Some generalizations of Abhyankar lemma.

K.N.Ponomaryov

Abstract

We prove some generalizations of Abhyankar lemma about tamely ramified extensions of discretely valued fields for an arbitrary (not finite, not tamely ramified) extension of valued fields.

The paper is based on the article [1]. We use notations from this paper and prove some generalizations of theorem 2 for an arbitrary extension of discretely valued fields of a nonzero characteristic p (may be not finite and not separable extension).

We have to extend the notion of strongly solvable extension from the paper. Let us include in the class of strongly solvable extension the class of unseparable finite extensions. We call a finite extension F/K as strongly solvable in wide sense extension if it is a strongly solvable separable extension over some finite purely unseparable extension.

Let us note that classes of strongly solvable extensions, of purely unseparable extensions and of finite extensions are marked classes of field extensions. So the class of strongly solvable in wide sense extensions is a marked class too (see [6]).

Any strongly solvable in wide sense extension K_{2}/K_{1} can be presented as a tower of quasicyclic extensions K_{2} = L_{n}/.../L_{0} = K_{1} of the height n. Any extension L_{i+1}/L_{i} is a decomposition field of a polynomial f(X) − a, a ∈ L_{i}. We denote by f or a separable monomial X^{m}, (m, p) = 1 or an unseparable monomial X^{p} or Artin - Schreier polynomial ℘(X) = X^{p} − X.

We assume that any field under consideration contains all roots from unit. We use the notion of strongly solvable extension in the wide sense only.

Let us remember that we denote a kernel of perfection ∩l^{p^{n}} as l^{e}. We denote an algebraic closure of a field l as l^{a}. The field l^{e} and any of its algebraic extension are perfect field. So any algebraic closure of a kernel of perfection is a separable closure l^{ea} = l^{es}.

A key statement of the paper is the following extension of theorem 1 from [1].

Theorem 1 Let L/K be some extension of fields. Let L_{0} be an intermediate discretely valued field with the property that extension L/L_{0} is a finite extension and L_{0}/K is a tamely unramified extension.

Let l_{0}/k be a corresponding extension of residue fields for L_{0}/K. Let we have an inclusion l_{0}^{ea} ⊆ k.

Then there is a solvable totally ramified extension K′/K in algebraic closure of K with the following property: the extension LK′/K′ is a tamely unramified extension for any continuation of the valuation from L_{0} to a composite of extensions LK′.

It is easy to see that in the proof of theorem 1 from [1] statement of theorem above was proved in a particular case of finite and wild ramified Galois extension L/L_{0}. We reduce the proof of theorem to this particular case.

We have in mind to apply the theorem to finite coverings of algebraic varieties. So we deduce some generalization of theorem 2 from [1].

Corollary 1 Let L be some finite extension of a discretely valued field K and a residue field of K be a functional field.

Then there is a totally ramified solvable extension K′/K in algebraic closure of K with the property that extension LK′/K′ is a tamely unramified extension for any continuation of valuation of K to a composite of extensions LK′.

Abhyankar lemma can be reformulated in the general form for an arbitrary (not finite) extension of discretely valued fields. It reduces any extension of valued field to a wild ramified extension (see proposition 2). From theorem 1 we deduce a generalization of Abhyankar lemma.

Theorem 2 Let L/K be some extension of discretely valued fields.

Let l/k be an extension of residue fields corresponding to L/K and we have an inclusion l^{ea} ⊆ k.

Then there is a solvable totally ramified extension K′/K in algebraic closure of K that the extension LK′/K′ is a tamely unramified extension for any continuation of valuation of L to LK′.

This theorem takes a final form when the extension of the residue fields l/k is of finite type , i.e. when l is a finitely generated extension of k. This is a suitable form for an application of the statement to an investigation of arbitrary coverings of algebraic varieties (not necessarily finite).

Corollary 2 Let L/K be an extension of discretely valued fields and a residue field of K be a functional field. Let a corresponding extension of residue fields be an extension of finite type.

Then there is a solvable totally ramified extension K′/K in algebraic closure of K with the property that the extension LK′/K′ is a tamely unramified extension for any continuation of the valuation of L to LK′.

Let us note that the statement of theorems above can be treated as some refinement of Epp theorem (see [4]).

1 Purely unseparable extensions.

We use approach of the paper [1] (see §3) and give a description of types of purely unseparable extensions of discretely valued fields. Let us remember that a valuation of K can be extended to a purely unseparable extension by unique way. We denote R(K) valuation ring, k residue field, and^{−} : R(K) → k natural projection of this field. Let U(K) be a group of units and π be a uniformizing parameter of K.

