What are some examples of morphisms of schemes which are not quasi separated?



Non-quasi separated morphisms
Suppose $U\hookrightarrow X$ is a non-quasi-compact open immersion. Then you can glue two copies of $X$ together along $U$ (effectively doubling up the complement of $U$) to get a non-quasi-separated scheme $Y$.


By the assumption that $U\hookrightarrow X$ is not quasi-compact, there is some open affine $W$ of $X$ such that the intersection of $W$ and $U$ is not quasi-compact. So there are two copies of $W$ sitting inside of $Y$ (one for each copy of $X$). The intersection of these two is exactly $U\cap W$, which is not quasi-compact. So we found two open affines in $Y$ whose intersection is not quasi-compact, which shows that $Y$ is not quasi-separated.


Now we just have to find some non-quasi-compact open immersions. The complement of the origin in $\mathbb A^\infty$ is one (so $\mathbb A^\infty$ with a doubled origin is a non-quasi-separated scheme).


**Edit:** Here's another one that I don't completely understand, but gives a finite-dimensional example (zero-dimensional in fact). Consider $X=Spec(\overline{\mathbb Q}\otimes\_{\mathbb Q}\overline{\mathbb Q})$. Topologically, $X=Gal(\overline{\mathbb Q}/\mathbb Q)$, with the profinite topology (perhaps somebody could explain how to see this in a comment). In particular, any point is closed, but the complement $U$ is not quasi-compact, so we get another example of a non-quasi-compact open immersion, so $X$ with a doubled point is non-quasi-separated.


