## 115.23 Intersection theory

Lemma 115.23.1. Let $b : X' \to X$ be the blowing up of a smooth projective scheme over a field $k$ in a smooth closed subscheme $Z \subset X$. Picture

$\xymatrix{ E \ar[r]_ j \ar[d]_\pi & X' \ar[d]^ b \\ Z \ar[r]^ i & X }$

Assume there exists an element of $K_0(X)$ whose restriction to $Z$ is equal to the class of $\mathcal{C}_{Z/X}$ in $K_0(Z)$. Then $[Lb^*\mathcal{O}_ Z] = [\mathcal{O}_ E] \cdot \alpha ''$ in $K_0(X')$ for some $\alpha '' \in K_0(X')$.

Proof. The schemes $X$, $X'$, $E$, $Z$ are smooth and projective over $k$ and hence we have $K'_0(X) = K_0(X) = K_0(\textit{Vect}(X)) = K_0(D^ b_{\textit{Coh}}(X)))$ and similarly for the other $3$. See Derived Categories of Schemes, Lemmas 36.38.1, 36.38.4, and 36.38.5. We will switch between these versions at will in this proof. Consider the short exact sequence

$0 \to \mathcal{F} \to \pi ^*\mathcal{C}_{Z/X} \to \mathcal{C}_{E/X'} \to 0$

of finite locally free $\mathcal{O}_ E$-modules defining $\mathcal{F}$. Observe that $\mathcal{C}_{E/X'} = \mathcal{O}_{X'}(-E)|_ E$ is the restriction of the invertible $\mathcal{O}_ X$-module $\mathcal{O}_{X'}(-E)$. Let $\alpha \in K_0(X)$ be an element such that $i^*\alpha = [\mathcal{C}_{Z/X}]$ in $K_0(Z)$. Let $\alpha ' = b^*\alpha - [\mathcal{O}_{X'}(-E)]$. Then $j^*\alpha ' = [\mathcal{F}]$. We deduce that $j^*\lambda ^ i(\alpha ') = [\wedge ^ i(\mathcal{F})]$ by Weil Cohomology Theories, Lemma 45.13.1. This means that $[\mathcal{O}_ E] \cdot \alpha ' = [\wedge ^ i\mathcal{F}]$ in $K_0(X)$, see Derived Categories of Schemes, Lemma 36.38.8. Let $r$ be the maximum codimension of an irreducible component of $Z$ in $X$. A computation which we omit shows that $H^{-i}(Lb^*\mathcal{O}_ Z) = \wedge ^ i\mathcal{F}$ for $i \geq 0, 1, \ldots , r - 1$ and zero in other degrees. It follows that in $K_0(X)$ we have

\begin{align*} [Lb^*\mathcal{O}_ Z] & = \sum \nolimits _{i = 0, \ldots , r - 1} (-1)^ i[\wedge ^ i\mathcal{F}] \\ & = \sum \nolimits _{i = 0, \ldots , r - 1} (-1)^ i[\mathcal{O}_ E] \lambda ^ i(\alpha ') \\ & = [\mathcal{O}_ E] \left(\sum \nolimits _{i = 0, \ldots , r - 1} (-1)^ i \lambda ^ i(\alpha ')\right) \end{align*}

This proves the lemma with $\alpha '' = \sum _{i = 0, \ldots , r - 1} (-1)^ i \lambda ^ i(\alpha ')$. $\square$

Lemma 115.23.2. Let $(S, \delta )$ be as in Chow Homology, Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $X$ be integral and $n = \dim _\delta (X)$. Let $a \in \Gamma (X, \mathcal{O}_ X)$ be a nonzero function. Let $i : D = Z(a) \to X$ be the closed immersion of the zero scheme of $a$. Let $f \in R(X)^*$. In this case $i^*\text{div}_ X(f) = 0$ in $A_{n - 2}(D)$.

Proof. Special case of Chow Homology, Lemma 42.30.1. $\square$

Remark 115.23.3. This remark used to say that it wasn't clear whether the arrows of Chow Homology, Lemma 42.23.2 were isomorphisms in general. However, we've now found a proof of this fact.

