aa r X i v : . [ m a t h . N T ] O c t LOCAL-TO-GLOBAL EXTENSIONS TO WILDLY RAMIFIED COVERS OFCURVES
RENEE BELL Introduction
Throughout this paper, k is an arbitrary field of characteristic p , and G is a finite p -group.Let Y be a smooth proper curve over k and y ∈ Y ( k ). We define a “ y -ramified G -cover of Y ” tobe a Galois cover of curves q : X → Y with Galois group G , totally ramified over y and unramifiedon the complement Y ′ := Y − { y } . By the Cohen structure theorem, we can choose uniformizers t and u i such that Q x i ∈ q − ( y ) \ O X,x i ∼ = Q x i ∈ q − ( y ) k [[ u i ]] and [ O Y,y ∼ = k [[ t ]]. After localization, weobtain a G -Galois ´etale algebra L := Q x i ∈ q − ( y ) k (( u i )) over k (( t )). We say that L arises from the G -action on X .Thus, for each curve Y and point y ∈ Y , we obtain a functor ψ Y,y : n y -ramified p -group coversof Y o → n Galois ´etale algebrasover k (( t )) with Galoisgroup a p -group o . Understanding the functor ψ Y,y allows us to use the geometry of Galois covers of curves to classifyautomorphisms of k [[ t ]] as in [1]. Conversely, it allows us to use extensions of k (( t )) in order toclassify filtrations of ramification groups of Galois covers of curves with Galois group a p -group asnoted in the survey [4].Questions about ψ Y,y can be approached by turning to ´etale cohomology. Throughout thispaper, for a scheme S and a (not necessarily abelian) group G , we denote by H ( S, G ) the ˇCechcohomology ˇ H et ( S, G ) of the constant sheaf of groups with coefficients in G with respect to the´etale site on X ; this cohomology set parameterizes principal G -bundles on X [9]. The inclusionof the ring of regular functions on Y ′ into the Laurent series field O ( Y ′ ) ֒ → k (( t )) induces a mapSpec k (( t )) → Y ′ which we can think of as inclusion of the formal deleted neighborhood around y into Y ′ . Hence, we obtain a map H ( Y ′ , G ) → H ( k (( t )) , G ) which we denote by Ψ Y,y,G . We notethat Ψ
Y,y,G is induced from ψ Y,y by restricting to G -covers and passing to isomorphism classes.We pose some basic questions about Ψ Y,y,G . Question 1.1.
When is Ψ Y,y,G surjective?
This is equivalent to asking when every G -Galois extension of k (( t )) extends to a global Galoiscover of Y . In [3], Harbater showed that if the ground field k is algebraically closed, then Ψ Y,y,G is surjective for any p -group G . In this paper, we provide an answer to Question 1.1 over a moregeneral field k , not necessarily algebraically or even separably closed, in the following theorem.Notation: for any ring R of characteristic p , let ℘ : R → R denote the Artin–Schreier map f f p − f , and let U n ( R ) denote the group of upper triangular n × n matrices with entries in R such that all diagonal entries are 1. Theorem 1.2.
Let G be a nontrivial finite p -group. Then the following are equivalent: (1) The equality k (( t )) = ℘ ( k (( t ))) + O ( Y ′ ) holds. (2) The map Ψ Y,y,U n ( F p ) is surjective for all n > . (3) The map Ψ Y,y,G is surjective.
We can also ask when any lift of an extension of k (( t )) to a global Galois cover of Y is uniqueup to isomorphism. Question 1.3.
When is Ψ Y,y,G injective?
An answer to this over k algebraically closed was given as well by Harbater in [3]. In fact, hecalculates the size of the fiber of Ψ Y,y,G as p r , where r is the p -rank of Y . We extend the answerto question 1.3 to a more general field k , which may not be separably closed. Theorem 1.4.
