aa r X i v : . [ m a t h . AG ] J u l ´ETALE COVERS AND LOCAL ALGEBRAIC FUNDAMENTALGROUPS CHARLIE STIBITZ
Abstract.
Let X be a normal noetherian scheme and Z ⊆ X a closed subset of codi-mension ≥
2. We consider here the local obstructions to the map ˆ π ( X / Z ) → ˆ π ( X ) being an isomorphism. Assuming X has a regular alteration, we prove the equivalence ofthe obstructions being finite and the existence of a Galois quasi-´etale cover of X , wherethe corresponding map on fundamental groups is an isomorphism. Introduction
Suppose that X is a normal variety over C and Z ⊆ X is a closed subset of codimension2 or more. Then a natural question to pose is whether the surjective map of fundamentalgroups π ( X / Z ) → π ( X ) is an isomorphism. For general normal schemes we can askthe same question for ´etale fundamental groups. For a regular scheme the Zariski-Nagatatheorem on purity of the branch locus implies the above map on ´etale fundamental groupsis an isomorphism (see [10], [8]). For a general normal scheme however this map need notbe an isomorphism so that ´etale 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 anisomorphism. Restricting any cover to a neighborhood of a point, we see that in order forit to be ´etale, it must restrict to an ´etale cover locally. Hence each point of Z gives riseto a possible obstruction determined by the image of the local ´etale fundamental groupinto the ´etale fundamental group of X / Z . Assuming these all vanish, the above map willbe an isomorphism.Even if they do not vanish, we can still hope for something in the case where all theobstructions are finite. A first guess might be that this would imply that the kernel of theabove map is finite, yet Example 1 of the singular Kummer surface shows that this canbe far from true in general. What is true however is that after a finite cover that is ´etalein codimension 1 the corresponding map is an isomorphism. The main theorem here isthat this is is fact equivalent to finiteness of the obstructions and a couple other similarconditions: Theorem 1.
Suppose that X is a normal noetherian scheme of finite type over an excellentbase B of dimension ≤ . Let Z = Sing ( X ) ⊆ X . Then the following are equivalent.(i) For every geometric point x ∈ Z the image G x ∶= im [ π ´et ( X x / Z x ) → π ´et ( X / Z )] isfinite (see § for definitions of X x and Z x ).(ii) There exists a finite index closed normal subgroup H ⊆ π ´et ( X / Z ) such that G x ∩ H is trivial for every geometric point x ∈ X .(iii) For every tower of quasi-´etale Galois covers of XX ← X ← X ← X ← ⋯ X i + → X i are ´etale for i sufficiently large.(iv) There exists a finite, Galois, quasi-´etale cover Y → X by a normal scheme Y suchthat any ´etale cover of Y reg extends to an ´etale cover of Y . As hinted to above, we can rephrase the question of the π ´et1 ( X / Z ) → π ´et1 ( X ) being anisomorphism as a purity of the branch locus statement of X . Both are equivalent to thefact that any ´etale cover of X / Z extends to an ´etale cover of X . By purity of the branchlocus for X reg , any ´etale cover of X / Z will at least extend to X reg . Hence it is enough toconsider the case where Z = X reg , as we have done in the theorem. From this point ofview, (iv) says that we can obtain purity after a finite, Galois, quasi-´etale cover.The following example of the singular Kummer surface then elucidates what is goingon in the above theorem. Example 1.