### 115.23.4 Blowing up lemmas

In this section we prove some lemmas on representing Cartier divisors by suitable effective Cartier divisors on blowups. These lemmas can be found in [Section 2.4, F]. We have adapted the formulation so they also work in the non-finite type setting. It may happen that the morphism $b$ of Lemma 115.23.11 is a composition of infinitely many blowups, but over any given quasi-compact open $W \subset X$ one needs only finitely many blowups (and this is the result of loc. cit.).

Lemma 115.23.5. Let $(S, \delta )$ be as in Chow Homology, Situation 42.7.1. Let $X$, $Y$ be locally of finite type over $S$. Let $f : X \to Y$ be a proper morphism. Let $D \subset Y$ be an effective Cartier divisor. Assume $X$, $Y$ integral, $n = \dim _\delta (X) = \dim _\delta (Y)$ and $f$ dominant. Then

$f_*[f^{-1}(D)]_{n - 1} = [R(X) : R(Y)] [D]_{n - 1}.$

In particular if $f$ is birational then $f_*[f^{-1}(D)]_{n - 1} = [D]_{n - 1}$.

Proof. Immediate from Chow Homology, Lemma 42.26.3 and the fact that $D$ is the zero scheme of the canonical section $1_ D$ of $\mathcal{O}_ X(D)$. $\square$

Lemma 115.23.6. Let $(S, \delta )$ be as in Chow Homology, Situation 42.7.1. Let $X$ be locally of finite type over $S$. Assume $X$ integral with $\dim _\delta (X) = n$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Let $s$ be a nonzero meromorphic section of $\mathcal{L}$. Let $U \subset X$ be the maximal open subscheme such that $s$ corresponds to a section of $\mathcal{L}$ over $U$. There exists a projective morphism

$\pi : X' \longrightarrow X$

such that

1. $X'$ is integral,

2. $\pi |_{\pi ^{-1}(U)} : \pi ^{-1}(U) \to U$ is an isomorphism,

3. there exist effective Cartier divisors $D, E \subset X'$ such that

$\pi ^*\mathcal{L} = \mathcal{O}_{X'}(D - E),$
4. the meromorphic section $s$ corresponds, via the isomorphism above, to the meromorphic section $1_ D \otimes (1_ E)^{-1}$ (see Divisors, Definition 31.14.1),

5. we have

$\pi _*([D]_{n - 1} - [E]_{n - 1}) = \text{div}_\mathcal {L}(s)$

in $Z_{n - 1}(X)$.

Proof. Let $\mathcal{I} \subset \mathcal{O}_ X$ be the quasi-coherent ideal sheaf of denominators of $s$, see Divisors, Definition 31.23.10. By Divisors, Lemma 31.34.6 we get (2), (3), and (4). By Divisors, Lemma 31.32.9 we get (1). By Divisors, Lemma 31.32.13 the morphism $\pi$ is projective. We still have to prove (5). By Chow Homology, Lemma 42.26.3 we have

$\pi _*(\text{div}_{\mathcal{L}'}(s')) = \text{div}_\mathcal {L}(s).$

Hence it suffices to show that $\text{div}_{\mathcal{L}'}(s') = [D]_{n - 1} - [E]_{n - 1}$. This follows from the equality $s' = 1_ D \otimes 1_ E^{-1}$ and additivity, see Divisors, Lemma 31.27.5. $\square$

Definition 115.23.7. Let $(S, \delta )$ be as in Chow Homology, Situation 42.7.1. Let $X$ be locally of finite type over $S$. Assume $X$ integral and $\dim _\delta (X) = n$. Let $D_1, D_2$ be two effective Cartier divisors in $X$. Let $Z \subset X$ be an integral closed subscheme with $\dim _\delta (Z) = n - 1$. The $\epsilon$-invariant of this situation is

$\epsilon _ Z(D_1, D_2) = n_ Z \cdot m_ Z$

where $n_ Z$, resp. $m_ Z$ is the coefficient of $Z$ in the $(n - 1)$-cycle $[D_1]_{n - 1}$, resp. $[D_2]_{n - 1}$.

