Timeline for General cohomology groups and motives
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
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| Nov 24, 2011 at 14:38 | comment | added | Daniel Litt | @James: There are plenty of books and papers about the subjects in your question, and even just a quick search of MO would reveal many references. For $\ell$-adic cohomology, try Freitag-Kiehl and Brian Conrad's notes; for motivic stuff there have been several MO questions with lists of references; for Langlands stuff there are many surveys, some written from a motivic point of view--really all of the aspects of your question have pretty rich literature surrounding them. | |
| Nov 24, 2011 at 14:04 | comment | added | James D. Taylor | @Daniel: While I agree that this seems to have no clear answer and is therefore worthy of closing down, I completely disagree that MO should not be the first place to go to with this sort of question. It is precisely the collective knowledge of mathematicians that I want to tap into. There is clearly no book about this; if there are papers about it they would be hard to find; and if individuals don't know the answer it doesn't mean that an answer doesn't exist. This is precisely what mathoverflow is for. | |
| Nov 24, 2011 at 8:15 | comment | added | David Hansen | I vote to close as well. | |
| Nov 24, 2011 at 7:23 | comment | added | Daniel Litt | @S. Carnahan: I agree, MO should not be the first place one comes to for this sort of question. Voting to close. | |
| Nov 24, 2011 at 4:36 | comment | added | S. Carnahan♦ | Okay, I am inclined to close this question. It looks like a fishing expedition. | |
| Nov 24, 2011 at 1:02 | comment | added | James D. Taylor | @Daniel: Alas, I did not. My understanding of Langlands is rudimentary at best. I was trying to place Langlands, which is a statement about Galois actions on certain groups coming from schemes, in the context of general Galois actions on groups coming from schemes. | |
| Nov 24, 2011 at 0:23 | comment | added | Daniel Litt | @James: I don't really know anything about that side of things, so I can't tell you (at least at finite level, one can find a stable lattice and reduce mod some power of the uniformizer in $\mathbb{Z}_p$ to get a $\mathbb{Z}/p^n\mathbb{Z}$-rep, but I don't see what this buys you). I was hoping you had some particular result in mind you were thinking to generalize. | |
| Nov 24, 2011 at 0:07 | comment | added | James D. Taylor | @Daniel: that's a very good point. Does $H^i(X,\mathbb{Q}_p)$ being automorphic imply anything of any content about the representation $H^i(X,\mathbb{Z}/p^n\mathbb{Z})$? | |
| Nov 24, 2011 at 0:00 | comment | added | Daniel Litt | @James: It's probably worth working through the usual $\ell$-adic theory yourself before trying to get fancy with non-abelian stuff--even doing things properly with torsion coefficients (e.g. the constant sheaf $\mathbb{Z}/n\mathbb{Z}$) is pretty contentful. Also, I feel like "what can we say about this" is a bit broad--could you give a (true) statement about some easier case, e.g. $\mathbb{Z}/n\mathbb{Z}$ coefficients, so we know what sort of thing you're really looking for? | |
| Nov 23, 2011 at 23:56 | comment | added | James D. Taylor | I'm not sure if lisse l-adic sheaf is really what I would be looking for. Ideally, I would like a statement that would also make sense for $\mathcal{F}$ a linear algebraic group scheme (which would restrict us to non-abelian cohomology), although that might be asking too much. For now it would suffice to restrict ourselves to sheaves into abelian groups. | |
| Nov 23, 2011 at 23:55 | comment | added | Daniel Litt | (cont.) $\ell$-adic definition does work. | |
| Nov 23, 2011 at 23:54 | comment | added | Daniel Litt | The computation of "naive etale cohomology" as you give it here is on page 118 of my copy of Freitag-Kiehl; as David Hansen states, with $\mathbb{Q}_p$ coefficients, all higher cohomology vanishes. The right thing here is not etale cohomology, but rather $\ell$-adic cohomology. FK also gives the naive computation for $\mathbb{Z}_p$ coefficients, which is pretty interesting, actually. One reason to think that this inverse limit gives the "right thing" is that one expects $H^1$ to be dual to the $\ell$-adic Tate module for an elliptic curve, and the naive definition doesn't work here, whereas | |
| Nov 23, 2011 at 23:53 | comment | added | James D. Taylor | Hmm... I was careless with my speech. You are right that I meant $\mathcal{F}$ as sheaf on the etale site on $X$ rather than $X$. (I viewed that as implicit by the fact that I look at $H_{et}^*(X,\mathcal{F})$.) Your other comment is more substantive -- is it true that $H^i_{et}(X,\mathbb{Q}_p)$ where $\mathbb{Q}_p$ is the constant sheaf on the etale site over $X$ is different from the inverse limit of the $H^i_{et}(X,\mathbb{Z}/p^i\mathbb{Z})$? | |
| Nov 23, 2011 at 23:44 | comment | added | David Hansen | The setup here, strictly speaking, is not correct. The sheaves which give rise to etale cohomology are sheaves on the etale site of X, not on X itself. Furthermore, etale cohomology with $\mathbf{Q}_p$-coefficients is defined as the inverse limit of etale cohomology groups with coefficients in $\mathbf{Z}/p^i\mathbf{Z}$, tensored with $\mathbf{Q}_p$. I believe that $\mathbf{Q}_p$- or $\mathbf{C}$-coefficients yield groups which are usually identically zero. The analogue of "nonconstant sheaf" in this setting might be something like "lisse $\ell$-adic sheaf", but I am far from an expert. | |
| Nov 23, 2011 at 22:51 | history | asked | James D. Taylor | CC BY-SA 3.0 |