Another question to add to the above is, also for local stalks?
In detail:
For affine schemes $X = \operatorname{Spec}(R)$ with structure sheaf $\mathcal{O}$ we have the following general facts:
- for open $U\subseteq X$,
we have that the ring $\mathcal{O}(U)$ is identified with (in a way extending to a sheaf isomorphism)
$\Gamma_{\text{global}}(\mathcal{O}) = R$ localized by (the multiplicative set of) all $f \in R$ which are nowhere-vanishing in $U$. - for a point $x\in X$, the local stalk $\mathcal{O}_x$ is identified with
$R$ localized by (the multiplicative set of) all $f \in R$ which are non-vanishing at $x$.
My questions are:
-
Do these also hold for general schemes $(X,\mathcal{O})$, with $R$ replaced by $\Gamma_{\text{global}}(\mathcal{O})$?
-
If the answer to 1 is yes: does it also hold for locally ringed spaces in general?
Best Answer
No. For example, $X = \mathbb P^1_{k}$, and take $U = \mathbb A^1_k\subset X$ which has $\Gamma(U, O_X|_U) = k[x]$ so that the stalks at any point of $U$ (or $X$ for that matter) will be way larger than $k$ so could not possibly be a localization of the global sections.