Étale Covers and Local Algebraic Fundamental Groups
Charlie Stibitz
math.AG



Let  X be a normal noetherian scheme and Z ⊆ X a closed subset of codimension ≥ 2. We consider here the local obstructions to the map π̂_1(X\ Z) →π̂_1(X) being an isomorphism. Assuming X has a regular alteration, we prove the equivalence of the obstructions being finite and the existence of a Galois quasi-étale cover of X, where the corresponding map on fundamental groups is an isomorphism.




Department of Mathematics, Princeton University, Princeton, NJ 08544, USA
cstibitz@math.princeton.edu
Étale Covers and Local Algebraic Fundamental Groups
    Charlie Stibitz
    
===================================================



§ INTRODUCTION
 Suppose that X is a normal variety over ℂ and Z ⊆ X is a closed subset of codimension 2 or more. Then a natural question to pose is whether the surjective map of fundamental groups π_1(X Z) →π_1(X) is an isomorphism. For general normal schemes we can ask the same question for étale fundamental groups.  For a regular scheme the Zariski-Nagata theorem on purity of the branch locus implies the above map on étale fundamental groups is an isomorphism (see <cit.>, <cit.>). For a general normal scheme however this map need not be an isomorphism so that étale covers of X Z need not extend to all of X. 

The next question to ask is what are the obstructions to the above map being an isomorphism. Restricting any cover to a neighborhood of a point, we see that in order for it to be étale, it must restrict to an étale cover locally. Hence each point of Z gives rise to a possible obstruction determined by the image of the local étale fundamental group into the étale fundamental group of X Z. Assuming these all vanish, the above map will be an isomorphism. 
 
Even if they do not vanish, we can still hope for something in the case where all the obstructions are finite. A first guess might be that this would imply that the kernel of the above map is finite, yet Example 1 of the singular Kummer surface  shows that this can be far from true in general. What is true however is that after a finite cover that is étale in codimension 1 the corresponding map is an isomorphism. The main theorem here is that this is is fact equivalent to finiteness of the obstructions and a couple other similar conditions:  

 Suppose that X is a normal noetherian scheme of finite type over an excellent base B of dimension ≤ 2. Let Z = (X) ⊆ X. Then the following are equivalent. 

    
  * For every geometric point x ∈ Z the image G_x := [π_1^(X_x Z_x) →π^_1(X Z)] is finite (see 2 for definitions of X_x and Z_x).
    
  * There exists a finite index closed normal subgroup H ⊆π_1^(X Z) such that G_x∩ H is trivial for every geometric point x ∈ X. 
    
  * For every tower of quasi-étale Galois covers of X
   
    X ← X_1← X_2← X_3←⋯

    X_i+1→ X_i are étale for i sufficiently large.
   
  * There exists a finite, Galois, quasi-étale cover Y → X by a normal scheme Y such that any étale cover of Y_ extends to an étale cover of Y.



As hinted to above, we can rephrase the question of the π_1^(X Z) →π_1^(X) being an isomorphism as a purity of the branch locus statement of X. Both are equivalent to the fact that any étale cover of X Z extends to an étale cover of X. By purity of the branch locus for X_reg, any étale cover of X Z will at least extend to X_reg. Hence it is enough to consider the case where Z = X_, as we have done in the theorem. From this point of view, (iv) says that we can obtain purity after a finite, Galois, quasi-étale cover. 

The following example of the singular Kummer surface then elucidates what is going on in the above theorem. 
 



Consider the quotient π:A → A/± = X, where A is an abelian surface and X is a singular Kummer surface over ℂ. Away from the 16 2-torsion points this map is étale, but at the 2-torsion points it ramifies. Each of these 2-torsion points gives rise to a nontrivial ℤ/2ℤ obstruction. In particular any étale cover of A will give a cover of X not satisfying purity of the branch locus, and in particular there are infinitely many such covers. On the other hand the fundamental group of X is trivial, which can be seen as X will be diffeomorphic to a standard Kummer surface given as a singular nodal quartic in ℙ^3 with the maximum number of nodes. In particular its étale fundamental group is trivial, and there are no étale covers of X. Note that on the other hand as A is smooth we  obtain purity on a finite cover of X that is étale away from a set of codimension 2 as in (iv) of the above theorem. In this sense, although the kernel of the map π̂_1(X_) →π̂_1(X) is large it is not far away from satisfying purity. 


