Timeline for How should a homotopy theorist think about sheaf cohomology?
Current License: CC BY-SA 2.5
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| Jul 26, 2010 at 2:36 | history | edited | Sean Tilson | CC BY-SA 2.5 |
added what might be what i am looking for
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| Jul 25, 2010 at 19:22 | history | edited | Sean Tilson | CC BY-SA 2.5 |
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| Jul 21, 2010 at 2:20 | history | edited | Sean Tilson | CC BY-SA 2.5 |
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| Jul 21, 2010 at 1:41 | answer | added | Greg Friedman | timeline score: 6 | |
| Jul 20, 2010 at 22:35 | comment | added | algori | Andrea -- you probably want the subspace to be closed, or else the cohomology may well change. | |
| Jul 20, 2010 at 22:20 | answer | added | algori | timeline score: 11 | |
| Jul 20, 2010 at 21:42 | comment | added | Kevin H. Lin | For (3), look at, for instance, the sections on Cech cohomology and cohomology of projective space in Chapter III of Hartshorne. Eisenbud's commutative algebra book probably has lots more good examples and exercises. | |
| Jul 20, 2010 at 21:27 | comment | added | Kevin H. Lin | You might be interested in this: mathoverflow.net/questions/32287/… | |
| Jul 20, 2010 at 21:25 | comment | added | Daniel Litt | @Andrea Ferretti: This is less the case if one considers sheaves as functors out of some well-behaved subcategory of Top, with the usual Grothendieck topology. In this way we can talk about "the same sheaf" on different spaces. | |
| Jul 20, 2010 at 21:13 | answer | added | Dustin Clausen | timeline score: 16 | |
| Jul 20, 2010 at 21:00 | comment | added | Andrea Ferretti | A thing that you may want to keep in mind is that often sheaf cohomology tells you more about the sheaf than about the space. For example, a sheaf $\mathcal{F}$ on a subspace $X \subset Y$ can be extended to $0$ on $Y$ and cohomology will not change. | |
| Jul 20, 2010 at 20:37 | comment | added | Daniel Litt | 2) and 3) Bredon's Sheaf Theory has some of the things you're looking for, and is dense but readable. As for toy examples, computing the cohomology of some simple sheaves via the Cech complex is illuminating, as is playing with sheaves on posets. I am by no means an expert, though. | |
| Jul 20, 2010 at 20:30 | comment | added | Tyler Lawson | Sheaf cohomology is in some sense orthogonal to ordinary homotopy theory. What this perspective does is allow you to study coefficient systems that vary in a nontrivial way over a base space (or topos). Homotopy theory works instead by sticking to plain spaces (where your "base space" is a point), and trying to study in depth the complexities that can occur in that situation. These both come together when studying something like sheaves of spaces or simplicial sets or spectra, where you might have families of cohomology theories varying nontrivially. | |
| Jul 20, 2010 at 20:19 | history | asked | Sean Tilson | CC BY-SA 2.5 |