Three ways of looking at a local system: Introduction and connection to cohomology theories

Suppose we have a space X. There are a lot of ways to describe the cohomology of X — algebraic geometers will know about étale cohomology, crystalline cohomology and (algebraic) deRham cohomology; topologists would add singular cohomology and simplicial cohomology to the list. Recently, thanks in large part to MIT’s K-theory seminar, I’ve come to understand how to tie the first three together. Each cohomology theory comes from a different way of looking at local systems.

Roughly, a local system consists of a vector bundle V on X, together with some additional data which gives isomorphisms between different fibers of X. In algebraic geometry, there are many different ways to make this idea precise.

In this series of posts, I want to present several of these ideas in the case of a smooth manifold X. In each case, if you pursued the idea far enough, you would get to a major tool of modern algebraic geometry.

I said above that a local system includes the data of isomorphisms between different fibers V_x and V_y of X. This data can depend on the choice of a path \gamma between x and y. Our three theories will be distinguished by how short the path \gamma is required to be.

We could work with an arbitrary \gamma. This would lead to étale sheaves

We could work with an infinitesimal \gamma (also known as an \infty-jet). This would lead to crystals 

We could work with a \gamma which is so small that it becomes a tangent vector. This would lead to \mathcal{D}-modules

These give rise to étale cohomology, crystalline cohomology and deRham cohomology. I should point out that, for modern applications, one usually wants to work on stacks, with singularities and in arbitrary characteristic. I won’t be addressing any of those issues; I just want to give the intuition behind each theory.

In this post, I will explain how to build a cohomology theory, given a notion of local system. I will then follow up with three more posts, one for each of the specific approaches above.

As usual, this series comes with a disclaimer: These are tools that are relevant to my work, but their inner workings are not my expertise. If you want to see experts discussing this sort of thing, it looks like that conversation is going on at Urs’ journal club.

How to build a cohomology theory

Let’s suppose we have something that acts like a category of local systems. In this category, there will be a trivial object \mathbb{R}_X. This will consist of the trivial line bundle over X and, for any two points x and y, the trivial isomorphism between the fibers (\mathbb{R}_X)_x and (\mathbb{R}_X)_y.

Also, there will be a functor \Gamma from local systems to vector spaces. For any local system \mathcal{V}, with underlying vector bundle V, \Gamma(\mathcal{V}) will be the vector space of sections \sigma of V such that the isomorphism(s) between V_x and V_y take \sigma(x) to \sigma(y).

In order to build a cohomology theory, you must find a way to embed your category into an abelian category (or a triangulated one). Then you will be able to talk about derived functors. The cohomology groups of X will be (R^i \Gamma)(\mathbb{R}_X).  Those who are used to computing with derived functors will know that it is very important, therefore, to find explicit resolutions of \mathbb{R}_X in our various categories. This task will, roughly speaking, become easier as we move down our list of theories.

This sounds discussion probably sounds very abstract; one of the reasons that the different perspectives above are useful is that they provide different ways to make this concrete.

Many of the difficulties in these subjects come about in constructing the larger abelian category. This is another important issue that I will gesture at, but not address.


42 thoughts on “Three ways of looking at a local system: Introduction and connection to cohomology theories

  1. On the topic of derived functors, there is something that I have been wondering about for a long time. In various situations not just chain complexes, but also double complexes are increasingly important. Perhaps the most important one is the Dolbeault complex.

    So how do you best complete this analogy, or is there even a reasonable way: Chain complexes are to derived functors, as double complexes are to ???.

  2. It looks like you’ll be focusing on cohomology theories arising specifically from algebraic geometry. Fine. Maybe, at the end of this series, you can add an appendix for those not acquainted with the Etale topology. For example, does it reduce to Euclidean topology on smooth manifolds?

  3. A technicality which I am not clear on from reading:

    Does a section \sigma in \Gamma(\mathcal{V}) have to satisfy \Phi(\sigma(x))=\sigma(y) for all isomorphisms \Phi (in the local system) between V_x and V_y, or for some such isomorphism \Phi?

    Other possibilities is that a system of paths (and hence a system of isomorphisms) is specified as an argument to \Gamma, or that \Gamma is a functor which gives you not a vector space of sections, but a vector space of functions from {path specifications} to {vector spaces of sections}.

