Are the hom sets in the category of varieties abelian groups?

Question

This is supposedly (though I know of the proof bud haven't read it) for the Hom sets of noetherian schemes. Since every variety can be thought of as a noetherian scheme then it seems right... when only considering morphisms of varieties, is the subset $\mathrm{Hom}_{\mathsf{Var}}$ a subgroup of $\mathrm{Hom}_{\mathsf{Csh}}$?
Many thanks in advance :)

Answer 1

The claim is surely false even for noetherian affine schemes. Indeed, let $\emptyset$ be the empty scheme and let $X$ be any non-empty scheme; then the set of scheme morphisms $X \to \emptyset$ is empty and therefore not an abelian group. Or if that's too contrived, you could consider $\operatorname{Spec} \mathbb{F}_p \to \operatorname{Spec} \mathbb{F}_{p'}$ for two different primes $p, p'$...

Even if you restrict to the category $\mathcal{C}$ non-empty varieties over an algebraically closed field $k$ (so that hom-sets are always non-empty), there is still no good reason to say that the hom-sets are abelian groups: while it is true that any non-empty set can be equipped with the structure of an abelian group, it doesn't mean $\mathcal{C}$ is a preadditive category! If it were, then the terminal object in $\mathcal{C}$ would also have to be an initial object, but this is clearly not the case.