On some finiteness properties for schemes - MathOverflow

On some finiteness properties for schemes

2012-11-06T21:36:54Z

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).

Answer by Piotr Achinger for On some finiteness properties for schemes

2012-11-07T05:43:05Z

A implies B. True, as you said, because a finitely generated ring is Noetherian, and $X$ is glued from finitely many spectra of such.

A implies C. True (argument as above).

B implies A. False, e.g. $X = \mathrm{Spec }\ \mathbb{Q}$.

B implies C. False (I believe). There are rings $R$ whose spectrum is homeomorphic to the topological space $\{1, 2, \ldots \}$ with open sets $\{n, n+1, \ldots\}$, which is Noetherian but of infinite Krull dimension. I think something like $\mathrm{Spec }\ k[x_1, x_1 x_2, x_1 x_2 x_3, \ldots]$ should work, but I didn't check the details. EDIT. This is nonsense - see the comments below and Fred Rohrer's answer.

C implies A. False, e.g. $X= \mathrm{Spec }\ \mathbb{Q}$.

C implies B. False, e.g. $X = \mathrm{Spec}\ k[x, x^{1/2}, x^{1/3}, \ldots]$.

Answer by Fred Rohrer for On some finiteness properties for schemes

2012-11-07T07:15:42Z

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.