As any purely unseparable extension can be presented as a tower of extensions of a prime degree we consider an extension L/K of a prime degree.

The extension L/K is a field of decomposition of some polynomial of the form X^{p} − a for some a ∈ K\K^{p}.

This extension can be defined by any element from the set aK^{p} + K^{p} = (a + K^{p})K^{p}. So we can choose a from the ring R(K). Let us call these elements as representative elements of the extension.

Lemma 1 Let L/K be a purely unseparable extension of discretely valued fields of a prime degree p.

Then this extension is or an immediate extension or a wild purely ramified extension. In the second case there are two possibilities.

(a) There is a representative element a ∈ R(K) and ord(a) = 1. Then the extension is a totally ramified extension.

(b) There is a representative element a ∈ U(K) and \overline{a} ∉ k^{p}. Then the extension is a tamely unramified extension.

Let us prove our lemma. Any immediate extension of K can be embedded in a maximal immediate extension, in the completion \overline{K}. As \overline{K}^{p} is an open subset in \overline{K} and K is a dense subset of \overline{K} then a purely unseparable immediate extension of a prime degree is an extension that can be presented by some representative element from the field \overline{K}^{p} ∩ K.

Let us assume that the extension is not immediate extension. Then a ∉ \overline{K}^{p} ∩ K.

The extension is finite and we have that the extension \overline{L} = L · \overline{K} is a wild extension of a prime degree p over \overline{K}.

Let us choose some field of representatives in \overline{K}. Let us identify this field with a residue field k by natural isomorphism.

We can perform an element a from the complete field \overline{K} by power series: a = a_{0} + a_{1}π + ..., a_{i} ∈ k.

As the extension is a purely ramified wild extension we have a ∈ K, a ∉ \overline{K}^{p}. So there is an index n that or a_{n} ≠ 0 and (n, p) = 1 or a_{n} ∉ k^{p}. Let us choose a minimal number n for these properties.

Field K is dense in \overline{K} so there is b ∈ K and a = b^{p} + a′_{n}π^{n} + ... = b^{p} + c, c = a′_{n}π^{n} + .... Let us choose c instead of a.

Case 1: (n, p) = 1. Let s, m ∈ Z and ns + pm = 1. Let us prove that we can choose c^{s}π^{pm} as representative element. This element has order 1.

Indeed, let x be a root of a polynomial X^{p}−c, so L = K(x). Then x^{ps} = c^{s} and element y = x^{s}π^{m} is a root of irreducible polynomial Y^{p} − c^{s}π^{pm}. We have K(y) ⊆ K(x) so K(x) = K(y).

Case 2: p|n but a_{n} ∉ k^{p}. Let n = pm then we can choose b = cπ^{−mp} = c · (π^{−m})^{p} as a representative element . This element belongs to U(R) and \overline{b} = a_{n} ∉ k^{p}.

We have proved the lemma.

Let us remember lemma 7 from [1]:

Lemma 2 Let L/K be some tamely unramified extension and K_{1}/K be a totally ramified extension. Let these extensions are contained in some general discretely valued field. Then the extension LK_{1}/K_{1} is a tamely unramified extension and the extension LK_{1}/L is a totally ramified extension.

Proposition 1 Let L/L_{0} be a purely unseparable extension of a degree p^{n} and L_{0}/K be some tamely unramified extension of discretely valued fields.

Let K′/K be a totally ramified purely unseparable extension of a degree p^{m}. Then if m ≥ n then the extension LK′/K′ is a tamely unramified extension for any continuation of valuation of K.

Let us prove the proposition. Let us assume L_{0} be a maximal tamely unramified subextension of L/K. If L_{0} = L then the extension is a tamely unramified extension. We can use lemma 2. So let us assume L ≠ L_{0}.

In the purely unseparable extension K′/K there is a subextension K_{n}/K of a degree p^{n} . By lemma 2 we can prove the proposition for K_{n}/K instead of K′/K. So let us assume n = m.

Let us present our two extensions by towers of extensions of a prime degree p: L = L_{n}/.../L_{0} = K K′ = K_{n}/.../K_{0} = K. We will prove that a composite L_{i}K_{i} is a tamely unramified extension over K_{i} by induction on i = 0, ..., n .

If i = 0 then nothing to prove. This gives us a first step of induction. Let we have proved that extension L_{i−1}K_{i−1}/K_{i−1} is a tamely unramified

extension. Extension L_{i}K_{i−1}/L_{i−1}K_{i−1} is a purely unseparable extension of a prime degree p and extension L_{i−1}K_{i−1}/K_{i−1} is a tamely unramified extension. The extension K_{i}/K_{i−1} is a totally ramified purely unseparable extension of a prime degree. By lemma 2 we can restrict ourselves to a case |K′ : K| = |L : L_{0}| = p, n = m = 1.

