A scheme $X$ is separated if the daiganoal morphism $\Delta:X \rightarrow X \times_{\Bbb Z} X$ is a closed immersion. I know how to show that all affine schemes are separated. So
Are open subschemes of affine schemes separated? If not, what is an example?
For example, I would be interested to know if the open subscheme
$$ D= Spec\, k[x.y] \setminus \{0\} \rightarrow Spec \,k[x.y] $$
is separated.
Best Answer
They are. In fact, any open subscheme of a separated scheme is separated.
Let $U$ be an open subscheme of X. Base change $X\to X\times_\mathbb{Z}X$ by the map from $U\times_\mathbb{Z}U$ induced by the inclusion $U\to X$. The fiber product is just $U$, and the induced morphism is the diagonal of $U$. Since closed immersions are stable under base change, $U$ is also separated.