# Elimination of Ramiﬁcation since Abhyankar and Epp

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## Elimination of Ramification since Abhyankar and Epp

Franz-Viktor Kuhlmann

Krakow, May 2015

Franz-Viktor Kuhlmann Elimination of Ramification

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## Smoothness and the Implicit Function Theorem

A point on an algebraic variety issmoothif and only if the Implicit Function Theorem holds at this point.

Given a point on an algebraic variety which is not smooth, we would like to find another birationally equivalent variety on which the corresponding point is smooth.

This task is a local version of Resolution of Singularities and is calledlocal uniformization.

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## Smoothness and the Implicit Function Theorem

A point on an algebraic variety issmoothif and only if the Implicit Function Theorem holds at this point.

Given a point on an algebraic variety which is not smooth,

we would like to find another birationally equivalent variety on which the corresponding point is smooth.

This task is a local version of Resolution of Singularities and is calledlocal uniformization.

Franz-Viktor Kuhlmann Elimination of Ramification

(4)

## Smoothness and the Implicit Function Theorem

A point on an algebraic variety issmoothif and only if the Implicit Function Theorem holds at this point.

Given a point on an algebraic variety which is not smooth, we would like to find another birationally equivalent variety on which the corresponding point is smooth.

This task is a local version of Resolution of Singularities and is calledlocal uniformization.

(5)

## Smoothness and the Implicit Function Theorem

A point on an algebraic variety issmoothif and only if the Implicit Function Theorem holds at this point.

Given a point on an algebraic variety which is not smooth, we would like to find another birationally equivalent variety on which the corresponding point is smooth.

This task is a local version of Resolution of Singularities

and is calledlocal uniformization.

Franz-Viktor Kuhlmann Elimination of Ramification

(6)

## Smoothness and the Implicit Function Theorem

A point on an algebraic variety issmoothif and only if the Implicit Function Theorem holds at this point.

Given a point on an algebraic variety which is not smooth, we would like to find another birationally equivalent variety on which the corresponding point is smooth.

This task is a local version of Resolution of Singularities and is calledlocal uniformization.

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## Two questions

• Implicit Function Theorem with respect towhich topology?

• What constitutes the correspondencebetween the given point and the corresponding point on the new variety?

Franz-Viktor Kuhlmann Elimination of Ramification

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## Two questions

• Implicit Function Theorem with respect towhich topology?

• What constitutes the correspondencebetween the given point and the corresponding point on the new variety?

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## Places

A point on a variety can be understood as a homomorphism of the coordinate ring of the variety.

This can be extended to a placeof its quotient field, the function field of the variety, where we allow the values of a place to include∞.

The place can single out the corresponding point on the new variety, because a birationally equivalent variety has the same function field.

Franz-Viktor Kuhlmann Elimination of Ramification

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## Places

A point on a variety can be understood as a homomorphism of the coordinate ring of the variety. This can be extended to a placeof its quotient field, the function field of the variety,

where we allow the values of a place to include∞.

The place can single out the corresponding point on the new variety, because a birationally equivalent variety has the same function field.

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## Places

A point on a variety can be understood as a homomorphism of the coordinate ring of the variety. This can be extended to a placeof its quotient field, the function field of the variety, where we allow the values of a place to include∞.

The place can single out the corresponding point on the new variety, because a birationally equivalent variety has the same function field.

Franz-Viktor Kuhlmann Elimination of Ramification

(12)

## Places

A point on a variety can be understood as a homomorphism of the coordinate ring of the variety. This can be extended to a placeof its quotient field, the function field of the variety, where we allow the values of a place to include∞.

The place can single out the corresponding point on the new variety,

because a birationally equivalent variety has the same function field.

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## Places

A point on a variety can be understood as a homomorphism of the coordinate ring of the variety. This can be extended to a placeof its quotient field, the function field of the variety, where we allow the values of a place to include∞.

The place can single out the corresponding point on the new variety, because a birationally equivalent variety has the same function field.

Franz-Viktor Kuhlmann Elimination of Ramification

(14)

## Valuations

Every place P gives rise to avaluationv=vP, and vice versa.

The valuation of a field L, in turn, induces a topology on L. Further, it associates with L an ordered abelian group vL, the value group, and a field Lv, theresidue field.

