The Stacks project

Proof. The proof is just that we will locally blow up $X$ in the product ideals $I_ i(M^\bullet , f)$ for any affine chart $(U, A, f, M^\bullet )$. The first few lemmas in More on Algebra, Section 15.96 show that this is well defined. The universal property (2) then follows from the universal property of blowing up. The details can be found below.

Let $U, A, f, M^\bullet$ be an affine chart for $(X, D, M)$. All but a finite number of the ideals $I_ i(M^\bullet , f)$ are equal to $A$ hence it makes sense to look at

and this is a finitely generated ideal of $A$. Denote

the blowing up of $U$ in $I$. Then $b_ U^{-1}(U \cap D)$ is defined by Divisors, Lemma 31.32.11. Recall that $f^{r_ i} \in I_ i(M^\bullet , f)$ and hence $b_ U$ is a $(U \setminus D)$-admissible blowing up. By Divisors, Lemma 31.32.12 for each $i$ the morphism $b_ U$ factors as $U' \to U'_ i \to U$ where $U'_ i \to U$ is the blowing up in $I_ i(M^\bullet , f)$ and $U' \to U'_ i$ is another blowing up. It follows that the pullback $I_ i(M^\bullet , f)\mathcal{O}_{U'}$ of $I_ i(M^\bullet , f)$ to $U'$ is an invertible ideal sheaf, see Divisors, Lemmas 31.32.11 and 31.32.4. It follows that $(U', b^{-1}D, Lb^*M|_ U)$ is a good triple, see Lemma 38.43.2 for the behaviour of the ideals $I_ i(-,-)$ under pullback. Finally, we claim that $b_ U : U' \to U$ has the universal property mentioned in part (2) of the statement of the lemma. Namely, suppose $h : Y \to U$ is a morphism of schemes such that the pullback $E = h^{-1}(D \cap U)$ is defined and $(Y, E, Lh^*M)$ is a good triple. Then $Y$ is covered by affine charts $(V, B, g, N^\bullet )$ such that $I_ i(N^\bullet , g)$ is an invertible ideal for each $i$. Then $g$ and the image of $f$ in $B$ differ by a unit as they both cut out the effective Cartier divisor $E \cap V$. Hence we may assume $g$ is the image of $f$ by More on Algebra, Lemma 15.96.2. Then $I_ i(N^\bullet , g)$ is isomorphic to $I_ i(M^\bullet \otimes _ A B, g)$ as a $B$-module by More on Algebra, Lemma 15.96.1. Thus $I_ i(M^\bullet \otimes _ A B, g) = I_ i(M^\bullet , f)B$ (Lemma 38.43.2) is an invertible $B$-module. Hence the ideal $IB$ is invertible. It follows that $I\mathcal{O}_ Y$ is invertible. Hence we obtain a unique factorization of $h$ through $b_ U$ by Divisors, Lemma 31.32.5.

Let $\mathcal{B}$ be the set of affine opens $U \subset X$ such that there exists an affine chart $(U, A, f, M^\bullet )$ for $(X, D, M)$. Then $\mathcal{B}$ is a basis for the topology on $X$; details omitted. For $U \in \mathcal{B}$ we have the morphism $b_ U : U' \to U$ constructed above which satisfies the universal property over $U$. If $U_1 \subset U_2 \subset X$ are both in $\mathcal{B}$, then $b_{U_1} : U'_1 \to U_1$ is canonically isomorphic to

by the universal property. In other words, we get an isomorphism $U'_1 \to b_{U_2}^{-1}(U_1)$ over $U_1$. These isomorphisms satisfy the cocycle condition (again by the universal property) and hence by Constructions, Lemma 27.2.1 we get a morphism $b : X' \to X$ whose restriction to each $U$ in $\mathcal{B}$ is isomorphic to $U' \to U$. Then the morphism $b : X' \to X$ satisfies properties (1) and (2) of the statement of the lemma as these properties may be checked locally (details omitted).

We still have to prove the final assertion of the lemma. Let $W \subset X$ be a quasi-compact open. Choose a finite covering $W = U_1 \cup \ldots \cup U_ T$ such that for each $1 \leq t \leq T$ there exists an affine chart $(U_ t, A_ t, f_ t, M_ t^\bullet )$. We will use below that for any affine open $V = \mathop{\mathrm{Spec}}(B) \subset U_ t \cap U_{t'}$ we have (a) the images of $f_ t$ and $f_{t'}$ in $B$ differ by a unit, and (b) the complexes $M_ t^\bullet \otimes _ A B$ and $M_{t'} \otimes _ A B$ define isomorphic objects of $D(B)$. For $i \in \mathbf{Z}$, set

Then $N_ t - \sum \nolimits _{j \geq i} (-1)^{j - i} rk(M_ t^ j) \geq 0$ and we can consider the ideals

It follows from More on Algebra, Lemmas 15.96.2 and 15.96.1 that the ideals $I_{t, i}$ glue to a quasi-coherent, finite type ideal $\mathcal{I}_ i \subset \mathcal{O}_ W$. Moreover, all but a finite number of these ideals are equal to $\mathcal{O}_ W$. Clearly, the morphism $X' \to X$ constructed above restricts to the blowing up of $W$ in the product of the ideals $\mathcal{I}_ i$. This finishes the proof. $\square$