TITLE: Why is the definition of l-adic sheaves so complicated?
QUESTION [11 upvotes]: I find the definition of constructible $\bar{\mathbb Q}_l$-sheaves (or their derived category) on a variety of positive characteristic quite involved and ad Hoc. Roughly it goes as follows:
First one defines constructible sheaves modules over torsion rings in the naive way, then over finite $\mathbb Z_l$ extensions as certain projektive systems of sheaves with bigger and bigger torsion coefficients. To pass from finite $\mathbb Z_l$ extensions to finite $\mathbb Q_l$ extensions one tensors the hom spaces in our category with the quotient field. Finally $\bar{\mathbb Q}_l$ sheaves are certain inductive systems of sheaves over finite $\mathbb Q_l$ extensions.
Is there a good reason a priori to make the definition as it is? Why are more naive definitions flawed?

REPLY [11 votes]: One reason why one needs a large coefficient group like $\overline{\mathbb{Q}_\ell}$ instead of $\mathbb{Q}$ is that a reasonable cohomology theory should be a functor.  Sometimes varieties over finite fields admit morphisms that cannot be captured by those smaller coefficients.
The standard example due to Serre is that of a supersingular elliptic curve $E$ over a finite field.  $E$ has an endomorphism ring that is a quaternion order over $\mathbb{Z}$, but elliptic curves over $\mathbb{C}$ have 2-dimensional $H^1$.  If you want the cohomology theory to behave like de Rham or singular cohomology, you need an action of the quaternion order on a two-dimensional vector space such that pullback along the multiplication-by-$n$ isogeny acts as multiplication by $n$.  The only way to get a quaternion algebra to act on a 2-dimensional vector space is to make sure the quaternion algebra splits over the coefficient field of the vector space, i.e., after taking a tensor product, it becomes isomorphic to a ring of $2 \times 2$ matrices.  The splitting is guaranteed over algebraically closed fields, but in this case, it fails to happen over $\mathbb{Q}$, or even $\mathbb{Q}_p$, where $p$ is the characteristic of the field of definition.
As it happens, you can get away with $\mathbb{Q}_\ell$ coefficients much of the time, but if you want to examine eigenvalues of the Frobenius acting on cohomology, it helps to have solutions to polynomial equations on hand.
As some comments on the question explained, one reason why $\mathbb{Q}_\ell$ coefficients are not given by the naïve construction is that the naïve construction will not yield a comparison with singular or de Rham cohomology.  Torsion-free coefficients will not see finite covers, and finite covers are the essence of étale site.