Timeline for Why are algebraic schemes called algebraic?
Current License: CC BY-SA 4.0
19 events
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| Sep 15, 2019 at 13:36 | comment | added | François Brunault | @red_trumpet It depends on whether "finite type" means as field extension or ring extension. In the second case we have algebraicity by the Nullstellensatz, but not in the first case. | |
| Sep 15, 2019 at 10:47 | comment | added | red_trumpet | @FrançoisBrunault Is it not a variant of the Nullstellensatz that a finite type field extension is finite, hence algebraic? | |
| Jun 10, 2019 at 0:25 | comment | added | Qfwfq | @GLe: yep, sorry. I personally woldn't call "variety" something nonreduced, but as you say it's just terminology.. | |
| Jun 9, 2019 at 23:15 | comment | added | display llvll | @Qfwfq Yes, the one you cite appears to be the most common definition. I was following the definition of alg. variety given in Qing Liu's book. It doesn't change anything for the purpose of answering the question. Algebraic varieties (whatever you want them to be) and algebraic schemes share the "algebraic" adjective because they can be covered by finitely many spectra of finitely generated algebras. | |
| Jun 9, 2019 at 19:35 | comment | added | Qfwfq | "finite type schemes over a field are exactly the schemes which correspond to algebraic varieties" - No: they also have to be reduced and separated (and some people also assume irreducible). | |
| Jun 9, 2019 at 16:36 | comment | added | display llvll | The reason is that an algebraic scheme over a field can be covered by a finite number of spectra of finitely generated algebras. Those affine schemes that arise as the spectrum of a finitely generated algebra over a field are called algebraic, and the intuition behind this is that they are defined by a finite set of polynomials in a finite number of symbols. I think this terminology and point of view is common in the context of group schemes, see for example the notes of J.S. Milne | |
| Jun 9, 2019 at 14:14 | comment | added | user141498 | @SamirCanning OK, then change the question. Why are "algebraic preschemes" called algebraic? | |
| Jun 9, 2019 at 14:13 | comment | added | Samir Canning | Varieties are separated, but so were schemes back when this terminology was introduced. Then, schemes=preschemes+separated. | |
| Jun 9, 2019 at 13:04 | comment | added | user141498 | @GLe interesting theory indeed. Where I come from, varieties are usually separated, though I understand it can be different elsewhere. | |
| Jun 9, 2019 at 12:59 | comment | added | display llvll | finite type schemes over a field are exactly the schemes which correspond to algebraic varieties, so algebraic scheme = algebraic variety. I think that is the reason for the name. | |
| S Jun 9, 2019 at 11:47 | history | suggested | Ali Taghavi |
I add a tag.
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| Jun 9, 2019 at 11:46 | review | Suggested edits | |||
| S Jun 9, 2019 at 11:47 | |||||
| Jun 7, 2019 at 17:03 | comment | added | François Brunault | In the context of field extensions, finite type does not imply algebraic. | |
| Jun 7, 2019 at 8:00 | history | edited | user141498 | CC BY-SA 4.0 |
added 305 characters in body
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| Jun 7, 2019 at 7:57 | comment | added | YCor | But actually now that I'm looking, I can see in Grothendieck (1960) numdam.org/article/SB_1958-1960__5__193_0.pdf the use of "schéma algébrique sur $A$" (algebraic scheme over $A$), where $A$ does not have to be a field (yet it seems to mean finite type). | |
| Jun 7, 2019 at 7:54 | comment | added | user141498 | @YCor reassuring to see some people of similar opinion. | |
| Jun 7, 2019 at 7:54 | comment | added | YCor | It's not a great terminology, and I actually never saw it (or didn't pay attention). +1 if you prefer "scheme of finite type over a field". | |
| Jun 7, 2019 at 7:48 | comment | added | user141498 | @NajibIdrissi that is kind of funny, I did not think of that. Maybe so, maybe so. | |
| Jun 7, 2019 at 7:34 | history | asked | user141498 | CC BY-SA 4.0 |