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Let $A$ be a Noetherian domain of dimension 1. Let $K$ be its field of fractions. Let $B$ be the integral closure of $A$ in $K$. Suppose $B$ is finitely generated as an $A$-module. It is well-known that $B$ is a Dedekind domain. Let $I(A)$ be the group of invertible fractional ideals of $A$. Let $P(A)$ be the group of principal fractional ideals of $A$. Similarly we define $I(B)$ and $P(B)$.

Then there exists the following exact sequence of abelian groups(Neukirch, Algebraic number theory p.78).

$0 \rightarrow B^*/A^* \rightarrow \bigoplus_{\mathfrak{p}} (B_{\mathfrak{p}})^*/(A_{\mathfrak{p}})^* \rightarrow I(A)/P(A) \rightarrow I(B)/P(B) \rightarrow 0$

Here, $\mathfrak{p}$ runs over all the maximal ideals of $A$.

Since we use this result to prove this, it'd be nice that we have the proof here(I don't understand well Neukirich's proof).

EDIT Since someone wonders what my question is(though I think it is obvious), I state it more clearly: How do you prove it?

EDIT[July 11, 2012] May I ask the reason for the downvote so that I could improve my question?

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    I noticed that someone serially upvoted for my questions. While I appreciate them, I would like to point out that serial upvotes are automatically reversed by the system.2013-11-27

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Lemma 1 Let $A$ be a Noetherian domain of dimension 1. Let $K$ be its field of fractions. Then there exists the following exact sequence of abelian groups.

$0 \rightarrow K^*/A^* \rightarrow \bigoplus K^*/(A_{\mathfrak{p}})^* \rightarrow I(A)/P(A) \rightarrow 0$

Here, $\mathfrak{p}$ runs over all the maximal ideals of $A$.

Proof: By this, $I(A)$ is canonically isomorphic to $\bigoplus I(A_{\mathfrak{p}})$. By this, $I(A_{\mathfrak{p}})$ is the group of principal fractional ideals of $A_{\mathfrak{p}}$. Hence $I(A_{\mathfrak{p}})$ is canonically isomorphic to $K^*/(A_{\mathfrak{p}})^*$.

On the other hand, $P(A)$ is canonically isomorphic to $K^*/A^*$. QED

Lemma 2 Let $A$ be a Noetherian domain of dimension 1. Let $K$ be its field of fractions. Let $B$ be the integral closure of $A$ in $K$. Suppose $B$ is finitely generated as an $A$-module. Then there exists the following exact sequence of abelian groups.

$0 \rightarrow K^*/B^* \rightarrow \bigoplus K^*/(B_{\mathfrak{p}})^* \rightarrow I(B)/P(B) \rightarrow 0$

Here, $\mathfrak{p}$ runs over all the maximal ideals of $A$.

Proof: This follows immediately from the proposition of this. QED

The proof of the exactness of the title sequence There exists a canonical morphism from the exact sequence of Lemma 1 to that of Lemma 2. The exactness of the title sequence follows immediately by snake lemma. QED