Timeline for Books on reductive groups using scheme theory
Current License: CC BY-SA 2.5
17 events
| when toggle format | what | by | license | comment | |
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Mar 13, 2010 at 20:24 | comment | added | Harry Gindi | Ah, that explains it. Thanks for the response and "setting me straight", as it were. | |
| Mar 13, 2010 at 18:02 | comment | added | BCnrd | @fpqc: that defn of q-c for alg spaces does not make sense for "general" functors on rings (cf. functorial criteria for lfp, smooth, proper). Since alg spaces are functors by defn (no loc. ringed space), it is a tautology that any definition for alg spaces is literally expressed in terms of its "functor of points" since that's all it is! The crux is whether the defn/criterion is meaningful for a functor not known to be an algebraic space. For q-c (and irred) the answer is "no". I put "useful non-tautologous way" in my previous comment to head off a reply as you wrote. | |
| Mar 11, 2010 at 5:31 | comment | added | Harry Gindi | Although to be honest, I have not given anything past cours 3 a serious read. I wouldn't be surprised if there were a way to define irreducibility only using functors of points. (And a minor correction above, the dimension of a finite type scheme over a field. What I said above did not make sense.) | |
| Mar 11, 2010 at 5:10 | comment | added | Harry Gindi | I'm using definitions from Toen's master course on stacks, which I'm not sure are standard, but everything that I'll mention is defined in terms of functors of points: An algebraic space is called quasi-compact if there exists a finite atlas of affines for it. A morphism of algebraic spaces $X\to Y$ is quasicompact if for all affine schemes Z, $X \times_Y Z$ is a quasicompact algebraic space. There is also a definition of dimension of a finite type scheme using this language (last page cours 4-2), but it appears that at least irreducibility is an inherently topological property. | |
| Mar 11, 2010 at 4:46 | comment | added | BCnrd | @fpqc: even in SGA3 the definition rests on a criterion with geometric fibers. Reductivity is fundamentally a geometric notion: over an imperfect field $k$ we cannot detect it using just $k$-subgroups. That is the whole point of the pseudo-reductive stuff. Functor of points is a convenient notion, even powerful for working with group objects and related constructions, but not everything can be shoehorned into it. Can you define (in a useful non-tautologous way...) the dimension of a finite type scheme over a field or quasi-compactness or irreducibility using functor of points? | |
| Mar 11, 2010 at 1:37 | vote | accept | Harry Gindi | ||
| Mar 11, 2010 at 1:15 | answer | added | JS Milne | timeline score: 33 | |
| Mar 10, 2010 at 13:09 | comment | added | Kevin Buzzard | On the way to work I thought of an analogy. When I was a grad student I was very enthusiastic about finite flat group schemes, and knew several good references that treated these gadgets over bases like spec(complete DVR) and so on. However of course I would sometimes invoke arguments from abstract group theory as a matter of course when reading and understanding proofs. So I don't know why I expected to be able to understand reductive group schemes without first understanding reductive groups over an alg closed field base. | |
| Mar 10, 2010 at 8:53 | comment | added | Harry Gindi | Is there a simple definition of a reductive group scheme using the functor of points approach? | |
| Mar 10, 2010 at 8:05 | comment | added | Kevin Buzzard | Even though, when I was a graduate student, I was firmly a "scheme" person, and looked upon varieties with some disdain, I realised later on that for the theory of reductive groups it's well worth setting up the basics over an alg closed field first, and then over a general field, and then over a general scheme. This is not often the case---e.g. I'm not sure what advantage there would be in defining a proper morphism of varieties and then a proper morphism of schemes---but for alg groups there is a definitely a case for it, esp if you're interested in Langlands type stuff. | |
| Mar 10, 2010 at 3:08 | vote | accept | Harry Gindi | ||
| Mar 11, 2010 at 1:37 | |||||
| Mar 10, 2010 at 0:06 | history | edited | Harry Gindi | CC BY-SA 2.5 |
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| Mar 9, 2010 at 23:48 | answer | added | BCnrd | timeline score: 15 | |
| Mar 9, 2010 at 23:29 | answer | added | Steven Sam | timeline score: 5 | |
| Mar 9, 2010 at 22:19 | history | edited | Harry Gindi | CC BY-SA 2.5 |
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| Mar 9, 2010 at 22:07 | history | asked | Harry Gindi | CC BY-SA 2.5 |