Talk:Glossary of scheme theory
Could someone give further explanation of these definitions?
Are the links to uncreated necessary if this is a glossary? Or will there be more detail on those pages?
It's all work-in-progress; not so different from the rest of Wikipedia. If you have a request for a topic, try Wikipedia:Requested articles/mathematics.
Charles Matthews 12:28, 26 May 2004 (UTC)
I've changed the definition of quasi-finite, which looked like it came from Hartshorne. I've replaced it with th EGA definition. Joeldl 17:07, 17 February 2007 (UTC)
Hi, I would suggest to rearrange the list of morphism properties, such that the relation of the several notions becomes clearer and also try to avoid (more) difficult terms (for example affine morphisms can be defined more easily). Do you agree? -- Jakob.scholbach 03:57, 1 October 2006 (UTC)
- Go ahead. Charles Matthews 08:52, 1 October 2006 (UTC)
The notion of finitely presented morphism should be added. I was redirected from another page, but could not find the definition. —Preceding unsigned comment added by 18.104.22.168 (talk) 09:39, 18 June 2008 (UTC)
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Table of properties of schemes
Two points about that table: (a) The concept "connected scheme" is used in the row treating integral schemes but was not defined in Wikipedia. So I added it to the table, following Hartshorne's Algebraic Geometry. (b) The column titled "definition" contains definitions but also further basic properties of the concept in question (e.g. the statement about irreducible components). This is useful but maybe one should move this, say, to another column? Well, maybe not. Ninte (talk) 07:38, 11 June 2009 (UTC)
- Yes, not explicitly. When it says about closed immersions, it is defining closed subschemes, but tacitly only. So some more words needed there. And open subschemes aren't explicitly defined, though it is a remark (perhaps a little more than just a remark) that restricting to an open subset you get a locally ringed space that is a scheme. Charles Matthews (talk) 15:12, 30 October 2009 (UTC)
Geometric description or explanation
When I learned algebraic geometry, I found it extremely helpful to have the different terms from scheme theory explained with a (complex) geometric intuition as well as formal algebraic definition. I also know some people who disagree with this because the importance of some of these ideas is especially prevalent when not working of the complex field.
Nonetheless, I think it would be really helpful for most visitors to include an explanation of why, for example, unramified is an interesting idea geometrically, and not just some random definition. Also, it would make this page more 'Wikipedia-like' and less 'Wikitionary-like'. Even though the title is 'Glossary of scheme theory'. I would volunteer, but I don't really think about this kind of stuff as much anymore, and I'm sure there are more able wikipedians.
There are some objects in this Glossary that are neither defined here, nor somewhere else. For example, a smooth scheme (or variety). The Glossary article uses smooth varieties without defining them. Other articles using smooth varieties (e.g. the article on D-modules) link to the Glossary.
- You are welcome to add definitions, or create an article. In the case of smooth schemes/varieties, the definition is implicitly there in that a "smooth morphism" is defined. Then, a smooth scheme is simply a scheme X such that the (unique) morphism from X to Spec Z is smooth. A scheme X over another scheme S is called "smooth over S" if the structure morphism X → S is smooth. As for varieties, the definition of variety V in full generality can vary, but usually is something like an integral, separated scheme of finite type over an algebraically closed field k; given a definition of variety V over k, the variety is called smooth if the structure morphism V → Spec k is smooth. RobHar (talk) 13:33, 17 August 2011 (UTC)
Scheme hausdorff iff zero-dimensional?
In the section about separated and proper morphisms, one line reads as follows:
"In algebraic geometry, the above formulation is used because a scheme is a Hausdorff space if and only if it is zero-dimensional."
I'm struggling about that. Is that really true? I've found no such result or proof of it in Google. Of course, this statement is well known to be correct for affine schemes and also for noetherian schemes (in which case it is equivalent to the scheme being an artinian scheme). Moreover, a scheme is zero-dimensional iff it is a topological T1-space. But is this really equivalent to the Hausdorff property in general? Is there someone who knows of a proof or a counterexample? — Preceding unsigned comment added by 22.214.171.124 (talk) 15:58, 22 October 2012 (UTC)
- The claimed equivalence is not correct. To find a counterexample, start with a non quasi-separated scheme X, and let Xcons be the set X endowed with the constructible topology on X (EGA IV.1.9.13). Then Xcons is not Hausdorff (separated) as a topological space (EGA, IV.1.9.15(ii)). It turns out that Xcons is actually the underlying topological space of some scheme of dimension 0 (EGA, IV.1.9.16, without proof. Or google with "constructible topology 1.9.16"). UL (talk) 22:14, 23 October 2012 (UTC)
"which coincides with the usual notion of étale map in differential geometry."
I have never head the term étale map in differential geometry, and the only reference for it I can find is on the n-lab: http://ncatlab.org/nlab/show/formally+etale+morphism#in_differential_geometry_30.
I suspect the "cohesive (infinity, 1)-topos of smooth infinity-groupoids" and the fact that the category of smooth manifolds embeds into this one are not common knowledge, and this ought to be noted. There are two possible appraoches:
The first would be something along the lines of "There exists a generalization of this notion which, in the category of smooth manifolds, coincides precisely with local diffeomorphisms." together with appropriate citation (of n-lab or elsewhere).
Alternatively, we could forget this broader notion (as it's kind of beside the point in this context) and simply change it to "which parallels the usual notion of local homeomorphisms in differential geometry."