# A lemma on the integral closure of a Noetherian domain of dimension 1

I need to prove the following lemma(?) which is motivated by this and this.

Lemma Let $A$ be a Noetherian domain of dimension 1. Let $K$ be the field of fractions of $A$. Let $B$ be the integral closure of $A$ in $K$. Suppose $B$ is finitely generated as an $A$-module. Let $\mathfrak{p}$ be a maximal ideal of $A$. Let $B_{\mathfrak{p}}$ be the localization of B with respect to the multiplicative subset $A - \mathfrak{p}$. Then $K^*/(B_{\mathfrak{p}})^*$ is isomorphic to $\bigoplus K^*/(B_{\mathfrak{P}})^*$, where $\mathfrak{P}$ runs over all the maximal ideals of $B$ lying over $\mathfrak{p}$.

EDIT[Jun 26, 2012] Using this lemma, we can prove the following result.

Proposition Let $A$ be a Noetherian domain of dimension 1. Let $K$ be the field of fractions of $A$. Let $B$ be the integral closure of $A$ in $K$. Suppose $B$ is finitely generated as an $A$-module. Let $I(B)$ be the group of invertible fractional ideals of $B$. Then $I(B)$ is canonically isomorphisc to $\bigoplus_{\mathfrak{p}} K^*/(B_{\mathfrak{p}})^*$. Here, $\mathfrak{p}$ runs on all the maximal ideals of $A$.

Proof: Since $B$ is a Noetherian domain of dimension 1, by this, $I(B)$ is canonically isomorphisc to $\bigoplus_{\mathfrak{P}} I(B_{\mathfrak{P}})$, where ${\mathfrak{P}}$ runs over all the maximal ideals of $B$. Since $B_{\mathfrak{P}}$ is a local domain, by this, $I(B_{\mathfrak{P}})$ is the group of principal fractional ideals of $B_{\mathfrak{P}}$. Hence $I(B_{\mathfrak{P}})$ is canonically isomorphic to $K^*/(B_{\mathfrak{P}})^*$. Hence $I(B)$ is canonically isomorphisc to $\bigoplus_{\mathfrak{P}} K^*/(B_{\mathfrak{P}})^*$. Hence by the above lemma, $I(B)$ is canonically isomorphisc to $\bigoplus_{\mathfrak{p}} K^*/(B_{\mathfrak{p}})^*$. QED

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## 1 Answer

Since $B_{\mathfrak{p}}$ is an integrally closed Noetherian domain of dimension 1, it is a Dedekind domain. Since it has only finitely many maximal ideals, it is a PID. Let $R = B_{\mathfrak{p}}$. By this, $K^*/R^*$ is canonically isomorphic to $\bigoplus_M K^*/(R_M)^*$, where $M$ runs on all the mximal ideals of $R$. Since $M$ is of a form $\mathfrak{P}R$, where $\mathfrak{P}$ is a maximal ideal of $B$ lying over $\mathfrak{p}$, $R_M$ is canonically isomorphic to $B_{\mathfrak{P}}$

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