Mixiu

    I will talk about 'étale fundamental group' , which Grothendieck defines on a scheme . This is analogy of the fundamental group of a topological space and  a generalization of the Galois group of a field. 

Section 1 

    We started with a topological space. For a good topological space X , we can defines the fundamental group π_1(X,x) as the homotopy class of loops start from a given point x , and there is a theorem says:" the category of covering space of X is equivalence to the category of π_1(X,x) sets"  

     In number theory, there is  Galios theorem  which is very similar to the previous theorem . Pick a field k , let G be the Galois group of k^(s)/k, the automorphism group of k^(s) that fixes k, here k^(s) is the separable closure of k. Galois theorem says there is a correspondence between the open subgroup of the G and the finite separable extension of k. 

   We formulate is in a categorical way: From a open subgroup H of G, we can get a continuous action of the group G on the finite right cosets space G/H. And given a finite continuous transitive action of G on a set φ , when we fix a element λ in φ. The subgroup of G that fix this elements is a open subgroup of G. Change the element, as the action is transitive, you get a open subgroup conjugate to H. Up to conjugate, the open subgroup of G corespondence to a finite set with a transitive continuous G-action. We can reformulate Galois theorem  as :" the category of finite etale covering of k is equivalence to the category of finite sets with a continues G-action " ( I will mention what etale cover later, but here the etale covering of k is just unions of finite separable extension of k with the projection map to k, and finite continuous G-sets are disjoint union of finite transitive G-sets, so this is just reformulate of Galois theorem ) 

      Grothendieck generalize this to a scheme X (you can think of a complex variety if not familiar with scheme), show that the there exist a group G, called the etale fundamental group of X, such that the category of continuous finite G-sets are equivalent to the category of finite etale cover of X. I will discuss the proof of this in this talk. 

       A finite étale cover of a scheme X is a scheme Y with a morphism to X to be finite and étale. Being finite means it is closed and being étale means it is open, so this is a subjective map, hence a cover map in the usual sense. I will not formally define what is a étale morphism , it involves too many commutative algebra :(  But I will give you the intuition and many examples. Intuitively ,  Étale morphism is a analogy of local isomorphism of manifolds in differential geometry, a covering of Riemann surface with no branch poInts in complex geometry and an unramified extension of a field in algebraic number theory(this correspondence to a étale map between ring of integers in two field) . And for map between two smooth complex variety, between a étale map is Just to be a local isomorphism. Example to  bear in mind are the covering map of Riemann surface with no branch point and unramified extension of a field. 

Section 2 

    First notice that the loop construction in topology space is not valid in scheme, there is no proper concept likes loops in scheme , instead, we can mimic the definition of fundamental group as Aut_X(Y), Y is the universal cover of X. 

    We then first reformulate theorem of fundamental group in the universal cover settings and see how it fits into the picture of scheme.  Take X be a good topological space , pick a point x in X, we get a functor F from the category of covering of X to the category of sets , by sending every covering Y of X to the fibre F(x) over the point x. Theses functor is usually called the fibre functor. The exists of universal covering Z provided that this functor is representable by Z. 

     The fundamental group G of X is the deck transformation of Z, namely Aut_X(Z) ,the homomorphism of Z to Itself that preserve the projection to X. In that case Hom_X(Z,Y) becomes a G sets by composition Aut_X(Z) with Hom_X(Z,Y). This fibre functor becomes a functor from the category of covering space of X to the category of G-sets, and the theorem says that : for a nice topological space, like locally simply connected and path connected, this functor is an equivalence of two category, this is just reformulate the theorem in section 1. 

    For a schemes X, pick a geometric point X of X (geometric meanings x=Spec(k) for k separable closed) , for simplicity you can think of X is a complex variety and x is really a point on X, consider the functor 

   This is mimic the fibre functor in previous, send each finite etale cover Y to the fibers of Y over x. We can wonder if this functor is representable by a scheme Z ? If so, this scheme is like the universal cover of X, take the Aut_X(Z) as the étale fundamental group of X , then we get the same case as the topological setting. Unfortunately here is not the case , we make a example to see why it falts. 

