Ten constructions of the cohomology of varieties
When talking about “the” cohomology of mathematical objects, we do not always explicitly mention which cohomology is used, because it is obvious (in cases there is only one possible definition), or because we really don’t care (since as we will see, it is frequent that different definitions lead to equivalent results). The case of differentiable manifolds or algebraic varieties is particularly impressive, since there were a lot of equivalent cohomology theories defined during last century in order to simplify proofs or allow generalisations. Most cohomology theories, if not all, are defined as the cohomology of a complex : i.e. a sequence of vector spaces or modules with a boundary map . The kernel of is called the set of cycles, while the images by are called boundaries : the cohomology is then the quotient of cycles in by the subspace of boundaries.
Classical topologists, for example, will use preferably (see MacLane, Homology or the book of Allen Hatcher) :
- simplicial (co)homology : it is defined for a triangulated space, i.e. the manifold is cut by curves, surfaces, etc. which make it isomorphic to a sort of polyhedron (a complex); simplicial homology describe non-triviality (holes) in the combinatorial structure of this polyhedron; here the boundary map is really the boundary map.
- singular (co)homology : a more abstract version of simplicial homology; now we consider the set of all possible simplexes (curves, polygons, polyhedra, and their generalisations…) drawn on the manifold; this definition was given by Eilenberg; I don’t know who first defined simplicial homology, but MacLane mentions that Poincaré and Noether gaves important contributions to this theory.
Then come sheaves, which were explicitly defined by Leray (tales for young mathematicians help remember that Leray made considerable efforts as a prisoner in concentration camps to focus his work on especially “useless” subjects to avoid helping Nazis).
- de Rham cohomology is defined on differentiable manifolds by means of the complex of differntial forms, and exterior derivative : this should be related to the theory of integration on submanifolds, and Stokes theorem; the precise relation to singular cohomology is given by de Rham’s theorem, which states these are isomorphic when working on a compact manifold, it is also known that de Rham worked on distributions and their analogues in differential forms (currents) : results by Laurent Schwartz and Grothendieck allow to formulate de Rham’s theorem and Poincaré duality in this new language;
- cohomology of constant sheaves : the cohomology of the sheaf of locally constant functions (in its Čech presentation) can be identified to de Rham’s cohomology by a Mayer-Vietoris argument, the corresponding abstract property is the fact that de Rham’s complex of (sheaves of) differential forms is an (acyclic) resolution of the constant sheaf; the Čech cohomology of the constant sheaf, calculated with a contractible cover, reduces to the simplicial cohomology of the nerve of the cover
- holomorphic de Rham cohomology : the de Rham complex has a holomorphic analogue (for holomorphic manifolds), which is the complex of holomorphic differential forms, which is again a resolution of the constant sheaf, its hypercohomology (the sheaf of holomorphic forms have nontrivial Čech cohomology, here we need to mix Čech resolutions with the de Rham complex) is again the same as the previous ones; in the context of Kähler manifolds, this leads to natural Hodge structures on cohomology.
- algebraic de Rham complex : the algebraic de Rham complex of a projective variety is the complex of differential forms with polynomial coefficients; this not at all a resolution of a constant sheaf (at least for the Zariski topology); however, it is nevertheless true that it gives the same cohomology as above (when considering the holomorphic variety associated to the algebraic one); this is Serre’s GAGA theorem. It allows defining a cohomology theory for algebraic varieties while staying in the framework of Zariski topology.
Then comes Grothendieck and its categorical generalisation of topologies (sites) : it allows to find new cohomologies isomorphic to the classical ones, mostly in algebraic geometry. The problem with algebraic varieties is that Zariski topology does not allow computing the cohomology of constant sheaves using a suitable open cover (moreover, the result of the computation is probably very unpleasant). Several stories about cohomology of algebraic varieties can be found in Motives (proceedings of a conference held in 1991, see the MathSciNet entry here).
- the étale cohomology is defined using the cohomology of constant sheaves on the étale site, which replaces Zariski open sets (with horrible topology) by étale covers (which tend have simpler topologies): it was the first cohomology theory working in characteristic p for the proof of Weil conjectures. The Secret Blogging seminar has a post about this.
- crystalline cohomology is another algebraic way of defining a site with a sheaf whose cohomology is the usual cohomology of the variety, which should fill some gaps in the properties of étale cohomology
Morse (co)homology uses Morse theory to construct a complex whose cohomology coincides the standard cohomolgy.
There also seems to be a relationship between the cohomology of a manifold and the Hochschild (co)homology and/or the cyclic (co)homology of its algebra of smooth functions, but I don’t know how it works. It is supposed to provide a definition of cohomology in a non-commutative setting.