  4. Alex,

    We demand that the isomorphisms on fibers only depend on the homotopy classes of the paths. If we choose a basepoint, we can define the space of global sections on that connected component to be the subspace of the fiber that is invariant under the monodromy action of the fundamental group.

  5. Greg,

    I don’t think I can completely answer your question, but you can sometimes run into a double complex when composing two derived functors. You get resolutions in different directions, and the Grothendieck spectral sequence compares the results with the derived functor of the composite.

  6. I enjoy seeing this post. Maybe its a coincidence, but it looks right-on like a follow-up to that “Journal Club” discussion (if it lives up to its name) we are having, in particular the exchange David Ben-Zvi and myself had here.

    If you look at that entry, you’ll see that I am expressing the hope that it will not only lead us to learn stuff, but also to accumulate what we learn into distilled wisdom, or approximations thereof, in our wiki called the nLab.

    Since the very useful expositional writing which you tradionally enjoy having here on the Secret Bloggin Seminar would fit perfectly with the aim of this wiki, I am wondering if I could get you interested in adding some of the expositional material which you are posting here to the nLab. The intended advantage is that all this expositional material would be arxived coherently with links to and from lots of related expositions and would in this way be of higher value for future readers than any isolated blog entry can be.

    To coax you into having a try, I created a stub entry nLab: local system. Maybe you’d enjoy hitting “edit page” on the bottom and playing around a bit with the possibilities.

    (You add formulas as usual in LaTeX.)

  7. At least to me… Chain complexes are to derived functors, as Double chain complexes are to compositions of derived functors. Every time a double complex comes up, or the associated spectral sequence, there seems to be a hidden composition of derived functors.

  8. It is true that a double complex has a “stupid” filtration and thus an associated spectral sequence. I can also believe that that sort of structure is associated to a composition of derived functors.

    But there is more going on than that. A double complex has not one, but two stupid filtrations. Moreover, the category of bounded double complexes, say over a field, has information that is not detected by either of the associated spectral sequences.

    More precisely, I learned (from Michael Khovanov, although it is not originally his result) that every bounded double complex is a direct sum of indecomposables with unique summands, and that the indecomposables are all 2×2 squares and antidiagonal zigzags. An antidiagonal zigzag can be called a “dot”; so for instance the Hodge theorem for a compact Kahler manifold says that its Dolbeault complex is entirely squares and dots. The spectral sequences reveal the even-length zigzags, moreover each of the spectral sequences reveals only one of two types of even zigzags. Neither spectral sequence notices the odd zigzags.

    If spectral sequences are related to compositions of derived functors, then is there a reasonable notion of a commuting pair of derived functors, that would always give you a semi-bounded double complex? Now I get the feeling that if it exists, then maybe I could work out what it would look like. It would be a matter of understanding the derived functor construction and applying it to some kind of commuting square of functors among four abelian categories. So maybe my question is a bit lazy.

    Well, the way to complete the line of thought would be to not only write down axioms, but then to argue that Dolbeault cohomology is the result of deriving a commuting square of functors.

    And yeah, the Grothendieck spectral sequence looks relevant too. Hmm…

  9. I apologize for the pedantic remark, but, strictly speaking,


    is not true: being a local system is a restriction on the bundle
    (the restriction of being locally constant ), not extra data.
    But I guess that’s why you have “roughly” there.

  10. The link between double complexes and composites of derived functors may be explainable partially in terms of the theory of homotopy coherent functors. Ten years ago Cordier and I wrote a paper (TAMS, if anyone wants to look it up) in which we showed how some of the derived functors were homotopy coherent Kan extensions. We also looked at compositions of homotopy coherent morphisms between simplicially enriched categories (the composition is only defined up to coherent choices) BUT the resulting composition depended on a filling algorithm of the subdiagonal of a cosimplicial simplicial set. This algorithm is related in this context to Eilenberg Zilber type results and to the Artin-Mazur codiagonal. Note we did not look at the bicomplex case as such as we were more concerned with the simplicial situation, but many of the arguments would seem to be not only valid, but easier in the algebraic case with chain complexes etc.

    Given the links between homotopy coherent Kan extensions and Grothendieck’s Dérivateurs, together with the work done by Maltsiniotis, Cisinski and others on those gadgets, it may be worth looking at that stuff toshed some new light on this idea.

