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.
Best Answer
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 $X=\mathbb P ^1, U=\mathbb A^1, x=(t-a)$