The recent history of studying these two problems started with a paper of Xu <cit.> showing that the local obstructions are finite for all klt singularities over ℂ. From this Greb, Kebekus, and Peternell <cit.> were able to show the global statements of (iii) and (iv) in the above theorem for klt singularities. Their proof used essentially the existence of a Whitney stratification, which allowed them to check only finitely many strata to prove a cover is étale. Then in positive characteristic, Caravajal-Rojas, Schwede, and Tucker <cit.> proved again that the local obstructions are finite for strongly F-regular singularities (which are considered a close analogue of klt singularities in positive characteristic). Using a bound on the size of the local fundamental groups from this paper, Bhatt, Carvajal-Rojas, Graf, Tucker and Schwede <cit.> were able to construct a stratification that enabled them to run a similar local to global argument to deduce statements of the form (ii), (iii), and (iv) for strongly F-regular varieties. It is also worth noting in the recent preprint of Bhatt, Gabber, and Olsson <cit.> they are able to reprove the results in characteristic 0 by spreading out to characteristic p. 

The proof of Theorem 1 can be broken down essentially into two parts. One is the construction of a stratification that allows us to deal with only finitely many obstructions. The second is a completely group theoretic fact about profinite groups, namely if we have a finite collection of finite subgroups of a profinite group, then there exists a closed finite index subgroup which intersects all of these groups trivially. Note that the assumption that the covers are Galois and the finite index subgroup is normal is essential. In fact due to the choice of basepoint that we have suppressed above (note that x is not the basepoint of these fundamental groups), G_x is actually only a conjugacy class of a finite subgroup inside of the group π_1^(X Z). 

Finally it is worth noting that in Xu's paper a different local fundamental group, π_1^(X_x{x}) was shown to be finite. If we instead defined our obstruction groups G_x as Xu did then in fact the theorem is false, as will be shown in the following example. There are two ways around this for klt singularities: either showing the larger fundamental groups π_1^(X_x Z_x) are finite in the klt case, or show that a similar implication (i)  (ii),(iii) will hold as long as the smaller local fundamental groups of all covers étale in codimension 1 remain finite, which is the case for klt singularities.  We will discuss this issue further in the last section. 



 Consider  X = CS the cone over a Kummer surface S = A/± where A is an abelian surface. Then there are three types of singular points: 

 First consider the case where x is the generic point of the cone over one of the nodes. Then X_x has a regular double quasi-étale cover ramifying at x (note that since we have localized there is no difference between the two possible fundamental groups). This shows π̂_1^(X_x{x}) = π̂_1^(X_x Z_x) ≅ℤ/2ℤ, and there is no ambiguity in which definition we choose. In general this will work for the generic point of any irreducible component of the singular locus. 

 The next type of point where we start to see a difference is when  x is a closed point in the cone over a node of S. Then in this case π̂_1^(X_x{x}) is trivial while π̂_1^(X_x(X)_x) ≅ℤ/2ℤ. Although they are different they are at least both finite. On the other hand if we desire for these groups to behave well under specialization it is clear that π̂_1^(X_x(X)_x) is the better choice. 

The last type of point, where the real problem occurs, is the cone point x ∈ CS.  First consider the fundamental group π̂_1^(X_x{x}). Then this will be isomorphic to π̂_1(S) ≅ 0 by the Lefschetz hyperplane theorem. In particular all the local fundamental groups defined in this sense are finite. So if this version of the theorem were true then any tower as above would stabilize. On the other hand π̂_1(S_) is infinite, giving an infinite tower of cones X=CS ← CS_1← CS_2←⋯ Galois over X and quasi-étale. In particular finiteness of all the local fundamental groups π̂_1^(X_x{x}) does not imply finiteness of the local fundamental groups π̂_1^(X_x(X)_x). Note that in this case the singularity at the origin is not klt.






Given a finite morphism f:Y → X the branch locus, written (f), is the locus over which f fails to be étale. A finite morphism f:Y → X is quasi-étale if it is étale in codimension 1 or in other words the branch locus has codimension ≥ 2. By purity of the branch locus for normal schemes quasi-étale is equivalent to being étale over the regular locus.


 I would like to thank my advisor János Kollár for his constant support. Also I would like to thank Ziquan Zhuang for several useful discussions. 




§ LOCAL FUNDAMENTAL GROUPS


In this section we review some basic facts and definitions about the fundamental groups  we will be considering. 

