Ordinary vs generalized cohomology theories

(Post updated on 7 Jan 2010, based on some useful feedback from Tobias Barthel)

Introduction

When trying to classify or organize cohomology theories “found in nature” within (commutative) algebraic geometry, one realizes that there is a fundamental divide between generalized cohomology theories and the more restrictive notion of ordinary cohomology theories. This can result in confusion, for example when speaking of “universal” cohomology theories. Today I will try to give some explanation of what the difference is, although I am still very far from understanding this completely. These two notions of cohomology can be given precise definitions in topology, and there are several ways one could imagine making them precise also in algebraic geometry.

First a word about terminology: The phrase “generalized cohomology” has been used in several different ways in the literature. For example, the excellent article Motivic sheaves and filtrations on Chow groups by Jannsen, in the Motives volumes, uses a definition of generalized cohomology which corresponds to what I want to call ordinary cohomology. My choice of terminology seems more common nowadays, and it also corresponds to well-established practice in topology.

Cohomology theories in algebraic topology

In topology, a cohomology theory is by definition a sequence of functors on a suitable category of topological spaces, which satisfy the Eilenberg-Steenrod axioms. If we include the so called dimension axiom, we get the definition of ordinary cohomology, and if we exclude it, we get the definition of generalized cohomology. The classical Brown representability theorem says essentially that a generalized cohomology in the above sense is exactly the same thing as a functor represented by a spectrum. For precise statements, and details on the Eilenberg-Steenrod axioms and Brown representability, see Kono and Tamaki: Generalized cohomology (Google Books link), or May: A concise course in algebraic topology (pdf available here). Recall that for any abelian group G, we can define singular cohomology with coefficients in G; these cohomology theories are the only examples of ordinary cohomology theories in algebraic topology. There are many examples of generalized (non-ordinary) cohomology theories, for example complex cobordism, elliptic cohomology/topological modular forms, Brown-Peterson cohomology, and various forms of K-theory. Stable homotopy theory is the study of the category of spectra, i.e. the objects representing such generalized cohomology theories.

A very important class of generalized cohomology theories is the class of oriented theories. Roughly speaking, these are the cohomology theories which admit a reasonable theory of characteristic classes of line bundles. To any oriented cohomology theory one can associate a formal group, and there is a converse to this for formal groups which are “Landweber exact”. There is a “universal” oriented cohomology theory, namely complex cobordism. Some other examples: Singular cohomology with coefficients in any ring (or maybe Q-algebra) corresponds to the additive formal group. Complex K-theory corresponds to the multiplicative formal group. An elliptic cohomology theory corresponds to a formal group law coming from an elliptic curve. See Kono and Tamaki, or Lurie’s Survey of elliptic cohomology, for definitions and more details.

(I feel I should apologize for being so brief in this section, but there are already good references for algebraic topology, and today I want to get to some interesting algebraic geometry before I die. Hopefully I will find time to post in the future on things such as background on formal groups, and various approaches to defining the category of spectra.)

Algebraic geometry background

Although I did not give precise definitions in the algebraic topology discussion, such definitions can be found in the references, and with these definitions it is completely clear what one means by “ordinary” and “generalized” cohomology. In algebraic geometry, the same distinction clearly exists “in nature”, but to give precise definitions is more difficult. Roughly speaking, ordinary cohomology theories can be understood using only homological algebra, while generalized cohomology theories need the more flexible language of homotopical algebra.

Some examples of ordinary cohomology theories: Etale cohomology, Deligne cohomology, motivic cohomology, crystalline cohomology, de Rham cohomology,  Betti cohomology. Most ordinary cohomology theories are defined as the sheaf cohomology of some sheaf of abelian groups (or more generally in terms of hypercohomology of some complex of sheaves of abelian groups). Here the notion of sheaf depends on some choice of Grothendieck topology. Many ordinary cohomology theories come in pairs of an “absolute” and a “geometric” theory (more about this in a future post!). Any Weil cohomology theory is ordinary, as well as any Bloch-Ogus theory. In algebraic topology, we have mentioned that ordinary theories correspond to abelian groups. The picture in algebraic geometry is more complicated, partly because we want to apply our cohomology theories to categories which are more complicated and more varied (see earlier posts on varieties and Weil cohomology for some examples).