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´etale, but at the 2-torsion points it ramifies. Each of these 2-torsion points gives rise toa nontrivial Z / Z obstruction. In particular any ´etale cover of A will give a cover of X not satisfying purity of the branch locus, and in particular there are infinitely many suchcovers. 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 quarticin P with the maximum number of nodes. In particular its ´etale fundamental group istrivial, and there are no ´etale covers of X . Note that on the other hand as A is smooth weobtain purity on a finite cover of X that is ´etale away from a set of codimension 2 as in(iv) of the above theorem. In this sense, although the kernel of the map ˆ π ( X reg ) → ˆ π ( 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 [9] show-ing that the local obstructions are finite for all klt singularities over C . From this Greb,Kebekus, and Peternell [5] were able to show the global statements of (iii) and (iv) in theabove theorem for klt singularities. Their proof used essentially the existence of a Whit-ney stratification, which allowed them to check only finitely many strata to prove a coveris ´etale. Then in positive characteristic, Caravajal-Rojas, Schwede, and Tucker [3] provedagain that the local obstructions are finite for strongly F -regular singularities (which areconsidered a close analogue of klt singularities in positive characteristic). Using a boundon the size of the local fundamental groups from this paper, Bhatt, Carvajal-Rojas, Graf,Tucker and Schwede [1] were able to construct a stratification that enabled them to runa 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 [2] they are able to reprove the results in characteristic 0 by spreadingout to characteristic p .The proof of Theorem 1 can be broken down essentially into two parts. One is the con-struction 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 havea finite collection of finite subgroups of a profinite group, then there exists a closed finiteindex subgroup which intersects all of these groups trivially. Note that the assumptionthat the covers are Galois and the finite index subgroup is normal is essential. In fact due ETALE COVERS AND LOCAL ALGEBRAIC FUNDAMENTAL GROUPS 3 to the choice of basepoint that we have suppressed above (note that x is not the basepointof these fundamental groups), G x is actually only a conjugacy class of a finite subgroupinside of the group π ´et1 ( X / Z ) .Finally it is worth noting that in Xu’s paper a different local fundamental group, π ´et1 ( 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 fundamen-tal groups π ´et1 ( 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 ´etale incodimension 1 remain finite, which is the case for klt singularities. We will discuss thisissue further in the last section. Example 2.
Consider X = CS the cone over a Kummer surface S = A / ± where A is anabelian 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-´etale cover ramifying at x (note that since we havelocalized there is no difference between the two possible fundamental groups). This showsˆ π loc1 ( X x /{ x }) = ˆ π loc1 ( X x / Z x ) ≅ Z / Z , and there is no ambiguity in which definition wechoose. In general this will work for the generic point of any irreducible component ofthe singular locus.The next type of point where we start to see a difference is when x is a closedpoint in the cone over a node of S . Then in this case ˆ π loc1 ( X x /{ x }) is trivial whileˆ π loc1 ( X x / Sing ( X ) x ) ≅ Z / Z . Although they are different they are at least both finite. Onthe other hand if we desire for these groups to behave well under specialization it is clearthat ˆ π loc1 ( X x / Sing ( X ) x ) is the better choice.The last type of point, where the real problem occurs, is the cone point x ∈ CS . Firstconsider the fundamental group ˆ π loc1 ( X x /{ x }) . Then this will be isomorphic to ˆ π ( S ) ≅ π ( S reg ) is infinite, giving an infinite tower ofcones X = CS ← CS ← CS ← ⋯ Galois over X and quasi-´etale. In particular finitenessof all the local fundamental groups ˆ π loc1 ( X x /{ x }) does not imply finiteness of the localfundamental groups ˆ π loc1 ( X x / Sing ( X ) x ) . Note that in this case the singularity at theorigin is not klt. Notation.
Given a finite morphism f ∶ Y → X the branch locus, written Branch ( f ) , isthe locus over which f fails to be ´etale. A finite morphism f ∶ Y → X is quasi-´etale if it is´etale in codimension 1 or in other words the branch locus has codimension ≥
2. By purityof the branch locus for normal schemes quasi-´etale is equivalent to being ´etale over theregular locus.
Acknowledgement.
I would like to thank my advisor J´anos Koll´ar for his constantsupport. Also I would like to thank Ziquan Zhuang for several useful discussions.
CHARLIE STIBITZ Local Fundamental Groups
In this section we review some basic facts and definitions about the fundamental groupswe will be considering.
Definition.
Suppose that ( R, m ) is a strictly Henselian local normal domain and that Z ⊆ Spec ( R ) is a closed subset of codimension ≥
2. Then we define the algebraic localfundamental with respect to Z to be π ´et1 ( Spec ( R )/ Z ) . If x is a normal geometric point ofan irreducible scheme X and Z ⊆ X has codimension ≥
2, we define the local space X x = Spec (O sh X,x ) , the spectrum of the strict Henselization of the local ring. This comes with amap ι ∶ X x → X and we define Z x = ι − ( Z ) . We then define the local fundamental group at x with respect to Z to be the algebraic local fundamental group of the strict Henselizationof O X,x with respect to the closed set Z and use the notation ˆ π loc1 ( X x / Z x ) . For anygeometric point x ∈ Z we define G x ∶= im [ ˆ π loc1 ( X x / Z x ) → ˆ π ( X / Z )] , the obstructionsoccurring in theorem 1. Note.