Take X= Spec(C[t, 1/t] ) , which is C/{0} , complex place without the origin point, in this case all the etale covers are 

     We could not find a biggest étale covers here. As a complex manifold, this X_n is a n-fold cover of C/{0} , and C/{0} has a universal cover C in which the cover map is given by exponential map , the problem with scheme(variety) is that this exponential map is not algebraic, so this is not a map between schemes(variety) .

     But even this functor is not representable, it is pseudo-representable, namely there exist a projective system 

These family of covering maps indexed by a directed sets serves as the same function as universal covers. Moreover, we can choose X_i to be Galois, i.e. of degree equal to [Aut_X(X_i)]. And we define the étale fundamental group G to be 

Here the group is a inverse limit of finite group , we put the profinite topology to make it a profinite group. Then as we did for your case of topological space, the fiber functor then can be views as a functor from the category of étale covering of X to the category of finites sets with a continues G-action. And the theorem says that this is an equivalence of categories.  

We give some examples : 

    Example 1 : C/{0} then have étale fundamental group Z^{hat} , which is the inverse limits of Z/nZ by the naturally index. 

    Example 2: Take a compact Riemann surface S of genius g , we know there always exist a algebraic structure on S make it into a complex variety, in that case , the étale fundamental group is just the profinite completion of the topological fundamental group ,namely take all the finite quotient and take the inverse limit of them by the natural index. For elliptic curves, the fundamental group is Z/times Z and the étale fundamental group is Z^{hat} /times Z^{hat} . 

     Example 3 : We give some examples from number theory. 

The étale fundamental group of a field is its Galois group . For ring of integers O_k inside a number field K , to be a étale cover says to be ring of integers O_L inside a unramified extension of L of K, the étale fundamental group of Spec(Z) is trivial because of Minkowski's theorem, which says the discriminant of any algebraic number field L not equal to Q has discriminant big than one , and L ramifies at some prime number divide its discriminant . It is generally not easy to get the fundamental group of the O_k, as this is the Galois group of maximum unramified extension of K . By class field theory, we get that the abelianization of étale fundamental group is isomorphic to the class group of K.

Remark : see you from here we can view class group as the first homology group of the number field. 

    Example 4: The most interesting example comes from arithmetic scheme. Like if X is a connected algebraic variety over Q, let X_{Q^} be its base change to Q^ . x be a geometric point in X. Then we have a exact sequence

     The first exact sequence come from the functoriality of the étale fundamental group. And the second isomorphism comes from the two category's FEt/X_Q^ and FEt/X_C are equivalent. 

      So π_1(X_Q^,x) is a profinite-completion of the fundamental group of X(C). Which we can see topologically. From the exact sequence, We have Gal(Q^/Q) acts on π_1(X_Q^,x) by lifting and conjugating, which is well defined module the inner automorphism of π_1(X_Q^,x) . Moreover , If x can be choose to be come from a Q point, we get the sequence split and we get a really action of Gal(Q^/Q) on π_1(X_Q^,x) . There are intensive research on this action  and still a lot of unsolved conjecture here. 

     We give an example: X is a projective line over Q with three points 0, 1, \infinity removed. Where π_1(X_Q^,x) is the profinite completion of group generated by loops γ_0,γ_1,γ_2 around three delete points with singe relation γ_0.γ_1.γ_2=1 . Grothendieck was first to study this group and this is  very interesting.  Material for these can be find in "Geometric Galois actions, 1. Around Grothendieck's Esquisee d'un progromme " for this. 

Remark : 

1 For  the exact sequence here, it is similar to such case in topology : suppose you have a covering map Y--X with deck transformation G , then we have a exact sequence 1--π^1(Y)--π^1(X)--G--1.  Here Scheme(X_Q^ {ét})  can be see as a cover of Scheme(X_{ét}) with the Galois group Gal(Q^/Q). 

