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I am trying to understand the multiplicities of Cartier divisors in Liu’s book on the relevant chapter. Actually I cannot understand the following conclusion of this remark.

Now let $ $ be an open everywhere dense subset of $$ such that $_{}=0$. Then: Any $∈$ of codimension 1 such that $mult()≠0$ is a generic point of $−. $

THEN: The above implies that in an open affine subset of $X$, there are only finite many codimension 1 points such that the corresponding multiplicity are non-zero. I’ve seen this question but I really need to understand the full conclusion, namely the last statement.

T. Wildwolf
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  • You need to provide more background here: what specific place in Liu's book is this from? What assumptions are involved here? What is $X$, specifically? Next, you probably have a typo in the last paragraph: you want "only finitely many codimension 1 points such that the corresponding multiplicity is *non-zero*", not zero as currently said. – KReiser Dec 12 '22 at 20:15

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Actually i found out. We restrict to an open affine $V$ and the closure of the intersection $U\cap V$ in $V$ is $V$. Hence we can apply the first claim, since the affine $V$ is noetherian and hence has finitely many irreducible components.

T. Wildwolf
  • 687