Timeline for How much of scheme theory can you visualize?
Current License: CC BY-SA 2.5
11 events
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| Feb 15, 2017 at 17:01 | comment | added | Ingo Blechschmidt | @Harry: You can also construct the structure sheaf of $\operatorname{Spec} A$ by localizing the constant sheaf $\underline{A}$ at the universal filter (the subsheaf given by $U \mapsto \{ f \in \underline{A}(U) \,|\, \text{$f(\mathfrak{p}) \not\in \mathfrak{p}$ for all $\mathfrak{p} \in U$} \}$). I would argue that this is an even better way of constructing the structure sheaf than your B-sheaf proposal, for instance because it makes it easy to verify the universal property of $\operatorname{Spec} A$ and because it's very natural from the point of view of the internal language. | |
| Feb 9, 2010 at 6:12 | comment | added | S. Carnahan♦ | I think Hartshorne approached his book from a very pragmatic standpoint: to me it seems he wanted to introduce just enough machinery to discuss the problems that interested him, and no more. Based on my interactions with him, I think he is simply uninterested in foundational baggage if he doesn't personally find it useful. If you flip through the book with this in mind, you can see why he made certain seemingly ad hoc choices, e.g., the role of the empty set his definition of presheaf. This pedagogical strategy may be disagreeable to you, but you shouldn't label it as intrinsically bad. | |
| Feb 7, 2010 at 22:02 | comment | added | Harry Gindi | They are literally cross-sections of the bundle defined by the presheaf. This is in Mac Lane and Moerdjik's sheaves in geometry and logic. Further, Hartshorne introduces sheaves of abelian groups as the fundamental example. In reality, the fundamental idea of sheaf theory is precisely a sheaf of sets. A sheaf of rings is actually a ring object in the category of sheaves of sets. An $\mathcal{O}_X$-module is a module of a ring object in this enriched setting. It's all very natural and organic, but Hartshorne's constructions miss all of the naturality and replace it with ad-hoc constructions | |
| Feb 7, 2010 at 21:56 | comment | added | Harry Gindi | Well, I tried to learn about schemes from Hartshorne, and my main objection was that all of his constructions were completely ad-hoc. Hartshorne's construction of the structure sheaf is also pretty bad. The correct approach is to construct a B-sheaf on the base of the topology. Then instead of "rigging the stalks", they come out of the very natural analogy between "taking smaller neighborhoods" and localization of the ring. Also, sheafification is properly motivated by considering the elements of the sheafification to be sections of the projection of the espace etale of the presheaf. | |
| Feb 7, 2010 at 20:40 | comment | added | S. Carnahan♦ | @Harry: The question was about visualizing schemes, not learning "modern" approaches to them. As far as I can tell, EGA and SGA were not written with this goal in mind at all. I have my own problems with Hartshorne (e.g., definitions of formal scheme, elliptic curve, etc.), but I found the illustrations quite compelling. Given the level of your experience with learning and teaching about schemes, you might want to temper your declarations to appear more subjective, e.g., "I don't think Hartshorne is very good for learning about schemes..." | |
| Feb 7, 2010 at 20:16 | comment | added | S. Carnahan♦ | @Arne: Thanks for pointing out the Red book. I had not looked at it much, so I didn't think of it. | |
| Feb 7, 2010 at 18:52 | comment | added | Wanderer | I won't say that SGA and EGA are bad (on the contrary). But for building geometric intuition, they are clearly not the best out there... | |
| Feb 7, 2010 at 18:32 | comment | added | Harry Gindi | In particular, SGA4 is quite good. It has many of the modern approaches to scheme theory, and many of its results readily generalize to stacks etc. | |
| Feb 7, 2010 at 18:31 | comment | added | Harry Gindi | Hartshorne is a bad book for learning about schemes. I'm not saying this because it's hard or easy or anything like that. All of Hartshorne's constructions are ad-hoc (they're motivated by sheaf theory, of course) and hard to follow. EGA and SGA are really much better for this. | |
| Feb 7, 2010 at 11:06 | comment | added | Wanderer | Mumford does very nice things in his "Red book of varieties and schemes". You will learn to "see" the $K$-points (for different fields of definition $K$) and the Galois actions, the map $\text{Spec}(O_{X,x}) \to X$, et cetera. In fact, as for geometric intuition, I learned more from Mumford than from Hartshorne or Eisenbud/Harris. | |
| Feb 7, 2010 at 8:10 | history | answered | S. Carnahan♦ | CC BY-SA 2.5 |