Lemma 115.23.8. Let $(S, \delta )$ be as in Chow Homology, Situation 42.7.1. Let $X$ be locally of finite type over $S$. Assume $X$ integral and $\dim _\delta (X) = n$. Let $D_1, D_2$ be two effective Cartier divisors in $X$. Let $Z$ be an open and closed subscheme of the scheme $D_1 \cap D_2$. Assume $\dim _\delta (D_1 \cap D_2 \setminus Z) \leq n - 2$. Then there exists a morphism $b : X' \to X$, and Cartier divisors $D_1', D_2', E$ on $X'$ with the following properties

1. $X'$ is integral,

2. $b$ is projective,

3. $b$ is the blowup of $X$ in the closed subscheme $Z$,

4. $E = b^{-1}(Z)$,

5. $b^{-1}(D_1) = D'_1 + E$, and $b^{-1}D_2 = D_2' + E$,

6. $\dim _\delta (D'_1 \cap D'_2) \leq n - 2$, and if $Z = D_1 \cap D_2$ then $D'_1 \cap D'_2 = \emptyset$,

7. for every integral closed subscheme $W'$ with $\dim _\delta (W') = n - 1$ we have

1. if $\epsilon _{W'}(D'_1, E) > 0$, then setting $W = b(W')$ we have $\dim _\delta (W) = n - 1$ and

$\epsilon _{W'}(D'_1, E) < \epsilon _ W(D_1, D_2),$
2. if $\epsilon _{W'}(D'_2, E) > 0$, then setting $W = b(W')$ we have $\dim _\delta (W) = n - 1$ and

$\epsilon _{W'}(D'_2, E) < \epsilon _ W(D_1, D_2),$

Proof. Note that the quasi-coherent ideal sheaf $\mathcal{I} = \mathcal{I}_{D_1} + \mathcal{I}_{D_2}$ defines the scheme theoretic intersection $D_1 \cap D_2 \subset X$. Since $Z$ is a union of connected components of $D_1 \cap D_2$ we see that for every $z \in Z$ the kernel of $\mathcal{O}_{X, z} \to \mathcal{O}_{Z, z}$ is equal to $\mathcal{I}_ z$. Let $b : X' \to X$ be the blowup of $X$ in $Z$. (So Zariski locally around $Z$ it is the blowup of $X$ in $\mathcal{I}$.) Denote $E = b^{-1}(Z)$ the corresponding effective Cartier divisor, see Divisors, Lemma 31.32.4. Since $Z \subset D_1$ we have $E \subset f^{-1}(D_1)$ and hence $D_1 = D_1' + E$ for some effective Cartier divisor $D'_1 \subset X'$, see Divisors, Lemma 31.13.8. Similarly $D_2 = D_2' + E$. This takes care of assertions (1) – (5).

Note that if $W'$ is as in (7) (a) or (7) (b), then the image $W$ of $W'$ is contained in $D_1 \cap D_2$. If $W$ is not contained in $Z$, then $b$ is an isomorphism at the generic point of $W$ and we see that $\dim _\delta (W) = \dim _\delta (W') = n - 1$ which contradicts the assumption that $\dim _\delta (D_1 \cap D_2 \setminus Z) \leq n - 2$. Hence $W \subset Z$. This means that to prove (6) and (7) we may work locally around $Z$ on $X$.

Thus we may assume that $X = \mathop{\mathrm{Spec}}(A)$ with $A$ a Noetherian domain, and $D_1 = \mathop{\mathrm{Spec}}(A/a)$, $D_2 = \mathop{\mathrm{Spec}}(A/b)$ and $Z = D_1 \cap D_2$. Set $I = (a, b)$. Since $A$ is a domain and $a, b \not= 0$ we can cover the blowup by two patches, namely $U = \mathop{\mathrm{Spec}}(A[s]/(as - b))$ and $V = \mathop{\mathrm{Spec}}(A[t]/(bt -a))$. These patches are glued using the isomorphism $A[s, s^{-1}]/(as - b) \cong A[t, t^{-1}]/(bt - a)$ which maps $s$ to $t^{-1}$. The effective Cartier divisor $E$ is described by $\mathop{\mathrm{Spec}}(A[s]/(as - b, a)) \subset U$ and $\mathop{\mathrm{Spec}}(A[t]/(bt - a, b)) \subset V$. The closed subscheme $D'_1$ corresponds to $\mathop{\mathrm{Spec}}(A[t]/(bt - a, t)) \subset U$. The closed subscheme $D'_2$ corresponds to $\mathop{\mathrm{Spec}}(A[s]/(as -b, s)) \subset V$. Since “$ts = 1$” we see that $D'_1 \cap D'_2 = \emptyset$.