Some examples of generalized (non-ordinary) cohomology theories: algebraic cobordism, algebraic K-theory, Witt groups. Theories of this kind are never defined in terms sheaf cohomology of abelian sheaves as above. However, there are more general notions of sheaf cohomology which do apply in most cases, using simplicial sheaves/presheaves in some form.

Remark 1: To speak of “universal” cohomology theories, it is necessary to specify what one means by “cohomology theory”. If we want to talk about all generalized cohomology theories, I guess the only thing that could be universal is some good notion of “stable homotopy type”. However, when it comes to more restrictive notions of cohomology, I am quite sure the following three statements can be made precise: (1) Algebraic cobordism is universal among oriented theories. (2) Pure motives is the universal ordinary (Weil) cohomology theory for smooth projective varieties over a field. (3) Motivic cohomology is the universal ordinary (Bloch-Ogus) cohomology for general varieties over a field.

Edit: Algebrac K-theory also has a universal property, at least when regarded as a functor on symmetric monoidal categories, but I am not sure about the details of this. There are some hints in Tyler Lawson’s answer to this MathOverflow question, and some more details in another answer of Clark Barwick.

Remark 2: It seems like the noun “motive” is used exclusively in connection with the world of ordinary cohomology theories (pure motives, mixed motives, triangulated categories of motives, etc). However, the adjective “motivic” is used in settings related to ordinary as well as generalized cohomology, e.g. motivic homotopy theory (generalized), motivic cohomology (ordinary).

Remark 3: There is a general heuristic principle which says that “any cohomology becomes ordinary after tensoring with Q” (i.e. killing all torsion). Some examples: (1) Algebraic K-theory cannot be defined in terms of homological algebra/abelian sheaf cohomology, but after tensoring with Q this becomes possible. (2) Homotopy groups in topology are very hard to compute in general, and homological algebra doesn’t help you at all, but after tensoring with Q, everything can be described in terms of (differential graded) homological algebra, thanks to rational homotopy theory. (3) The classical Grothendieck school in the 60s never really bothered about homotopical algebra – this seems related to the fact that they were always studying cohomology theories with coefficients in Q-algebras only. (4) In topology, and maybe also in algebraic geometry, any generalized cohomology theory has an associated Atiyah-Hirzebruch spectral sequence, which relates the ordinary cohomology with coefficients in to itself, and I believe this spectral sequence tends to be complicated, but degenerate after tensoring with Q.

Edit: For oriented cohomology theories, this phenomenon is probably related to the fact that any formal group law is isomorphic to the additive one over the rationals.

Characterizing ordinary cohomology theories

I am not aware of any completely satisfactory definition of generalized cohomology theory in algebraic geometry (the best candidate would be “something represented by a spectrum in the sense of motivic homotopy theory”). However, we could pretend for a moment that such a definition exists, and then ask for a characterization/definition of ordinary cohomology theories. I can imagine four approaches to this question, but I have no idea if any of them can be made precise in any reasonable way.

Candidate 1: “A cohomology theory is ordinary if it is functorial not only with respect to maps in the classical sense, but also with respect to correspondences”. Recall that a correspondence from to is a cycle on , a special case being the graph of a map . In his Motives volume article mentioned above, Jannsen remarks that any cohomology satisfying his axioms is functorial with respect to correspondences.  Voevodsky also discusses this in the introduction to Homology of schemes I , claiming that in topology this functoriality property actually characterizes ordinary cohomology theories among all generalized theories. This surrounding discussion on what Voevodsky calls his underlying “simple topological intuition” was unfortunately not included in the preprint but only in the published version.