If v=vP, then Lv∪ {}is the image of L under P.

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## Valuations

Every place P gives rise to avaluationv=vP, and vice versa.

The valuation of a field L, in turn, induces a topology on L.

Further, it associates with L an ordered abelian group vL, the value group, and a field Lv, theresidue field.

If v=vP, then Lv∪ {}is the image of L under P.

Franz-Viktor Kuhlmann Elimination of Ramification

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## Valuations

Every place P gives rise to avaluationv=vP, and vice versa.

The valuation of a field L, in turn, induces a topology on L.

Further, it associates with L an ordered abelian group vL, the value group,

and a field Lv, theresidue field. If v=vP, then Lv∪ {}is the image of L under P.

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## Valuations

Every place P gives rise to avaluationv=vP, and vice versa.

The valuation of a field L, in turn, induces a topology on L.

Further, it associates with L an ordered abelian group vL, the value group, and a field Lv, theresidue field.

If v=vP, then Lv∪ {}is the image of L under P.

Franz-Viktor Kuhlmann Elimination of Ramification

(18)

## Valuations

Every place P gives rise to avaluationv=vP, and vice versa.

The valuation of a field L, in turn, induces a topology on L.

Further, it associates with L an ordered abelian group vL, the value group, and a field Lv, theresidue field.

If v=vP, then Lv∪ {}is the image of L under P.

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## Valued function fields

We take a variety defined over an arbitrary field K.

We denote by F the function field of our variety.

We choose a point on the variety and some place P associated with it (in general, there are many).

We consider F together with the valuation v=vP.

Further, we choose a transcendence basis T of F|K from the generators of the coordinate ring; then F is a finite extension of the rational function field L=K(T).

Franz-Viktor Kuhlmann Elimination of Ramification

(20)

## Valued function fields

We take a variety defined over an arbitrary field K.

We denote by F the function field of our variety.

We choose a point on the variety and some place P associated with it (in general, there are many).

We consider F together with the valuation v=vP.

Further, we choose a transcendence basis T of F|K from the generators of the coordinate ring; then F is a finite extension of the rational function field L=K(T).

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## Valued function fields

We take a variety defined over an arbitrary field K.

We denote by F the function field of our variety.

We choose a point on the variety and some place P associated with it

(in general, there are many).

We consider F together with the valuation v=vP.

Further, we choose a transcendence basis T of F|K from the generators of the coordinate ring; then F is a finite extension of the rational function field L=K(T).

Franz-Viktor Kuhlmann Elimination of Ramification

(22)

## Valued function fields

We take a variety defined over an arbitrary field K.

We denote by F the function field of our variety.

We choose a point on the variety and some place P associated with it (in general, there are many).

We consider F together with the valuation v=vP.

Further, we choose a transcendence basis T of F|K from the generators of the coordinate ring; then F is a finite extension of the rational function field L=K(T).

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## Valued function fields

We take a variety defined over an arbitrary field K.

We denote by F the function field of our variety.

We choose a point on the variety and some place P associated with it (in general, there are many).

We consider F together with the valuation v=vP.

Further, we choose a transcendence basis T of F|K from the generators of the coordinate ring; then F is a finite extension of the rational function field L=K(T).

Franz-Viktor Kuhlmann Elimination of Ramification

(24)

## Valued function fields

We take a variety defined over an arbitrary field K.

We denote by F the function field of our variety.

We choose a point on the variety and some place P associated with it (in general, there are many).

We consider F together with the valuation v=vP.

Further, we choose a transcendence basis T of F|K from the generators of the coordinate ring;

then F is a finite extension of the rational function field L=K(T).

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## Valued function fields

We take a variety defined over an arbitrary field K.

We denote by F the function field of our variety.

We choose a point on the variety and some place P associated with it (in general, there are many).

We consider F together with the valuation v=vP.

Further, we choose a transcendence basis T of F|K from the generators of the coordinate ring; then F is a finite extension of the rational function field L=K(T).

Franz-Viktor Kuhlmann Elimination of Ramification

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## Implicit Function Theorem and ramification

Ramificationin the valued field extension(F|K(T), vP) expresses the failure of the Implicit Function Theorem at the point that is determined by P.