Suppose we have a prime $\mathfrak q \subset A[s]/(as - b)$ of height one with $s, a \in \mathfrak q$. Let $\mathfrak p \subset A$ be the corresponding prime of $A$. Observe that $a, b \in \mathfrak p$. By the dimension formula we see that $\dim (A_{\mathfrak p}) = 1$ as well. The final assertion to be shown is that

$\text{ord}_{A_{\mathfrak p}}(a) \text{ord}_{A_{\mathfrak p}}(b) > \text{ord}_{B_{\mathfrak q}}(a) \text{ord}_{B_{\mathfrak q}}(s)$

where $B = A[s]/(as - b)$. By Algebra, Lemma 10.124.1 we have $\text{ord}_{A_{\mathfrak p}}(x) \geq \text{ord}_{B_{\mathfrak q}}(x)$ for $x = a, b$. Since $\text{ord}_{B_{\mathfrak q}}(s) > 0$ we win by additivity of the $\text{ord}$ function and the fact that $as = b$. $\square$

Definition 115.23.9. Let $X$ be a scheme. Let $\{ D_ i\} _{i \in I}$ be a locally finite collection of effective Cartier divisors on $X$. Suppose given a function $I \to \mathbf{Z}_{\geq 0}$, $i \mapsto n_ i$. The sum of the effective Cartier divisors $D = \sum n_ i D_ i$, is the unique effective Cartier divisor $D \subset X$ such that on any quasi-compact open $U \subset X$ we have $D|_ U = \sum _{D_ i \cap U \not= \emptyset } n_ iD_ i|_ U$ is the sum as in Divisors, Definition 31.13.6.

Lemma 115.23.10. Let $(S, \delta )$ be as in Chow Homology, Situation 42.7.1. Let $X$ be locally of finite type over $S$. Assume $X$ integral and $\dim _\delta (X) = n$. Let $\{ D_ i\} _{i \in I}$ be a locally finite collection of effective Cartier divisors on $X$. Suppose given $n_ i \geq 0$ for $i \in I$. Then

$[D]_{n - 1} = \sum \nolimits _ i n_ i[D_ i]_{n - 1}$

in $Z_{n - 1}(X)$.

Proof. Since we are proving an equality of cycles we may work locally on $X$. Hence this reduces to a finite sum, and by induction to a sum of two effective Cartier divisors $D = D_1 + D_2$. By Chow Homology, Lemma 42.24.2 we see that $D_1 = \text{div}_{\mathcal{O}_ X(D_1)}(1_{D_1})$ where $1_{D_1}$ denotes the canonical section of $\mathcal{O}_ X(D_1)$. Of course we have the same statement for $D_2$ and $D$. Since $1_ D = 1_{D_1} \otimes 1_{D_2}$ via the identification $\mathcal{O}_ X(D) = \mathcal{O}_ X(D_1) \otimes \mathcal{O}_ X(D_2)$ we win by Divisors, Lemma 31.27.5. $\square$

Lemma 115.23.11. Let $(S, \delta )$ be as in Chow Homology, Situation 42.7.1. Let $X$ be locally of finite type over $S$. Assume $X$ integral and $\dim _\delta (X) = d$. Let $\{ D_ i\} _{i \in I}$ be a locally finite collection of effective Cartier divisors on $X$. Assume that for all $\{ i, j, k\} \subset I$, $\# \{ i, j, k\} = 3$ we have $D_ i \cap D_ j \cap D_ k = \emptyset$. Then there exist

1. an open subscheme $U \subset X$ with $\dim _\delta (X \setminus U) \leq d - 3$,

2. a morphism $b : U' \to U$, and

3. effective Cartier divisors $\{ D'_ j\} _{j \in J}$ on $U'$

with the following properties:

1. $b$ is proper morphism $b : U' \to U$,

2. $U'$ is integral,

3. $b$ is an isomorphism over the complement of the union of the pairwise intersections of the $D_ i|_ U$,

4. $\{ D'_ j\} _{j \in J}$ is a locally finite collection of effective Cartier divisors on $U'$,

5. $\dim _\delta (D'_ j \cap D'_{j'}) \leq d - 2$ if $j \not= j'$, and

6. $b^{-1}(D_ i|_ U) = \sum n_{ij} D'_ j$ for certain $n_{ij} \geq 0$.

Moreover, if $X$ is quasi-compact, then we may assume $U = X$ in the above.

