Timeline for Weights on equivariant cohomology?
Current License: CC BY-SA 3.0
18 events
| when toggle format | what | by | license | comment | |
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| Jul 1, 2013 at 9:07 | comment | added | Jan Weidner | The map $X/G\rightarrow pt/G$ is easy, while $X/G\rightarrow pt$ is more problematic. For example suppose $X=pt$. Then the push forward along the first map is just the identity functor, while push forward along the second map is taking equivariant hypercohomology (of a point). | |
| Jun 20, 2013 at 23:26 | comment | added | Reladenine Vakalwe | Regarding the discussion on proper maps: What is your favorite (non trivial) example of a proper map of stacks $X/G \to pt/G$ that doesn't come from a proper $X \to pt$ and such that the induced maps on approximations aren't proper? Sorry I am not very fluent with stacks. | |
| Jun 20, 2013 at 22:48 | comment | added | Reladenine Vakalwe | (contd. 2) that $H^1$ and $H^2$ of strata vanish (basically to give an induction that Ext groups in your triangulated categories coincide). This sort of situation excludes several cases (for instance the Harish-Chandra case) when you have non trivial local systems. This isn't proof but I think some jiggling of this line of argument will show that nice ABC approximations can't exist if you have non trivial equivariant local systems. It would be nice to get around this, since this is basically the 'obstruction' to construct a tilting generator (with Hodge package) to get equivariant formality | |
| Jun 20, 2013 at 22:38 | comment | added | Reladenine Vakalwe | Regarding ABC approximations: This is exactly the issue I was referring to when I mentioned things blowing up if you have non-trivial local systems on the orbits/strata. However, just to clarify: (working non-equivariantly) the constructs le derived category is always equivalent to the derived category of perverse sheaves. Things screw up only if you fix a stratification. Then the constructive (with respect to the fixed stratification) is not necessarily equivalent to the derived category of perverse sheaves smooth along that stratification. The ABC stratifications essentially require (contd.) | |
| Jun 18, 2013 at 8:36 | comment | added | Jan Weidner | For $f_*=f_!$ you are right, but only in the case where $f$ is a map between two $G$-varieties. However in the most interesting examples this is not the case! For example the relevant map for equivariant cohomology $X//G\rightarrow pt$ does NOT fit into this framework! | |
| Jun 18, 2013 at 8:36 | comment | added | Jan Weidner | Thank you very much for your detailed discussion! By approxiamtions, I meant nice approximations, called "ABC" Olaf. Their most crucial property is, that they admit a nice stratification, such that the derived category of perverse sheaves equals the constructible derived category. It suffices to construct them for the group acting on a point and as you said, they are given by Stiefel varieties, if $G=GL_n$. Now the problem is, if you just take a subgroup of $GL_n$ and let it act on the Stiefel varieties, the resulting approximation won't be nice! | |
| Jun 17, 2013 at 15:52 | comment | added | Reladenine Vakalwe | I seem to have slipped under the rug that $G\backslash(X\times V_k)$ is a well behaved variety. I am not so sure about this. I think additional hypothesis are required. I would have to think a bit more about this. | |
| Jun 17, 2013 at 15:37 | comment | added | Reladenine Vakalwe | (contd. 1) In this process for nothing to blow up in your face it becomes convenient to at least have each strata being affine and contractible. All of this is not necessary if the goal is just to have a well behaved (minus formality) mixed equivariant derived category. If you are interested in generalizing Olaf's thesis, I basically don't know how one would proceed if the orbits admit non trivial equivariant local systems. Once more the ideal person to ask would be Wolfgang. He has surely thought a lot about the Harish-Chandra situation | |
| Jun 17, 2013 at 15:25 | comment | added | Reladenine Vakalwe | Finally, regarding Olaf's thesis: much more is done there than just checking that the mixed equivariant derived category makes sense and behaves. There (if I remember correctly), one sets up a Hodge package to get formality of the $Ext$ algebra. For this one needs (approximations) to projectives, Hodge structures on them etc. The construction of projectives is inductive in the style of Beilinson-Ginzburg-Soergel. Again if I remember correctly, basically a compatible family of projectives on each approximation is constructed. (contd. below) | |
| Jun 17, 2013 at 15:19 | comment | added | Reladenine Vakalwe | Regarding constructing approximations: if we are limiting ourselves to linear algebraic groups, then it suffices to have then for $GL_n$. Take the $k$-th approximation to be the Steifel variety of $n$-tuples of independent vectors in $\mathbb{C}^{n+k}$. | |
| Jun 17, 2013 at 15:15 | comment | added | Reladenine Vakalwe | I should add that I am assuming above that the approximations being used above (this is relevant for the argument showing that choice of approximation is irrelevant and nothing messes with weights) is smooth. | |
| Jun 17, 2013 at 15:13 | comment | added | Reladenine Vakalwe | Regarding pushforward: so we are given a proper equivariant map $f\colon X\to Y$, then the pushforwards $f_!$ and $f_*$ are basically going to be a family of pushforwards (on the approximations) given by base change. Each of these will be proper, so $f_* = f_!$. Am I missing/misunderstanding something? | |
| Jun 17, 2013 at 15:06 | comment | added | Reladenine Vakalwe | Regarding choice of approximation: say $X$ was the $G$-variety; if $V_k'$ and $V_k''$ are both approximations for (the classifying space) for $G$ then so is $V_k = V_k' \times V_k''$. Pulling back along $G\backslash (X\times V_k) \to G\backslash (X\times V_k')$ and similarly for $V_k''$ should give an equivalence showing that your category doesn't depend on the approximation. (I think I am just remembering the argument from Bernstein-Lunts, sorry am traveling right now so can't be more comprehensive). | |
| Jun 17, 2013 at 8:57 | comment | added | Jan Weidner | I have looked into the other references you suggest and one thing I am missing (though it could be hidden somewhere, I don't understand them very well...) would be the formula $f_!=f_∗$ for proper $f$. Without it, I can't even deduce that $H_G(pt)$ is pure. Do you know if $f_!=f_∗$ is in there or follows easily? | |
| Jun 17, 2013 at 8:54 | comment | added | Jan Weidner | Thanks for your comprehensive answer! One problem when working with approximations is, that you need to construct them. For example I would not know how to generalize O. Schnurer's thesis to the $P\G/Q$ case since I don't see how to construct the relevant approximations. Also approximations involve choices and choices usually lead to trouble in the long run. | |
| Jun 15, 2013 at 17:51 | history | edited | Reladenine Vakalwe | CC BY-SA 3.0 |
Added links/references, cleaned up language to be more coherent
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| Jun 15, 2013 at 5:46 | history | edited | Ricardo Andrade | CC BY-SA 3.0 |
correctedd tex
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| Jun 14, 2013 at 17:48 | history | answered | Reladenine Vakalwe | CC BY-SA 3.0 |