Candidate 2: “A cohomology theory is ordinary if it is oriented and its associated formal group is the additive formal group”. This definition makes sense if we have a notion of oriented theory in algebraic geometry, as well as a correspondence with formal group laws. There are definitions of oriented theories which might be suitable for our purpose here, but I am not sure about the formal groups bit. For closely related ideas, see for example this preprint of Naumann, Østvær, and Spitzweck.

Candidate 3: “A cohomology theory is ordinary if it factors through some suitable triangulated category of motives”. One reason for thinking that this might be a reasonable definition is that any theory I know of with this property (i.e. any theory which corresponds to a “realization functor”) should be thought of as being ordinary. Another reason is some very vague feeling that for a nice base scheme S, maybe the triangulated category DM(S) of motives relates to the motivic stable homotopy category SH(S) roughly as the derived category of abelian groups relates to the classical stable homotopy category (this might be complete nonsense though).

Edit: The right way of making this precise might be by using the motivic Eilenberg-MacLane functor. I hope to come back to this in a future post.

Candidate 4: Urs Schreiber advocates the viewpoint that every cohomology theory can be expressed in terms of a mapping space Map(X,K) in some higher topos in the sense of Lurie. Whenever this point of view is valid, one could probably define the cohomology to be ordinary if the target object K is in the essential image of some kind of Dold-Kan correspondence. I believe this idea is excellent, but it is not always clear which higher categories to work with in a concrete algebro-geometric problem. See the nLab pages on cohomology and generalized cohomology for more details.

Homotopical categories and simplicial sheaves

(This is an expanded version of the 2nd part of a talk I gave last month. For the first part, see this post.)

Homotopical categories

The topic for this post is “homotopical categories”, and their role in algebraic geometry. I want to emphasize that I am very much in the process of learning about all these things, so this post is based more on interest and enthusiasm than actual knowledge. I hope to convey some of the main ideas and why they could be interesting, and come back to the details in many future posts, after having learned more. I apologize for not defining everything carefully, and for brushing the “stable” aspects of the theory, i.e. spectra and sheaves of spectra, under the carpet.

There are many different ways to speak of “homotopical categories”, and I only use this expression because I don’t know of a better thing to call them. The most well-known approach is the language of model categories, invented by Quillen and developed by many others. There are many excellent online introductions, for example Dwyer-Spalinski, Goerss-Schemmerhorn, and appendix A2 of Jacob Lurie’s book on higher topos theory, available on his webpage. Other languages are given by the many different approaches to higher categories; see the nLab page and the survey of Bergner. Still other languages include Segal categories, A-infinity categories, infinity-stacks, and homotopical categories in the precise sense of Dwyer-Hirschhorn-Kan-Smith.

Although I don’t want to go into the details of all these different homotopical/higher-categorical subtleties, I will try to list some of the basic features that “homotopical” categories typically have.

• A homotopical category should behave like a nice category of topological spaces.
• In particular, there should be a class of morphisms called weak equivalences, and:
• To any homotopical category , one should be able to associate a “homotopy category” and a functor which is universal among functors sending weak equivalences to isomorphisms. Morally, is obtained from by “formally inverting the weak equivalences”.
• A homotopical category should admit all limits and colimits, and also homotopy limits and homotopy colimits.
• A homotopical category should be enriched over some kind of spaces, i.e. for any two objects , the set should be a “space” in some sense, for example a simplicial set, a topological space, or a chain complex of abelian groups.

Simplicial objects

Before talking about algebraic geometry, we need to recall some “simplicial language”. The category is defined as follows. Objects are the finite ordered sets of the form . Morphisms are order-preserving functions , i.e. functions such that . If is any category, we define the category of simplicial -objects to be the category in which the objects are the contravariant functors from to , and the morphisms are the natural transformations of functors. There is a functor from to given by sending an object of to the corresponding constant functor, i.e. the functor sending all objects to and all morphisms to the identity morphisms of .

