Timeline for Can one glue De Rham cohomology classes on a differential manifolds?
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| May 14, 2022 at 16:09 | history | edited | Georges Elencwajg | CC BY-SA 4.0 |
added 454 characters in body
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| May 14, 2022 at 14:01 | answer | added | Georges Elencwajg | timeline score: 4 | |
| May 13, 2022 at 19:51 | answer | added | Georges Elencwajg | timeline score: 4 | |
| May 13, 2022 at 16:11 | vote | accept | Georges Elencwajg | ||
| May 12, 2022 at 18:59 | history | became hot network question | |||
| May 12, 2022 at 18:02 | answer | added | Dmitri Pavlov | timeline score: 11 | |
| May 12, 2022 at 14:48 | comment | added | M.G. | Dear @DavidESpeyer, thanks for the very clear explanation! For whatever reasons I had never given much thought to the De Rham complex in terms of sheaves. I guess there is always a first! | |
| May 12, 2022 at 14:24 | comment | added | Georges Elencwajg | Dear @M.G., amusingly one of the brilliant geometers I allude to in my question made a very similar comment. Great minds think alike! Unfortunatately I had to tell him, as I am telling you, that I have only a very rudimentary knowledge of derived categories... | |
| May 12, 2022 at 14:24 | comment | added | David E Speyer | @M.G. The Poincare lemma (ncatlab.org/nlab/show/Poincar%C3%A9+lemma) says that, on a smooth manifold, the de Rham complex of sheaves is exact, so the image sheaf of $d : \Omega^{q-1} \to \Omega^q$ is the same as the kernel of $d : \Omega^{q} \to \Omega^{q+1}$ and the cohomology sheaves are zero. Concretely, this image/kernel is the sheaf of closed $q$-forms: $Z^q(U)$ is the vector space of closed $q$-forms on $U$. It is easy to see this using the description as the kernel of $d : \Omega^{q} \to \Omega^{q+1}$, since kernel is the same for sheaves and presheaves. | |
| May 12, 2022 at 12:08 | comment | added | M.G. | Just a comment. Your first observation sounds like a good point to be included in introductions to derived categories. Incidentally, this also makes me wonder about a description/interpretation of the cohomology sheaves of the De Rham complex in the smooth category (i.e. the quotient sheaves of the respective kernel and image sheaves under $d$). I've actually never thought about it! | |
| May 12, 2022 at 11:57 | answer | added | Will Sawin | timeline score: 18 | |
| May 12, 2022 at 10:53 | history | asked | Georges Elencwajg | CC BY-SA 4.0 |