stalk of projective variety in terms of the coordinate ring

Question

Let $X \subseteq \mathbb{P}^n$ be an embedded projective variety over some field $k$ with its corresponding homogeneous coordinate ring $R = k[X_0,\dots,X_n]/I(X)$. Let further $X = \bigcup_{i=0}^n U_i$ be the standard covering of $X$ by affine varieties, and $p \in X$ some point. When we want to consider the stalk of $X$ at $p$ we can reduce to the affine case since $\mathcal{O}_{X,p} \cong \mathcal{O}_{U_i,p}$ if $p \in U_i$. Because of this the stalk at $p$ can be described in terms of the coordinate ring by dehomogenizing : Let for example $p = (1 : p_1 : \dots : p_n) \in U_0$ then $\mathcal{O}_{X,p} \cong (R/(X_0-1))_{(X_1-p_1, \dots, X_n-p_n)}$.

This description however depends on the choice of the particular affine open sub-variety $U_i$ which contains $p$. Is there another (unifying) way to give $\mathcal{O}_{X,p}$ in terms of $R$ which does not depend on the location of $p$ in $X$?

Thank you in advance!

Answer 1

Given a completely arbitrary graded ring $R$ (in particular not supposed to be a domain) and its associated projective scheme $X=\text{Proj}R$, the stalk of the structure sheaf $\mathcal O_X$ at $\mathfrak p\in X$ is $$\mathcal O_{X,\mathfrak p}=R_{(\mathfrak p)}$$ Here $R_{(\mathfrak p)}\subset R_{\mathfrak p}$ is the subring consisting of fractions $\frac r\pi$ where $r\in R, \pi\in R\setminus \frak p$ are homogeneous elements of the same positive degree.

References:
Hartshorne Proposition 2.5 (a), page 76 or EGA II Proposition (2.5.5), page 31 or Miyanishi Lemma 5.3(3), page 106.
Non-references:
After a superficial check, I noted that the result seems to be missing from such textbooks as Bosch, Eisenbud-Harris, Görtz-Wedhorn, Qing Liu.
This is no criticism of these fine references: obviously, writing mathematical books implies tough choices on what to omit.

Answer 2

Often the structure sheaf of $\mathrm{Proj}(A)$ (where $A$ is a commutative graded ring) is constructed via gluing of the affine pieces $D_+(f) \cong \mathrm{Spec}(A_{(f)})$. But actually, we can write it down explicitly. More generally, if $M$ is some graded $A$-module, we can write down the sheaf $\widetilde{M}$ explicitly. The idea is to make the same as in the affine case, but replace localizations by homogeneous localizations.

Recall that $\mathrm{Proj}(A)$ is the space of homogeneous prime ideals of $A$ not containing $A_+$. Let $M$ be a graded $A$-module. Let $U \subseteq \mathrm{Proj}(A)$ be an open subset. We define $\widetilde{M}(U)$ as a subset of $\prod_{\mathfrak{p} \in U} M_{(\mathfrak{p})}$ (here $M_{(\mathfrak{p})}$ denotes the homogeneous localization of $M$ at $\mathfrak{p}$) consisting of those tuples $(s_\mathfrak{p})$ such that for all $\mathfrak{p} \in U$ there is some $\mathfrak{p} \in V \subseteq U$ open and some homogeneous elements $m \in M$, $r \in A$ of the same degree such that for all $\mathfrak{q} \in V$ we have $r \notin \mathfrak{q}$ and $\frac{m}{r} = s_\mathfrak{q}$ in $M_{(\mathfrak{q})}$.

From this description it is quite clear that $\widetilde{M}_\mathfrak{p} = M_{(\mathfrak{p})}$.

After writing this answer, I realize that this is also done in Hartshorne's book.