Some examples:

• Take ,  the category of sets. The above construction gives us the category  of simplicial sets. This category is “sort of the same as the category of topological spaces”. The precise statement is that there is a pair of adjoint functors which make  and into Quillen equivalent model categories; in particular, their homotopy categories are equivalent (as categories). For the purposes of algebraic topology, we can work with any of these categories. For example, we can define homotopy groups and various generalized homology and cohomology groups of a simplicial set. The inclusion of into corresponds to viewing a set as a discrete topological space. A weak equivalence between two simplicial sets is a morphism inducing isomorphisms on all homotopy groups.
• Take , the category of abelian groups. There is a forgetful functor from to the category , induced by the forgetful functor from to. The Dold-Kan correspondence tells us that there is an equivalence between and the category of (non-negatively graded) chain complexes of abelian groups. Under this equivalence, homotopy groups of a simplicial abelian group correspond to homology groups of a chain complex.
• Take, the category of-algebras for a commutative ring. Then there is some kind of Dold-Kan correspondence between simplicial algebras and DG-algebras. See Schwede-Shipley for precise statements.
• Take , the category of sheaves of sets on some topological space or site. Then is the category of simplicial sheaves. This category can also be viewed as the category of sheaves of simplicial sets on the site. Any category of simplicial sheaves is a “homotopical category” (I am not making this precise here). For example, one way of defining weak equivalences is to say that a morphism of simplicial sheaves is a weak equivalence iff it induces weak equivalences of simplicial sets on all stalks. 

Homotopical categories in algebraic geometry

Now to algebraic geometry. Through a few examples I want to argue that homotopical categories (in particular categories of simplicial sheaves) provide a useful and natural setting for certain aspects of algebraic geometry.

Firstly, let’s consider the general problem of viewing a cohomology theory as a representable functor. In algebraic topology, the Brown representability theorem says that any generalized cohomology group is representable, when viewed as a functor on the homotopy category of topological spaces. In other words, there is a space such that the cohomology of a space is given by , where the is taken in the homotopy category. Examples include the Eilenberg-MacLane spaces , which represent the singular cohomology groups , and the space , which represents K-theory. The existence of a long exact sequence relating the cohomology groups for various corresponds to the fact that the different Eilenberg-MacLane spaces fit together to form a so called spectrum. The Brown representability theorem is best expressed using the language of spectra, i.e. stable homotopy theory, but I want to postpone a discussion of this to a future post. An interesting aspect of Brown representability for singular cohomology is that by identifying the coefficient group with the corresponding Eilenberg-MacLane space, the two arguments of a singular cohomology group , namely the space and the coefficient group , suddenly are on equal footing. By this I mean that they both live in the same category of topological spaces, rather than in the two separate worlds of topological spaces and abelian groups, respectively.  

In classical algebraic geometry, there is no analogue of Brown representability. Most cohomology theories are of the form , where is some kind of variety, and is a sheaf of abelian groups. One may ask if there is a way to express such a cohomology group as a representable functor. In order to obtain a picture parallell to the topological picture above, a necessary requirement is to have a homotopical category in which the variety and the sheaf both live as objects, “on equal footing”. One possibility for such a category is some category of simplicial sheaves. In order to explain how this works, let us fix some category of varieties, for example the category smooth varieties over some base field . Let us also fix some Grothendieck topology on this category, for example the Zariski topology, the Nisnevich topology, the etale topology, or some flat topology. This defines a site, and we can speak of sheaves on this site, i.e. contravariant functors on , satisfying a “glueing” or “descent” condition with respect to the given topology.

Since Grothendieck, we are familiar with the idea of identifying a variety with the sheaf of sets that it represents, by the Yoneda embedding. We mentioned earlier that for any category , there is a functor . Taking to be the category of sheaves of sets, we get a functor from sheaves of sets to simplicial sheaves. In particular, any variety can be viewed as a simplicial sheaf, by composing the Yoneda embedding with the canonical functor from sheaves of sets to simplicial sheaves. 