    For if the base point choose to be a Q point , the exact sequence split, write the proof late...

   π^1(X,x) can be see as a sheaf of groups on Scheme(Spec(Q)_ét) , write it later...

2 Grothendieck did a lot of work here and also make conjectures concerns the étale fundamental group. See' anabelian geometry', 'section conjecture'  and also'  dessin d'enfants ' for details. 

And Grothendieck actually not only consider the étale fundamental group, let the base point change he get a fundamental groupoid, which carries more information  and seems to be the right object to study. 

3 The étale fundamental group can be view as the first homotophy group , so you will wonder are there definition of higher homotophy group for schemes.  See 'étale homotophy type' 

Section 3 The proof and Tannakian duality

     We discuss the proof of the existence of étale fundamental group and see how it fits to the generally philosophy of Tannakian duality. 

      We first introduce the Tannakian duality, it is very naturally consideration, say we have a symmetric object G, like a group or a Hopf algebra, and study the representation of this object in some category D , Examples are : for groups, in the category of sets , we get permutation representation of group. In category of k-vector space, we get the linear representation of the groups. The Tannakian duality says that we can reconstruct that symmetry object from the endomorphism of the forgetful functor F : Rep_G(D) ---D . the simple case is C/iso Aut(F) . 

The forgetful forget F is called fiber functor. 

    Example 1 : group and its permutation representation. 

    In this case the Tannakian duality follows from the Yoneda lemma, we state this theorem and give a detail proof. For this is the model for the étale fundamental group. 

    Theorem : Let G be a finite group and Rep_G be the category of G-sets.Let F be the forgettful functor : Rep_G ----Sets . Which sends a representation to its underlying sets. Then Aut(F) \iso G . 

     Here Aut(F) is the group of invertible natural transformation from F to itself. 

Proof: the functor F is representable by the G-sets G, which we denote G^ for distinguishing. F(X)= Hom (G^, X) . So by Yoneda lemma, Nat(F,F) /Iso Hom (G^,G^) which is G as a set, and the composition of natural transformation correspondence to the multiplication in G. 

 (Remark 1 : if you view a group G as a category C with one object and morphism says elements in the group, then a permutation representation of the group G is just a functor from the category C to the category of sets, then the proof use twice yoneda lemma. Can you see it ? 

   Remark 2.: this theorem is also true if we replace the finite group by profinite group and consider category of finite set with continues G-action. And these is the case appears in étale fundamental group. 

    Example 2 : say G is a compact Lie group and Rep_(G) be the category of finite dimensional representation of G in C.  

     This is the original Tannakian-krein duality. 

Back to the étale fundamental group , we want to proof the category FEt/X is equivalent to a continues finite G-set category. 

What Grothendieck's did is an axiomatic characterization of categories that are equivalent to G-set for G profinite and then proof FEt/X satisfy that property. 

Here is the definition of Galois category

Grothendieck proof that Galois category is equivalence to a finite continuous G-set category and explicit , Aut(F) gives that group G. Apples to  category FEt/X , the fundamental functor to be the fiber functor defines previous, all remains is to proof this satisfy the axiom of Galois category , which is done to hand check. For details , see Lentra 'Galois theorem for schemes' 

Section 4 

   We can have a Sheaf theoretic treating . Remembers that a covering map for topological space X correspondence to a locally constant sheaf of sets on X. And the category of locally constant sheaf on X is equivalence to the Cov/X. 

   For schemes, we can also view a étale cover of X as a locally constant sheaf on the Sites Scheme_ X_{ét) : the scheme over X with the étale topology. And we get the category of locally constant sheaf on that Sites is equivalence to FEt/X , hence equivalent to the category of finite sets with continuous π_1(X, x) acting. 

Thanks