On some finiteness properties for schemes - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T05:28:46Z http://mathoverflow.net/feeds/question/111685 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/111685/on-some-finiteness-properties-for-schemes On some finiteness properties for schemes Anonymous 2012-11-06T21:36:54Z 2012-11-08T16:59:05Z <p>Consider the following properties of scheme $X$:</p> <p>A: $X$ is of finite type over $\mathbb{Z}$</p> <p>B: $X$ is Noetherian</p> <p>C: $X$ is of finite Krull dimension</p> <p>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?</p> <p>(Apologies if this question is "trivial", but I'm not an expert in algebraic geometry.)</p> <p>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).</p> http://mathoverflow.net/questions/111685/on-some-finiteness-properties-for-schemes/111704#111704 Answer by Piotr Achinger for On some finiteness properties for schemes Piotr Achinger 2012-11-07T05:43:05Z 2012-11-08T16:59:05Z <p><strong>A implies B.</strong> True, as you said, because a finitely generated ring is Noetherian, and $X$ is glued from finitely many spectra of such.</p> <p><strong>A implies C.</strong> True (argument as above).</p> <p><strong>B implies A.</strong> False, e.g. $X = \mathrm{Spec }\ \mathbb{Q}$.</p> <p><strong>B implies C.</strong> 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. <strong>EDIT.</strong> This is nonsense - see the comments below and Fred Rohrer's answer.</p> <p><strong>C implies A.</strong> False, e.g. $X= \mathrm{Spec }\ \mathbb{Q}$.</p> <p><strong>C implies B.</strong> False, e.g. $X = \mathrm{Spec}\ k[x, x^{1/2}, x^{1/3}, \ldots]$. </p> http://mathoverflow.net/questions/111685/on-some-finiteness-properties-for-schemes/111709#111709 Answer by Fred Rohrer for On some finiteness properties for schemes Fred Rohrer 2012-11-07T07:15:42Z 2012-11-08T08:16:19Z <p>An example of a noetherian ring of infinite dimension can be found in Nagata's <em>Local Rings</em>, Appendix A1, Example 1.</p> <p><strong>Edit:</strong> 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 <a href="http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.hmj/1206136627" rel="nofollow"><em>Infinite dimensional Noetherian Hilbert domains</em></a>, Hiroshima Math. J. 5 (1975), 181–185. </p>