We therefore wish toeliminate ramificationby finding a different transcendence basis T0of F|K such that the new extension(F|K(T0), vP)has no ramification. This means that we choose a newmodelfor the function field F|K and thus a new variety. But this variety is birationally equivalent to the one we started with.

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## Implicit Function Theorem and ramification

Ramificationin the valued field extension(F|K(T), vP) expresses the failure of the Implicit Function Theorem at the point that is determined by P. We therefore wish toeliminate ramification

by finding a different transcendence basis T0of F|K such that the new extension(F|K(T0), vP)has no ramification. This means that we choose a newmodelfor the function field F|K and thus a new variety. But this variety is birationally equivalent to the one we started with.

Franz-Viktor Kuhlmann Elimination of Ramification

(28)

## Implicit Function Theorem and ramification

Ramificationin the valued field extension(F|K(T), vP) expresses the failure of the Implicit Function Theorem at the point that is determined by P. We therefore wish toeliminate ramificationby finding a different transcendence basis T0of F|K such that the new extension(F|K(T0), vP)has no ramification.

This means that we choose a newmodelfor the function field F|K and thus a new variety. But this variety is birationally equivalent to the one we started with.

(29)

## Implicit Function Theorem and ramification

Ramificationin the valued field extension(F|K(T), vP) expresses the failure of the Implicit Function Theorem at the point that is determined by P. We therefore wish toeliminate ramificationby finding a different transcendence basis T0of F|K such that the new extension(F|K(T0), vP)has no ramification.

This means that we choose a newmodelfor the function field F|K

and thus a new variety. But this variety is birationally equivalent to the one we started with.

Franz-Viktor Kuhlmann Elimination of Ramification

(30)

## Implicit Function Theorem and ramification

Ramificationin the valued field extension(F|K(T), vP) expresses the failure of the Implicit Function Theorem at the point that is determined by P. We therefore wish toeliminate ramificationby finding a different transcendence basis T0of F|K such that the new extension(F|K(T0), vP)has no ramification.

This means that we choose a newmodelfor the function field F|K and thus a new variety.

But this variety is birationally equivalent to the one we started with.

(31)

## Implicit Function Theorem and ramification

Ramificationin the valued field extension(F|K(T), vP) expresses the failure of the Implicit Function Theorem at the point that is determined by P. We therefore wish toeliminate ramificationby finding a different transcendence basis T0of F|K such that the new extension(F|K(T0), vP)has no ramification.

This means that we choose a newmodelfor the function field F|K and thus a new variety. But this variety is birationally equivalent to the one we started with.

Franz-Viktor Kuhlmann Elimination of Ramification

(32)

## What is ramification?

Naively speaking, an extension(F|L, v)isramifiedif vL6=vF.

Unfortunately, if char Lv>0, then there is more to ramification than just the change of the value group.

To see in which algebraic extensions of valued fields

ramification occurs, and thus the Implicit Function Theorem fails, we take any valued field(L, v)and extend the valuation to the algebraic closure Lacof L.

We set p=char Lv if it is positive, and p=1 otherwise.

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## What is ramification?

Naively speaking, an extension(F|L, v)isramifiedif vL6=vF.

Unfortunately, if char Lv>0, then there is more to ramification than just the change of the value group.

To see in which algebraic extensions of valued fields

ramification occurs, and thus the Implicit Function Theorem fails, we take any valued field(L, v)and extend the valuation to the algebraic closure Lacof L.

We set p=char Lv if it is positive, and p=1 otherwise.

Franz-Viktor Kuhlmann Elimination of Ramification

(34)

## What is ramification?

Naively speaking, an extension(F|L, v)isramifiedif vL6=vF.

Unfortunately, if char Lv>0, then there is more to ramification than just the change of the value group.

To see in which algebraic extensions of valued fields ramification occurs,

and thus the Implicit Function Theorem fails, we take any valued field(L, v)and extend the valuation to the algebraic closure Lacof L.

We set p=char Lv if it is positive, and p=1 otherwise.

(35)

## What is ramification?

Naively speaking, an extension(F|L, v)isramifiedif vL6=vF.

Unfortunately, if char Lv>0, then there is more to ramification than just the change of the value group.

