## 38.41 Blowing up perfect modules

This section tries to find normal forms for perfect modules of tor dimension $\leq 1$ after blowups. We are only partially successful.

Lemma 38.41.1. Let $X$ be a scheme. Let $\mathcal{F}$ be a perfect $\mathcal{O}_ X$-module of tor dimension $\leq 1$. For any blowup $b : X' \to X$ we have $Lb^*\mathcal{F} = b^*\mathcal{F}$ and $b^*\mathcal{F}$ is a perfect $\mathcal{O}_ X$-module of tor dimension $\leq 1$.

Proof. We may assume $X = \mathop{\mathrm{Spec}}(A)$ is affine and we may assume the $A$-module $M$ corresponding to $\mathcal{F}$ has a presentation

$0 \to A^{\oplus m} \to A^{\oplus n} \to M \to 0$

Suppose $I \subset A$ is an ideal and $a \in I$. Recall that the affine blowup algebra $A[\frac{I}{a}]$ is a subring of $A_ a$. Since localization is exact we see that $A_ a^{\oplus m} \to A_ a^{\oplus n}$ is injective. Hence $A[\frac{I}{a}]^{\oplus m} \to A[\frac{I}{a}]^{\oplus n}$ is injective too. This proves the lemma. $\square$

Lemma 38.41.2. Let $X$ be a scheme. Let $\mathcal{F}$ be a perfect $\mathcal{O}_ X$-module of tor dimension $\leq 1$. Let $U \subset X$ be a scheme theoretically dense open such that $\mathcal{F}|_ U$ is finite locally free of constant rank $r$. Then there exists a $U$-admissible blowup $b : X' \to X$ such that there is a canonical short exact sequence

$0 \to \mathcal{K} \to b^*\mathcal{F} \to \mathcal{Q} \to 0$

where $\mathcal{Q}$ is finite locally free of rank $r$ and $\mathcal{K}$ is a perfect $\mathcal{O}_ X$-module of tor dimension $\leq 1$ whose restriction to $U$ is zero.

Lemma 38.41.3. Let $X$ be a scheme. Let $\mathcal{F}$ be a perfect $\mathcal{O}_ X$-module of tor dimension $\leq 1$. Let $U \subset X$ be an open such that $\mathcal{F}|_ U = 0$. Then there is a $U$-admissible blowup

$b : X' \to X$

such that $\mathcal{F}' = b^*\mathcal{F}$ is equipped with two canonical locally finite filtrations

$0 = F^0 \subset F^1 \subset F^2 \subset \ldots \subset \mathcal{F}' \quad \text{and}\quad \mathcal{F}' = F_1 \supset F_2 \supset F_3 \supset \ldots \supset 0$

such that for each $n \geq 1$ there is an effective Cartier divisor $D_ n \subset X'$ with the property that

$F^ i/F^{i - 1} \quad \text{and}\quad F_ i/F_{i + 1}$

are finite locally free of rank $i$ on $D_ i$.

Proof. Choose an affine open $V \subset X$ such that there exists a presentation

$0 \to \mathcal{O}_ V^{\oplus n} \xrightarrow {A} \mathcal{O}_ V^{\oplus n} \to \mathcal{F} \to 0$

for some $n$ and some matrix $A$. The ideal we are going to blowup in is the product of the Fitting ideals $\text{Fit}_ k(\mathcal{F})$ for $k \geq 0$. This makes sense because in the affine situation above we see that $\text{Fit}_ k(\mathcal{F})|_ V = \mathcal{O}_ V$ for $k > n$. It is clear that this is a $U$-admissible blowing up. By Divisors, Lemma 31.32.12 we see that on $X'$ the ideals $\text{Fit}_ k(\mathcal{F})$ are invertible. Thus we reduce to the case discussed in the next paragraph.

