TITLE: Dimension of Tensor Product for Flat Extensions
QUESTION [5 upvotes]: Suppose that $A,B,$ and $C$ are commutative unital rings, $A\to B$ is flat, and $A\to C$ is any map.  I am trying to determine whether 
$$
\dim B\otimes_AC=\dim B+\dim C-\dim A
$$
Any counterexamples or references?  I am taking the Krull dimension of the zero ring to be -1 (in case a tensor product is 0).
Thanks for any feedback.
Edit: By $\dim$ I mean Krull dimension.

REPLY [4 votes]: Take $A=\mathbb Z,$ and $B=C=\mathbb Q$. Then $B\otimes_AC=\mathbb Q$ and your formula gives: $0=0+0-1$, absurd.