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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.

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how has it been two years, and no-one has pointed out the typo in the title? –  name Mar 23 '13 at 16:49
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2 Answers

up vote 5 down vote accepted

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.

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That is very nice. I reallize how silly I was to use the espace \'etale to construct the functor in the topological case, I should also construct coverings by constant sheaves and glue these together using compatiblity conditions. Then this generalize to the \'etale case. Thank you very much, Angelo! –  Lei Jul 4 '11 at 9:11
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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.

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Thanks Carnahan! I am not quite clear about the first part of your explaination yet, it seems your answer is neater while the essential idea maybe the same as Angelo's. By $U\mapsto O_U^{F(U)}$ do you really mean you take the sheaf $O_U^{F(U)}$ instead of taking the gobal sections of it? What do you mean by taking the relative spetrum of the etale sheaf, do you mean you restrict it to Zariski site? For the second part could you give me any reference for that? I roughly know how one should set up the functors but not clear yet, if this is written clearly somewhere, it would save me sometime. –  Lei Jul 4 '11 at 10:36
    
I am also interested in the "fundamental groupoid" thing you mentioned. That should be the groupoid in Deligne's sense, (ex. in "le groupe fondamental de la droite projective moins trois points" section 10) right? How that comes up? our etale fundamental group is an abstract group. Do you get the groupoid by some non-neutral tannakian category? –  Lei Jul 4 '11 at 10:54
    
Sorry for the sloppiness; I meant to take global sections. Regarding fundamental groups and groupoids, I mostly learned about it be talking to others, so I don't know which references are good. Perhaps some work of Minhyong Kim, or de Jong's paper "Etale fundamental groups of non-archimedean analytic spaces" has more information. –  S. Carnahan Jul 5 '11 at 2:42
    
Thank you very much, Carnahan! –  Lei Jul 6 '11 at 7:40
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