An exact sequence on the ideal class group of a Noetherian domain of dimension 1 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?
 A: 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