  11. I haven’t really worked this out in detail, but some operations can be understood by viewing double complexes as representations of the Tannakian supergroup (\mathbb{G}_a^{odd} \times \mathbb{G}_a^{odd}) \rtimes \mathbb{G}_m^2. Totalization comes from restricting to the diagonal subgroup, forgetting differentials comes from restricting to the torus, and taking homology involves some kind of integration.

  12. Scott: Well, yes, that’s one of the motivations for my question, and for the classification of bounded double complexes. You can think of a chain complex over a field k as a graded representation of the 2-dimensional Hopf superalgebra \Lambda^*(k). Likewise a double complex is a graded representation of the 2-dimensional Hopf superalgebra \Lambda^*(k^2). In the former case, taking the homology consists of throwing away the projective summands; and this is something that you can also do in the latter case. Throwing away projective summands in general can be called passing to the homotopy category. And of course forgetting differentials and totalization come from restrictions. Moreover, the classification of indecomposable modules can be interpreted as that classification question for a certain quiver algebra. So that all works so far.

    What that framework does not accommodate is the very nice theory of derived functors. Hence my question.

    In fact, now that I think of it, a double complex over k has even a little more structure that I would not know how to express with derived functors. Namely, it has an action of GL(2,k) = Aut(\Lambda^*(k^2)).

  13. John Mangual: “Maybe, at the end of this series, you can add an appendix for those not acquainted with the Etale topology. For example, does it reduce to Euclidean topology on smooth manifolds?”

    The answer is literally “no”, but morally “yes”. For any nonzero integer N, the cohomology groups H_{et}^i(X, \mathbb{Z}/N) and H^i_{top}(X, Z/N) are canonically isomorphic. One defines the \ell-adic cohomology of X, for \ell a prime, to be the inverse limit of H(X, \mathbb{Z}/\ell^i) and this is also the same whether considered in the analytic or the etale topology. There is also an etale fundamental group and, for any finite group G, there is a canonical bijection between Hom(\pi_1^{top}(X), G) and Hom(\pi_1^{et}(X), G). As a rule of thumb, one can say that the etale topology gives the same results as the analytic topology when the coefficient objects are finite.

    However, H^i_{et}(X, \mathbb{Z}) is completely different from H^i_{top}(X, \mathbb{Z}). In general, H^i_{top}(X, \mathbb{Z}) really is a topological object, which can’t be recovered by algebraic techniques. (This might be a post of its own, one day.)

    According to the philosophy of this post, I should be talking about etale sheaves, not their cohomology. There is an analytification functor, from the category of sheaves on the etale topology to the category of sheaves on the analytic topology. This is fully faithful, and preserves cohomology. I think that the image is precisely the constructible sheaves, but I can’t find a reference, and my collaborator just showed up.

    Responses to the other comments this evening!

  14. Greg, I don’t think the GL_2 action preserves the grading, but if you restrict to the torus, you get the group I mentioned above.

  15. @9: Actually, Alf, you seem to be misreading things. Being a local system is an intrinsic property of a sheaf, but what David was saying is that local systems typically arise as the sheaf of flat sections of a vector bundle (a locally free coherent sheaf) with flat connection, which is the extra data he always had in mind.

  16. I think saying


    while true in a sense, is a tiny bit misleading. Most crude information about H^i_{top}(X, \mathbb{Z}), like its isomorphism type, can be gotten from the etale world. The torsion part can be gotten as the p-torsion part of homology with p-adic integer coefficients, and the rank is the same over the integers or p-adic integers. But passing to the p-adic destroys more sensitive topological information like intersection forms and cup products, except what you can keep by passing to characteristic p coefficents.

  17. I’ll just mention, the importance of double complexes is one of the weird mysteries of knot homology (hence the interest of Michael Khovanov, mentioned above). In an act of shameless self-promotion, I’ll also note that my recent work with Geordie Williamson actually shows that for HOMFLY homology, at least, the source of these bicomplexes is basically the Doulbeault complexes on certain non-compact complex manifolds.


    You’re right that it does not preserve the bigrading; it does preserve the total grading though.


    Well that’s very interesting!

  19. I’m baffled by comment 17. H^1(X,Z), where X is the affine line minus a point over the ring of complex numbers and where H is computed in the etale topology, is zero.