As L_{0}/K is a maximal tamely ramified extension then the extension L/L_{0} is a totally ramified extension. As far as a completion of an extension preserves ramification index we can take completions of a fields. Let us assume our fields are complete fields. We choose some field of representatives in K. We can identify this field with residue field by some canonical isomorphism.

We use lemma 1 and present extension K′/K as a decomposition field of a polynomial X^{p} − π for some uniformizing parameter π. Then for some ϵ ∈ U(K) the extension L/L_{0} is a decomposition field of a polynomial Y^{p}−πϵ. So LK′/L_{0}K′ is a prime extension of L_{0}K′/K′ by some root of a polynomial Z^{p} − ϵ. By lemma 2 it is a tame ramified extension.

Let us take some representation of ϵ by power series π: ϵ = α_{0} + α_{1}π + ..., α_{i} ∈ k. Let τ ∈ K′ and τ^{p} = π then τ is an uniformizing parameter of K′. So the element ϵ in the field LK′ can be represented as ϵ = α_{0}+α_{1}τ^{p}+..., α_{i} ∈ k.

If for any i we have α_{i} ∈ k^{p} then ϵ ∈ K^{′p} and LK′ = L_{0}K′. Theorem is proved.

Let us consider second case α_{i} ∉ k^{p}. The extension LK′/L_{0}K′ is a tamely unramified extension. This extension defines a purely unseparable extension of the residue field k. This extension is a primary extension by p - root of the element α_{i} with smallest index i.

We have proved the theorem.

2 Abhyankar lemma.

In the 50th S.S.Abhyankar have proved the following proposition (see [8], Exp.10, Lemme 3.6, P.279).

Lemma 3 (Abhyankar lemma) Let L/K and K′/K be finite Galois extensions of a discretely valued field K in some separable closure of K. Let these extensions are tame ramified extensions with ramification indices n and m. Then if a natural number n is a divisor of a natural number m then extension LK′/K′ is an unramified extension for any continuation of the valuation of K to LK′.

Proof see in [8].

We can generalize this statement.

Proposition 2 Let L′/L_{0} be a finite separable extension of a tamely unramified extension of discretely valued fields L_{0}/K. Then there is prime to p natural number n = n(L/L_{0}) ∈ N with the following property.

For any totally ramified and tamely ramified extension K′/K such that n|e_{r}(K′/K) the extension LK′/L_{0}K′ is a wild extension. Besides e_{r}(LK′/L_{0}K′) is a divisor of e_{r}(L/K).

Let us prove proposition. We use lemma 2 and reduce the proof of proposition to a particular case L_{0} = K. Indeed any totally ramified and tame ramified extension K′/K gives us a totally ramified and a tamely ramified extension K′L_{0}/L_{0}. It follows from multiplicative property of ramification index that e_{r}(K′/K) = e_{r}(K′L_{0}/L_{0}). From now on let us assume K = L_{0}.

As subextension of a wild extension is a wild extension too we can consider L/K as Galois extension.

The extension K′/K is a totally ramified extension so the valuation of K can be continued to K′ by a unique way. But any continuation of a valuation K to LK′ is defined by continuation of the valuation K′ to LK′. As extension LK′/K′ is Galois extension the whole set of continuations is conjugated by Galois group. These continuations have equals ramification indexes. So we have to prove proposition for only one continuation of valuation of K to LK′. From now on let us choose some continuation of valuation of K to LK′.

Let K_{V} be a ramification field of Galois extension L/K. This is a maximal tame ramified extension, this is Galois extension and (e_{r}(K_{V} /K), p) = 1.

The extension K/K_{V} is purely wild Galois p - extension, this is a solvable extension. Let n = e_{r}(K_{V} /K).

We take any totally ramified and tamely ramified separable extension K′/K of a degree n, n|e(K′/K). By Abhyankar lemma extension K_{V} K′/K′ is tamely unramified extension and extension LK′/K_{V} K′ is a wild extension by marked property of the class of wild solvable extensions (see proposition 2 in [1]).

Proof of the proposition follows from a calculation of ramification indexes. Remark. Totally ramified and tamely ramified extension of index m

can be obtained as a decomposition field of a polynomial X^{m} − τ for any uniformizing parameter τ of K ([2].1 §8,proposition 8.1). Such extensions are cyclic extensions.

3 Proof of theorem 1.

Let us prove theorem 1. Let us assume that the conditions of theorem are satisfied.

