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The Stacks project

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


Comments (5)

Comment #7190 by on

It seems too fast for me to deduce from . I understand it as follows:

The localization map is injective as the prime ideal is the inverse image of the maximal ideal . By dévissage, we deduce that is also injective. Since , we have . Thus, for any , we write for some . We may assume that the concerning effective Cartier divisor are pairwise different, so that is a unit in for . Comparing and at for , we still have for . Repeating the previous process, we inductively write for any . In conclusion, for any .

I will appretiate if anyone can share his or her faster ideas.

Comment #7191 by on

Very good. I will add this when I next go through all the comments. Thanks!

Comment #9555 by Branislav Sobot on

Are you assuming in the statement that there are finitely many divisors ? If so, I suggest you add it. If not, then I don't understand what would the sum of these divisors be since we only defined finite sums of (effective Cartier) divisors. It also doesn't hurt to add to the statement that are nonnegative integers.

Comment #9556 by Branislav Sobot on

Also, for the proof of the final statement, what happens with the irreducible components of which have codimension zero in ? For instance, what if is irreducible and , how do you find ?


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