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    • CommentRowNumber1.
    • CommentAuthorhilbertthm90
    • CommentTimeNov 11th 2010
    I've been reading through the Cech cohomology and the Abelian Sheaf Cohomology pages (and several of the links off of them) and trying to figure out when they give the same thing in certain situations. It came up recently in a student seminar I'm a part of that we wanted to do a sheaf cohomology computation on a DM stack using Cech cohomology, but no one seemed to know if this would give the right answer.

    I'm getting a better idea of what's going on after reading these pages, but the terms are still quite unfamiliar. So I guess my question is, is there some nice statement that a group of algebraic geometers who don't know the language of (oo, 1)-toposes can take away?

    As a group, our guess was that we could use a Cech cover if it was DM, but maybe Artin stacks it wouldn't necessarily always work. Urs already explained to me the general theory of when it works, but I don't know the terms well enough to unwind whether DM stacks are in this class. I was hoping someone would just know. Thanks!
    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeNov 12th 2010
    • (edited Nov 12th 2010)

    I promised to reply in more detail to this question when you come here, and actually we should write an nLab entry about it. But right now it is late at night here for me and need to call it quits and tomorrow I’ll be very busy. So for the moment just a little bit.

    First of all, I would slightly reformulate the question, just so that one better sees what the problem is:

    the general issue is that we have chosen some site CC and declared that a morphism between complexes of sheaves on that site is a weak equivalence if it is stalkwise a quasi-ismorphism (assuming that the sheaf topos on the site has enough points).

    By Dwyer-Kan theory, that alone already completely fixes what the correct notion of cohomology is: for complexes of sheaves XX and A[n]A[n], we are to somehow get hold of the derived hom-space

    H(X,A[n]) \mathbf{H}(X,A[n])

    and its connected components is the correct H n(X,A)H^n(X,A).

    So now the question is how to get hold of that. In practice we usually invoke Quillen who says that if we manage to put a model category structure on the situation, then there is a comparatively simpler algorithm for computing H(X,A[n])\mathbf{H}(X,A[n]).

    In the usual definition of abelian sheaf cohomology, one more or less implicitly picks one of the model structure on chain complexes of sheaves where most evverything is cofibrant. In such a situation the hom-space is computed by forming a fibrant resolution of A[n]A[n] and then homming into it. This is what the traditional description of sheaf cohomology as the “right derived global section functor” does in the case that XX is the terminal sheaf.

    So in a way sheaf cohomology is by definition correct because it tells us: apply one of the algorithms to obtain a fibrant replacement.

    The trouble with “Cech cohomology” is that it is not by definition correct . Cech cohomology more or less implicitly assumes a dual model structure where most everything is fibrant, and hence offers a presciption for how to resolve XX instead of AA. But the prescription is just a prescription, with no guarantee “make sure that the result is cofibrant”.

    That’s why there is an issue at all with Cech- vs derived-global-section functor cohomology. The former is one concrete algorithm that sometimes produces the right resolution, whereas the latter is kind of by definition the right algorithm for producing the right resolution.

    So the question is really: when does the Cech-algorithm produce the correct answer. And using model category tools we can say this more precisely:

    suppose on our site most everything is fibrant. In particular the A[n]A[n] that we care about. Then the whole question is: is there a cover of XX such that the Cech nerve C(U)C(U) is cofibrant?

    If it is, then Quillen guarantees you that Cech cohomology produces the correct cohomology and hence in particular coinides with the derived-global section cohomology.

    What can go wrong is that there does not exist a single choice of cover {U iX}\{U_i \to X\} such that the Cech nerve C(U)C(U) is cofibrant. Instead, it may happen that you have to keep refining this Cech nerve itself. By hypercovers. That old theorem by Kenneth Brown at BrownAHT – building on Verdier’s hypercovering theorem – tells us that refining the Cech-prescription by hypercovers is a gives an algorithm that is guaranteed to compute the correct cohomology.

    And it may happen that there is just no way around this, that there is just no site for your problem such that Cech covers of UU are cofibrant.

