Lemma 31.15.8. Let X be a Noetherian scheme. Let Z \subset X be a closed subscheme. Assume there exist integral effective Cartier divisors D_ i \subset X and a closed subset Z' \subset X of codimension \geq 2 such that Z \subset Z' \cup \bigcup D_ i set-theoretically. Then there exists an effective Cartier divisor of the form
D = \sum a_ i D_ i \subset Z
such that D \to Z is an isomorphism away from codimension 2 in X. The existence of the D_ i is guaranteed if \mathcal{O}_{X, x} is a UFD for all x \in Z or if X is regular.
Proof.
Let \xi _ i \in D_ i be the generic point and let \mathcal{O}_ i = \mathcal{O}_{X, \xi _ i} be the local ring which is a discrete valuation ring by Lemma 31.15.4. Let a_ i \geq 0 be the minimal valuation of an element of \mathcal{I}_{Z, \xi _ i} \subset \mathcal{O}_ i. We claim that the effective Cartier divisor D = \sum a_ i D_ i works.
Namely, suppose that x \in X. Let A = \mathcal{O}_{X, x}. Let D_1, \ldots , D_ n be the pairwise distinct divisors D_ i such that x \in D_ i. For 1 \leq i \leq n let f_ i \in A be a local equation for D_ i. Then f_ i is a prime element of A and \mathcal{O}_ i = A_{(f_ i)}. Let I = \mathcal{I}_{Z, x} \subset A be the stalk of the ideal sheaf of Z. By our choice of a_ i we have I A_{(f_ i)} = f_ i^{a_ i}A_{(f_ i)}. We claim that I \subset (\prod _{i = 1, \ldots , n} f_ i^{a_ i}).
Proof of the claim. The localization map \varphi : A/(f_ i) \to A_{(f_ i)}/f_ iA_{(f_ i)} is injective as the prime ideal (f_ i) is the inverse image of the maximal ideal f_ iA_{(f_ i)}. By induction on n we deduce that \varphi _ n : A/(f_ i^ n)\to A_{(f_ i)}/f_ i^ nA_{(f_ i)} is also injective. Since \varphi _{a_ i}(I) = 0, we have I \subset (f_ i^{a_ i}). Thus, for any x \in I, we may write x = f_1^{a_1}x_1 for some x_1 \in A. Since D_1, \ldots , D_ n are pairwise distinct, f_ i is a unit in A_{(f_ j)} for i \not= j. Comparing x and x_1 at A_{(f_ i)} for n \geq i > 1, we still have x_1 \in (f_ i^{a_ i}). Repeating the previous process, we inductively write x_ i = f_{i + 1}^{a_{i + 1}}x_{i + 1} for any n > i \geq 1. In conclusion, x \in (\prod _{i = 1, \ldots n} f_ i^{a_ i}) for any x \in I as desired.
The claim shows that \mathcal{I}_ Z \subset \mathcal{I}_ D, i.e., that D \subset Z. Moreover, we also see that D and Z agree at the \xi _ i, which proves that D \to Z is an isomorphism away from codimension 2 on X.
To see the final statements we argue as follows. A regular local ring is a UFD (More on Algebra, Lemma 15.121.2) hence it suffices to argue in the UFD case. In that case, let D_ i be the irreducible components of Z which have codimension 1 in X. By Lemma 31.15.7 each D_ i is an effective Cartier divisor.
\square
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