I'm reading A Royal Road to Algebraic Geometry by Holme. The book defines the local ring as follows:
The local ring of $K$ at $P=(a,b)$ is the ring $$\mathcal{O}_{K,P}=\Gamma(K)_{\mathfrak{m}(a,b)},$$ where $\Gamma(K)$ is the coordinate ring of the curve $K$.
I know there must be something obvious I am missing. I am very confused with this definition. I understand that essentially the local ring at a point $P$ is the set of functions $\frac{f}{g}$ where $f,g\in \Gamma(K)$, and $g(P)\ne 0$.
But in this definition, the symbol $\Gamma(K)_{\mathfrak{m}(a,b)}$ means the set of the functions of the form $\frac{f}{g}$, $f\in \Gamma(K), g\in \mathfrak{m}(a,b)$. So this means $g(P)$ must be zero.
Could someone explain what I am missing? Thank you for your help!