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