Assume $\text{Fit}_ k(\mathcal{F})$ is an invertible ideal for $k \geq 0$. If $E_ k \subset X$ is the effective Cartier divisor defined by $\text{Fit}_ k(\mathcal{F})$ for $k \geq 0$, then the effective Cartier divisors $D_ k$ in the statement of the lemma will satisfy

$E_ k = D_{k + 1} + 2 D_{k + 2} + 3 D_{k + 3} + \ldots$

This makes sense as the collection $D_ k$ will be locally finite. Moreover, it uniquely determines the effective Cartier divisors $D_ k$ hence it suffices to construct $D_ k$ locally.

Choose an affine open $V \subset X$ and presentation of $\mathcal{F}|_ V$ as above. We will construct the divisors and filtrations by induction on the integer $n$ in the presentation. We set $D_ k|_ V = \emptyset$ for $k > n$ and we set $D_ n|V = E_{n - 1}|_ V$. After shrinking $V$ we may assume that $\text{Fit}_{n - 1}(\mathcal{F})|_ V$ is generated by a single nonzerodivisor $f \in \Gamma (V, \mathcal{O}_ V)$. Since $\text{Fit}_{n - 1}(\mathcal{F})|_ V$ is the ideal generated by the entries of $A$, we see that there is a matrix $A'$ in $\Gamma (V, \mathcal{O}_ V)$ such that $A = fA'$. Define $\mathcal{F}'$ on $V$ by the short exact sequence

$0 \to \mathcal{O}_ V^{\oplus n} \xrightarrow {A'} \mathcal{O}_ V^{\oplus n} \to \mathcal{F}' \to 0$

Since the entries of $A'$ generate the unit ideal in $\Gamma (V, \mathcal{O}_ V)$ we see that $\mathcal{F}'$ locally on $V$ has a presentation with $n$ decreased by $1$, see Algebra, Lemma 10.102.2. Further note that $f^{n - k}\text{Fit}_ k(\mathcal{F}') = \text{Fit}_ k(\mathcal{F})|_ V$ for $k = 0, \ldots , n$. Hence $\text{Fit}_ k(\mathcal{F}')$ is an invertible ideal for all $k$. We conclude by induction that there exist effective Cartier divisors $D'_ k \subset V$ such that $\mathcal{F}'$ has two canonical filtrations as in the statement of the lemma. Then we set $D_ k|_ V = D'_ k$ for $k = 1, \ldots , n - 1$. Observe that the equalities between effective Cartier divisors displayed above hold with these choices. Finally, we come to the construction of the filtrations. Namely, we have short exact sequences

$0 \to \mathcal{O}_{D_ n \cap V}^{\oplus n} \to \mathcal{F} \to \mathcal{F}' \to 0 \quad \text{and}\quad 0 \to \mathcal{F}' \to \mathcal{F} \to \mathcal{O}_{D_ n \cap V}^{\oplus n} \to 0$

coming from the two factorizations $A = A'f = f A'$ of $A$. These sequences are canonical because in the first one the submodule is $\mathop{\mathrm{Ker}}(f : \mathcal{F} \to \mathcal{F})$ and in the second one the quotient module is $\mathop{\mathrm{Coker}}(f : \mathcal{F} \to \mathcal{F})$. $\square$

Lemma 38.41.4. Let $X$ be a scheme. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a homorphism of perfect $\mathcal{O}_ X$-modules of tor dimension $\leq 1$. Let $U \subset X$ be a scheme theoretically dense open such that $\mathcal{F}|_ U = 0$ and $\mathcal{G}|_ U = 0$. Then there is a $U$-admissible blowup $b : X' \to X$ such that the kernel, image, and cokernel of $b^*\varphi$ are perfect $\mathcal{O}_{X'}$-modules of tor dimension $\leq 1$.

Proof. The assumptions tell us that the object $(\mathcal{F} \to \mathcal{G})$ of $D(\mathcal{O}_ X)$ is perfect. Thus we get a $U$-admissible blowup that works for the cokernel and kernel by Lemmas 38.40.2 and 38.41.1 (to see what the complex looks like after pullback). The image is the kernel of the cokernel and hence is going to be perfect of tor dimension $\leq 1$ as well. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).