Consider the following properties of scheme $X$:

A: $X$ is of finite type over $\mathbb{Z}$

B: $X$ is Noetherian

C: $X$ is of finite Krull dimension

What implications are there between these three? I believe that B and C are independent of each other (although I can't find a reference right now), and it follows from EGA I, 6.3.7 that A implies B. But does A imply C?

(Apologies if this question is "trivial", but I'm not an expert in algebraic geometry.)

As an aside, I would also be interested if any of these properties can be related to some notion of cohomological dimension (not sure what kind of topologies would be relevant for this).