To see in which algebraic extensions of valued fields

ramification occurs, and thus the Implicit Function Theorem fails,

we take any valued field(L, v)and extend the valuation to the algebraic closure Lacof L.

We set p=char Lv if it is positive, and p=1 otherwise.

Franz-Viktor Kuhlmann Elimination of Ramification

(36)

## What is ramification?

Naively speaking, an extension(F|L, v)isramifiedif vL6=vF.

Unfortunately, if char Lv>0, then there is more to ramification than just the change of the value group.

To see in which algebraic extensions of valued fields

ramification occurs, and thus the Implicit Function Theorem fails, we take any valued field(L, v)and extend the valuation to the algebraic closure Lacof L.

We set p=char Lv if it is positive, and p=1 otherwise.

(37)

## What is ramification?

Naively speaking, an extension(F|L, v)isramifiedif vL6=vF.

Unfortunately, if char Lv>0, then there is more to ramification than just the change of the value group.

To see in which algebraic extensions of valued fields

ramification occurs, and thus the Implicit Function Theorem fails, we take any valued field(L, v)and extend the valuation to the algebraic closure Lacof L.

We set p=char Lv if it is positive, and p=1 otherwise.

Franz-Viktor Kuhlmann Elimination of Ramification

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## Absolute ramification theory

Gd Gi Gr 1 Galois group

Lh Li Lr Lsep

Lac field

vL vL

1 p0vL

fvL fvL value group

Lv (Lv)sep (Lv)sep (Lv)ac (Lv)ac

residue field

Gal Lv Char

Galois, defectless abelian Galois p0-extension, defectless Galois p-extension purely inseparable

division by p

division prime to p

Galois purely inseparable

absolute decomposition field (= henselization)

absolute inertia

field absolute ramification

field separable-

algebraic closure

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## Absolute ramification theory

where

Gal L=Aut Lac|L is the absolute Galois group of L,

fvL=vLac is the divisible hull of vL,

1

p0vL denotes the p0-divisible hull of vL, Char denotes the character group

Hom

vLr/vLi,(Liv)×.

Franz-Viktor Kuhlmann Elimination of Ramification

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## Absolute ramification theory

where

Gal L=Aut Lac|L is the absolute Galois group of L,

fvL=vLac is the divisible hull of vL,

1

p0vL denotes the p0-divisible hull of vL, Char denotes the character group

Hom

vLr/vLi,(Liv)×.

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## Absolute ramification theory

where

Gal L=Aut Lac|L is the absolute Galois group of L,

fvL=vLac is the divisible hull of vL,

1

p0vL denotes the p0-divisible hull of vL,

Char denotes the character group Hom

vLr/vLi,(Liv)×.

Franz-Viktor Kuhlmann Elimination of Ramification

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## Absolute ramification theory

where

Gal L=Aut Lac|L is the absolute Galois group of L,

fvL=vLac is the divisible hull of vL,

1

p0vL denotes the p0-divisible hull of vL, Char denotes the character group

Hom

vLr/vLi,(Liv)×.

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## Elimination of ramification

For a valued function field(F|K, v), elimination of ramification means to find a transcendence basis T

such that F ⊂ K(T)i.

Franz-Viktor Kuhlmann Elimination of Ramification

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## Elimination of ramification

For a valued function field(F|K, v), elimination of ramification means to find a transcendence basis T such that

F ⊂ K(T)i.

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## Tame and wild ramification

From Li to Lris the area oftame ramification,

and from Lrto Lac is the area ofwild ramification.

Wild ramification does not necessarily mean that the value groups change. It can also mean that we have a purely

inseparable extension of the residue fields. It also refers to the case where neither value group nor residue fields change, but there is a unique extension of the valuation; in this case we have nontrivialdefect.

Note that Lr=Lacif p=1, that is, if char Lv=0. In this case, all ramification is tame.

Franz-Viktor Kuhlmann Elimination of Ramification

(46)

## Tame and wild ramification

From Li to Lris the area oftame ramification, and from Lrto Lac is the area ofwild ramification.

Wild ramification does not necessarily mean that the value groups change. It can also mean that we have a purely

inseparable extension of the residue fields. It also refers to the case where neither value group nor residue fields change, but there is a unique extension of the valuation; in this case we have nontrivialdefect.