Let G be a nontrivial finite p -group. Then the following are equivalent: (1) The equality ℘ ( k (( t ))) ∩ O ( Y ′ ) = ℘ ( O ( Y ′ )) holds. (2) The map Ψ Y,y,G is injective. (3)
The map Ψ Y,y, Z /p Z is injective. Combining our answers to Question 1.1 and 1.3, without assuming that the base field is alge-braically or even separably closed, gives a criterion on Y for ψ Y,y to be an equivalence of categories.This generalizes the result of Katz in [6], which is that over any field of characteristic p , the functor ψ P , ∞ is an equivalence of categories. Curves satisfying the criteria of theorems 1.4 and 1.2 areparticularly useful for relating the geometry of the curve and its covers to properties of k (( t )) andits extensions. In Section 4 of this paper, we give another explicit example of a class of such curves.Our proofs of these theorems use new and more explicit methods. Proofs in previous work,as in [6], have reduced the problem to the case in which G is abelian. In this case, one can usethe vanishing of certain H groups or a characterization of abelian p -group field extensions usingWitt vector theory, as noted in [4]. However, in this paper, we prove our results using a differentmethod: describing and working with an explicit characterization of G -Galois ´etale algebras for G not necessarily abelian. This characterization, which we will call the Inaba classification, is ageneralization of a theorem of Inaba in [5], which extends Artin–Schreier–Witt theory to nonabelianGalois ´etale algebras. Theorem 1.5.
Let G be a finite p -group, and fix an injective homomorphism Λ : G → U n ( F p ) forsome suitable n . Let R be a ring of characteristic p such that Spec R is connected, and let L/R bea Galois ´etale algebra with Galois group G . i The R -algebra L is generated by elements a ij ∈ L for ≤ i < j ≤ n such that the unipotentmatrix A := ( a ij ) satisfies A ( p ) = M A for some M ∈ U n ( R ) . We also have that for σ ∈ G , σ A = A Λ( σ ) , where σ acts entry-wise on A . ii Given two algebras
L, L ′ ∈ H ( Spec
R, G ) , if we choose ( A, M ) for L and ( A ′ , M ′ ) for L ′ ,then L, L ′ are isomorphic if and only if M = C ( p ) M ′ C − for some C ∈ U n ( R ) . Inaba Classification of p -group Covers We now provide generalizations of Artin-Schreier theory to non-abelian groups. We begin witha lemma about U n ( F p )-extensions. Lemma 2.1.
Let R be a ring of characteristic p such that Spec R is connected. Then the finiteGalois ´etale algebras over Spec R with Galois group U n ( F p ) are the algebras R [ X ] / ( X ( p ) = M X ) where M ranges over all matrices in U n ( R ) , and the Galois action is given by matrix multiplication OCAL-TO-GLOBAL EXTENSIONS TO WILDLY RAMIFIED COVERS OF CURVES 3 X X · g . Two such Galois algebras defined by matrices M, M ′ are isomorphic as R -algebras with U n ( F p ) -action if and only if M = C ( p ) M ′ C − for some C ∈ U n ( R ) .Proof. Let U n be the F p group scheme representing the functor which sends a ring A to U n ( A ), andlet U n ( F p ) be the constant group scheme. We have a sequence U n ( F p ) → U n L −→ U n where L is the morphism (which is not a group homomorphism) B B ( p ) B − . By Lang’s theorem[8], L is surjective and identifies U n /U n ( F p ) with U n . Since Spec R is connected, we know that H (Spec R, U n ( F p )) = U n ( F p ), so by Proposition 36 of [12], we have an exact sequence of pointedsets 1 → U n ( F p ) → U n ( R ) L −→ U n ( R ) δ −→ H (Spec R, U n ( F p )) → H (Spec R, U n )where δ sends a matrix