Let $R$ be a discrete valuation ring. Then $R$ has only two prime ideals: $0$ and the maximal ideal $\mathfrak{m}$. It is said in Hartshorne, page 74, Example 2.3.2 that the localization of $R$ at $0$ is $K$, where $K$ is the field of fractions of $R$ and the localization of $R$ at $\mathfrak{m}$ is $R$. In the book, $0$ is denoted by $t_2$ ($t_2=0$ is open) and $\mathfrak{m}$ is denoted by $t_1$ ($t_1=\mathfrak{m}$ is closed).
But according to Wikipedia it seems that $R_{\mathfrak{m}}$ is also $K$ since $R-\mathfrak{m}$ does not contain $0$.
I am confused. Could you explain this? Thank you very much.
