Let $X = Spec(\mathbb{Z}[t])$ and $U = D(2)\cup D(t)$. I want to show that the open subscheme $(U,\mathcal{O}_X|_U)$ is not affine. There is a hint to determine $\mathcal{O}_X(U)$. The question has already been discussed here.
But I am lost trying to show that $\mathcal{O}_X(U) = \mathcal{O}_X(D(2))\cap \mathcal{O}_X(D(t))$. First, I wanted to know if there was some general relation $\mathcal{F}(U\cup V) \subset \mathcal{F}(U)\cap \mathcal{F}(V)$ for scheaves of sets, abelian groups,…. And if so, how to prove it?
If not, then this has to do with the special structure of $\mathcal{O}_X$ and I need to consider the projective/inverse limit
$\mathcal{O}_X(U) = \underset{\underset{D(f)\subset U}{\leftarrow}}{\lim} \mathcal{O}_X(D(f)) =\underset{\underset{D(f)\subset U}{\leftarrow}}{\lim}\mathbb{Z}[t]_f$, where $U = D(2)\cup D(t) = X – \{(2,t)\}$.
But I wasn't able to compute it yet. Maybe it's trivial and I am using overkill here.
Best Answer
By the gluing condition on sheaves, $\mathcal{O}_X(U \cup V)$ is isomorphic to $\mathcal{O}_X(U) \times_{\mathcal{O}_X(U\cap V)} \mathcal{O}_X(V) = \{ (f, g) \in \mathcal{O}_X(U) \times \mathcal{O}_X(V) \mid f |_{U\cap V} = g |_{U\cap V} \}$. However, in this case, the restriction maps $\mathcal{O}_X(D(2)) \to \mathcal{O}_X(D(2, t))$ and $\mathcal{O}_X(D(t)) \to \mathcal{O}_X(D(2, t))$ are injective; and in general, the fibered product of two injective maps is isomorphic to the intersection of the images.