In algebraic geometry, it is not completely clear what a cohomology theory really is, or rather, there are many different useful definitions of “cohomology theory”. There are several issues here:
Different kinds of cohomology theories are defined on different subcategories of the category of schemes. The most friendly such category is probably the category of smooth projective varieties over a nice field, for example a field admitting resolution of singularities. In algebraic topology, there are also several natural domain categories for cohomology (finite CW complexes, CW complexes, compactly generated topological spaces, …) but the situation are still far simpler than in algebraic geometry.
While in algebraic topology, there is essentially only one list of axioms (Eilenberg-Steenrod), in algebraic geometry there are many different lists of axioms in the literature. Examples include Weil cohomology axioms, Bloch-Ogus axioms, and Oriented cohomology axioms. These differ substantially, and each list gives rise to its own “universal” theory (which generates quite a lot of interesting mathematics.)
Remark: Jannsen discusses other variants to the Bloch-Ogus axioms, by Gillet and by Beilinson (see p 37-38).
Rigidity - see paper by Hornbostel.
Descent properties!!
In algebraic geometry, every cohomology theory is represented by a spectrum, and vice versa, so one could define a cohomology theory to be something represented by an object in the stable homotopy category of spectra. In algebraic geometry, things are more complicated. There are many different stable homotopy categories, and it is not at all clear that every cohomology theory is represented by a spectrum.
Some notes by Joshua containing a very interesting axiomatization. See also this Inventiones article containing some results on pseudo pretheories, by Ostvaer and Rosenschon.
Kth76 (1995): Voevodsky discusses algebraic Morava K-theory, and defines there a cohomology theory by four axioms: Exactness, Nisnevich descent, and two types of homotopy invariance.
Voevodsky: BK conjecture for Z mod 2 coeffs and algebraic Morava K-theories. File in Voevodsky folder. Discussion of BK conj and related conjectures. P 32: axioms for cohomology theories on simplicial schemes. Proof that existence of algebraic Morava K-theories satisfying certain properties would imply the BK conjecture.
There are some kind of axioms for generalized cohomology in Jannsen Motives vol article. See also the ref Ja3 in this article for some possibly different axioms.
Fulton and MacPherson: Categorical framework for the study of singular spaces. In Geometry-Various folder. This seems to be the original source for bivariant theories in general, with quite a lot of material.
See Pretheory
In the letter to Beilinson, Voevodsky formulates axioms for a homology theory. He considers as an (n-1)-dim sphere, write also S for the 1-dim sphere in this sense. Let Sch/k be the cat of separated schemes of finite type over a base k. Then a homological theory is a functor from Sch/k together with a family of natural isos . This functor should satisfy some conditions: Morally, homotopy invariance, MV exact triangle, an exact triangle for blowups, and transfer for flat finite morphisms. Get a 2-cat of homological theories over . Examples: Algebraic K-th with rational coeffs, l-adic homology, Hodge homology ass to a complex embedding. Thm: There is an initial object in this cat, which we call the triang cat of eff mixed motives over k. Notion of reduced homological theory, and reduced motive of a scheme. Any motive in the above sense is of the form , where we may assume affine and . Tate object and comparison with K-theory. Bigger cat which contains the previous as a full triang subcat, but admits a more explicit description rather than just the universal property. Can also be viewed as the closure of the previous, wrt direct sums and inductive limits. Need the h-topology, in particular coverings including surjective blowups, finite surjetive maps, etale coverings. Various filtrations on (homotopy canonical, geometrical, motivic canonical, weight). The weight filtr should be related to pure numerical motives.
In the first chapter of Levine-Morel, there is a nice discussion of axioms, and the relation between oriented theories and theories represented by spectra. Recommended: read the overview in the introduction, and the first two chapters. Another nice list is in Bloch’s article on higher Chow groups.
In Levine, K-theory handbook, there is a description of “axiomatic” properties of the triangulated category of motives (which should be te derived category of Beilinson’s abelian category of mixed motives). These axioms are: Additivity, Homotopy, Mayer-Vietoris, Kunneth, Gysin DT, Cycle classes, Unit, Motivic cohomology, and Projective bundle formula. See page 470.
In Levine there is also a notion of geometric cohomology theory. Should search the whole database for all notions of arithmetic and geometric CTs.
Should read Huber and Wildeshaus: Classical…
See http://www.ams.org/mathscinet-getitem?mr=1440308 for a many standard properties such as cohomological purity, excision, Gysin maps, finiteness, etc (for rigid cohomology).
Kahn, Coll-Th and Hoobler has a paper (Bloch-Ogus-Gabber theorem) where they discuss axioms in a rather nice way. This paper is removed from K-th archive and published in Fields Inst Comm.
There are some axiom sets in Deglise-Cisinski: Mixed Weil cohom, and maybe also in Triang cats, in the stable homotopy theory section.
Various notions in a paper by Naumann? Spitzweck? et al?, maybe the paper on Landweber exactness. Graded, nongraded etc.
Thomason: Le principe de scindage (Thomason folder). He shows roughly that any theory for schemes satisfying certain axioms and mapping to Quillen K-theory, must be surjective when evaluated on the ground field . This implies that Milnor K-theory, if required to satisfy these axioms, cannot be extended to the category of varieties over . The axioms are: should be a map from smooth quasi-projective k-varieties, to “multiplicative spectra”, satisfing homotopy, Mayer-Vietoris, units axiom, and weak localization. See paper for details.
There is something by Jannsen and Saito where they write down axioms for Borel-Moore homology I think.
arXiv:1208.2586 Invariants of upper motives from arXiv Front: math.KT by Olivier Haution Let H be a homology theory for algebraic varieties over a field k. To a complete k-variety X, one naturally associates an ideal of the coefficient ring H(k). We show that, when X is regular, this ideal depends only on the upper Chow motive of X. This generalises the classical results asserting that this ideal is a birational invariant of smooth varieties for particular choices of H, such as the Chow group. When H is the Grothendieck group of coherent sheaves, we obtain a lower bound on the canonical dimension of varieties. When H is the algebraic cobordism, we give a new proof of a theorem of Levine and Morel. Finally we discuss some splitting properties of geometrically unirational field extensions of small transcendence degree.
nLab page on Axioms in algebraic geometry