Timeline for What is the Zariski topology good/bad for?
Current License: CC BY-SA 2.5
16 events
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| Apr 17, 2010 at 4:24 | comment | added | Harry Gindi | Dear Emerton, I agree, by the way, that it's an excellent example of your point. I was just explaining my initial response. | |
| Apr 16, 2010 at 16:42 | comment | added | Harry Gindi | Dear Emerton, I see in Toën's notes a way to define the open complement of a closed subscheme of an affine scheme (cours 4), but I don't know if you can define the closed complement of an open subscheme, or if this definition generalizes to general non-affine schemes. Is this the problem you are referring to? With regards to irreducibility, there is a definition that is true, but I don't know if it is useful: A sheaf $F$ is irreducible if the fiber product of any two open immersions $A\to F$, $B\to F$ is isomorphic to $spec(0)$ if and only if either $A$ or $B$ is isomorphic to $Spec(0)$. | |
| Apr 16, 2010 at 15:50 | comment | added | Emerton | Dear Harry, My point about open and closed sets is that when you have a topology, you can take complements of open sets to obtain closed sets, and that these satisfy an important formalism, which (as per my added example) captures the notion of special position. As far as I know, you cannot (in any naive way) take the complement of an etale morphism. The etale topology captures certain intuitions, but it doesn't directly capture the basic geometric notions that the Zariski topology does. (The example of irreducibility, given by Brian above, is a good one.) | |
| Apr 16, 2010 at 13:51 | comment | added | Harry Gindi | Dear Brian, the point of my original comment in this post was that it's not really significantly harder to define "Zariski-open/closed" in the étale-topology I was responding to what Emerton had posted (before his edit). I absolutely agree that the Zariski topology is useful and not made obsolete by the étale topology, just that this is not one of those cases. I am not arguing for the point that you explained to me in the main body of the post. I agree with you there. My point was that the defns of open/closed are not exclusively a selling point of the Zariski topology. | |
| Apr 16, 2010 at 12:37 | comment | added | BCnrd | Dear Harry: We know how these defns go for etale top. But a defn not saying "Zariski topology" doesn't allow do much interesting without many results proved in Zar. top. Many results for alg. spaces are proved via reduction to schemes with Zar. topology. Real test is not making defns, but proving nontrivial thms. For lots in geometry of varieties, need Zariski-local rings, not henselian local rings (e.g., function field is local ring at generic point for Zar. top., and irreducibility is not etale local.) You would be well-served studying deeper theorems instead of more formalism. | |
| Apr 16, 2010 at 9:18 | comment | added | Harry Gindi | A morphism of sheaves (on the affine étale site) $F\to G$ is called a closed immersion if given any affine scheme $X=Spec(A)$ and any morphism $X\to G$, $F \times_G X$ is affine, and the canonical projection $Spec(B)\cong F \times_G X\to X\cong spec(A)$ corresponds to a surjective map of rings $A\to B$. | |
| Apr 16, 2010 at 8:22 | comment | added | Harry Gindi | Dear Brian and Pete, I feel that my position has been unfairly caricatured. There are ways to make the definition I've given just as intuitive as the standard definition using the Zariski topology. | |
| Apr 16, 2010 at 7:31 | comment | added | Harry Gindi | Dear Emerton, if one is familiar with étale ring maps and descent, it is at least as easy as the definition for the Zariski topology. | |
| Apr 16, 2010 at 5:05 | history | edited | Emerton | CC BY-SA 2.5 |
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| Apr 16, 2010 at 4:55 | history | edited | Emerton | CC BY-SA 2.5 |
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| Apr 16, 2010 at 4:51 | comment | added | Emerton | Dear Harry, I'm not sure that "just as easily" has the seem meaning for me as for you. | |
| Apr 16, 2010 at 4:15 | comment | added | BCnrd | Pete, even though that characterization of non-negativity has consequences, I think it's a perfectly good illustration of the last sentence of your comment. To be honest, I doubt it is even logically feasible in the sense that one couldn't really develop elementary number theory on the basis of such an absurd definition. | |
| Apr 16, 2010 at 4:11 | comment | added | Pete L. Clark | @Brian: I think your example, if anything, makes too much sense. (It is known to have some model-theoretic consequences.) How about this? We don't need to define the real (or complex, or p-adic numbers) explicitly, because they can be defined as quotient rings of the adele ring of a global field. This is presumably possible (non-circularly, I mean) but is clearly looking through the wrong end of the telescope. Just because something is logically possible doesn't make it a good approach to thinking about the subject, at any level. | |
| Apr 16, 2010 at 3:22 | comment | added | BCnrd | Harry, here's a "definition" of non-negativity in $\mathbf{Z}$: being a sum of 4 perfect squares. Do you think it is reasonable to develop number theory based on that definition? Suggesting one can develop the notion of closedness via etale topology by avoiding the Zariski topology makes about as much sense as that. | |
| Apr 16, 2010 at 2:56 | comment | added | Harry Gindi | The notions of open and closed can be just as easily defined in the étale topology, where open subschemes are precisely étale monomorphisms. There is a definition of closed that also avoids the Zariski topology. | |
| Apr 16, 2010 at 2:29 | history | answered | Emerton | CC BY-SA 2.5 |