The definition above depends on a choice of both strict Henselization (requiringa choice of separable closure of the residue field of x ) and a choice of base point. Inparticular the choice of base point implies that the groups G x are only defined up toconjugacy. It is for this reason that we need to take Galois morphisms in the maintheorem. For counterexamples when the morphisms are not Galois see [5]. Also note thedifference between the definition here and that used in [9].The following basic lemma shows the purpose in using Henselizations when definingthe local fundamental group. Lemma 1 ([1] Claim 3.5) . Suppose that f ∶ Y → X is a quasi-´etale morphism of normalschemes, that is ´etale away from some subset Z of codimension ≥
2. Then f is ´etale overa geometric point x ∈ X if and only if the pull back of the map to U x ∶= Xx / Z x is trivial. Proof.
It is enough to prove that the map is ´etale once we pull back to the strict Henseliza-tion of the local ring. Now in this case if the map f is ´etale then it induces a trivial coverof Spec (O shX,x ) 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 Spec (O shX,x ) since the varieties are normal. Hence the morphism is´etale. (cid:3) The Branch Locus of a Quasi- ´Etale Morphism
In this section we consider the question of where a quasi-´etale cover of a normal variety X branches. We will see that there are finitely many locally closed subsets of X suchthat any branch locus is the union of some subcollection of these subsets. Moreover wewill show that such a stratification is possible to compute in terms of any alteration. Westart with the criterion for telling if a morphism is ´etale from an alteration. Alterations. [4] A map π ∶ ˜ X → X is an alteration if it is proper, dominant and generi-cally finite. A regular alteration will be an alteration where ˜ X is regular. In his work, deJong showed that regular alterations exist for noetherian schemes of finite type over anexcellent base of dimension ≤
2. We will say that a divisor E ⊂ ˜ X is exceptional if π ( E ) ETALE COVERS AND LOCAL ALGEBRAIC FUNDAMENTAL GROUPS 5 has codimension at least 2 on X . Lemma 2.
Suppose that X is normal Noetherian scheme, with a regular alteration π ∶ ˜ X → X . Let f ∶ Y → X be a finite morphism of Noetherian schemes, and denote by˜ f ∶ ˜ Y → ˜ X the normalized fiber product of the maps. Then f is ´etale if and only if ˜ f is´etale and for any geometric point x ∈ X , ˜ f induces a trivial cover of π − ( x ) . Proof.
First suppose that f ∶ Y → X is ´etale. Then the base change Y × X ˜ X → ˜ X is ´etale,so that the fiber product was already normal. Hence it follows ˜ f is ´etale. Then since ˜ Y is just the fiber product and f is ´etale, for any of the d points q ∈ Y mapping to p ∈ X wesee that σ − ( q ) ≅ π − ( p ) × k ( p ) k ( q ) . Therefore ˜ f is just the trivial degree d cover on everyfiber π − ( p ) .Now suppose that ˜ f is ´etale and induces a trivial cover on every fiber of π . Then inparticular for any x ∈ X , f − ( x ) will have deg ( f ) geometric connected components. Inparticular it must be ´etale at x ([7], V.7). (cid:3) Example 3.
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-´etale doublecover f ∶ A → X . Blowing up the origins gives a map of the normalized fiber-productsBL A → BL X , 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´etale cover E ′ → E , which will induce a quasi-´etale cover X ′ → X of their cones. Afterblowing up the origins we obtain the map of normalized fiber products which is ´etale, butinduces a nontrivial cover of the exceptional divisors.Using the above lemma we can check whether f is ´etale based on a single regular al-teration. Our next goal will be to show that based on this alteration we really only needto check that f is ´etale at finitely many points. To identify what are the points we needto check we require the following condition, which roughly says that the reduced fibers ofa 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 withgeometrically reduced fibers.We now show that there exists a stratification of X such that π will satisfy the abovecondition ∗ over each of the strata. Lemma 3.
Let π ∶ ˜ X → X be a morphism of Noetherian schemes. Then there existsa stratification X = ⋃ S i , where the S i are irreducible locally closed subsets, such that π − ( S i ) → S i satisfies condition ( ∗ ) for all i . Proof.