 Suppose that (R,m) is a strictly Henselian local normal domain and that Z ⊆(R) is a closed subset of codimension ≥ 2. Then we define the algebraic local fundamental with respect to Z to be π_1^((R) Z). If x is a normal geometric point of an irreducible scheme X and Z ⊆ X has codimension ≥ 2, we define the local space X_x = (𝒪_X,x^), the spectrum of the strict Henselization of the local ring. This comes with a map ι:X_x→ X and we define Z_x = ι^-1(Z). We then define the local fundamental group at x with respect to Z to be the algebraic local fundamental group of the strict Henselization of  𝒪_X,x with respect to the closed set Z and use the notation π̂_1^(X_x Z_x). For any geometric point x ∈ Z we define G_x := [π̂_1^(X_x Z_x) →π̂_1(X Z)], the obstructions occurring in theorem 1. 




The definition above depends on a choice of both strict Henselization (requiring a choice of separable closure of the residue field of x) and a choice of base point. In particular the choice of base point implies that the groups G_x are only defined up to conjugacy. It is for this reason that we need to take Galois morphisms in the main theorem. For counterexamples when the morphisms are not Galois see <cit.>. Also note the difference between the definition here and that used in <cit.>. 
 

The following basic lemma shows the purpose in using Henselizations when defining the local fundamental group. 


 Suppose that f:Y → X is a quasi-étale morphism of normal schemes, that is étale away from some subset Z of codimension ≥ 2. Then f is étale over a geometric point x ∈ X if and only if the pull back of the map to U_x: = Xx Z_x is trivial. 


It is enough to prove that the map is étale once we pull back to the strict Henselization of the local ring. Now in this case if the map f is étale then it induces a trivial cover of (𝒪_X,x^sh) and hence of the open set V. On the other hand if the cover of U_x is trivial, then so is the cover of (𝒪_X,x^sh) since the varieties are normal. Hence  the morphism is étale. 








§ THE BRANCH LOCUS OF A QUASI-ÉTALE MORPHISM


In this section we consider the question of where a quasi-étale cover of a normal variety X branches. We will see that there are finitely many locally closed subsets of X such that any branch locus is the union of some subcollection of these subsets. Moreover we will show that such a stratification is possible to compute in terms of any alteration. We start with the criterion for telling if a morphism is étale from an alteration. 


Alterations. <cit.> A map π:X̃→ X is an alteration if it is proper, dominant and generically finite. A regular alteration will be an alteration where X̃ is regular. In his work, de Jong showed that regular alterations exist for noetherian schemes of finite type over an excellent base of dimension ≤ 2. We will say that a divisor E ⊂X̃ is exceptional if π(E) has codimension at least 2 on X. 


 Suppose that X is normal Noetherian scheme, with a regular alteration π:X̃→ X. Let f:Y → X be a finite morphism of Noetherian schemes, and denote by f̃:Ỹ→X̃ the normalized fiber product of the maps. Then  f is étale if and only if f̃ is étale and for any geometric point x ∈ X, f̃ induces a  trivial cover of π^-1(x). 


First suppose that f:Y → X is étale. Then  the base change Y ×_XX̃→X̃ is étale, so that the fiber product was already normal. Hence it follows f̃ is étale. Then since Ỹ is just the fiber product and f is étale,  for any of the d points q ∈ Y mapping to p ∈ X we see that σ^-1(q)  ≅π^-1(p) ×_k(p) k(q). Therefore f̃ is just the trivial degree d cover on every fiber π^-1(p). 

Now suppose that f̃ is étale and induces a trivial cover on every fiber of π. Then in particular for any x ∈ X,  f^-1(x) will have (f) geometric connected components. In particular it must be étale at x (<cit.>, V.7).




Each of the two conditions in the lemma above are easily seen to be necessary.  For example we can let X be the cone over a smooth conic. This has a quasi-étale double cover f:𝔸^2→ X. Blowing up the origins gives a map of the normalized fiber-products _0𝔸^2→_0X, that will ramify along the exceptional divisors.  

On the other hand we can take the X to be the cone over an elliptic curve E. Take an étale cover E' → E, which will induce a quasi-étale cover X' → X of their cones. After blowing up the origins we obtain the map of normalized fiber products which is étale, but induces a nontrivial cover of the exceptional divisors. 



Using the above lemma we can check whether f is étale based on a single regular alteration. Our next goal will be to show that based on this alteration we really only need to check that f is étale at finitely many points. To identify what are the points we need to check we require the following condition, which roughly says that the reduced fibers of a morphism fit together in a flat family.  


Condition ∗. Suppose that g:Z → S is a proper  morphism of Noetherian schemes with S integral. Then g satisfies this condition if there exists a purely inseparable morphism i:S' → S such that if Z' = Z×_S S' → S' is the base change, then Z_red' → S' is flat with geometrically reduced fibers. 



We now show that there exists a stratification of X such that π will satisfy the above  condition ∗ over each of the strata. 


