There are four so-called classical Weil cohomology theories:


The proofs of the axioms for Betti cohomology and de Rham cohomology are comparatively easy and classical. For ${\displaystyle \ell }$-adic cohomology, for example, most of the above properties are deep theorems.


The vanishing of Betti cohomology groups exceeding twice the dimension is clear from the fact that a (complex) manifold of complex dimension n has real dimension 2n, so these higher cohomology groups vanish (for example by comparing them to simplicial (co)homology).


The de Rham cycle map also has a down-to-earth explanation: Given a subvariety Y of complex codimension r in a complete variety X of complex dimension n, the real dimension of Y is 2n−2r, so one can integrate any differential (2n−2r)-form along Y to produce a complex number. This induces a linear functional ${\displaystyle \textstyle \int _{Y}\colon \;H_{\text{dR}}^{2n-2r}(X)\to \mathbf {C} }$. By Poincaré duality, to give such a functional is equivalent to giving an element of ${\displaystyle H_{\text{dR}}^{2r}(X)}$; that element is the image of Y under the cycle map.