M ∈ U n ( R ) to the class of the ´etale algebra L M . We also see that the actionof U n ( R ) on U n ( R ) via L is given as follows. Since L is surjective, there is some ´etale R -algebra S and some N ∈ U n ( S ) such that M = N ( p ) N − . So the action of C ∈ U n ( k ) on M sends M to L ( CN ) = C ( p ) N ( p ) N − C − = C ( p ) M C − .Next, since H ( X, O X ) = 1 for affine schemes and U n has a composition series whose factorsare G a , we see by induction that H (Spec R, U n ) = 1, so the map U n ( R ) → H (Spec R, U n ( F p ))is surjective and expresses H (Spec R, U n ( F p )) as quotient of U n ( R ) by the left action of U n ( R )via the map L . So every U n ( F p ) ´etale algebra is isomorphic to some L M with the condition forequivalence as stated in the lemma. (cid:3) Now we look at G a general p -group and fix an embedding Λ : G → U n ( F p ). Proof of Theorem 1.5 . First, we note that the inclusion Λ : G → U n ( F p ) induces a map H (Spec R, G ) → H (Spec R, U n ( F p ))sending L Y G \ U n ( F p ) L =: ˜ L with the following left U n ( F p )-action. Let u , ..., u r be coset representatives for G \ U n ( F p ), with u = e being the identity element. Then we can write any element of Q G \ U n ( F p ) L as ( ℓ i ) ri =1 with ℓ i ∈ L . For each u ∈ U n ( F p ), there exist g i ∈ G such that u i u = g i u j ( i ) , where j ( i ) is the index ofthe coset of u i u . Then u · ( ℓ i ) i = ( g j − i ) ℓ j − ( i ) ) i .Now consider the map π : ˜ L → L which is projection onto the first component and note that π (( P ℓ i g i ) h ) = h ℓ , so π is a map of ´etale G -algebras. But by lemma 2.1, ˜ L ∼ = R [ X ] / ( X ( p ) = M X )as G -algebras, so the surjection π expresses L as R [ A ], where A is the matrix with ij -coordinateequal to π ( x ij ). And since π is compatible with the action of G , the original G -action on L agrees with the action coming from matrix multiplication by Λ( G ). Also, since we now know that H (Spec R, G ) ֒ → H (Spec R, U n ( F p )), we see that the condition for equivalence is the same as inthe preceding lemma. 3. Reduction to Z /p Z case We now show that properties of the map Ψ
Y,y,G can be checked at G = Z /p Z , or equivalently itsuffices to know the behavior of ℘ on k (( t )) and O ( Y ′ ). We begin with a lemma about the structureof U n ( R ). RENEE BELL
Lemma 3.1.
Let R be a ring of characteristic p . Suppose M = ( m ij ) , M ′ = ( m ′ ij ) ∈ U n ( R ) are p -equivalent, so M = B ( p ) M ′ B − for some B = ( b ij ) ∈ U n ( R ) . Then for each pair i, j ,there exists an element C of the Z -algebra generated by { m i ′ j ′ , b i ′ j ′ , m ′ i ′ j ′ | i ′ − j ′ < i − j } such that m ij = ℘ ( b ij ) + m ′ ij + C . That is, m ij = ℘ ( b ij ) + m ′ ij modulo the elements on the lower diagonals.Proof. Consider two matrices W = ( w ij ) , Z = ( z ij ) ∈ U n ( R ). We compute that the ( i, j ) entryof W Z is P k w ik z kj , but since these matrices are in U n , this actually equals 1 · z ij + w ij · P i We first show that (i) implies (ii), so suppose k (( t )) = O ( Y ′ ) + ℘ ( k (( t ))).Let L be a U n ( F p )-Galois ´etale k (( t ))-algebra, so by Lemma 2.1, L ∼ = L M for some M ∈ U n ( k (( t ))).We want to find a matrix M ′ ∈ U n ( O ( Y ′ )) which is p -equivalent over k (( t )) to M . Suppose thatnot all entries of M are in O ( Y ′ ). Let m ij be such an entry on the lowest diagonal not having allentries in O ( Y ′ ). By the assumption in (i), there exists b ∈ k (( t )) such that m ij + ℘ ( b ) ∈ O ( Y ′ ).Let B be the matrix in