Timeline for What should be learned in a first serious schemes course?
Current License: CC BY-SA 2.5
14 events
| when toggle format | what | by | license | comment | |
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| Aug 22, 2023 at 12:02 | comment | added | Elías Guisado Villalgordo | Another good reference for the equivalence between sheaves and sheaves defined on a basis is Tag 009H. | |
| Dec 22, 2010 at 9:02 | history | made wiki | Post Made Community Wiki | ||
| Jun 19, 2010 at 8:10 | comment | added | Kevin H. Lin | BCnrd: You're right that the full "Nike Lemma" is not explicitly stated there. But the idea is there, no? | |
| Jun 18, 2010 at 23:05 | comment | added | BCnrd | Kevin, the curious thing is that in the link you give, it is only said that there's some open in the overlap that is simultaneous basic open in each piece (which is not useful), rather than that there's an entire open cover by such doubly-distinguished opens (which as you know is quite useful in getting things off the ground). | |
| Jun 18, 2010 at 15:13 | comment | added | Kevin H. Lin | I learned that fact from here: www-math.mit.edu/~kedlaya/18.726-spr05/virtual.html | |
| Jun 18, 2010 at 13:39 | comment | added | Ravi Vakil | He's told me since that he heard this from Nike Vatsal, who later forgot about it and was surprised to hear someone call it "Nike's Lemma". (Of course no one would claim that it was unknown before Nike.) | |
| Jun 18, 2010 at 13:38 | comment | added | Ravi Vakil | On a related note (also related to Allen's discussion of schemes as gluing together affines): I have found that using the fact that schemes are affines glued together, rather than just ringed spaces, makes showing facts we care about regarding quasicoherent sheaves much easier. In particular, in graduate school, it was a revelation when a visiting grad school (who now goes under the pseudonym "BCnrd") pointed out to me the simple fact that the intersection of 2 affines is a union of affines simultaneously distinguished in them both. This turned a host of Hartshorne ideas from hard to easy. | |
| Jun 18, 2010 at 0:39 | comment | added | BCnrd | David Speyer: another reference is 3.2 in EGA 0$_{\rm{I}}$. Strange that it is not in Godement's book (as best I can tell). | |
| Jun 18, 2010 at 0:00 | comment | added | Ravi Vakil | Definitely! In fact I don't see how to do without this, without going through extreme contortions. | |
| Jun 17, 2010 at 21:12 | comment | added | Kevin H. Lin | I think Chapter I of Eisenbud-Harris is really great in general. | |
| Jun 17, 2010 at 18:48 | comment | added | David E Speyer | Wow, a reference for that lemma! Thanks, Kevin. | |
| Jun 17, 2010 at 18:25 | comment | added | BCnrd | My first (limited) exposure to schemes was in a course from Joe Harris which used a draft copy of the book with Eisenbud, and most of which went completely over my head. But when I eventually "graduated" to reading Hartshorne I knew to ignore his mysterious construction of the structure sheaf and followed the "B-sheaf" method from Eisenbud-Harris instead, with the help of Exercise 23 in Chapter 3 of Atiyah-MacDonald. | |
| Jun 17, 2010 at 17:59 | comment | added | Harry Gindi | The fact that this is not stated in Hartshorne is one of the reasons why his construction of the structure sheaf of an affine scheme is so ad-hoc. | |
| Jun 17, 2010 at 17:53 | history | answered | Kevin H. Lin | CC BY-SA 2.5 |