Timeline for Building algebraic geometry without prime ideals
Current License: CC BY-SA 4.0
20 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 25, 2021 at 20:00 | comment | added | xuq01 | (not an expert) but this smells to me as something potentially related to the internal language version of algebraic geometry being developed by Blechschmidt and others. | |
| Dec 18, 2020 at 9:42 | vote | accept | Anton Mellit | ||
| Dec 18, 2020 at 9:04 | vote | accept | Anton Mellit | ||
| Dec 18, 2020 at 9:42 | |||||
| Dec 3, 2020 at 17:20 | vote | accept | Anton Mellit | ||
| Dec 18, 2020 at 9:04 | |||||
| Dec 3, 2020 at 17:20 | vote | accept | Anton Mellit | ||
| Dec 3, 2020 at 17:20 | |||||
| Dec 2, 2020 at 9:44 | answer | added | Daniel Loughran | timeline score: 14 | |
| Dec 2, 2020 at 9:32 | history | became hot network question | |||
| Dec 2, 2020 at 8:42 | comment | added | Anton Mellit | @darijgrinberg I am still going to use the structure sheaf to capture the nilpotents, i.e. a scheme is still a ringed space, the proposal only concerns the underlying set. The problem with taking points with values in arbitrary $k$-algebras comes when you want to construct gluing. If you glue $X$ and $Y$ along some open $U$ which sits inside both, you want the points of the result to be the union of the points of $X$ and the points of $Y$. Or, you use your notion of points to define what it means to be an open cover. You don't get the right notion for points with values in arbitrary rings. | |
| Dec 2, 2020 at 8:34 | comment | added | Anton Mellit | @BrianShin yes, these are two different points, topologically indistinguishable of course, as Sam points out. | |
| Dec 2, 2020 at 8:27 | comment | added | Leo Alonso | @Asvin This approach may give you a topological space, the trouble arises when one tries to construct a sheaf of rings whose global sections recover the original ring. | |
| Dec 2, 2020 at 8:25 | answer | added | Leo Alonso | timeline score: 29 | |
| Dec 2, 2020 at 8:23 | comment | added | Asvin | I thought of this a while back. Interestingly, this generalizes to a notion of spectrum for non commutative rings (you replace fields by matrix algebras, more generally, what we really want to consider is the "set" of all possible linear representations of your algebra). I don't know if this is a fruitful generalization. | |
| Dec 2, 2020 at 6:45 | comment | added | Faris | When you take fiber products you might have to increase the set of test fields, seems like rather unpleasant indexing. | |
| Dec 2, 2020 at 2:42 | comment | added | Benjamin Steinberg | @darijgrinberg the spectrum doesn’t see the nilpotents. You need the structure sheaf for that | |
| Dec 2, 2020 at 1:07 | comment | added | darij grinberg | Why take points over fields when you can take points over arbitrary commutative $k$-algebras? (Aren't you losing nilpotents with your approach, too?) Marc Nieper-Wißkirchen's Algebraische Geometrie lecture notes (used to be on his website; now I can only find them on libgen) take a functor-of-points approach; from what I recall at least one version of EGA tried the same -- but I guess both are above the level at which you're trying to teach. | |
| Dec 2, 2020 at 1:04 | comment | added | Sam Hopkins | @BrianShin: isn't that exactly the $T_0$ issue mentioned in 2.? | |
| Dec 2, 2020 at 0:50 | comment | added | Brian Shin | How do you deal with field extensions? If we have a point $R \to \mathbb{R}$, then we get another point by inclusion into $\mathbb{C}$. Are these just two different points now? | |
| Dec 2, 2020 at 0:32 | history | edited | Todd Trimble |
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| Dec 2, 2020 at 0:24 | history | edited | LSpice | CC BY-SA 4.0 |
\DeclareMathOperator
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| Dec 2, 2020 at 0:12 | history | asked | Anton Mellit | CC BY-SA 4.0 |