Most often than not, the sheaves appearing in algebraic geometry (with the Zariski topology) are $\mathcal{O}_X$-modules, instead of simple abelian sheaves.
Now, when dealing with topological spaces (for example, in Verdier duality) and in étale cohomology, it seems that abelian sheaves have the main role.
I wonder why. For example, Verdier duality works just fine as a statement about $\mathcal{O}_X$-modules (as in Spaltenstein's Resolution of unbounded complexes) and we recover the standard topological statements setting $\mathcal{O}_X=\underline{\mathbb{Z}}$. Even more, since ringed spaces always have fibered products (and the underlying topological spaces are the usual fibered products), base change also works well.
(I'm not sure about the étale case. I would love to know more about it.)
Is there some fundamental reason not to consider $\mathcal{O}_X$-modules in these cases?
Best Answer
We come to the question of what $\mathcal{O}_X$ should mean, when $X$ is a manifold or topological space. If $\mathcal{O}_X$ is smooth (or continuous) real valued functions, then we will always wind up studying $H^{\ast}(X, \mathbb{R})$. There is nothing wrong with this and, in fact, de Rham cohomology makes lots of use of sheaves of $\mathcal{O}_X$-modules, such as the sheaf of differential $k$-forms. But you might want to study $H^{\ast}(X, \mathbb{Z})$.
Alternatively, one could decide that $\mathcal{O}_X$ is the sheaf of locally constant $\mathbb{Z}$-valued functions. In this case, abelian sheaves are equivalent to $\mathcal{O}_X$-modules, so it is just a matter of terminology which one you say you are studying.
In algebraic geometry, the sheaf of locally constant $\mathbb{Z}$-valued functions is flasque in the Zarsiki topology (if your underlying space is irreducible), so you always wind up either working in the etale topology, or else studying some variant of de Rham cohomology (D-modules, crystals, etc). It is only in the topological world that you get to work with locally constant sheaves of abelian groups in the ordinary topology and learn something interesting.