Proof. Let us first prove this in the quasi-compact case, since it is perhaps the most interesting case. In this case we produce inductively a sequence of blowups

$X = X_0 \xleftarrow {b_0} X_1 \xleftarrow {b_1} X_2 \leftarrow \ldots$

and finite sets of effective Cartier divisors $\{ D_{n, i}\} _{i \in I_ n}$. At each stage these will have the property that any triple intersection $D_{n, i} \cap D_{n, j} \cap D_{n, k}$ is empty. Moreover, for each $n \geq 0$ we will have $I_{n + 1} = I_ n \amalg P(I_ n)$ where $P(I_ n)$ denotes the set of pairs of elements of $I_ n$. Finally, we will have

$b_ n^{-1}(D_{n, i}) = D_{n + 1, i} + \sum \nolimits _{i' \in I_ n, i' \not= i} D_{n + 1, \{ i, i'\} }$

We conclude that for each $n \geq 0$ we have $(b_0 \circ \ldots \circ b_ n)^{-1}(D_ i)$ is a nonnegative integer combination of the divisors $D_{n + 1, j}$, $j \in I_{n + 1}$.

To start the induction we set $X_0 = X$ and $I_0 = I$ and $D_{0, i} = D_ i$.

Given $(X_ n, \{ D_{n, i}\} _{i \in I_ n})$ let $X_{n + 1}$ be the blowup of $X_ n$ in the closed subscheme $Z_ n = \bigcup _{\{ i, i'\} \in P(I_ n)} D_{n, i} \cap D_{n, i'}$. Note that the closed subschemes $D_{n, i} \cap D_{n, i'}$ are pairwise disjoint by our assumption on triple intersections. In other words we may write $Z_ n = \coprod _{\{ i, i'\} \in P(I_ n)} D_{n, i} \cap D_{n, i'}$. Moreover, in a Zariski neighbourhood of $D_{n, i} \cap D_{n, i'}$ the morphism $b_ n$ is equal to the blowup of the scheme $X_ n$ in the closed subscheme $D_{n, i} \cap D_{n, i'}$, and the results of Lemma 115.23.8 apply. Hence setting $D_{n + 1, \{ i, i'\} } = b_ n^{-1}(D_ i \cap D_{i'})$ we get an effective Cartier divisor. The Cartier divisors $D_{n + 1, \{ i, i'\} }$ are pairwise disjoint. Clearly we have $b_ n^{-1}(D_{n, i}) \supset D_{n + 1, \{ i, i'\} }$ for every $i' \in I_ n$, $i' \not= i$. Hence, applying Divisors, Lemma 31.13.8 we see that indeed $b^{-1}(D_{n, i}) = D_{n + 1, i} + \sum \nolimits _{i' \in I_ n, i' \not= i} D_{n + 1, \{ i, i'\} }$ for some effective Cartier divisor $D_{n + 1, i}$ on $X_{n + 1}$. In a neighbourhood of $D_{n + 1, \{ i, i'\} }$ these divisors $D_{n + 1, i}$ play the role of the primed divisors of Lemma 115.23.8. In particular we conclude that $D_{n + 1, i} \cap D_{n + 1, i'} = \emptyset$ if $i \not= i'$, $i, i' \in I_ n$ by part (6) of Lemma 115.23.8. This already implies that triple intersections of the divisors $D_{n + 1, i}$ are zero.

OK, and at this point we can use the quasi-compactness of $X$ to conclude that the invariant

115.23.11.1
$$\label{obsolete-equation-invariant} \epsilon (X, \{ D_ i\} _{i \in I}) = \max \{ \epsilon _ Z(D_ i, D_{i'}) \mid Z \subset X, \dim _\delta (Z) = d - 1, \{ i, i'\} \in P(I)\}$$

is finite, since after all each $D_ i$ has at most finitely many irreducible components. We claim that for some $n$ the invariant $\epsilon (X_ n, \{ D_{n, i}\} _{i \in I_ n})$ is zero. Namely, if not then by Lemma 115.23.8 we have a strictly decreasing sequence

$\epsilon (X, \{ D_ i\} _{i \in I}) = \epsilon (X_0, \{ D_{0, i}\} _{i \in I_0}) > \epsilon (X_1, \{ D_{1, i}\} _{i \in I_1}) > \ldots$

of positive integers which is a contradiction. Take $n$ with invariant $\epsilon (X_ n, \{ D_{n, i}\} _{i \in I_ n})$ equal to zero. This means that there is no integral closed subscheme $Z \subset X_ n$ and no pair of indices $i, i' \in I_ n$ such that $\epsilon _ Z(D_{n, i}, D_{n, i'}) > 0$. In other words, $\dim _\delta (D_{n, i}, D_{n, i'}) \leq d - 2$ for all pairs $\{ i, i'\} \in P(I_ n)$ as desired.

