Timeline for What should be learned in a first serious schemes course?
Current License: CC BY-SA 2.5
13 events
| when toggle format | what | by | license | comment | |
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| Dec 22, 2010 at 9:02 | history | made wiki | Post Made Community Wiki | ||
| Jun 25, 2010 at 14:26 | comment | added | Quetzalcoatl | Speaking of Mumford and arithmetic approaches, it seems the fabled "Mumford-Lang" type-script has been subsumed by notes of Mumford/Oda (Lang is not mentioned as a coauthor here). In these notes they discuss Galois theory for schemes. Cf. especially Chapter 4 (Part I) of math.upenn.edu/~chai/624_08/math624_08.html For a student with a background in algebraic number theory, Grothendieck's version of Galois theory may provide for added motivation to learn about schemes | |
| Jun 18, 2010 at 19:37 | comment | added | Georges Elencwajg | Obligatory indeed, Kevin. Whoever reads my old friend Lieven Le Bruyn's brilliant exegesis will be delighted by his detailed explanations. As a bonus one gets a picture of $\mathbb A^1_{\mathbb Z}$ and one of Mumford on the same page. | |
| Jun 18, 2010 at 18:51 | comment | added | Kevin H. Lin | Obligatory: neverendingbooks.org/index.php/mumfords-treasure-map.html | |
| Jun 18, 2010 at 14:36 | comment | added | BCnrd | Ravi & Georges: in the same spirit, the pictures in Mumford's book on abelian varieties are not as good as the ones in EGA. :) | |
| Jun 18, 2010 at 14:24 | comment | added | Georges Elencwajg | @BCnrad: you are right. I have noticed that if I think hard enough, sometimes surprises in algebraic geometry start to seem less mysterious. But unfortunately this doesn't happen very often to me. Maybe it is a general phenomenon: understanding means you are no longer surprised ? | |
| Jun 18, 2010 at 14:15 | comment | added | Georges Elencwajg | I am very proud of your endorsement, Ravi: thank you. Contrariwise, my heart missed a beat when I read your sentence "I strongly disagree with you about EGA" but fortunately, reading on, I realized that the rich resources of the empty set were coming to my rescue :-) | |
| Jun 18, 2010 at 13:53 | comment | added | Ravi Vakil | I strongly disagree with you about EGA. Every single picture in EGA is incredibly enlightening, and beautifully rendered. :-) [Anyone who has looked at EGA will realize I'm actually agreeing with Georges, but that my "Every single picture" comment is also true...] | |
| Jun 18, 2010 at 13:51 | comment | added | Ravi Vakil | Agreed. And further, it seems essential (in a literal sense, not just polemic sense) to know that maps to projective space are what you say. And helpful (and nearly essential) to recognize this as a functorial description of projective space (by universal property). Also related to your discussion: I see little advantage to restricting to working over an algebraically closed field, or even a field. It makes very few arguments simpler. It is true that working over $\overline{k}$ can be help reduce anxiety. But comfort levels can be stretched by working with beautiful curves like Spec Z. | |
| Jun 18, 2010 at 13:21 | comment | added | BCnrd | Georges: the common "surprise" about points of projective space or of affine space minus 0-section valued in a ring (or scheme) has always seemed best to explain by analogy with how the same issue comes up in differential geometry, or even alg. geom. using only varieties and not schemes. The meaning of a map from a manifold to real projective $n$-space works out exactly as with schemes, and likewise for affine space minus the 0-section, so it is good to stress to students that none of this is peculiar to working with schemes or is a phenomenon special to the "arithmetic" case. | |
| Jun 18, 2010 at 12:16 | comment | added | Georges Elencwajg | Spot on, Charles: it was indeed that question which motivated this answer | |
| Jun 18, 2010 at 12:00 | comment | added | Charles Siegel | In particular, it would help with the issue from this question: mathoverflow.net/questions/28485/integral-points-on-varieties | |
| Jun 18, 2010 at 11:39 | history | answered | Georges Elencwajg | CC BY-SA 2.5 |