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Saal Hardali
Saal Hardali
8:07 AM
Let X be an algebraic variety over C. Is the derived catevory of perfect complexes equivalent to the derived category of local systems (complexes of etale sheaves of C-vector spaces with locally constant cohomologies)?
I guess an easier pre-question would be: Does every vector bundle admit a filtration whose quotients admit flat connections
 
Pieter
Pieter
8:45 AM
Consider X=P^1, then D^perf(P^1) has a very easy description using the Kronecker quiver. Is there any reason why you'd expect this description for the derived category of local systems?
Given that I never work with local systems, I cannot give a good argument as to why they are different, but one thing for instance which comes to mind is that coherent sheaves have recollements going in the wrong way, whereas local systems (if they have recollements at all, because I'm always confused about non-torsion coefficients in the étale topology) should have recollements in the correct direction.
And maybe you want to look at the fact that coherent cohomology vanishes above degree 1, whereas that shouldn't be true for local systems (but again, I don't know what to expect here).
 
 
5 hours later…
nikitas
nikitas
1:24 PM
Does anyone know some references on the " Stable orbit category " ? I only know Schwede-Shipley "Stable model categories " that they mention it and Lewis Jr., J.P. May, M. Steinberger, Equivariant Stable Homotopy Theory, Lecture Notes in Mathematics
 
Saal Hardali
Saal Hardali
1:49 PM
@Pieter Good you brought up $P^1$ since It made realize what was wrong in my initial phrasing. What I actually want is not perfect complexes but the derived category of complexes with locally free finite cohomologies (vector bundles).
By the way i'm pretty sure that the derived category of constructible sheaves on $P^1$ has the same description in terms of kronecker delta hence equivalent to the derived category of perfect complexes (this was pretty much the motivation for the question).
 
Denis Nardin
Denis Nardin
@nikitas It is usually known as the Burnside category. I like this paper but there are plenty of references around.
 
Pieter
Pieter
2:10 PM
@SaalHardali The recollement argument I mentioned is bogus, because it is for the unbounded derived category, so ignore that. Can you give a reference for the derived category of constructible sheaves on P^1 being equivalent to the Kronecker quiver?
You want the subcategory of D^perf(P^1) spanned by objects whose cohomology sheaves are vector bundles?
That seems somewhat ill-defined if you look at it in a naive way: just consider any inclusion O(-1)\to O, which should be a morphism between two objects in your category, but their cone isn't an object in there anymore.
 
Saal Hardali
Saal Hardali
2:42 PM
I get what you're saying. I don't know about the statement on $P^1$ but I think giving a perverse sheaf with one singular point is the same as giving a representation of the said quiver and the derived category of perverse sheaves is equivalent to the derived category of constructible sheaves in favorable situations.
Now there's the issue of reducing to the case of 1 singular point. Here i have no idea.
the statement might even be wrong.
 
Bruno Stonek
Bruno Stonek
3:18 PM
quick question: consider simply connected spaces. then the operations of taking loop space and rationalizing commute, right? what's the cleanest way to see this?
 
Denis Nardin
Denis Nardin
3:39 PM
@BrunoStonek I think it follows from the Eilenberg-Moore spectral sequence: from that you get that C^*(ΩX,Q) is the loopspace of C^*(X;Q) in the category of E_∞-algebras over Q.
(I'm using the fact that C^*(-;Q) embeds nice rational spaces into the opposite category of E_∞-algebras over Q)
Oh sorry, I meant the suspension of C^*(X;Q), that is the loopspace in the opposite category (those damned ops...)
 
 
2 hours later…
Bruno Stonek
Bruno Stonek
5:43 PM
hmm. I feel like it should be more elementary? not using any E_oo stuff
 
Elden Elmanto
Elden Elmanto
6:02 PM
@SaalHardali these guys are of a different nature brings different struggles. For example the $K$-theory of $Perf_{P^1_S}$ splits as $K(S) \oplus K(S)$. Whereas the $K$-theory of the derived category of etale sheaves (by this I mean the $K$-theory of the (abelian category of, say) lisse sheaves on $P^1$) is the $K$-theory of the group ring of profinite etale pi_1 of $P^1$, which is 0 over char 0
conceptually I think, they are two seemingly different functors from schemes to $Cat$. For example one takes $X \times_S Y$ to $Perf_X \otimes_{Perf(S)} Perf_Y$ for suitable schemes, while such decomposition hardly ever exists in the constructible/locally constant world
 
Pieter
Pieter
6:23 PM
Ah good, an answer by someone who actually knows both sides of the story :).
 
Pieter
Pieter
6:34 PM
Ah, isn't it also a totally obvious thing to say that one is insensitive to infinitesimal thickenings?
 
Elden Elmanto
Elden Elmanto
@Pieter More like someone who's suffered both sides of the story. And yea I suppose so. I guess I just tend to K-things immediately :)
I'd summon Adeel if he's around to suffer with me
 
Denis Nardin
Denis Nardin
@BrunoStonek If you prefer you can say cdga over Q (obviously there's no difference, although I tend to think in terms of E_∞-algebras more easily). I don't think you can do much rational homotopy theory without involving cdga and this is really just a special case of the Eilenberg-Moore spectral sequence
 
Elden Elmanto
Elden Elmanto
7:09 PM
@Pieter there's also something that keeps me up at night. Here are some coincidences between the constructible world and the Perfect (pun intended) world.
1. H^*_{et} clearly belongs to the constructible world. But somehow H^1_{et} also belong to the second world since it classified line bundles. Now, while H^2_{et} (aka gerbes) are really constructible gadgets, we can also think of H^2_{et} as invertible objects in R-linear categories which should belong to a (higher version) of the Perfect world. There are speculations here that I don't want to make but you should feel free to.
anyway I guess the whole history of this world has been about breaking down the constructible pieces in terms of its perfect parts (hodge theory)...so I guess I'm not saying anything much.
 
Pieter
Pieter
The questions you raise are very interesting, but above my pay grade I'm afraid :).
 
Elden Elmanto
Elden Elmanto
yeah totally above mine too. Hopefully someone comes along one day..
 
 
2 hours later…
Mike Miller
Mike Miller
9:38 PM
@DenisNardin I agree that it follows from the Eilenberg-Moore SS, but mainly because it's obvious that (with Q coefficients) the map $X \to X \otimes \Bbb Q$ is an isomorphism on cohomology, and the EMSS runs from a tor over the cohomology of each to the cohomology of the loopspace; so the map $\Omega X \to \Omega(X \otimes \Bbb Q)$ is a rational equivalence, which is what I assumed Bruno wanted? I guess I'm using some basics of rational homotopy theory in here, which was your point?
 
Denis Nardin
Denis Nardin
@MikeMiller Yeah, I meant more or less that. I guess exactly how much rational homotopy theory you are using depends on what exactly you use as the definition of X⊗Q (I like Map(C^*(X;Q),Q), where the mapping space is in the category of cdga over Q )
My idea was to reduce to the fact that C^*(ΩX,Q)=Q⊗_{C^*(X;Q)} Q, which is basically the EMSS
(the tensor product is, of course, derived)
 
Mike Miller
Mike Miller
Ah I see. I don't think I was working with a functorial model of rationalization, just the universal property. So I guess I can give you an isomorphism but not a natural one.
if that's all you want might be a little bit easier to phrase, I guess, since you can avoid a little bit of algebra
 

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