This is a pretty basic question. Hartshorne defines "intersection multiplicity" for any two divisors on a surface. Fulton has an impressive framework of generalizing this in his book (my understanding of which is scant). But for whatever reason, in arithmetic texts one often sees intersection multiplicities defined only between a (general) divisor and a vertical divisor. What's really going on? Does this just make it easier to explain things, or is there an actual impediment to defining an intersection number between general divisors in the arithmetic setting? And if so, how does Fulton address it?
(just to be clear, when I say the "arithmetic setting" I mean in a scheme which has relative dimension 1 over a regular scheme of pure dimension 1)