Q:

geometric interpretation of a height inequality of prime ideals

Theorem 15.1 in Matsumura's Commutative Ring Theory states that if $f:A\rightarrow B$ is a homomorphism of Noetherian rings, $P \in \operatorname{Spec}B, P \cap A=p$, then $ht (P) \le ht(p) + \dim B_P/pB_P$. He proceeds to say that we can replace $A$ by $A_p$ and $B$ by $B_P$ and then the inequality becomes $\dim B \le \dim A + \dim B/mB$, where $A,B$ are now local rings and $m$ is the maximal ideal of $A$. Indeed, i can see why this is true.
He then states: "rewriting the original inequality in this form makes the geometrical content clear". My question is: what is the geometrical content?

A:

He is saying that if $P$ is a point on $Y = \operatorname{Spec}(B)$ and $p\in X = \operatorname{Spec}(A)$ is its image under the morphism $^{a}\varphi:Y\to X,$ then the inequality implies that the dimension of $Y$ at $P$ is less than or equal to the dimension of $A$ at $p$ plus the dimension of the fibre of $^a\varphi$ over $p,$ with equality in the case of a (locally) flat morphism.

