6
$\begingroup$

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.

  • 1
    This is a more general fact about the Proj of a graded ring R. At a homogeneous prime $\mathfrak{p}$, the stalk is isomorphic to $R_{(\mathfrak{p})}$. This is because the Proj can be defined by gluing together "basic" open sets of the form $\mathrm{Spec} R_{(f)}$ (for $f$ homogeneous), and the direct limit of $R_{(f)})$ for $f \notin \mathfrak{p}$ will be precisely what was claimed.2010-11-08
  • 0
    @Akhil: Dear Akhil, I think that you want to take degree $0$ parts of the various localizations in your comment. (Added: or maybe this is implicit in your notation?)2010-12-08
  • 0
    Dear @Matt E: Yes, this is what I mean (EGA uses the parentheses to denote elements of degree zero).2010-12-08
  • 0
    @Akhil: Dear Akhil, Thanks; I wondered if this was the case as I was posting my comment (hence my "Added" remark).2010-12-08

2 Answers 2