The Stacks project

113.21 Intersection theory

Lemma 113.21.1. 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 113.21.2. 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 42.29.1. $\square$

Remark 113.21.3. This remark used to say that it wasn't clear whether the arrows of Chow Homology, Lemma 42.22.2 were isomorphisms in general. However, we've now found a proof of this fact.

113.21.4 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 113.21.11 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 113.21.5. 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 42.25.3 and the fact that $D$ is the zero scheme of the canonical section $1_ D$ of $\mathcal{O}_ X(D)$. $\square$

Lemma 113.21.6. 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 42.25.3 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 113.21.7. 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 113.21.8. 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 10.123.1 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 113.21.9. 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 113.21.10. 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 42.23.2 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 113.21.11. 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 113.21.8 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 113.21.8. 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 113.21.8. 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

113.21.11.1
\begin{equation} \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)\} \end{equation}

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 113.21.8 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 113.21.8 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 113.21.8 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 (113.21.11.1). 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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