Timeline for Constructive algebraic geometry
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
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| Nov 16, 2017 at 20:33 | comment | added | Simon Henry | @AbdelmalekAbdesselam Not sure, names for this sort of things often comes from (one of) the semantics you have in mind for them, and I don't know an interesting semantics for this system. So in waiting for a good semantics (which as far I'm concern also mean " a reason to work in this system" ) I would call it just ZF+DC. There are various names that already exists if you add to this system some axioms which contradict full choice (the sort of things that are true in Solovay model) like "Dream mathematics" ncatlab.org/nlab/show/dream+mathematics | |
| Nov 16, 2017 at 20:25 | comment | added | Abdelmalek Abdesselam | @SimonHenry: what would be a better name then? | |
| Nov 16, 2017 at 19:20 | comment | added | Simon Henry | @AbdelmalekAbdesselam : Maybe, but 'Algorithmic' would be a really terrible name for something that accept the full law of excluded middle. Algoritmically speaking its fine to assume that for exemple elements of the ring you work with satisfies (a=b) or not (a=b), but if I gave you a sequence of element of the ring or a function P, decide if "there exist x such that $P(x)=0$" or "$P(x) \neq0$ for all x" is not algorithmic, the case where P is a polynomial in several integer variables is the typical algoritmically undecidable problem ! | |
| Nov 16, 2017 at 0:01 | comment | added | Abdelmalek Abdesselam | @ಠ_ಠ: the word "constructive" is not the exclusive property of the area technically known as "constructive mathematics" as in the ncatlab page. A more precise name for that religion is "mathematics which rejects the axiom of choice and the excluded middle". Whereas what I am talking about is "mathematics which rejects the full axiom of choice, is fine with the excluded middle and with the (countable) axiom of dependent choice". Maybe one could call that algorithmic mathematics. This is how algebraic geometers proved their theorems in 19th century like Gordan's finite generation thm. | |
| Nov 15, 2017 at 23:19 | comment | added | ಠ_ಠ | @AbdelmalekAbdesselam I haven't heard that term used for 19th century algebraic geometry before (I'm no expert though) but I'm using the word "constructive" in the modern sense of constructive mathematics, which is broadly speaking mathematics done without the law of excluded middle. | |
| Nov 15, 2017 at 22:49 | comment | added | Abdelmalek Abdesselam | I thought "constructive algebraic geometry" meant "19th century algebraic geometry" which uses resultants, elimination theory, classical invariant theory as described in the two books by Faa di Bruno and Salmon's "Higher Algebra"... | |
| Nov 15, 2017 at 22:07 | history | edited | ಠ_ಠ | CC BY-SA 3.0 |
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| Nov 15, 2017 at 20:20 | comment | added | Simon Henry | (read "whose close subspaces are radical ideals") | |
| Nov 15, 2017 at 20:01 | vote | accept | ಠ_ಠ | ||
| Nov 15, 2017 at 19:16 | comment | added | Simon Henry | @QiaochuYuan : Unless I'm mistaken It actually agrees with the locale whose closed subspace are ideal and whose points are prime ideal, but this locales might or might not be spatial. | |
| Nov 15, 2017 at 18:22 | comment | added | Qiaochu Yuan | Here is a way to talk about the Zariski spectrum without ever mentioning prime ideals: it is the right adjoint to the global sections functor from locally ringed spaces to commutative rings. If we replace locally ringed spaces with locally ringed locales, for the right definition of "locally ringed" (there is a definition which does not refer to stalks) then we should get a constructively meaningful construction, which may or may not agree with the locale whose closed sets are the ideals. | |
| Nov 15, 2017 at 12:36 | comment | added | Derek Elkins left SE | Lombardi's and Quitté's book Commutative algebra: Constructive methods. Finite projective modules gives a quite involved look at what the commutative algebra part looks like constructively. This may give you some idea of what takes the place of maximal ideals and such. It is not focused on algebraic geometry though. | |
| Nov 15, 2017 at 11:13 | answer | added | Simon Henry | timeline score: 55 | |
| Nov 15, 2017 at 10:05 | history | edited | ಠ_ಠ | CC BY-SA 3.0 |
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| Nov 15, 2017 at 9:47 | history | asked | ಠ_ಠ | CC BY-SA 3.0 |