Stalks on Projective Scheme

Question

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.

Answer 1

The stalk of the generic point is, of course, the rational function field $k(t)$. Now $\mathbb{P}^1$ is covered by two copies of $\mathbb{A}^1$. If a closed point $x$ corresponds to the maximal ideal generated by $(t-a)$ for some $a \in k$, then the stalk is

$O_{\mathbb{A}^1,x} = k[t]_{(t-a)}$

and this is isomorphic to $k[t]_{(t)}$.

Answer 2

For topological space $X$, open subset $U \subset X$ and point $x\in U$ we have for all sheaf $F$ on $X$ : $F_x = (F|U)_x$. Apply to $X=\mathbb P ^1, U=\mathbb A^1, x=(t-a)$