I am reading some books about etale cohomology theory. I found there are some theorems which are very similar with the topological theorems. For example, there are Poincare Duality theorems for both schemes and manifolds. So my first question is if there are relationships between the theorems of schemes and manifolds.
We know that for general topological spaces, we could consider the sheaf cohomology. So I wonder if we could consider etale coholomgy theories for manifolds? In other words, could we generalize etale cohomology for manifolds?
Now, when we consider the varieties over $\mathbb{C}$, we have three different cohomology theories: singular cohomology, sheaf cohomology and etale cohomology. I feel they are all equivalent, although I can not describe what the equivalence exactly means. So my next question is if there is any method to describe the equivalence between these three cohomology theory for complex varieties.
I also notice for manifolds we have singular homology theories. However, when I study algebraic geometry, there are all cohomology theories, such as sheaf cohomology, Cech cohomology, etale cohomology, etc. I wonder if there are some homology theories for schemes? Why we do not consider homology for schemes?
I think these are soft problems and may do not have exact answers. So could you explain some ideas or recommend some books for me to read? Thank you very much for your help.
Let's start with some historical context. Originally, etale cohomology was invented to give something that worked better because it was more like singular cohomology for manifolds. We start with an example which demonstrates a defect of sheaf cohomology compared to singular cohomology.
Let $X$ be a smooth irreducible complex projective variety, and let $\underline{\Bbb C}$ denote the constant sheaf with value $\Bbb C$. In the Zariski topology, we have that $\underline{\Bbb C}$ is flasque, and thus all higher cohomology on $X$ vanishes. On the other hand, if we consider the $\Bbb C$ points of $X$ with the analytic topology, we get that $H^{2\dim X}(X(\Bbb C),\underline{\Bbb C})=\Bbb C$. The defect is basically due to the fact that the Zariski topology doesn't have enough open sets, so we need to find a way to get more. There's no good solution in terms of literally putting more sets in the topology - instead, one has to consider more general versions of a covering, where instead of picking some open sets inside $X$, we consider a covering to be a collection of schemes $U_i$ and etale morphisms $\varphi_i:U_i\to X$ with $X$ the union of the images of the morphisms. This gives us access to the "more open sets" that we need. (We'll come back to applying these ideas to the manifold case at the end of the post.)
One can see that from the above, it's not exactly the case that sheaf cohomology, etale cohomology, and singular cohomology are all equivalent for varieties over $\Bbb C$. There are some situations where these things line up, though: the dimension of the cohomology groups defined for etale cohomology with coefficients in $\Bbb F_q$ agree with the dimension of the cohomology groups for singular cohomology with coefficients in $\Bbb F_q$ (this doesn't work for integer coefficients, though - $H^2_{et}(\Bbb P^1_\Bbb C,\Bbb Z)=0$, for instance). The de Rham theorem also shows that sheaf cohomology of the constant sheaf $\underline{\Bbb R}$ and singular cohomology with coefficients in $\Bbb R$ agree for smooth manifolds.
One of the reasons that defining homology is hard in algebraic geometry is mentioned by Geoff in the comments: there aren't enough projectives, so we can't resolve things in the direction we would need to use an algebraic definition of homology. This means instead that if we want to define a homology theory for general schemes, we would need to do this geometrically, and it turns out this is somewhat complicated. Two ideas to read about here are Chow groups and Borel-Moore homology (though, depending on your background, you may need some amount of prep work before really getting in to these).
There have been some attempts to turn this concept of Grothendieck topologies back around to the manifold world. The ones I'm aware of all involve Pierre Schapira, and a decent overview of some of what's been done is available in these lecture slides and this expository paper. Briefly, by putting together concepts from o-minimal geometry, Grothendieck topologies, homological algebra & derived categories, Schapira and his collaborators manage to sheafify a range of analytic constructions, such as constructing Sobolev sheaves (see their 2016 publication). This sort of thing is not wholly in my wheelhouse, but my impression of it is that even the idea of sheafifying something like this sort of construction is kind of wild. If you like algebraic analysis (D-modules, microlocal analysis, etc), then this sort of thing would probably be pretty up your alley.