The Stacks project

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

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

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

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

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