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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.

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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. –  Akhil Mathew Nov 8 '10 at 1:32
    
@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?) –  Matt E Dec 8 '10 at 6:19
    
Dear @Matt E: Yes, this is what I mean (EGA uses the parentheses to denote elements of degree zero). –  Akhil Mathew Dec 8 '10 at 13:21
    
@Akhil: Dear Akhil, Thanks; I wondered if this was the case as I was posting my comment (hence my "Added" remark). –  Matt E Dec 8 '10 at 14:04

2 Answers 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 $X=\mathbb P ^1, U=\mathbb A^1, x=(t-a)$

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@Mathnoob: $\mathbb{P}^1$ (or more generally the Proj of a graded ring) is defined in EGA precisely by gluing together open affines (and verifying that the gluing is sensible). In Hartshorne, I believe it is mentioned shortly after the definition that $Proj(R)$ is covered by $\mathrm{Spec}R_{(f)}$ for $f$ homogeneous. In the case of $\mathbb{P}^1 = Proj k[t, u]$, we have $\mathrm{Spec} k[t, u]_{(u)}$ as the open set, which is just $\mathrm{Spec} k[t]$. –  Akhil Mathew Nov 8 '10 at 1:35
    
Note incidentally that if $R$ is a graded ring, and $f \in R$ a homogeneous element of degree $d$, then the homogeneous localization $R_{(f)}$ is isomorphic as a ring to $R^{(d)}/(f-1) R^{(d)}$ where $R^{(d)}$ consists of the fraction of $R$ that sits in degree $d$ or multiples thereof. This is an easy way of noting that $k[t,u]_{(u)} \simeq k[t]$. (Cf. EGA 2.2.) –  Akhil Mathew Nov 8 '10 at 1:36

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 $k[t]_{(t)}$.

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@John: $\mathbb P^1$ is most easily defined as being two copies of $\mathbb A^1$ glued together in the usual way. So the formula for restrictions that you ask about is automatic, and is part of the construction. –  Matt E Dec 8 '10 at 6:22

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