  20. Re 17: Everything Ben says (in the new, edited version) is true. However, here is something you cannot get from the etale topology:

    Let E be a variety over \mathbb{C} and \sigma an automorphism of E. Then we have an abelian group H:=H_{top}^i(X, \mathbb{Z}) and an automorphism \sigma^* : H \to H. The isomorphism class of the pair (H, \sigma^*) can not be extracted by any algebraic means.

  21. This post is intended to clarify the points raised in 17, 20 and 21 above.

    If M is an abelian group, and X is a scheme, then H^1(X,M) (etale cohomology with coefficients in the constant etale sheaf given by M) is equal to the space of continuous homomorphisms from the etale fundamental group of X (which is naturally a profinite group) to M (with its discrete topology, which is really the only sensible topology that we can impose on an arbitrary abelian group).

    In particular, any such homomorphism has finite image (this is a general property of continuous maps from a profinite group to a discrete one), and thus if M is torsion free, then H^1(X,M) vanishes.

    This won’t necessarily be true of higher H^i: using the short exact sequence

    0 –> Z –> Q –> Q/Z –> 0,

    we see that H^1(X,Q/Z) (which need non vanish, since Q/Z does contain lots of finite subgroups), contributes classes to H^2(X,Z).

    Nevertheless, H^i(X,Z) doesn’t have much to do with the usual cohomolgy of the analytification of X, even though H^i(X,Z/n) agrees with the cohomology of the analytification with Z/n coefficients, for any n.

    The usual way (going back to Grothendieck, and, in the context of H^1 of abelian varieties, to Weil and Tate) that people use to get around this problem is the following: choose an M that has some other, non-discrete topology (a profinite topology would obviously be best), but which still looks somewhat similar to Z, and then redefine things so that for the H^1, one computes continuous Homs from \pi_1 to M, where M now has an interesting (non-discrete, maybe even profinite) topology.

    There are various ways to implement this: at the most elaborate, one can construct categories of complexes of sheaves where the sheaves have profinite topologies (or perhaps more precisely, are given as pro-objects) and one computes global sections, and hence derived global sections, via some limiting process that takes into account the topology on the sheaves.

    But one normally proceeds in the following more down-to-earth manner: one chooses a prime l (\ell in LaTeX), takes M to be Z_l, and defines

    H^i(X,Z_l) := inverse limit over n of H^i(X, Z/l^n)

    This is a *redefinition* of the left-hand side. It already had a meaning, as the etale cohomology with coefficients in the constant sheaf Z_l, but since Z_l is torsion-free, this original definition is pretty useless, as we saw above.

    So now we redefine it via the above equation. One then shows that the H^i(X,Z_l) give a good cohomology theory. (Roughly, one has too examine all the various long exact sequences that arise with Z/l^n coefficients, pass to the inverse limit over n, and check that all the R^1-lims vanish. It turns out that they do, essentially because of the basic finiteness theorems in the theory.)

    Now on the analytic side, it is clear that if we take the inverse limit of the usual cohomology of a complex variety with Z/l^n Z_l coefficients. (This is because the cohomology of a variety is finitely generated in all degrees.) And by universal coefficients, this is just the Z-cohomology tensored with Z_l.

    Thus, with the new definition of the etale H^i(X,Z_l), this gives a group that it is canonically isomorphic to the usual cohomology with Z_l coefficients. These isomorphisms respect all long exact sequences, intersection pairing/cup products, Poincare duality, cycle class maps, …, everything.

    Incidentally, whenever one sees H^i(X,Z_l) written in any book or paper, it will always mean the second of the above definitions (the one with the inverse limit over finite coefficients). And this is always what people mean when they say l-adic cohomology. (Either this, or the result of tensoring this with Q_l over Z_l. So when people write H^i(X,Q_l), they mean — by definition — Q_l tensor over Z_l with H^i(X,Z_l), where the latter is defined as the inverse limit of the Z/l^n cohomologies. They will *never* mean etale cohomology computed with constant Q_l-coefficients, since that will always just vanish above degree 0.)

    So using Z_l-etale cohomology in this new sense, one can compute all the basic numerical information inherent in the usual cohomology with Z-coefficients: the Betti numbers (use any Z_l), and the torsion. (The l-Sylow subgroup of the torsion in H^i(X,Z) is picked out by tensoring with Z_l, and so can be computed using H^i(X,Z_l) on the etale side.)

