Let $k$ be an algebraic closed field. Let $x$ be a point in $X=P_k^1$. What is $O_{X,x}$?
For example, if I have $x=(t-a)\in \text{Spec }k[t]$. Looking $x$ inside $P_k^1$, does $O_{X,x}=k[t]_{(t-a)}$? I'm confused when I have to deal with the sheaf of rings.