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Let $K(A)$ denote the fraction field of $A$. For context, I'm trying to prove $A = \bigcap_{\text{ht}(\mathfrak{p}) = 1} A_{\mathfrak{p}}$ for an integrally closed domain $A$, from Atiyah-Macdonald's corollary 5.22 which states that $A$ is the intersection of all valuation rings containing it. This would follow from the following conjecture: $R$ is a valuation ring of $K(A)$ iff $R = A_{\frak{p}}$ for some $\mathfrak{p} \subseteq A$ of height 1.

Before that, though, I want to prove an intermediate result that if $(R, \frak{m})$ is a local ring in $K(A)$ containing $A$, then $R = A_{\frak{p}}$ for some prime $\frak{p}$. The choice for $\mathfrak{p}$ is very natural, we take $\mathfrak{p} = A \cap \mathfrak{m}$. Then, if $x \in A \setminus \mathfrak{p}$, we would have that $x$ is a unit in $R$, since $x \notin \frak{m}$. Hence, we have an induced homomorphism $A_{\mathfrak{p}} \to R$. I think this is injective by definition, but I am not sure why it would be surjective.

Alternatively, I thought that $R$ may just satisfy the universal property of $A_{\mathfrak{p}}$ but this is still unclear. Is this just not true given the hypotheses or is there something I am missing?

Thanks!

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This is not true. For instance, let $k$ be a field and $A=k[x,y]$, and let $B=A[y/x]$. Note that $B=k[x,y,y/x]$ can be identified with a polynomial ring $k[x,z]$ by mapping $y$ to $xz$ and $y/x$ to $z$. In particular, $(x,y/x)$ is a maximal ideal of $B$; let $R$ be the localization of $B$ at this maximal ideal. I claim $R$ is not a localization of $A$. To see this, note that if $\mathfrak{m}$ is the maximal ideal of $R$, then $\mathfrak{m}\cap A=(\mathfrak{m}\cap B)\cap A=(x,y/x)\cap A$ is the ideal $(x,y)$ of $A$. So if $R$ is a localization of $A$, it must be the localization at $(x,y)$. But $y$ is not divisible by $x$ in that localization, whereas $y$ is divisible by $x$ in $B$.

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  • $\begingroup$ Great counterexample! I suppose my strategy for proving the theorem is probably not salvageable then. $\endgroup$
    – Daniel
    Commented Aug 11, 2021 at 17:10

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