Talk:Ringed space
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I removed the paragraph
- One can trace this back to the logical form of the definition of local ring. In a clean logical formulation, it can be put as 'for all r in R, either there is s with sr = 1 or t with t(1-r) = 1'. In topos theory it is shown how the theory therefore qualifies for a classifying topos, which parametrises local rings. The structure of a locally ringed space is equivalent to the right kind of morphism to this topos - which can also be identified via algebraic geometry.
While the classifying topos sentence is clear, I cannot make sense of the last sentence. A morphism to the classifying topos from where? From the topological space X considered as category? The word "which" in the last sentence: does it refer to "right kind of morphism" or to "morphism" or to "this topos"? How does algebraic geometry identify this? AxelBoldt 16:56, 13 Nov 2003 (UTC)
Ah - well, it is probably too much. If you ask a serious category theorist how to construct the theory of schemes, say schemes over Spec(R), you get an interesting answer, but it does assume the geometric morphism theory. What is happening now at scheme (mathematics) is OK - the traditional theory you could call it. What should happen is that the structure of X as a locally ringed space of R-algebras should be equivalent to a morphism to the classifying topos for local R-algebras, which is (known to algebraic geometers as) the Zariski topos for Spec(R). I think this deserves to be somewhere for NPOV, basically.
Charles Matthews 17:19, 13 Nov 2003 (UTC)
By the way - about PL functions. I remember reading Zeeman writing in some lecture notes that the sheaf of piecewise-linear functions was the invariant way to define a PL structure on a manifold. So it's quite an interesting concept. So, I guess the point is that f PL, f(x) non-zero implies f-1 continuous but not PL near 0. Isn't this interesting enough to make an example for this page?
Charles Matthews 13:54, 14 Nov 2003 (UTC)
[edit] wrong definition of coherency
the definition of quasi-coherent and coherent modules given at the end was wrong. i editted it.
[edit] Definition
It is unclear to me why the definition of Ringed Spaces is necessary. As stated, it is just a sheaf of rings. The only way I can see out of this conundrun is if the definition of morphism between ringed spaces is a particular type of the possible ways to define a morphism between sheaves. Hence, when somebody tells me that "X is a ringed space" then I will take it for a short word to X being a sheaf of rings where we consider a particular type of morphism between then. I am aware that one could say that a ringed space is not a sheaf of rings since it is a pair (M,F) where M is a topological space and F is a sheaf of rings over M. However, M is already an information we can get out of F. Therefore, it seems to me redundant to say that we have such a pair (M,F) to just saying that we have an F. Just to be more precise. To me to say that we have a ringed space means that we have an object of a very specific category. What makes this category special is not its objects, but the set of morphisms we allow between any two objects. 17:23, 16 January 2008 (UTC)
