Dimension of tensor product vs. fiber product

Question

This is a very fuzzy and ill-formed question, but: we usually think about the tensor product having dimension equal to the product of the dimensions of its factors (at an intuitive level, e.g. vector spaces). On the other hand the fiber product typically has dimension something like the sum of the dimensions of the factors (minus the dimension of whatever it's over). Maybe what I'm missing is something to account for the dimension of the base ring in the tensor product case, since the classical case is over fields, but it still seems like there's a mismatch between the additive and multiplicative structures, given that the fiber product and the tensor product are dual constructions. What's the intuition here?

Answer 1

You're confusing two different notions of dimension: one is dimension as a vector space (cardinality of a basis) and one is dimension as a ring (Krull dimension, aka length of longest chain of prime ideals). The first behaves multiplicatively, while the second behaves additively over a tensor product.