TITLE: On some finiteness properties for schemes
QUESTION [6 upvotes]: 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).

REPLY [8 votes]: An example of a noetherian ring of infinite dimension can be found in Nagata's Local Rings, Appendix A1, Example 1.
Edit: An interesting generalisation of Nagata's construction yielding noetherian rings of infinite dimension whose maximal ideals have prescribed heights was given by Fujita in his article Infinite dimensional Noetherian Hilbert domains, Hiroshima Math. J. 5 (1975), 181–185.