    The one thing you can’t get from etale cohomology is an actual functorially constructed Z-module isomorphic to H^i(X,Z). You just see the shadows of this lattice obtained by tensoring it with Z_l for each l in turn.

  22. With regard to 22, again, what one *can* get on the etale side is the Z_l-module with automorphism obtained by tensoring (H,\sigma^*) with Z_l, for any prime l.

    Whether this recovers (H,\sigma^*) depends on the automorphism \sigma^*. For simplicity, suppose that H is torsion free, hence free of finite rank, over Z. (This is not such a serious restriction, since as we saw above, the torsion is more directly observable on the etale side than the free rank.)

    If the free rank is r, then \sigma^* is an element of GL_r(Z), and the isomorphism class of (H,\sigma^*) is described by the conjugacy class of \sigma^* in GL_r(Z). So we are asking whether we can detect the conjugacy class of a particular element of GL_r(Z) by knowing its image in GL_r(Z_l) for every prime l. This depends on the conjugacy class, I guess. As a trivial example, if \sigma^* acts by a scalar, then we can certainly compute this scalar by passing to GL_r(Z_l), even for just a single l.

    More generally, we can compute the characteristic polynomial after passing to GL_r(Z_l) for a fixed l. So suppose, for simplicity, that this polynomial had distinct roots, so that it determined \sigma^* up to conjugacy in GL_r(Q). Then we would see that the etale cohomology would at least let use compute the isomorphism type of (H,\sigma^*) after tensoring up with Q over Z.

  23. A quick remark on paragraphs 2 and 3 of comment 23.

    I believe you need to make some kind of unibranch assumption on X. For example, if X is a nodal P^1 over the complex numbers, then (I’m pretty sure) the etale cohomology group H^1(X,Z) is amazingly actually Z. One can write down a Z-torsor over X by taking an infinite chain of P^1’s, with 0 on each glued transversely to infinity on the next. Perhaps we could define a variant of the etale site by requiring our covering morphisms to be not just etale (=formally etale and locally of finite presentation) but also quasi-compact, and maybe then the H^1 above would be 0.

    Not surprisingly, this issue also comes up with the algebraic fundamental group, i.e., whether you’re trying to classify finite etale covers or some kind of locally finite etale covers. So you get two algebraic fundamental groups, which agree under certain unibranch assumptions. The big one that classifies the locally finite covers sometimes has a French adjective prepended to it, maybe gros. I think I once heard there was a missing SGA expose about it. This big one is not always profinite. In the case of the nodal P^1, it is not surprisingly just Z.

    I should add that I while I’m pretty sure this is correct, I’ve never really gone through all this — it’s just what at some point I decided must be true based on things I’ve heard over the years and things I actually know. So repeat it at your own risk.

  24. Dear James,

    Thanks for your informative comment, which sounds quite reasonable. The phenomenon you bring up is not one I’d thought of before. Do you know where in SGA the (apparently only sometimes) isomorphism

    Hom(\pi_1(X), M) = H^1(X,M)

    is discussed? I would be interested to look and see what precise hypotheses are given there.

  25. James,
    Only allowing quasi-finite morphisms won’t make a difference. Forget covering spaces and think about sheaves. That the fundamental group is Z is saying that a local system can have any monodromy. In the usual topology, we can build local systems on the circle with any monodromy, just using open sets, which are quasi-finite. Similarly, if you take two copies of P^1 glued at two points, it has an open cover given by deleting each of the nodal points. The intersection of these open sets is disconnected, so you can build a Czech cocycle for this (Zariski!) cover with arbitrary monodromy, the ratio of the values on the two components.

  26. If I remember correctly, Voevodsky/Suslin’s singular homology of schemes gives the same as topolgical singular homology with Z coefficients.

  27. @16
    Ben, I am really making a trivial remark, it’s rather a question of phrasing…
    Loc maps onto a substack of Bun, not onto all of Bun.
    Not any locally free sheaf admits a flat connection. For the ones that admit, it’s an extra data.

  28. Ben: Thanks, good point. (I was confused about your comment until I realized you were speaking to the final sentence in my second paragraph.)

