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The Stacks project

Lemma 31.15.11. Let X be a Noetherian scheme. Let D \subset X be an effective Cartier divisor. Assume that there exist integral effective Cartier divisors D_ i \subset X such that D \subset \bigcup D_ i set theoretically. Then D = \sum a_ i D_ i for some a_ i \geq 0. The existence of the D_ i is guaranteed if \mathcal{O}_{X, x} is a UFD for all x \in D or if X is regular.

Proof. Choose a_ i as in Lemma 31.15.8 and set D' = \sum a_ i D_ i. Then D' \to D is an inclusion of effective Cartier divisors which is an isomorphism away from codimension 2 on X. Pick x \in X. Set A = \mathcal{O}_{X, x} and let f, f' \in A be the nonzerodivisor generating the ideal of D, D' in A. Then f = gf' for some g \in A. Moreover, for every prime \mathfrak p of height \leq 1 of A we see that g maps to a unit of A_\mathfrak p. This implies that g is a unit because the minimal primes over (g) have height 1 (Algebra, Lemma 10.60.11). \square


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