[Math] Dimension of tensor product of modules

Question

$A\rightarrow B$ a ring homomorphism of Noetherian rings, where $A$ is local. $M$, $N$ finitely generated and nonzero $A$- and $B$- modules, respectively. Then I seem to get $\mbox{dim}_ {B}(M\otimes_{A} N) = \mbox{dim}_ {B}N$. Could that be true? It seems a little strange that the dimension of $(M\otimes_{A} N)$ (as a $B$-module) is independent of $M$.

Answer 1

See Bruns and Herzog A.5(b) and A.11(b).