Timeline for Sheaf / de Rham cohomology of a stack with values in a complex of abelian sheaves
Current License: CC BY-SA 4.0
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| Apr 14, 2020 at 10:38 | history | edited | Praphulla Koushik | CC BY-SA 4.0 |
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| Apr 14, 2020 at 10:17 | history | edited | YCor | CC BY-SA 4.0 |
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| Apr 14, 2020 at 10:11 | history | edited | Praphulla Koushik | CC BY-SA 4.0 |
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| Jul 16, 2018 at 17:22 | answer | added | Liam Keenan | timeline score: 2 | |
| Jul 16, 2018 at 5:21 | comment | added | Praphulla Koushik | @user40276 I will see that :) thanks.. | |
| Jul 12, 2018 at 2:45 | comment | added | user40276 | The de Rham cohomology of a manifold can be computed by the same complex. The point is that in the manifold case, the sheaf of differential forms is fine and, therefore, acyclic. This allows one to compute de Rham cohomology (in the case of paracompact Hausdorff manifolds) by only using the global sections of the sheaf of differential forms. Take a look at Bott and Tu "Differential forms in algebraic topology" and look at the Cech-de Rham bicomplex. | |
| Jul 11, 2018 at 14:51 | comment | added | Praphulla Koushik | @KeerthiMadapusiPera Hello. Thanks for your comment. I am not aware of notion of derived category associated to an abelian category.. I know only little about derived functors that is also from Hartshorne.. Can you suggest some reference (I can search online but I might end up with something that is not written smoothly) for notion of derived category associated with an abelian category... Can you say little more about "the singular chain complex of a CW complex carries a lot more information than the bare homology groups." | |
| Jul 11, 2018 at 14:41 | comment | added | Keerthi Madapusi | Your question doesn't appear to have much to do with stacks but rather with the notion of a (bounded) derived category associated with an abelian category (here, the category of abelian sheaves on $\mathcal{X}$). I suggest reading up on derived categories first. They're a way to keep track of finer information than just homology or cohomology: For instance, the singular chain complex of a CW complex carries a lot more information than the bare homology groups. | |
| Jul 11, 2018 at 13:32 | history | asked | Praphulla Koushik | CC BY-SA 4.0 |