Let L^{n} be a normal closure of the extension L/L_{0}. If the statement of theorem is true for extension L^{n}/K then this statement is valid for any subextension. In particular for extension L/K this statement is true. So we will prove theorem for normal extension L/L_{0}.

Let us present a normal extension L/L_{0} as a tower L/L_{us}/L_{0}. Here L_{us}/L_{0} is a purely unseparable extension and L/L_{us} is Galois extension.

The valuation of L_{0} can be extended to valuation of purely unseparable extension L_{us}/L_{0} by a unique way. We use proposition 1 for L_{us}/K and find such totally ramified purely unseparable extension K_{1}/K that the extension L_{us}K_{1}/K_{1} is a tamely unramified extension for any continuation of valuation. As far as the extension L_{us}K_{1}/L_{0} is purely unseparable such continuation is a uniquely defined continuation.

Remark. As an example of an extension from above one can choose a decomposition field of a polynomial X^{p^{m}} − π where π is a uniformizing parameter, K_{1} = K(π^{p^{−m}}).

Now the extension LK_{1}/L_{us}K_{1} is finite Galois extension. Let us use proposition 2 for the tower LK_{1}/L_{us}K_{1}/K_{1} and find totally ramified and tamely ramified cyclic Galois extension K_{2}/K_{1} that the extension LK_{2}/L_{us}K_{2} is wild Galois extension for any continuation of the valuation of L_{0} to LK_{2}. By lemma 2 the extension L_{us}K_{2}/K_{2} is a tamely unramified extension.

Remark. As an example of the extension above we can choose a primary extension by a root of any uniformizing parameter of K_{1}, K_{2} = K_{1}(τ^{1/n}).

The extension K_{2}/K is a totally ramified extension and by lemma 2 the extension K_{2}L_{0}/L_{0} is a totally ramified extension too and there are trivial extensions of residue fields. So we can use the proof of theorem 1 from [1] for tower LK_{2}/L_{us}K_{2}/K_{2} and find a totally ramified wild and solvable extension K′/K_{2} that the extension LK′/K′ is a tamely unramified extension for any continuation of valuation K_{2}L_{us} to K_{2}L. Because the valuation of L_{0} can be extended to K_{2}L_{us} by a unique way we have that the extension LK′/K′ is a tamely unramified extension for any continuation of the valuation L_{0} to LK′. This proves theorem.

Corollary 1 follows from statement of theorem 1. We have L_{0} = K and we use remarks from §4 in [1]. From proposition 3 of this paper follows the inclusion l^{ea} ⊆ k.

4 Proof of theorem 2.

Let us prove theorem 2. At first, we prove it for complete discrete valued fields. At second, we prove it for arbitrary fields.

Let L/K be an extension of complete discretely valued fields and l/k be an extension of residue fields. By Cohen theorem [5] we choose in L some field of representatives l′. Let π be a uniformizing parameter of K.

We denote by k′_{1} = l′ ∩ K a proper subfield of the valuation ring of K. We define in K a complete subfield generated by completion of the field of fractions of a prime extension k′_{1}(π), F = k′_{1}((π)). This field is a complete field, it contains uniformizing parameter π and it contains a field of representatives k′_{1}. So extension K/F is a tamely unramified extension.

We note that the condition on the extension of the residue fields is satisfied already in extension L/F. Moreover the condition is satisfied already for fields of representatives. This follows from two lemmata.

Lemma 4 Let l/k be an extension of fields. Then the inclusion l^{ea} ⊆ k is equivalent to the conditions

1) l^{e} = k^{e},

2) k^{ea} ⊆ k.

Indeed, from the inclusion k ⊆ l it follows k^{e} ⊆ l^{e}. Because l^{ea} ⊆ k then l^{e} ⊆ k. So for any natural n we have l^{e} = (l^{e})^{p^{n}} =⊆ k^{p^{n}} and l^{e} ⊆ k^{e}. It follows l^{e} = k^{e}, l^{ea} = k^{ea} and k^{ea} ⊆ k.

Conversely, these two conditions give us the inclusions l^{ea} = k^{ea} ⊆ k. Lemma is proved.

Lemma 5 Let L be a complete discretely valued field and l be a field of representatives. Then L^{e} = l^{e} and l^{ea} ∩ L ⊆ l.

Indeed, first equality follows from the structure of complete discretely valued fields. Second equality follows from maximality of a field of representatives in a valuation ring of L.

Field l^{e} is an perfect field and extension l^{ea}/l^{e} is separable. So the extension (l^{ea} ∩ L) · l/l is a separable extension in L. This extension gives us a greater field of representatives. That is impossible.