U n ( k (( t ))) which has b in the ( i, j ) entry and is equal to the identity matrixelsewhere. Then the ( i, j ) entry of B ( p ) M B ( − is m ij + ℘ ( b ) + C with C ∈ O ( Y ′ ). And by Lemma3.1, the ( i, j ) entry and entries of all lower diagonals of B ( p ) M B ( − are in O ( Y ′ ), so we can iteratethis process until we have a matrix M ′ in U n ( O ( Y ′ )).Next, we show that (ii) implies (iii), so suppose Ψ Y,y,U n ( F p ) is surjective. Since U ( F p ) ∼ = Z /p Z ,we can also assume that Ψ Y,y, Z /p Z We proceed by induction on the order of G . Since G is a p -group, G has a central subgroup H which is isomorphic to Z /p Z . By Proposition 42 of [12] and Lemma1.4.3 of [6] (which states that both H ( k (( t )) , Z /p Z ) and H ( Y ′ , Z /p Z ) are zero), the commutative OCAL-TO-GLOBAL EXTENSIONS TO WILDLY RAMIFIED COVERS OF CURVES 5 diagram induced by the map Spec k (( t )) → Y ′ / / H ( Y ′ , Z /p Z ) ι Y / / Ψ Y,y, Z /p Z (cid:15) (cid:15) H ( Y ′ , G ) φ Y / / Ψ Y,y,G (cid:15) (cid:15) H ( Y ′ , G/H ) Ψ Y,y,G/H (cid:15) (cid:15) / / / / H ( k (( t )) , Z /p Z ) ι k (( t )) / / H ( k (( t )) , G ) φ k (( t )) / / H ( k (( t )) , G/H ) / / H ( Y ′ , G ) have the same image in H ( Y ′ , G/H ) if and onlyif they are in the same H ( Y ′ , Z /p Z )-orbit (and similarly for H ( k (( t )) , G )). And by the inductivehypothesis, Ψ Y,y, Z /p Z and Ψ Y,y,G/H are surjective. The surjectivity of Ψ Y,y,G is proved via thefollowing diagram chase.Let ˜ β be an element of H ( k (( t )) , G ), and let ˜ γ := φ k (( t )) ( ˜ β ). By the inductive hypothesis, ∃ γ ∈ H ( Y ′ , G/H ) such that Ψ Y,y,G/H ( γ ) = ˜ γ . Let β be an element of H ( Y ′ , G ) mappingto γ . Then by Proposition 42 of [12], there exists an element ˜ α ∈ H ( k (( t )) , Z /p Z ) such that ι k (( t )) (˜ α ) · Ψ Y,y,G ( β ) equals ˜ β . Now let α be an element of H ( Y ′ , Z /p Z ) mapping to ˜ α . We seethat i Y ( α ) · β maps to ˜ β under Ψ Y,y,G , so Ψ Y,y,G is surjective.Next, we show that (iii) implies (i), so suppose Ψ Y,y,G for finite some p -group G , and again let H be a nontrivial central subgroup of G isomorphic to Z /p Z . Again consider the aforementioneddiagram. Since φ k (( t )) ◦ Ψ Y,y,G is surjective, Ψ Y,y,G/H is surjective. Iterating this process showsthat Ψ Y,y, Z /p Z is surjective. Let f ∈ k (( t )), so k (( t ))[ x ] / ( x p − x − f ) is a Z /p Z -Galois ´etale algebraover k (( t )). Since Ψ Y,y, Z /p Z is surjective, Theorem 1.5 tells us that f is p -equivalent to an element g of O ( Y ′ ), so f = ℘ ( b ) + g for some b ∈ k (( t )). Thus the equality in (i) holds. (cid:3) We now give a concise reformulation of the criterion for when Ψ Y,y,G is a bijection. We denoteby F the Frobenius map O Y → O Y sending f f p . This induces a map F ∗ : H ( Y, O Y ) → H ( Y, O Y ); we also let ℘ ∗ := F ∗ − Id be the map on cohomology induced by Artin–Schreier. Thenour reformulation is as follows. Corollary 3.2. The map ψ Y,y is an equivalence of categories if and only if ℘ ∗ : H ( Y, O Y ) → H ( Y, O Y ) is a bijection.Proof. The natural open immersion i : Y ′ ֒ → Y gives the following exact sequence of sheaves on Y :0 → O Y → i ∗ O Y ′ → skysc y (cid:18) k (( t )) k [[ t ]] (cid:19) → y (cid:16) k (( t )) k [[ t ]] (cid:17) denotes the skyscraper sheaf at y , with value group k (( t )) /k [[ t ]] where thegroup structure is given by the additive structure on k (( t )). We see that H ( Y, i ∗ O Y ′ ) = H ( Y ′ , O Y ′ ) =0 since Y ′ is affine, so the induced long exact sequence in cohomology gives