Let $(R, \mathfrak m)$ be a Noetherian local ring of dimension $1$. Then the affine- scheme $X=\operatorname {Spec}(R)$ can be written as a set-theoretic union $\operatorname{Spec}(R)=Min(R)\cup \{\mathfrak m\}$.
So if $U=\operatorname{Spec}(R)\setminus \{\mathfrak m\}$ is the punctured spectrum, then $U$ is a finite set and every point is a closed point (under the subspace topology), even more, every Singleton set is an irreducible component. In particular, $\dim U=0$.
My question is: Is $(U, \mathcal O_X|_U) $ an affine- scheme ? If it is indeed an affine-scheme, then it is the Spectrum of an Artinian ring (as $U$ is a Noetherian scheme of dimension zero), in which case, can we identify the global section ring $\Gamma_U(U)$ ?
I'm willing to assume $R$ is reduced if that helps.
Best Answer
Note that $U$ is a discrete topological space, so the structural sheaf over $U$ is simply a product of the rings of stalks at each point. In particular, if $V \subset U$ is open, clearly $\mathcal{O}_U(V)=\prod_{x \in V}{\mathcal{O}_{U,x}}=\prod_{p \in V}{R_p}$.
But each such $R_p$ is a ring with a single prime ideal: so that $(U,\mathcal{O}_U)$ is the spectrum of the product of the $R_p$ over $p \in Min(A)$.