Dimension of polynomial rings.

Question

It is well known that if $A$ is a Noetherian ring, then $\dim A[T]=\dim A+1$ and $\dim A[[T]]=\dim A+1$. Now $1$ is the fibre dimension of the faithfully flat algebras $A\rightarrow A[T]$ (resp. $A[[T]]$?). Are there more general formulas for (flat) algebras $A\rightarrow B$ connecting $\dim A$, $\dim B$ and the fibre dimension? Also how is flatness related to fibre dimension?