We also want to show that a sheaf of abelian groups can be viewed as a simplicial sheaf. We can regard any abelian group as a chain complex, by placing it in degree zero, and placing the zero group in all other degrees. This gives an embedding of the category of abelian groups into the category of chain complexes, and by composing with the Dold-Kan equivalence we get a functor from abelian groups to simplicial sets. This induces a functor from sheaves of abelian groups to simplicial sheaves. More generally, any complex of sheaves of abelian groups can be viewed as a simplicial sheaf.

Now one could hope for an analogue of Brown representability, namely that the sheaf cohomology group could be expressed as , where the Hom is taken in the homotopy category of simplicial sheaves. It seems to be the case that something along these lines should be true. For example, this nLab page on cohomology seems to imply that all forms of cohomology should be of this form, at least sheaf cohomology groups of the type just described. Also, Hornbostel has proved a Brown representability theorem in the setting of motivic homotopy theory.

There are many other phenomena in algebraic geometry which also seem to indicate that categories of simplicial sheaves might be more natural to study than the smaller categories of schemes and varieties we typically consider. Some examples (longer explanations of these will have to wait until future posts):

• It seems to be the case that almost any geometric object generalizing the concept of a variety can be thought of as a simplicial sheaf. Examples: Simplicial varieties, stacks, algebraic spaces.
• Deligne’s groundbreaking work on Hodge theory in the 70s (see Hodge II and Hodge III) uses in a crucial way that the singular cohomology of a complex variety can be defined on the larger category of simplicial varieties. Simplicial varieties are special cases of simplicial sheaves, and I believe it should be true that functors on simplicial varieties can be extended to simplicial sheaves.
• Simplicial varieties/schemes also pop up naturally in other settings. For example, Huber and Kings need K-theory of simplicial schemes for their work on the motivic polylogarithm.
• As already indicated, simplicial sheaves appears to be the most natural domain of definition for many different kinds of cohomology theories.
• Morel and Voevodsky’s  A1-homotopy theory (also known as motivic homotopy theory) is based on categories of simplicial sheaves for the Nisnevich topology.
• Brown showed that Quillen’s algebraic K-theory can be thought of as “generalized sheaf cohomology”, where the coefficients is no longer a sheaf of abelian groups, but a simplicial sheaf.
• The work of Thomason relating algebraic K-theory and etale cohomology uses the language of simplicial sheaves.
• Simplicial sheaves provide a natural language for “resolutions”. For example, it gives a unified picture of the two methods for computing sheaf cohomology: Cech cohomology and injective resolutions.
• Simplicial sheaves seems to be the most natural language for descent theory.
• Toen‘s work on higher stacks can be formulated in terms of simplicial sheaves. 
• Homotopy categories of simplicial sheaves can be thought of a generalization of the more classical derived categories of sheaves. The homotopical point of view seems to clarify some unpleasant aspects of the classical theory of triangulated categories. 

See also the nLab entry on motivation for sheaves, cohomology, and higher stacks.

Questions

I hope to come back to many of these examples in detail. For now, I  just want to list a few questions which I find intriguing. 

• To define a category of simplicial sheaves, we must choose a Grothendieck topology. How does this choice affect the properties of the category we obtain? Morel and Voevodsky work with the Nisnevich topology, Huber and Kings work with the Zariski topology, and Toen (at least sometimes) works with some flat topology. For some purposes, it seems to be the case that we don’t need a topology at all, instead we can just work with simplicial presheaves. What is the role of the Grothendieck topology? 
• Most of the above examples are developed for varieties over a base field of characteristic zero. Based on the above, it seems reasonable to believe that simplicial sheaves are useful in this case, but what if the base scheme is field of characteristic p, a local ring, a Dedekind domain, or something even more general? Is it the case that simplicial sheaves is the most natural language for understanding cohomology theories for arithmetic schemes, such as schemes which are flat and of finite type over ? Are simplicial sheaves important in number theory/Arakelov theory/geometry over the field with one element? What are the obstacles to “doing homotopy theory over an arithmetic base”?

Obviously I hope that there will be interesting answers to these questions, but I am still completely in the dark as to what these answers might be.