The stalk of a point on a scheme is a localization of ring of affine open?

Question

Let $(X,\mathcal O_X)$ be a Noetherian scheme. For every affine open subset $U$ of $X$, it holds that $U=\text{Spec}(\mathcal O_X(U))$. Let $x \in X$, and let $U$ be an affine open subset of $X$ containing $x$. Then, do we have an isomorphism of rings $\mathcal O_{X,x}\cong \mathcal O(U)_x$ ?

Answer 1

This is true and at a larger level of generality than you state. To do this we'll use a sheaf theory lemma first for some heavy lifting, but the crux of the scheme theory sits in the last paragraph if you'd like to ignore the sheaf theory at first glance.

Lemma: Let $X$ be a topological space and let $\mathscr{F}$ be a sheaf on $X$ (so in particular $\mathscr{F}$ is a functor $\mathscr{F}:\mathbf{Open}(X)^{\operatorname{op}} \to \mathbf{Set}$). Then for any point $x \in X$ and any open $U \subseteq X$ with $x \in U$, $\mathscr{F}_x \cong \mathscr{F}|_{U,x}$.

Proof: I'll give a categorical proof of this because it's cute and shows how to do this sort of stuff formally. Let $i:U \to X$ be the inclusion and note that $\mathscr{F}|_{U} := i^{-1}\mathscr{F}$ as sheaves on $U$. Now recall the formula for the definition of $i^{-1}$ in terms of sheafification: this allows us to compute $i^{-1}\mathscr{F}$ as the composite, for any open $V \subseteq U$, $$ i^{-1}\mathscr{F}(V) := \left(\operatorname*{colim}_{\substack{W \subseteq X\, \text{open} \\ U \subseteq W}} \mathscr{F}(V \times_X W)\right)^{++} \cong \left(\operatorname*{colim}_{\substack{W \subseteq X\, \text{open} \\ U \subseteq W}} \mathscr{F}(V \cap W)\right)^{++} = \left(\operatorname*{colim}_{\substack{W \subseteq X\, \text{open} \\ U \subseteq W}} \mathscr{F}(V)\right)^{++} \cong \mathscr{F}(V) $$ where $(-)^{++}$ denotes sheafification; note that the last isomorphism follows because $\mathscr{F}$ is a sheaf. Thus $$ (\mathscr{F}|_{U})_x = \operatorname*{colim}_{\substack{V \subseteq U\,\text{open} \\ x \in V}} \mathscr{F}|_{U}(V) \cong \operatorname*{colim}_{\substack{V \subseteq U\,\text{open} \\ x \in V}}\mathscr{F}(V) \cong \operatorname*{colim}_{\substack{V \subseteq X\,\text{open} \\ x \in V}} \mathscr{F}(V) = \mathscr{F}_x. $$ Note that the last isomorphism uses that the set $\lbrace V \subseteq X \; | \; V \subseteq U, x \in V \rbrace$ is a cofinal family in the lattice of open sets of $X$ --- topologically this is just saying that if something happens really, really close to $x$ it may as well happen in an open that also sits inside $U$. I've shown this in a ``high-tech'' way, but it's something that's worth doing by hand at least once.

Now assume that $X = (\lvert X \rvert, \mathcal{O}_X)$ is a scheme (note that $\lvert X \rvert$ is the underlying space of $X$) and let $U$ be an affine open subscheme of $X$. Then by the lemma above we have that $$ \mathcal{O}_{X,x} \cong (\mathcal{O}_{X})|_{U, x} \cong \mathcal{O}_{U,x}. $$ Since $\mathcal{O}_A(\lvert \operatorname{Spec} A\rvert)_{\mathfrak{p}} \cong A_{\mathfrak{p}}$ for any commutative ring and affine scheme $\operatorname{Spec} A$, it follows that $\mathcal{O}_{U,x} \cong \mathcal{O}_{U}(\lvert U \rvert)_{x}$. Putting these observations together gives $$ \mathcal{O}_{X,x} \cong \mathcal{O}_{U,x} \cong \mathcal{O}_{U}(\lvert U \rvert)_{x}, $$ which is exactly what you wanted.