We will proceed by Noetherian induction on X . Take an irreducible component S of X . Consider the map π − ( S ) red → S . Taking an irreducible component W of π − ( S ) red if W → S is not separable, we can take some high enough power of the Frobenius sothat the pullback by the map is separable. Doing this for every irreducible component of CHARLIE STIBITZ π − ( 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 − ( U ) red → U is reduced and that this morphismis flat. Continuing on will give the desired stratification. (cid:3) Lemma 4 (e.g. [6] 7.8.6) . Suppose that g ∶ Z → S is a morphism of Noetherian schemessatisfying condition ( ∗ ) with S integral. Then the number of connected components ofgeometric fibers are constant. Proof.
We have a purely inseparable morphism S ′ → S such that Z ′ → S ′ is flat withgeometrically reduced fibers. Since S ′ → S is a universal homeomorphism, it follows that Z ′ → Z is a homeomorphism. Hence the number of connected components remains thesame, so we can assume from the beginning that Z → S is flat with geometrically reducedfibers.Now in this case we will show that the Stein factorization of g ∶ Z → S factors as Z → ˆ S → S where ˆ S → S is ´etale. Taking the strict Henselization of the local ring atany point we can reduce to the case where S is the spectrum of a strictly Henselian localring. In this case ˆ S is a product of finitely many local rings. Our goal is to show thatthese are isomorphic to S . Now consider a connected component W of Z , so that themap g ∶ W → S is flat and proper, with geometrically reduced fibers. Now since W isconnected and S is the spectrum of strictly Henselian ring, the special fiber W is alsoconnected. But then since W is reduced H ( W , O W ) = k ( ) . Hence we see by thetheorem of Grauert that O S → g ∗ O W is an isomorphism. This implies that ˆ S → S is thus´etale, so in particular the number of connected components of the geometric fibers areconstant. (cid:3) Theorem 2.
Suppose that X is a normal Noetherian scheme and π ∶ ˜ X → X a regularalteration. Then there exists a stratification X = ⋃ i ∈ I Z i into locally closed subsets suchthat for any f ∶ Y → X quasi-´etale, with Y a normal Noetherian scheme, Branch ( f ) = ⋃ i ∈ J ⊂ I Z i .Proof. The above lemma gives a stratification X = ⋃ i S i such that π − ( S i ) → S i satisfiescondition ∗ . 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 thento show that any branch locus of a quasi-´etale morphism is a union of these strata.Consider ˜ Y = ( ˜ X × X Y ) n the normalized fiber product which comes with a morphism˜ f ∶ ˜ Y → ˜ X that is ´etale away from the exceptional locus. Now by purity of the branch locusBranch ( ˜ f ) = ⋃ i E i where the E i are some subset of the exceptional divisors. In particularthe branch locus of f will include B = ⋃ i π ( 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 ˜ f is in fact ´etale. In particular ˜ f − ( π − ( S i )) → π − ( S i ) → S i satisfies condition ∗ . Hencethe number of connected components of the fibers are constant. This implies that for anypoint s ∈ S i that if the cover of π − ( s ) is geometrically trivial, then the correspondingcover for other point in S i is also trivial. Hence we see that the branch locus must be aunion of the strata. (cid:3) Remark.
In the proof of (i) implying (iii) of the main theorem it would be nice toapply this theorem directly on X . However when we take the normalized pullback of ETALE COVERS AND LOCAL ALGEBRAIC FUNDAMENTAL GROUPS 7 an alteration we may not get another alteration. To remedy this we will need to takealterations of varieties that are further along in the tower.4.
Proof of the Main Theorem
In this section we prove the different implications in the main theorem. (i) ⇒ (ii). Proof.
Consider a regular alteration π ∶ ˆ X → 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 thefinitely many finite groups G i = G η i . Then as π ´et1 ( U ) is profinite there exists some finiteindex closed normal subgroup H intersecting all of these G i trivially. This corresponds toa quasi-´etale cover γ ∶ Y → X that is ´etale over U . Moreover by our choice of stratificationfor any geometric point x we will also have that G x ∩ H is trivial. Hence such a finiteindex normal subgroup H can be taken uniformly for all x ∈ X . (cid:3) (ii) ⇒ (i). Proof.
Our assumption (ii) gives a closed finite index normal subgroup H ⊆ π ´et1 ( U ) suchthat G x ∩ H = { } . Then in particular G x ≅ G x / G x ∩ H ⊆ π ´et1 ( U )/ H which is finite. Hence G x is finite as well. (cid:3) (i) ⇒ (iii). Proof.