Next, we come to the general case where we no longer assume that the scheme $X$ is quasi-compact. The problem with the idea from the first part of the proof is that we may get and infinite sequence of blowups with centers dominating a fixed point of $X$. In order to avoid this we cut out suitable closed subsets of codimension $\geq 3$ at each stage. Namely, we will construct by induction a sequence of morphisms having the following shape

$\xymatrix{ X = X_0 \\ U_0 \ar[u]^{j_0} & X_1 \ar[l]_{b_0} \\ & U_1 \ar[u]^{j_1} & X_2 \ar[l]_{b_1} \\ & & U_2 \ar[u]^{j_2} & X_3 \ar[l]_{b_2} }$

Each of the morphisms $j_ n : U_ n \to X_ n$ will be an open immersion. Each of the morphisms $b_ n : X_{n + 1} \to U_ n$ will be a proper birational morphism of integral schemes. As in the quasi-compact case we will have effective Cartier divisors $\{ D_{n, i}\} _{i \in I_ n}$ on $X_ n$. At each stage these will have the property that any triple intersection $D_{n, i} \cap D_{n, j} \cap D_{n, k}$ is empty. Moreover, for each $n \geq 0$ we will have $I_{n + 1} = I_ n \amalg P(I_ n)$ where $P(I_ n)$ denotes the set of pairs of elements of $I_ n$. Finally, we will arrange it so that

$b_ n^{-1}(D_{n, i}|_{U_ n}) = D_{n + 1, i} + \sum \nolimits _{i' \in I_ n, i' \not= i} D_{n + 1, \{ i, i'\} }$

We start the induction by setting $X_0 = X$, $I_0 = I$ and $D_{0, i} = D_ i$.

Given $(X_ n, \{ D_{n, i}\} )$ we construct the open subscheme $U_ n$ as follows. For each pair $\{ i, i'\} \in P(I_ n)$ consider the closed subscheme $D_{n, i} \cap D_{n, i'}$. This has “good” irreducible components which have $\delta$-dimension $d - 2$ and “bad” irreducible components which have $\delta$-dimension $d - 1$. Let us set

$\text{Bad}(i, i') = \bigcup \nolimits _{W \subset D_{n, i} \cap D_{n, i'} \text{ irred.\ comp. with }\dim _\delta (W) = d - 1} W$

and similarly

$\text{Good}(i, i') = \bigcup \nolimits _{W \subset D_{n, i} \cap D_{n, i'} \text{ irred.\ comp. with }\dim _\delta (W) = d - 2} W.$

Then $D_{n, i} \cap D_{n, i'} = \text{Bad}(i, i') \cup \text{Good}(i, i')$ and moreover we have $\dim _\delta (\text{Bad}(i, i') \cap \text{Good}(i, i')) \leq d - 3$. Here is our choice of $U_ n$:

$U_ n = X_ n \setminus \bigcup \nolimits _{\{ i, i'\} \in P(I_ n)} \text{Bad}(i, i') \cap \text{Good}(i, i').$

By our condition on triple intersections of the divisors $D_{n, i}$ we see that the union is actually a disjoint union. Moreover, we see that (as a scheme)

$D_{n, i}|_{U_ n} \cap D_{n, i'}|_{U_ n} = Z_{n, i, i'} \amalg G_{n, i, i'}$

where $Z_{n, i, i'}$ is $\delta$-equidimensional of dimension $d - 1$ and $G_{n, i, i'}$ is $\delta$-equidimensional of dimension $d - 2$. (So topologically $Z_{n, i, i'}$ is the union of the bad components but throw out intersections with good components.) Finally we set