Note that Lr=Lacif p=1, that is, if char Lv=0. In this case, all ramification is tame.

(47)

## Tame and wild ramification

From Li to Lris the area oftame ramification, and from Lrto Lac is the area ofwild ramification.

Wild ramification does not necessarily mean that the value groups change. It can also mean that we have a purely inseparable extension of the residue fields.

It also refers to the case where neither value group nor residue fields change, but there is a unique extension of the valuation; in this case we have nontrivialdefect.

Note that Lr=Lacif p=1, that is, if char Lv=0. In this case, all ramification is tame.

Franz-Viktor Kuhlmann Elimination of Ramification

(48)

## Tame and wild ramification

From Li to Lris the area oftame ramification, and from Lrto Lac is the area ofwild ramification.

Wild ramification does not necessarily mean that the value groups change. It can also mean that we have a purely

inseparable extension of the residue fields. It also refers to the case where neither value group nor residue fields change, but there is a unique extension of the valuation;

in this case we have nontrivialdefect.

Note that Lr=Lacif p=1, that is, if char Lv=0. In this case, all ramification is tame.

(49)

## Tame and wild ramification

From Li to Lris the area oftame ramification, and from Lrto Lac is the area ofwild ramification.

Wild ramification does not necessarily mean that the value groups change. It can also mean that we have a purely

inseparable extension of the residue fields. It also refers to the case where neither value group nor residue fields change, but there is a unique extension of the valuation; in this case we have nontrivialdefect.

Note that Lr=Lacif p=1, that is, if char Lv=0. In this case, all ramification is tame.

Franz-Viktor Kuhlmann Elimination of Ramification

(50)

## Tame and wild ramification

From Li to Lris the area oftame ramification, and from Lrto Lac is the area ofwild ramification.

Wild ramification does not necessarily mean that the value groups change. It can also mean that we have a purely

inseparable extension of the residue fields. It also refers to the case where neither value group nor residue fields change, but there is a unique extension of the valuation; in this case we have nontrivialdefect.

Note that Lr=Lacif p=1, that is, if char Lv=0.

In this case, all ramification is tame.

(51)

## Tame and wild ramification

From Li to Lris the area oftame ramification, and from Lrto Lac is the area ofwild ramification.

Wild ramification does not necessarily mean that the value groups change. It can also mean that we have a purely

inseparable extension of the residue fields. It also refers to the case where neither value group nor residue fields change, but there is a unique extension of the valuation; in this case we have nontrivialdefect.

Note that Lr=Lacif p=1, that is, if char Lv=0. In this case, all ramification is tame.

Franz-Viktor Kuhlmann Elimination of Ramification

(52)

## Zariski’s local uniformization

In 1940, O. Zariski proved that every place on every function field over a ground field of characteristic 0 admits local uniformization.

He used this result to prove resolution of singularities for surfaces in characteristic 0. For this, he had

“only” to eliminate tame ramification.

In positive characteristic, one also has to eliminate wild ramification (explicitly or implicitly). This explains why the case of positive characteristic is so much harder. Indeed, neither local uniformization nor resolution of singularities has been proved in positive characteristic for dimensions>3.

(53)

## Zariski’s local uniformization

In 1940, O. Zariski proved that every place on every function field over a ground field of characteristic 0 admits local uniformization. He used this result to prove resolution of singularities for surfaces in characteristic 0.

For this, he had

“only” to eliminate tame ramification.

In positive characteristic, one also has to eliminate wild ramification (explicitly or implicitly). This explains why the case of positive characteristic is so much harder. Indeed, neither local uniformization nor resolution of singularities has been proved in positive characteristic for dimensions>3.

Franz-Viktor Kuhlmann Elimination of Ramification

(54)

## Zariski’s local uniformization

In 1940, O. Zariski proved that every place on every function field over a ground field of characteristic 0 admits local uniformization. He used this result to prove resolution of singularities for surfaces in characteristic 0. For this, he had

“only” to eliminate tame ramification.

In positive characteristic, one also has to eliminate wild ramification (explicitly or implicitly). This explains why the case of positive characteristic is so much harder. Indeed, neither local uniformization nor resolution of singularities has been proved in positive characteristic for dimensions>3.