We proceed by Noetherian induction. Consider our tower of finite morphismsdenoted by γ k ∶ X k + → X k , and consider the collection U of open sets U ⊆ X such thatwhen we restrict the tower over U the morphisms are eventually ´etale. The assumptionthat all the morphisms are quasi-´etale implies that X reg ∈ U . Since X is assumed to beNoetherian this collection has a maximal element and our goal is to show that this mustbe 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 = Spec (O sh X,η ) and restrict the tower of X i over X x to get a towerSpec (O sh X,η ) = X x, ← X x, ← X x, ← X x, ← ⋯ Now using the assumption (i) applied to the point x , it follows that eventually the coverswill be trivial when restricted over the regular locus and hence will be ´etale. This thenshows that there exists some N >> γ n is ´etale over η for n ≥ N and they are´etale over the open set U coming from Noetherian induction.Now take a regular alteration π ∶ ˆ X N → X N . Then using π we construct a stratification X n = ⋃ i Z i as before. Then any of the maps X N + k → X N must be ´etale over U and η . Butbecause the branch locus must be a union of strata it follows that these are all ´etale oversome open set U ′ ⊃ U with U ′ ∋ η . Hence such a larger U ′ ∈ U exists and by Noetherianinduction we see that X ∈ U . This proves property (iii). (cid:3) (iii) ⇒ (iv). CHARLIE STIBITZ
Proof.
Assuming that no such cover exists, we inductively construct a tower X ← X ← X ← X ← ⋯ as in (iii) of the main theorem using Galois closures, such that none ofthe X i + → X i are ´etale. This will contradict our assumption, so eventually every ´etalecover of one of the X i, reg will extend to an ´etale cover of X i . This gives the desired coversatisfying purity. (cid:3) (iv) ⇒ (i). Proof.
Consider a geometric point x of X . Take a cover f ∶ Y → X as in (iv), and ageometric 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 fundamentalgroups. ˆ π loc1 ( Y y / f − ( Z ) y ) ÐÐÐ→ ˆ π ( f − ( U ))×××Ö ×××Ö ˆ π loc1 ( X x / Z x ) ÐÐÐ→ ˆ π ( U ) Now the assumption on Y implies that the top map is zero. On the other hand, the imageof the map on the left is a finite index normal subgroup. Hence looking at the images inˆ π ( U ) , we see that G x has a trivial finite index subgroup and hence must be finite. (cid:3) Applications
Using our main theorem we can recover the results of [5] and [1].
Corollary 1.
Suppose that X is a normal klt variety over C . Then X satisfies thecondition (ii).Proof. We want to show that X satisfies condition (i). There are two issues to deal withif we wish to apply Xu’s result [9]. First is the problem that in this paper the localfundamental groups are defined in terms of links instead of the local spaces X x given byHenselization. The second is that Xu proves the finiteness of ˆ π loc1 ( X x /{ x }) and we sawthat 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 toprove the finiteness of algebraic local fundamental groups as considered in this paper. Inthe proof of his main theorem, Xu cuts down to a surface. It is then possible to consideronly quasi-´etale covers instead of ´etale covers of X x /{ x } , as these will agree after cuttingdown. Also you would need an equivalence of the algebraic local fundamental groupdefined in terms of links and Henselizations. Once this is done though (ii) will followimmediately from the main theorem. Note that also the recent result of [2] is strongenough to apply directly.The second way to prove this is to note that we can get around the issue of whichfundamental group we consider when we work in a class of normal varieties R satisfyingthe following. We want for every X ∈ R , and every quasi-´etale cover Y → X that Y ∈ R ,and also for every x a geometric point of X ∈ R that ˆ π loc1 ( X x / x } is finite. In particularklt singularities satisfy both these conditions by [9]. Then under these assumptions, thesame argument for (i) implies (ii) works with the fundamental groups ˆ π loc1 ( X x / x } . Thisapproach is used in [5]. (cid:3) ETALE COVERS AND LOCAL ALGEBRAIC FUNDAMENTAL GROUPS 9
Corollary 2.
Suppose that X is a normal F -finite strongly F -regular variety over a fieldof characteristic p . Then X satisfies the condition (ii).Proof. In this case the result of Carvajal-Rojas, Schwede, and Tucker [3] applies directlyto the main theorem without any changes. (cid:3)
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Department of Mathematics, Princeton University, Princeton, NJ 08544, USA
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