Finite union of affine schemes

Question

Let $X$ be a scheme and suppose that $X$ admits a finite open covering $(U_i)_{i\in I }$ of affine schemes, such that $X_i\cap X_j$ is an affine scheme, for all $i,j\in I$. In this case is it true that $X$ is an affine scheme?

Answer 1

This is not true. Any projective space $\Bbb P_k^n$ for a field $k$ gives a counterexample.

Answer 2

Here's another (horrible!) example: the union

$$ \mathbb{A}^2_R \setminus \{ (0,0) \} = \operatorname{Spec}(R[x, y, x^{-1}]) \cup \operatorname{Spec}(R[x, y, y^{-1}]) $$

is not an affine scheme (despite being a subscheme of an affine scheme!). The intersection of the two patches is

$$\operatorname{Spec}(R[x, y, x^{-1}]) \cap \operatorname{Spec}(R[x, y, y^{-1}]) = \operatorname{Spec}(R[x, y, x^{-1}, y^{-1}]) $$