us an isomorphism H (cid:16) Y, skysc y (cid:16) k (( t )) k [[ t ]] (cid:17)(cid:17) im( H ( Y, i ∗ O Y ′ )) = k (( t )) O ( Y ′ ) + k [[ t ]] ∼ −→ H ( Y, O Y ) . We also see that the map F ∗ : k (( t )) O ( Y ′ ) + k [[ t ]] → k (( t )) O ( Y ′ ) + k [[ t ]]maps ¯ f ¯ f p for f ∈ k (( t )).Now suppose ℘ ∗ is surjective. Then for every f ∈ k (( t )) there exist g ∈ k (( t )) , h ∈ O ( Y ′ ) , l ∈ k [[ t ]]such that f = g p − g + h + l . Let a be the constant term of l ; by setting h ′ := h + a and l ′ := l − a we can assume l has no constant term. So l = P ∞ i =1 a i t i . We define a power series ˜ l := P ∞ i =1 b i where b i = − a i for i not divisible by p and b np = b pn − a np . So f = ℘ ( g + ˜ l ) + h and Ψ Y,y,G issurjective. The converse, that surjectivity of Ψ Y,y,G implies surjectivity of ℘ ∗ , is clear.Next, suppose ℘ ∗ is injective, and consider f such that ℘ ( f ) ∈ O ( Y ′ ). Then f ∈ k [[ t ]] + O ( Y ′ ),so there exist g ∈ O ( Y ′ ) , l ∈ k [[ t ]] such that f = g + l . Again, we can assume that l has no constantterm, so l ∈ tk [[ t ]]. But then ℘ ( g ) + ℘ ( l ) ∈ O ( Y ′ ), which implies ℘ ( l ) ∈ O ( Y ′ ). But since a nonzero l ∈ tk [[ t ]] would have a zero at y , it could not come from a regular function on Y ′ , so we musthave l = 0 and so in fact ℘ ( f ) ∈ ℘ ( O ( Y ′ )) and Ψ Y,y,G is injective. Conversely, suppose Ψ Y,y,G isinjective, and consider f ∈ k (( t )) such that ℘ ( f ) = g + l for some g ∈ O ( Y ′ ) , l ∈ tk [[ t ]]. We canwrite l = ℘ (˜ l ) for ˜ l as above, so ℘ ( f − ˜ l ) ∈ O ( Y ′ ), which implies f − ˜ l ∈ O ( Y ′ ) by the hypothesis.Then ¯ f = ¯0 in k (( t )) / ( k [[ t ]] + O ( Y ′ )), which is what we wanted to show. (cid:3) Examples In this section, we apply the previous results to study Ψ E for E an elliptic curve over F p . Theoperator F ∗ on H ( X, O X ) acts as multiplication by some a ∈ F p , and in fact E ( F p ) = p + 1 − a .We say that E is anomalous if E ( F p ) = p . Theorem 4.1. For an elliptic curve E over F p , the following are equivalent: (1) E is not anomalous. (2) The map Ψ E is injective. (3) The map Ψ E is surjective. (4) The map Ψ E gives an equivalence of categories.Proof. We let e denote a generator of H ( E, O E ) as an F p -vector space, so F ∗ maps e a · e . Thenfor x ∈ F p , we have xe x p ae . But since x ∈ F p , the map is xe xae , so ℘ ∗ ( xe ) = x ( a − e .Then clearly the map ℘ ∗ : H ( E, O E ) → H ( E, O E ) is surjective if and only if a = 1 and it’sinjective if and only if a = 1. Applying Corollary 3.2 gives the result. (cid:3) As indicated in [10], anomalous curves are, as their name suggests, uncommon, and so Theorem4.1 provides us with a broad class of curves whose p -group Galois covers correspond nicely to p -group k (( t )) extensions. References [1] F.M. Bleher, T. Chinburg, B. Poonen and P. Symonds. Automorphisms of Harbater-Katz-Gabber curves, Math-ematische Annalen (2017) 368: 811.[2] I. Bouw and R. Pries. Rigidity, reduction, and ramification. Mathematische Annalen (2003) 326: 803.[3] D. Harbater. Moduli of p-covers of curves. Communications in Algebra Natural Science Report,Ochanomizu University Vol. 12, No.2 (1961), 26-36.[6] N. Katz. Local-to-global extensions of representations of fundamental groups Annales de l’institut Fourier , tome36, no 4 (1986), 69-106.[7] S. Lang. Algebra Graduate Texts in Mathematics, volume 211. Springer-Verlag, New York, third edition, 2002.[8] S. Lang. Algebraic Groups Over Finite Fields. 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