locally constant constructible sheaves and finite etale coverings

Question

Maybe it is well known to experts or maybe it is just a stupid idea, but I will ask any way.

We know that if $X$ is a topological space, then there is an equivalence of categories between the category of locally constant sheaves (of sets) on $X$ and the category of covers (sous-entendu local homoemorphism) of $X$.

The equivalence is given by "$\Rightarrow$" using the "espace \'etal\'e" of sheaves, "$\Leftarrow$" taking the sheaf of sections.

Now I replace $X$ by a scheme (locally noetherien or something?), and I think there is an equivalence of categories between the category of locally constant constructible sheaves of sets (By constructible I mean the constant values should be finite) on the \'etale site of $X$ and the category of finite \'etale coverings of $X$.

I tried to construct the functor "$\Leftarrow$": Given $Y\rightarrow X$ finite \'etale, we associate to any $T\to X$, the set of sections $T\to Y\times_XT$. This is a locally constant constructible sheaf if $X$ is locally noetherian: You decompose $X$ into connect components and by SGA1 corollary 5.3 then you can see easily that on each connected component the association is a constant sheaf with a finite constant value.

Is this an equivalence? If it is how one constructs the quasi-inverse?

Furthermore, do we have any formulation like the finite representations of $\pi_1^{\text{et}}(X,x)$ is equivalent to the category of locally constant constructible \'etale sheaves (of vector spaces) on $X$. If this is true it should be a direct consequence of Grothendieck's main theorem on $\pi_1^{\text{et}}$ and the above statement.

Answer 1

Consider your functor from étale coverings to locally constant constructible sheaves. It is fully faithful, by Yoneda's lemma. The fact that it is essentially surjective follows from descent theory. If $F$ is a locally constant constructible sheaf, take an étale cover $\{U_i \to X\}$ such that the restriction of $F$ to $U_i$ is constant; call $A_i$ a finite set such that $F_i := F\mid_{U_i}$ is represented by $U_i \times A_i := \bigsqcup_{a \in A_i}U_i$. The sheaf $F$ gives descent data $\mathrm{pr}_2^*F_j \simeq \mathrm{pr}_1^*F_i$ on the fibered products $U_i \times_X U_j$; by faithful flatness, these give descent data for the covers $U_i \times A_i \to U_i$, yielding a finite étale cover of $X$ that represents $F$.

[Edit] I should have pointed out that descent for étale covers works because étale covers are affine maps; ultimately, this relies on descent for quasi-coherent sheaves, a version of which is used in Scott's answer.

Answer 2

To construct a quasi-inverse, you may use the equivalence of categories between affine morphisms and sheaves of quasicoherent algebras described in EGA2 Chapter 1. Given a locally constant constructible sheaf $F$ of sets, you can take the étale sheafification of the presheaf of algebras $U \mapsto \mathscr{O}_U^{F(U)}$ on the small étale site of $X$. The relative spectrum of this sheaf is the finite étale cover you want. It looks like you need some descent to prove this, so this construction is more or less a disguised version of Angelo's.

If you have a pointed connected scheme $(X,x)$, then there is an equivalence between finite representations of the fundamental group on a vector space, and locally constant constructible étale sheaves of vector spaces. In one direction, a sheaf $F$ is taken to the fiber over $x$. In the other direction, you pass to a trivializing cover, take a constant sheaf of appropriate dimension, and apply the associated sheaf construction (which is just descent). If your scheme is not connected, you replace the fundamental group with the fundamental groupoid.