From these remarks we see that under conditions above we have l^{′e} = L^{e} ⊇ K^{e} = k^{′e}. From the condition l^{ea} ⊆ k of theorem 2 we have equality l^{e} = k^{e}. In particular we have l^{′e} = k^{′e}. So l^{′ea} = k^{′ea} ⊆ k′ ⊆ K ⊆ L. It follows l^{′ea} = l^{′ea} ∩ L ⊆ l′. So l^{′ea} ⊆ K ∩ l′ = k′_{1}. We see that for extension L/F all the conditions of theorem 2 are satisfied.

Let us note that we have to prove theorem 2 for extension L/F only. Indeed, it follows from lemma 2 that any totally ramified and solvable extension defines a totally ramified and solvable extension F′K/K. If the extension LF′/F′ is a tamely unramified extension then the subextension LF′ = LKF′/KF′ is a tamely unramified extension too. Then for the extension K′ = KF′ all the conditions of the theorem are satisfied. From now on we suppose K = F = k′_{1}((π)).

We denote a completion of a field of fractions of a prime extension l′(π) by L_{0}. By structure theorem we have L_{0} = l′((π)) and a residue field of L_{0} is equals to l.

Because the extension of the residue fields in the extension L/L_{0} is trivial so e_{r}(L/L_{0}) = |L : L_{0}| and extension L/L_{0} is a totally ramified and a finite extension. We use theorem 1 for tower L/L_{0}/K and find an extension required K′/K. This proves theorem 2 for complete fields. Now let us consider arbitrary fields.

Let L/K be some extension of discretely valued fields. We take a completion of L and define extension of complete fields \overline{L}/\overline{K}.

We use the above statement for this extension and find a totally ramified solvable extension \overline{K}′/\overline{K} that the extension \overline{L}\overline{K}′/\overline{K}′ is a tamely unramified extension.

We use theorem 1 and find totally ramified solvable extension K′/K in the field \overline{K}′ that \overline{K}′ = K′ · \overline{K}. Then extension LK′/K′ is contained in the tower of tamely unramified extensions \overline{L}K′/\overline{K}′/K′, it is a tamely unramified extension. The statements of theorem 2 are satisfied for this extension and for continuous extension of valuation v of the field L to the field LK′.

Indeed, from the proof of theorem 1 it follows that the extension \overline{K}′/\overline{K} can be presented as a tower \overline{K}′ = \overline{K_{3}}/\overline{K_{2}}/\overline{K_{1}}/\overline{K}. Where the extension \overline{K_{1}}/\overline{K} is a decomposition field of a polynomial X^{p^{n}} −π for some uniformizing parameter π ∈ \overline{K}. We can choose uniformizing parameter from the field K. Then we have a totally ramified solvable extension K_{1}/K and K_{1}\overline{K} = \overline{K_{1}}. The extension \overline{K_{1}}/K_{1} is an unramified extension and the field K_{1} is dense in \overline{K_{1}}.

The extension \overline{K_{2}}/\overline{K_{1}} is a decomposition field of polynomial X^{n} − τ for some uniformizing parameter τ ∈ \overline{K_{1}}. We can choose τ ∈ K_{1} and define such a field K_{2} that K_{2}\overline{K} = \overline{K_{2}}. The extension \overline{K_{2}}/K_{2} is an unramified and immediate extension.

At last, the extension \overline{K_{3}}/\overline{K_{2}} is a separable totally ramified extension. By proposition 1 from [1] there is a totally ramified solvable extension K_{3}/K_{2} and K_{3}\overline{K} = \overline{K_{3}} = \overline{K}′.

So for a given extension of the valuation we can take K′ = K_{3}. For any continuation of valuation the statement follows from the remark.

Lemma 6 Let L/K be some extension of valued fields and K′/K be some finite extension.

Then a reduced ramification index of the extension e_{r}(LK′/K′) does not depend on the continuation of valuation of L to LK′.

Let \overline{L} be a completion of L. Then the completion of K in \overline{L} gives us a completion \overline{K} of K. We have the extension of complete fields \overline{L}/\overline{K}.

We choose some extension of valuation of L to LK′. Completion of a field preserves a group of valuation so it preserves reduced ramification index. So e_{r}(LK′/K′) = e_{r}(\overline{LK′}/\overline{K′}). But the second extension is a finite extension of a complete field \overline{L}. Any complete valuation can be extended to any finite extension by a unique way. So completions \overline{LK′} (and \overline{K′}) does not depend on extension of valuation. This proves our lemma and theorem.

The statement of corollary 2 follows from the statement of theorem. We have use §4 from [1].

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