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 $\mathop{\mathrm{Coker}}(f : \mathcal{F} \to \mathcal{F})$.
$\square$
Comments (0)