    Matt: I would guess that we always have Hom(\pi_1(X), M) = H^1(X,M), as long as we take the big fundamental group. This is probably because they both classify M-torsors, which is probably true almost by the definition of M-torsor and the defining property of \pi_1. Regarding the equality between the big pi_1 and the profinite pi_1 under sufficient unibranch hypotheses, someone once told me about a missing document discussing it. I think it might be a missing SGA expose, but that was in 2000, so I can’t be too sure I remember correctly. I’ll poke around a bit and see what I can find another reference.

  29. Actually, I probably was misremembering, because there is a perfectly extant expose in SGA 3 that discusses this a bit (and took about a minute and a half to find by searching for ‘unibranch’). It’s expose X, by Grothendieck, especially section 6. What I was calling the big fundamental group is there called the enlarged fundamental group (“le groupe fondemantal \’elargi”). The relation to etale cohomology is of course not discussed. Both the example I mentioned and the one Ben mentioned are discussed (p. 112, in the Springer edition).

  30. Here is another point of view on James’ example (no. 27 above). This examples considers X = P^1 glued at two points (i.e. a nodal rational curve). In this case one has (computing the usual cohomology of the complex points) that H^1(X,Z) = Z.

    The surprising thing is that if one now computes etale cohomology, one again finds that H^1(X,Z) = Z ! (At least, this is surprising if one adopts the point of view, implicit in my post no. 23 above, that etale cohomology with Z coefficients computes the wrong thing in degrees greater than 0.)

    How does this happen, i.e. what makes this example tick? Here is one answer: although H^1(X,Z) is in cohomological degree 1, it is in motivic weight 0, and this is why etale cohomology can detect it.

    What does this mean? This is not the time (for me, at least) to go into the details of motivic weights, but here is a rough sketch: all cohomology of all varieties can be assembled out of cohomology classes attached to smooth projective varieties. But if X is singular, and/or non-projective, than a class in H^i(X,Z) may not come from degree i cohomology of a smooth variety, but some other degree. This other degree is called the weight of the class.

    The point is that in the above paragraph, “assembled” means, in slightly more precise terms, “assembled via various usual cohomology long exact sequences”. And in these long exact sequences, cohomology classes can be shifted into different degrees via boundary maps.

    The general yoga of weights, due to Grothendieck and Deligne, is developed carefully in Deligne’s three papers on Hodge theory. But rather than continuing with generalities, let us consider the case of the nodal curve X. To compute its H^1 in terms of smooth, projective varieties, it is better to consider a smooth P^1 (the normalization of X), let P and Q be the two points on P^1 lying over the node of X, and observe that there is an isomorphism

    H^1(X,Z) = H^1(P^1,{P,Q}; Z) ,

    where on the right I mean cohomology of the pair. We now consider the long exact sequence of the pair, and find that

    H^1(P^1,P{P,Q}; Z) = H^0({P,Q},Z)/H^0(P^1,Z) = Z,

    and this Z “comes from” the H^0 of the smooth variety {P,Q}. Thus it is in motivic weight 0.

    We can make the exact same computation in etale cohomology and get the same answer:


    The point is that this Z in H^1 is explained by some other H^0 via a completely tautological long exact sequence, valid for any cohomology theory, and etale cohomology can certainly compute the correct value of H^0, even with Z coefficients.

    So one should be able to make lots of other examples like James’, by choosing other singular varieties that have positive degree cohomology classes which never the less lie in motivic weight 0.

  31. Oh, that’s nice! Is there a weight filtration on the enlarged fundamental group? And if so, is the weight-zero piece a finitely generated discrete subgroup (say for varieties of finite type over an algebraically closed field), and is the quotient by the weight-zero piece the profinite fundamental group? I’m going to guess the answer to all three is yes.

  32. The previous questions were a bit confused. The pure weight-zero piece should be a quotient of the fundamental group, not a subgroup, and it might agree with the weight-zero piece of the profinite fundamental group, not the whole thing. Perhaps they both agree with the profinite fundamental group of a suitable normalization, though normalizations of connected spaces are often disconnected, so even if it’s morally true, some care might be required to express it properly.

  33. I can’t do anything right. The weight-*non*zero part of the two fundamental groups might agree, and they might also agree with that of a normalization. Please let there be no more sign errors…

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