### Question

Given an abelian variety $V$ and an integer $n$, is there a natural abelian category with a natural object $X$ and natural coefficients $F$ so that $V\simeq H^n (X,F)$?

### Motivation

Studying abelian varieties is awesome. Studying objects in long exact sequences is awesome. How do (somewhat forcefully) combine these two? I mean without taking cohomology of the variety like everyone else does...

### Possible answers

The abelian variety is a $G$-module, where $G=Gal(\bar{k}/k)$, $k$ the field over which the variety is defined. So, maybe there is an interesting $G$-module that answers the above? The cases of abelian varieties over number fields and finite fields are the most interesting, so $G$ is assumed to be interesting as well (i.e. not trivial).

Maybe it arises as the $n$-th cohomology of some interesting sheaf of some interesting related variety?