In Weil cohomology, a nice curve has cohomology up to degree 2, or more generally a nice $n$-dimensional variety has cohomology up to degree $2n$.


I know that this was motivated at least in part by a desire to extend cohomology of complex manifolds to algebraic varieties over other fields, and since complex curves are two-dimensional manifolds, curves should have cohomology up to degree 2.


My question is if this is just an artifact of the motivation, or if there is a more 'intrinsic' reason why cohomology should work out that way? Or put differently, should one expect there to be cohomology theories where, say, curves act like they're five-dimensional?



Why should curves be two-dimensional?
This is a very subtle question, I think. First of all, the sense in which an $n$-dimensional algebraic variety $X$ acts as if it is "cohomologically $2n$-dimensional" is quite complicated--for example, unless one uses some notion of cohomology with compact support, an affine $n$-dimensional variety looks $n$-dimensional, not $2n$-dimensional (just as over $\mathbb{C}$, affine varieties have the homotopy types of $n$-dimensional CW complexes). So this is really something about proper varieties, or about the more complicated notion of cohomology with compact support.


Second, this is just a fact about algebraically closed fields. For example, if $k$ is a finite field and $X$ is an $n$-dimensional smooth projective $k$-variety, $X$ "looks" like it is $2n+1$-dimensional (indeed, it looks a lot like a $2n+1$-manifold fibered over a circle). For example, there is an arithmetic Poincare duality between $H^i$ and $H^{2n+1-i}$. (Of course the reason for this is that $k$ has cohomological dimension $1$ and has reasonable duality properties.)


Nonetheless, let me try my hand at some kind of explanation. I think the proof of Grothendieck's vanishing theorem gives a reasonable explanation of why *Zariski* cohomology of $X$ vanishes in degree greater than $\dim(X)$. Typically when one defines a fancier cohomology theory (e.g. etale or crystalline cohomology) one chooses a topology on some category of schemes which is finer than the Zariski topology. Call such a topology $\tau$. As $\tau$ is finer than the Zariski topology, there is a forgetful functor $f\_\*$ from sheaves on the topology $\tau$ to Zariski sheaves. Grothendieck teaches us to think of this as a morphism $$f: X\_\tau\to X\_\text{Zar}.$$


(Maybe a good analogy is that there is a natural continuous map of spaces $$g: X(\mathbb{C})^{\text{an}}\to X\_{\text{Zar}}$$ if $X$ is a finite-type $\mathbb{C}$-scheme.)


For all of the topologies $\tau$ which we like to use to define Weil cohomology theories, the morphism $f$ has cohomological dimension $n$, in the sense that the right-derived functor of $f\_\*$, denoted $Rf\_\*$, sends certain sheaves to complexes with cohomology concentrated in degrees $[0, n]$. For example, if $\tau$ is the etale topology, and we are working over an algebraically closed field, $Rf\_\*\mathcal{F}$ is concentrated in degrees $[0,n]$ if $\mathcal{F}$ is constructible. Or if $\tau$ is the crystalline site (over a field of characteristic zero), $$Rf\_\*\mathcal{O}\_X\simeq \Omega^\bullet\_{X, dR}$$
which is concentrated in degrees $[0,n]$ (and there is a similar formula with any flat vector bundle in place of $\mathcal{O}\_X$). So the fact that varieties in these topologies are $2n$-dimensional comes from the formula $$2n=n+n,$$ where the first $n$ comes from $Rf\_\*$ and the second comes from the cohomological dimension of the Zariski topological space. Likewise, the honest continuous map $g: X(\mathbb{C})^{\text{an}}\to X\_{\text{Zar}}$ has cohomological dimension $n$.


Of course (1) proving that $f$ has the claimed cohomological dimension is usually quite involved, and (2) we like these cohomology theories because they behave in ways which match our intuitions. So it's unclear to me whether this is really a non-anthropological answer to your question. But I think the pattern (of finding a site over the Zariski site which has relative cohomological dimension $n$) is ubiquitous enough to be worth commenting on.


