Maybe this is a dumb question, but better safe than sorry.

In Hartshorne, Exercise **I.3.15** and **I.3.16** we are asked to examine products of affine and projective varieties. **I.3.15** has been smooth sailing. The reader is asked to prove:

a) $X \times Y$ is irreducible

b) $A(X \times Y) = A(X) \otimes_k A(Y)$

c) $X \times Y$ is a categorical product

d) $\dim (X \times Y) = \dim X + \dim Y$

None of these have been particularly difficult.

In the next exercise, we are asked to prove some similar claims about projective varieties, but (d) does not have an analogous statement above. It would seem plausible to me that something like this is true. We have many analogous theorems about the dimension of a projective variety from **I.2**, so I'd be surprised if there was no such relationship.

Can we find a proof of this claim, or a counter-example?