - 1$ of the $\text{d}t_ j$ involved this finishes the proof. $\square$

Lemma 50.16.3. Let $E = 0(P)$ be the exceptional divisor of the blowing up $b$. For any locally free $\mathcal{O}_ X$-module $\mathcal{E}$ and $0 \leq i \leq n - 1$ the map

\[ \mathcal{E} \longrightarrow Rb_*(b^*\mathcal{E} \otimes _{\mathcal{O}_ L} \mathcal{O}_ L(iE)) \]

is an isomorphism in $D(\mathcal{O}_ X)$.

Proof. By the projection formula it is enough to show this for $\mathcal{E} = \mathcal{O}_ X$, see Cohomology, Lemma 20.54.2. Since $X$ is affine it suffices to show that the maps

\[ H^0(X, \mathcal{O}_ X) \to H^0(L, \mathcal{O}_ L) \to H^0(L, \mathcal{O}_ L(iE)) \]

are isomorphisms and that $H^ j(X, \mathcal{O}_ L(iE)) = 0$ for $j > 0$ and $0 \leq i \leq n - 1$, see Cohomology of Schemes, Lemma 30.4.6. Since $\pi $ is affine, we can compute global sections and cohomology after taking $\pi _*$, see Cohomology of Schemes, Lemma 30.2.4. If $n = 1$, then $L \to X$ is an isomorphism and $i = 0$ hence the first statement holds. If $n > 1$, then we consider the composition

\[ H^0(X, \mathcal{O}_ X) \to H^0(L, \mathcal{O}_ L) \to H^0(L, \mathcal{O}_ L(iE)) \to H^0(L \setminus E, \mathcal{O}_ L) = H^0(X \setminus Z, \mathcal{O}_ X) \]

Since $H^0(X \setminus Z, \mathcal{O}_ X) = H^0(X, \mathcal{O}_ X)$ in this case as $Z$ has codimension $n \geq 2$ in $X$ (details omitted) we conclude the first statement holds. For the second, recall that $\mathcal{O}_ L(E) = \mathcal{O}_ L(-1)$, see Divisors, Lemma 31.32.4. Hence we have

\[ \pi _*\mathcal{O}_ L(iE) = \pi _*\mathcal{O}_ L(-i) = \bigoplus \nolimits _{k \geq -i} \mathcal{O}_ P(k) \]

as discussed in More on Morphisms, Section 37.51. Thus we conclude by the vanishing of the cohomology of twists of the structure sheaf on $P = \mathbf{P}^{n - 1}_ Z$ shown in Cohomology of Schemes, Lemma 30.8.1. $\square$

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