let $ X $ be a separated scheme over an affine scheme $ S$. let $U$ and $V$ be open affine subsets of $X$.Then $U\cap V$ is also affine.and Give an example to to show that this is fails if $X$ is not separated.
This is an exercise 4.3(Chapter 2)of Hartshorne.
I know that closed subschme of an affine scheme is affine. But I don't know how to use this when $S$ is affine scheme.
Thank you in advance!
Best Answer
The affine plane with the doubled origin has open affine sets $U$ and $V$, each isomorphic to the affine plane. But $U \cap V$ is the affine plane with a point removed, which is not affine.