 Let π:X̃→ X be a morphism of Noetherian schemes. Then there exists a stratification X = ⋃ S_i, where the S_i are irreducible locally closed subsets,  such that π^-1(S_i) → S_i satisfies condition (∗) for all i. 


We will proceed by Noetherian induction on X. Take an irreducible component S  of X. Consider the map π^-1(S)_red→ S. Taking an irreducible component  W of π^-1(S)_red  if W → S is not separable, we can take some high enough power of the Frobenius so that the pullback by the map is separable. Doing this for every irreducible component of π^-1(S)_red we may assume that the general fiber is reduced. Then taking an open subset U of S we may assume that every fiber of π^-1(U)_red→ U is reduced and that this morphism is flat. Continuing on will give the desired stratification.


 Suppose that g:Z → S is a  morphism of Noetherian schemes satisfying condition (∗) with S integral. Then the number of connected components of geometric fibers are constant. 


We have a purely inseparable morphism S' → S such that Z' → S' is flat with geometrically reduced fibers. Since S' → S is a universal homeomorphism, it follows that Z' → Z is a homeomorphism. Hence the number of connected components remains the same, so we can assume from the beginning that Z → S is flat with geometrically reduced fibers. 

Now in this case we will show that the Stein factorization of g:Z → S factors as Z →Ŝ→ S where Ŝ→ S is étale. Taking the strict Henselization of the local ring at any point we can reduce to the case where S is the spectrum of a strictly Henselian local ring. In this case Ŝ is a product of finitely many local rings. Our goal is to show that these are isomorphic to S. Now consider a connected component W of Z, so that the map g:W → S is flat and proper, with geometrically reduced fibers. Now since W is connected and S is the spectrum of strictly Henselian ring, the special fiber W_0 is also connected. But then since W_0 is reduced H^0(W_0,𝒪_W_0) = k(0). Hence we see by the theorem of Grauert that 𝒪_S→ g_∗𝒪_W is an isomorphism. This implies that Ŝ→ S is thus étale, so in particular the number of connected components of the geometric fibers are constant.




Suppose that X is a normal Noetherian scheme and  π:X̃→ X a regular alteration. Then there exists a stratification X = ⋃_i∈ IZ_i into locally closed subsets such that for any f:Y → X quasi-étale, with Y a normal Noetherian scheme,  (f) = ⋃_i ∈ J ⊂ IZ_i. 


The above lemma gives a stratification X = ⋃_i S_i such that π^-1(S_i) → S_i satisfies condition ∗. Moreover we a finite number of exceptional divisors E_i giving closed subsets π(E_i) on X. Putting these together gives our desired stratification of X. Our goal is then to show that any branch locus of a quasi-étale morphism is a union of these strata. 

Consider Ỹ = (X̃×_X Y)^n the normalized fiber product which comes with a morphism f̃:Ỹ→X̃ that is étale away from the exceptional locus. Now by purity of the branch locus  (f̃) = ⋃_i E_i where the E_i are some subset of the exceptional divisors. In particular the branch locus of f will include B=⋃_iπ(E_i), which will be a union of some strata. Now looking on the complement of B, and replacing X by X B we can assume that f̃ is in fact étale. In particular  f̃^-1(π^-1(S_i)) →π^-1(S_i) → S_i satisfies condition ∗. Hence  the number of connected components of the fibers are constant. This implies that for any point s ∈ S_i that if the cover of π^-1(s) is geometrically trivial, then the corresponding cover for other point in S_i is also trivial. Hence we see that the branch locus must be a union of the strata. 


 In the proof of (i) implying (iii) of the main theorem it  would be nice to apply this theorem directly on X. However when we take the normalized pullback of an alteration we may not get another alteration. To remedy this we will need to take alterations of varieties that are further along in the tower. 




§ PROOF OF THE MAIN THEOREM



In this section we prove the different implications in the main theorem.   


(i)  (ii).

 Consider a regular alteration π:X̂→ X. This will give us a stratification X = ⋃_i Z_i. Now for each of the finitely many generic points η_i of the different strata consider the finitely many finite groups G_i = G_η_i. Then as π_1^(U) is profinite there exists some finite index closed normal subgroup H intersecting all of these G_i trivially. This corresponds to a quasi-étale cover γ:Y → X that is étale over U. Moreover by our choice of stratification for any geometric point x we will also have that G_x∩ H is trivial. Hence such a finite index normal subgroup H can be taken uniformly for all x ∈ X. 


(ii)  (i). 


