ag.algebraic geometry - Can Hom_gp(G,H) fail to be representable for affine algebraic groups?

Hom(G_a, G_m) is not representable.

Let R be a ring of characteristic zero. I claim that Hom(G_a, G_m)(Spec R) is {Nilpotent elements of R}. Intuitively, all homs are of the form x -> e^{nx} with n nilpotent.

More precisely, the schemes underlying G_a and G_m are
Spec R[x] and Spec R[y, y^{-1}] respectively. Any hom of schemes is of the form y -> sum f_i x^i for some f_i in R. The condition that this be a hom of groups says that
sum f_k (x_1+x_2)^k = (sum f_i x_1^i) (sum f_j x_2^j). Expanding this, f_{i+j}/(i+j)! = f_i/i! f_j/j!. So every hom is of the form f_i = n^i/i!, and n must by nilpotent so that the sum will be finite.

Now, let's see that this isn't representable. For any positive integer k, let R_k = C[t]/t^k. The map x -> e^{tx} is in Hom(G_a, G_m)(Spec R_k) for every k. However, if R is the inverse limit of the R_k, there is no corresponding map in Hom(G_a, G_m)(Spec R). So the functor is not representable.