$Z_ n = \bigcup \nolimits _{\{ i, i'\} \in P(I_ n)} Z_{n, i, i'} = \coprod \nolimits _{\{ i, i'\} \in P(I_ n)} Z_{n, i, i'},$

and we let $b_ n : X_{n + 1} \to X_ n$ be the blowup in $Z_ n$. Note that Lemma 115.23.8 applies to the morphism $b_ n : X_{n + 1} \to X_ n$ locally around each of the loci $D_{n, i}|_{U_ n} \cap D_{n, i'}|_{U_ n}$. Hence, exactly as in the first part of the proof we obtain effective Cartier divisors $D_{n + 1, \{ i, i'\} }$ for $\{ i, i'\} \in P(I_ n)$ and effective Cartier divisors $D_{n + 1, i}$ for $i \in I_ n$ such that $b_ n^{-1}(D_{n, i}|_{U_ n}) = D_{n + 1, i} + \sum \nolimits _{i' \in I_ n, i' \not= i} D_{n + 1, \{ i, i'\} }$. For each $n$ denote $\pi _ n : X_ n \to X$ the morphism obtained as the composition $j_0 \circ \ldots \circ j_{n - 1} \circ b_{n - 1}$.

Claim: given any quasi-compact open $V \subset X$ for all sufficiently large $n$ the maps

$\pi _ n^{-1}(V) \leftarrow \pi _{n + 1}^{-1}(V) \leftarrow \ldots$

are all isomorphisms. Namely, if the map $\pi _ n^{-1}(V) \leftarrow \pi _{n + 1}^{-1}(V)$ is not an isomorphism, then $Z_{n, i, i'} \cap \pi _ n^{-1}(V) \not= \emptyset$ for some $\{ i, i'\} \in P(I_ n)$. Hence there exists an irreducible component $W \subset D_{n, i} \cap D_{n, i'}$ with $\dim _\delta (W) = d - 1$. In particular we see that $\epsilon _ W(D_{n, i}, D_{n, i'}) > 0$. Applying Lemma 115.23.8 repeatedly we see that

$\epsilon _ W(D_{n, i}, D_{n, i'}) < \epsilon (V, \{ D_ i|_ V\} ) - n$

with $\epsilon (V, \{ D_ i|_ V\} )$ as in (115.23.11.1). Since $V$ is quasi-compact, we have $\epsilon (V, \{ D_ i|_ V\} ) < \infty$ and taking $n > \epsilon (V, \{ D_ i|_ V\} )$ we see the result.

Note that by construction the difference $X_ n \setminus U_ n$ has $\dim _\delta (X_ n \setminus U_ n) \leq d - 3$. Let $T_ n = \pi _ n(X_ n \setminus U_ n)$ be its image in $X$. Traversing in the diagram of maps above using each $b_ n$ is closed it follows that $T_0 \cup \ldots \cup T_ n$ is a closed subset of $X$ for each $n$. Any $t \in T_ n$ satisfies $\delta (t) \leq d - 3$ by construction. Hence $\overline{T_ n} \subset X$ is a closed subset with $\dim _\delta (T_ n) \leq d - 3$. By the claim above we see that for any quasi-compact open $V \subset X$ we have $T_ n \cap V \not= \emptyset$ for at most finitely many $n$. Hence $\{ \overline{T_ n}\} _{n \geq 0}$ is a locally finite collection of closed subsets, and we may set $U = X \setminus \bigcup \overline{T_ n}$. This will be $U$ as in the lemma.

Note that $U_ n \cap \pi _ n^{-1}(U) = \pi _ n^{-1}(U)$ by construction of $U$. Hence all the morphisms

$b_ n : \pi _{n + 1}^{-1}(U) \longrightarrow \pi _ n^{-1}(U)$

are proper. Moreover, by the claim they eventually become isomorphisms over each quasi-compact open of $X$. Hence we can define

$U' = \mathop{\mathrm{lim}}\nolimits _ n \pi _ n^{-1}(U).$

The induced morphism $b : U' \to U$ is proper since this is local on $U$, and over each compact open the limit stabilizes. Similarly we set $J = \bigcup _{n \geq 0} I_ n$ using the inclusions $I_ n \to I_{n + 1}$ from the construction. For $j \in J$ choose an $n_0$ such that $j$ corresponds to $i \in I_{n_0}$ and define $D'_ j = \mathop{\mathrm{lim}}\nolimits _{n \geq n_0} D_{n, i}$. Again this makes sense as locally over $X$ the morphisms stabilize. The other claims of the lemma are verified as in the case of a quasi-compact $X$. $\square$

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