Let $X$ be a Scheme and $f: X \to Y$ a morphism of schemes. If the relative diagonal $\Delta_{X/Y}: X \to X \times_Y X$ is affine, does this imply that the "absolute" diagonal $\Delta : X \to X \times_{\operatorname{Spec} \mathbb{Z}} X$ is affine?
I know the answer is positive if $Y$ is affine.
I know a theorem which lets you calculate the cohomology of a quasi-coherent sheaf on $X$ using Cech-Cohomology. One of its prerequisites is that $\Delta$ is affine, and I was wondering whether this theorem would also be useful in the relative case.
In general the answer is no: We can take $X$ to be a non-semiseparated scheme (for example the affine plane with doubled origin) and take $f$ to be the identity map $\operatorname{id}_X$.