I've started feeling this rather curious mystique coming from an unaddressed - at least in my experience - excessive presence of the number $2$ in a few different areas of maths. My curiosity really sparked the moment I realised there was a mismatch between different explanations lurking in my knowledge of cohomology:
Of course, the singular cohomology of a complex variety should vanish in degrees greater than double the dimension of the variety, since complex varieties are effectively of real dimension twice their complex dimension, and this is because the index of the field extension $\mathbf{C}/\mathbf{R}$ is two - singular co/homology is a theory fundamentally built from simplices which are inherently "real" objects.
Singular cohomology really isn't about simplices: these are just a way of accessing it concretely; singular cohomology is really just sheaf cohomology with the simplest possible choice of coefficients.
Even when you consider cohomologies which aren't quite modelled on the same sorts of spaces, such as the étale cohomology of schemes or that of spaces which locally look nothing like $\mathbf{C}$ or $\mathbf{R}$, like adic spaces or diamonds, somehow the main point is that these are formalisms which give back the same or analogous results you're supposed to get for the "complex varieties analogue" of whatever it is you're computing.
I felt particularly stumped when I learned that the ubiquitous number $2$ appearing in the representation theory of semisimple Lie algebras and algebraic groups can also be given a cohomological explanation, via the cohomology of flag varieties - somehow the $2$ appearing there corresponds to the intersection theory of their singular subvarieties (via their Chow groups/perverse sheaves) and the fact that "the intersection of a pair of (real) even-dimensional varieties gives back an even-dimensional variety".
It might be because I'm only somewhat acquainted with only a few of these theories, being still just a student, but I feel like there's something I'm not seeing here that all my teachers do: how can it possibly be that $\mathbf{P}^n_{\overline{\mathbf{Q}}}$ with coefficients in $\underline{\mathbf{Z}/n}\in \text{Sh}_{\text{Ab}}(\mathbf{P}^n_{\overline{\mathbf{Q}},\text{ét}})$ has exactly the same cohomology as $\mathbf{C}P^n$ with coefficients in $\mathbf{Z}/n$ if the latter theoretically depends on the equality $[\mathbf{C}:\mathbf{R}] = 2$ whereas the former has no mention of either the real or complex numbers...?
I apologise if my question is a little vague and rather soiled with my little experience; I'd really appreciate any piece of wisdom (however relevant) you might have.
Thanks for reading! :D