    So, then next we need to get hold of conditions and means to decide whether or not Cech nerves can be cofibrant in our situation. I mentioned on MO that this works with a theorem by Dan Dugger. This is reviewed at model structure on simplicial presheaves in the section on cofibrant replacement.

    This applies to the projective model structure. That’s the one where Cech cohomology tools play a role (because in the injective structure everything is already cofibrant and the whole work is instead in producing the fibrant replacement of A[n]A[n]).

    Dugger tells us that a sufficient condition for a Cech nerve to be cofibrant is that all finite non-empty intersection U i 0U i nU_{i_0} \cap \cdots U_{i_n} of the patches, formed as pullbacks of their represented presheaves, are again representable.

    You recognize in this general prescription a well known special case: if our site is CartSp, the site of open balls, then a cover {U iX}\{U_i \to X\} has a projectively cofibrant Cech nerve precisely if it is a good open cover. So we reproduce the well-known theorem that for computing Cech cohmology on a paracompact space, it is sufficient to compute it on a good cover.

    But the problem is that you are interested in an algebraic site. I haven’t thought much about these, and so here for the last step you are on your own. I can just describe the algorithm again for what you’d need to do and check:

    1. pass throughout to the functor-of-points perspective and regard all your DM stacks and everyhting in the game as stacks on some “gros” site of abstract algebraic spaces, such as all duals to finitely generated rings/algebra or the like;

    2. in fact, see if you can make that smaller: you want a site over which your coefficient sheaf satisfies descent, so that it is projective fibrant.

    3. Once you have that, check which covers of your base space object you can form and check, finally…

    4. if there are any such covers such that all finite non-empty intersections of patches are again objects of your site.

    If you have all that, then Cech cohomology on these good covers will compute the correct cohomology. And hence in particular coincide with derived-global-section cohomology.

  1. Cech cohomology and sheaf cohomology disagree even for schemes, let alone stacks. This has come up a couple times on MathOverflow, for example this question:

    Perhaps if your sheaf is of a very special sort?

    • CommentRowNumber4.
    • CommentAuthorhilbertthm90
    • CommentTimeNov 12th 2010
    We were quite aware of the fact that they disagree for schemes since Grothendieck has a nice easy example in Tohoku.

    I hid a lot of what we actually were discussing in order to ask this in as much generality as possible. We were actually only caring about two very specific things. We wanted to know if it worked for H^1, since it does for nice schemes, and we were also only really caring about the sheaf of invertible elements of the structure sheaf O*.

    The very, very first way this popped up was we were wondering if you could show that Pic of certain moduli spaces of curves was isomorphic to H^1(M, O*). So we wanted to do it the same way as for schemes, which requires taking a suitable Cech cover. After some discussion, we wondered how general we could make this.

    I guess the question shouldn't have "Cech" in it, but "hypercover"? Does this make is sound more reasonable? I think then they don't disagree for schemes (under some mild conditions?)
    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeNov 12th 2010

    It’s true that if you look just at H 1H^1 the situation drastically simplifies: for the computation of H 1H^1 there is no difference between a Cech cover and a hypercover, since the hypercover starts kicking in in degree 2 only by refining the double intersections of the Cech cover.

    • CommentRowNumber6.
    • CommentAuthorhilbertthm90
    • CommentTimeNov 17th 2010
    • (edited Nov 17th 2010)
    Sorry to keep bringing this up. My comments above (post 4) weren't well thought out. I was looking at SGA 4 expose V, and if you take the limit over all hypercovers it gives the same answer as sheaf cohomology. Only now did I realize that Urs actually already said that about half way through his post when talking about Kenneth Brown.

    So the interesting question really is about Cech cohomology (not hypercovers), and how mild of conditions can we assume to guarantee the correct answer. Urs' approach seems best (even though I haven't tried anything with it yet), since the standard spectral sequence approach seems to give conditions that are really hard to understand.

    On a side note, I don't really need this for anything, so maybe there is a good argument for why this question is completely uninteresting. Maybe there are much better ways of computing cohomology on DM stacks than trying to check weird conditions for Cech to work like just taking hypercoverings? Or maybe it just happens so rarely that it isn't worth figuring out?