Timeline for Why is continuity so crucial in arithmetic/algebraic geometry? [closed]
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| Jun 1, 2010 at 1:34 | history | closed |
S. Carnahan♦ Harry Gindi Andrew Stacey Anton Geraschenko |
not a real question | |
| Jun 1, 2010 at 1:33 | comment | added | Anton Geraschenko | There is a thread about this question at tea.mathoverflow.net/discussion/411. The question is pretty vague (I don't know what kind of answer you're looking for), the comment thread seems to have given up on the question and run off in another direction, and the OP has made no effort to clarify. So I'm voting to close until the question is clarified. @OP: please have a look at the "How to ask a good question" page: mathoverflow.net/howtoask | |
| May 26, 2010 at 16:13 | comment | added | BCnrd | @George: briefly, Springer sometimes assumes it suffices to check things with $F_ {\rm{sep}}$-points when it does not (e.g., surjectivity of a map of $F$-varieties, image of which could miss an inseparable point). This leads him to make some false statements in his discussion of $F$-reductive (= pseudo-reductive over $F$) groups. But now there's another reference on these matters. :) To tell you specific errors, it will be easier to discuss in person with a copy of the book. Will you be at the Gross conference next week? If so, we can chat then. | |
| May 26, 2010 at 16:09 | comment | added | BCnrd | @Pete: "alg. envelope" is "Zar.-closure of image of Galois", yes? Doesn't seem to be example of Zar. topology on rat'l points (over non-sep. closed field), since its purpose is to replace study of image of Galois (compact gp of rat'l pts in analytic topology) with algebraic variety, which has more structure than a set of rat'l pts (via structure theory of alg. gps). So I still don't see an example of why useful to consider Zar. topology on rat'l pts. For non-experts, also seems bad: why not use $\overline{K}^n$ with polys with coeffs in $K$, which is also concrete, more accurate, and works? | |
| May 26, 2010 at 12:44 | comment | added | George McNinch | @BCnrd: I haven't looked at that part of Springer's book carefully in quite a while, but I can imagine having students look at it. Could you say where the bugs are, so I can make a note? (Well, probably I can find them myself, but if you happen to know section numbers that would be great...) | |
| May 26, 2010 at 8:34 | comment | added | Harry Gindi | @OP, it's not exactly clear to me what you're asking, and forgive me, but it seems like it's not really clear to you either. | |
| May 26, 2010 at 7:53 | comment | added | Pete L. Clark | Pedagogically speaking, I have had some success teaching this kind of Zariski topology -- at least on affine n-space and its subsets -- to non-algebraic geometers. It is often the case that simply having "Zariski closure" in your vocabulary makes apparently nontrivial theorems transparent. For instance, the fact that for $n \times n$ matrices $A$ and $B$ over a field, $AB$ and $BA$ have the same char. poly. In this regard, finite fields are a bit of a pain -- no point in taking the closure in a discrete top.! -- but this can often be circumvented by base change to the algebraic closure. | |
| May 26, 2010 at 7:45 | comment | added | Pete L. Clark | P.S.: I am not saying that there is any situation in which this relatively naive language covers more ground than the scheme-theoretic language: it is an easy exercise in the "soberification" of a topological space to see that the two points of view are consistent. But there is some merit in not speaking of scheme-theoretic closure unless one really needs to: there are an order of magnitude more mathematicians who can understand "coarsest topology on $K^n$ making the polynomials continuous" than "the scheme-theoretic closure".... | |
| May 26, 2010 at 7:28 | comment | added | Pete L. Clark | Example: the algebraic envelope of an $\ell$-adic Galois representation on an abelian variety. (Unnecessarily snarky comment alluding to the half dozen papers I have written about varieties without rational points removed.) | |
| May 26, 2010 at 6:10 | comment | added | BCnrd | Pete, what is something useful done with Zariski topology on set of rational points over a field that is not separably closed? (Statements that rational points constitute a Zariski-dense set in the underlying $K$-scheme are important, but that's an entirely separate matter.) It sounds dangerous, since there can be plenty of smooth closed subvarieties of the $K$-scheme which have no $K$-points. (Springer's book makes some errors related to that.) What's an example where it's better to think in such terms instead of "rational points in a $K$-scheme"? | |
| May 26, 2010 at 5:43 | comment | added | Pete L. Clark | Yes, I mean the relative Zariski topology on the set of $K$-rational points. That certainly comes up sometimes, e.g. when one talking about linear algebraic groups. | |
| May 26, 2010 at 4:46 | comment | added | BCnrd | Pete, varieties of positive dimension over any field (finite or otherwise) never have the discrete topology; think of the generic points or even the affine line. Even if abusing terminology and ignoring non-closed points (which I assume you wouldn't do), it's still non-discrete (since inclusion of set of closed points with subspace topology is a "quasi-homeomorphism", as for any Jacobson scheme). Maybe you're referring to just the finite set of rational points (which has no useful geometric structure at all)? Not sure. | |
| May 26, 2010 at 3:47 | comment | added | Pete L. Clark | I don't really understand the question. For instance, the Zariski topology on a variety over a finite field is the discrete topology, so indeed (Zariski-)continuity is not important at all: all functions are automatically continuous. Are you perhaps trying to ask about etale cohomology? Do you realize that this is not a cohomology theory on topological spaces but rather a cohomology theory with respect to sheaves on a Grothendieck topology? (Here the fact that algebraic maps induce "continuous" morphisms on the etale sites is quite straightforward...) | |
| May 26, 2010 at 3:25 | comment | added | KConrad | Do you mean why are topological ideas so important? | |
| May 26, 2010 at 3:17 | history | asked | teil | CC BY-SA 2.5 |