 Our assumption (ii) gives a closed finite index normal subgroup H ⊆π_1^(U) such that G_x∩ H = {1}. Then in particular G_x≅ G_x/G_x∩ H ⊆π_1^(U)/H which is finite. Hence G_x is finite as well.




(i)  (iii). 


We  proceed by Noetherian induction. Consider our tower of finite morphisms denoted by γ_k:X_k+1→ X_k, and consider the collection 𝒰 of open sets U ⊆ X such that when we restrict the tower over U the morphisms are eventually étale. The assumption that all the morphisms are quasi-étale implies that X_∈𝒰. Since X is assumed to be Noetherian this collection has a maximal element and our goal is to show that this must be all of X. 

Therefore we need to show that if U ∈𝒰 and U ≠ X then we can find a larger U' ∈𝒰. To do this take any x a generic point of an irreducible component of X U. Consider X_x = (𝒪_X,η^) and restrict the tower of X_i over X_x to get a tower

    (𝒪_X,η^) = X_x,0← X_x,1← X_x,2← X_x,3←⋯

 Now using the assumption (i) applied to the point x, it follows that eventually the covers will be trivial when restricted over the regular locus and hence will be étale. This then shows that there exists some N >> 0 such that γ_n is étale over η for n ≥ N and they are étale over the open set U coming from Noetherian induction.

Now take a regular alteration  π:X̂_N→ X_N. Then using π we construct a stratification X_n = ⋃_iZ_i as before. Then any of the maps X_N+k→  X_N must be étale over U and η. But because the branch locus must be a union of strata it follows that  these are all étale over some open set U' ⊃ U with U' ∋η. Hence such a larger U' ∈𝒰 exists and by Noetherian induction we see that X ∈𝒰. This proves property (iii).





(iii)  (iv). 

Assuming that no such cover exists, we inductively construct a tower X ← X_1← X_2← X_3←⋯ as in (iii) of the main theorem using Galois closures, such that none of the X_i+1→ X_i are étale. This will contradict our assumption, so eventually  every étale cover of one of the X_i, will extend to an étale cover of X_i. This gives the desired cover satisfying purity. 


(iv)  (i).

Consider a geometric point x of  X. Take a cover f:Y → X as in (iv), and a geometric point y of Y mapping to x. Denote by U the regular locus of X and Z = X U the singular locus.  This gives rise to the following commutative diagram of fundamental groups. 

    π̂_1^(Y_y f^-1(Z)_y) @>>> π̂_1(f^-1(U)) 
     
    @VVV @VVV 
    π̂_1^(X_x Z_x) @>>> π̂_1(U)

Now the assumption on Y implies that the top map is zero. On the other hand, the image of the map on the left is a finite index normal subgroup. Hence looking at the images in π̂_1(U), we see that G_x has a trivial finite index subgroup and hence must be finite. 







§ APPLICATIONS


Using our main theorem we can recover the results of <cit.> and <cit.>. 



Suppose that X is a normal klt variety over ℂ. Then X satisfies the condition (ii). 


We want to show that X satisfies condition (i). There are two issues to deal with if we wish to apply Xu's result <cit.>. First is the problem that in this paper the local fundamental groups are defined in terms of links instead of the local spaces X_x given by Henselization. The second is that Xu proves the finiteness of π̂_1^(X_x{x}) and we saw that this is not enough to guarantee (ii) in general. 

There are two ways to get around this. The first is to strengthen the result of Xu to prove the finiteness of algebraic local fundamental groups as considered in this paper. In the proof of his main theorem, Xu cuts down to a surface. It is then possible to consider only quasi-étale covers instead of étale covers of X_x{x}, as these will agree after cutting down. Also you would need an equivalence of the algebraic local fundamental group defined in terms of links and Henselizations. Once this is done though (ii) will follow immediately from the main theorem. Note that also the recent result of <cit.> is strong enough to apply directly. 

The second way to prove this is to note that we can get around the issue of which fundamental group we consider when we work in a class of normal varieties ℛ satisfying the following. We want for every X ∈ℛ, and every quasi-étale cover Y → X that Y ∈ℛ, and also for every x a geometric point of X ∈ℛ that π̂_1^(X_xx} is finite. In particular klt singularities satisfy both these conditions by <cit.>. Then under these assumptions, the same argument for (i) implies (ii) works with the fundamental groups π̂_1^(X_xx}. This approach is used in <cit.>.



Suppose that X is a normal F-finite strongly F-regular variety over a field of characteristic p. Then X satisfies the condition (ii).


In this case the result of Carvajal-Rojas, Schwede, and Tucker <cit.> applies directly to the main theorem without any changes. 





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