Timeline for Why study Higher Sheaf Cohomology?
Current License: CC BY-SA 3.0
24 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Oct 31, 2016 at 0:47 | comment | added | finnlim | I see, thank you.That's one less weight off my heart (I always thought that I don't understand sheaf cohomology very well because I don't understand how to think of the canonical resolution using discontinuous sections). | |
| Oct 28, 2016 at 3:50 | answer | added | David E Speyer | timeline score: 8 | |
| Oct 27, 2016 at 20:44 | answer | added | ACL | timeline score: 11 | |
| Oct 27, 2016 at 19:52 | comment | added | ACL | The point with resolutions is that you do not need to bother too much about what they are. The interest of considering canonical resolutions is to have well defined cohomology objects (by opposition to objects well defined up to canonical isomorphisms, as is the case in homological algebra in general Abelian categories, for example). Once you have objects in hand, you can freely play with them and homological algebra gives you a lot of choices. | |
| Oct 27, 2016 at 18:09 | answer | added | Allen Knutson | timeline score: 18 | |
| Oct 27, 2016 at 4:35 | comment | added | finnlim | I suppose you are talking about the "canonical resolution" where the sheaf of discontinuous sections are taken interatively? I hope I don't sound like too much of a complainer but at second or third iteration already I found the resulting sheaf (and the cohomology) quite unintuitive... even though choices of the acyclic sheaves were canonical here. | |
| Oct 27, 2016 at 4:32 | vote | accept | finnlim | ||
| Oct 26, 2016 at 11:01 | comment | added | ACL | I suggest you study an introductory book on cohomology and sheaves, such as Godement's book, Topologie algébrique et théorie des faisceaux. Note that in this book, Godement canonically embeds a sheaf into the sheaf of its discontinuous sections; the latter sheaf is flasque and this procedure leads to a canonical resolution of the initial sheaf. | |
| Oct 26, 2016 at 0:41 | comment | added | finnlim | @DenisNardin For $H^2(X;G)=H^1(X;BG)$, I am more satisfied because this gives a canonical choice of "something else" to point at. Also, I am a student with quite much homotopy-theory interest too so I'd love to know more about them! Thank you. | |
| Oct 25, 2016 at 19:11 | comment | added | arsmath | Syzygies are well-explained in Cox, Little, and O'Shea. They're a pretty elementary idea. | |
| Oct 25, 2016 at 15:37 | answer | added | Will Sawin | timeline score: 41 | |
| Oct 25, 2016 at 15:30 | comment | added | Denis Nardin | I know very little about sygyzies so I really could not do anything apart from pointing you to Hartshorne's book on the subject. Also $H^2$ is used to classify gerbes banded by a certain abelian group, but this is in a sense related to the procedure you already dismissed (you write $H^2(X;G) = H^1(X;BG)$ where $BG$ is the stack in groups of torsors for $G$). All the references I know about this have a strong homotopy-theoretic slant and would bring you very far from your original concerns, so maybe someone can provide something more accessible. | |
| Oct 25, 2016 at 14:57 | answer | added | Steven Gubkin | timeline score: 2 | |
| Oct 25, 2016 at 13:39 | history | edited | finnlim | CC BY-SA 3.0 |
minor edit to prevent confusion
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| Oct 25, 2016 at 13:36 | answer | added | Jim Humphreys | timeline score: 10 | |
| Oct 25, 2016 at 13:35 | comment | added | finnlim | A friend pointed out to me that $H^2$ may be of use to classify stacks, supposedly for the reason precisely the same as $H^1(X,\mathcal O^\times)$ classifying line bundles by encoding transition maps. I really don't know about this business, so if anyone could verify if this is right (and point to right references...?) I'd be very grateful. | |
| Oct 25, 2016 at 13:31 | comment | added | finnlim | Also, I understand that cohomology theories have very interesting relationships between them, including Serre duality, Poincare duality, Hard Lefschetz, Hodge decomposition, Frolicher relations, Lefschetz Hyperplane theorem, ... but would the intrinsic structures of $H^3, H^4, \cdots$ ever be of use to something coming from outside of the formalism? | |
| Oct 25, 2016 at 13:28 | comment | added | finnlim | Thank you for the comment. Could you elaborate a little more about how knowledge of higher $H^i$ can help us compute $H^0$? For example, if we didn't know $H^0$ and knew $H^3, H^4, \cdots$, then can we say something about $H^0$? Also I'd greatly appreciate some references / expository articles about how cohomology relates to syzygy, and ultimately to combinatorics! | |
| Oct 25, 2016 at 12:47 | comment | added | Denis Nardin | I suppose you could say that we only care about $H^0$ and all the higher $H^i$'s are technical instruments that are used to reach at $H^0$. On the other hand cohomology has often an intrinsic structure worth studying in its own right (e.g. Serre and Poincarè dualities) and in a more algebraic setting has a nice relation to sygyzies that might appeal to the combinatorially minded. | |
| Oct 25, 2016 at 6:36 | review | Close votes | |||
| Oct 25, 2016 at 11:06 | |||||
| Oct 25, 2016 at 6:24 | history | edited | finnlim | CC BY-SA 3.0 |
typo corrected
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| Oct 25, 2016 at 6:16 | review | First posts | |||
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| Oct 25, 2016 at 6:14 | history | asked | finnlim | CC BY-SA 3.0 |