(55)

## Zariski’s local uniformization

In 1940, O. Zariski proved that every place on every function field over a ground field of characteristic 0 admits local uniformization. He used this result to prove resolution of singularities for surfaces in characteristic 0. For this, he had

“only” to eliminate tame ramification.

In positive characteristic, one also has to eliminate wild ramification (explicitly or implicitly).

This explains why the case of positive characteristic is so much harder. Indeed, neither local uniformization nor resolution of singularities has been proved in positive characteristic for dimensions>3.

Franz-Viktor Kuhlmann Elimination of Ramification

(56)

## Zariski’s local uniformization

In 1940, O. Zariski proved that every place on every function field over a ground field of characteristic 0 admits local uniformization. He used this result to prove resolution of singularities for surfaces in characteristic 0. For this, he had

“only” to eliminate tame ramification.

In positive characteristic, one also has to eliminate wild ramification (explicitly or implicitly). This explains why the case of positive characteristic is so much harder.

Indeed, neither local uniformization nor resolution of singularities has been proved in positive characteristic for dimensions>3.

(57)

## Zariski’s local uniformization

In 1940, O. Zariski proved that every place on every function field over a ground field of characteristic 0 admits local uniformization. He used this result to prove resolution of singularities for surfaces in characteristic 0. For this, he had

“only” to eliminate tame ramification.

In positive characteristic, one also has to eliminate wild ramification (explicitly or implicitly). This explains why the case of positive characteristic is so much harder. Indeed, neither local uniformization nor resolution of singularities has been proved in positive characteristic for dimensions>3.

Franz-Viktor Kuhlmann Elimination of Ramification

(58)

## Partial results

Resolution of singularities has been proved up to dimension 3, for all but finitely many characteristics by Abhyankar. Recently, Cossart and Piltant have removed the restrictions on the

characteristics.

Local uniformization has only been proved for all dimensions for certain well-behaved places, theAbhyankar places[Knaf-K 2006]. If one allows an extension of the function field

(alteration), then one has de Jongh’s Resolution by Alteration, a more precise local version [Knaf-K 2009], and Temkin’s

Inseparable Local Uniformization.

While all of these results explicitly or implicitly are instances of elimination of ramification, it has not been achieved in general.

(59)

## Partial results

Resolution of singularities has been proved up to dimension 3, for all but finitely many characteristics by Abhyankar. Recently, Cossart and Piltant have removed the restrictions on the

characteristics.

Local uniformization has only been proved for all dimensions for certain well-behaved places,

theAbhyankar places[Knaf-K 2006]. If one allows an extension of the function field

(alteration), then one has de Jongh’s Resolution by Alteration, a more precise local version [Knaf-K 2009], and Temkin’s

Inseparable Local Uniformization.

While all of these results explicitly or implicitly are instances of elimination of ramification, it has not been achieved in general.

Franz-Viktor Kuhlmann Elimination of Ramification

(60)

## Partial results

Resolution of singularities has been proved up to dimension 3, for all but finitely many characteristics by Abhyankar. Recently, Cossart and Piltant have removed the restrictions on the

characteristics.

Local uniformization has only been proved for all dimensions for certain well-behaved places, theAbhyankar places[Knaf-K 2006].

If one allows an extension of the function field

(alteration), then one has de Jongh’s Resolution by Alteration, a more precise local version [Knaf-K 2009], and Temkin’s

Inseparable Local Uniformization.

While all of these results explicitly or implicitly are instances of elimination of ramification, it has not been achieved in general.

(61)

## Partial results

Resolution of singularities has been proved up to dimension 3, for all but finitely many characteristics by Abhyankar. Recently, Cossart and Piltant have removed the restrictions on the

characteristics.

Local uniformization has only been proved for all dimensions for certain well-behaved places, theAbhyankar places[Knaf-K 2006]. If one allows an extension of the function field

(alteration), then one has de Jongh’s Resolution by Alteration,

a more precise local version [Knaf-K 2009], and Temkin’s

Inseparable Local Uniformization.

While all of these results explicitly or implicitly are instances of elimination of ramification, it has not been achieved in general.

Franz-Viktor Kuhlmann Elimination of Ramification

(62)

## Partial results

Resolution of singularities has been proved up to dimension 3, for all but finitely many characteristics by Abhyankar. Recently, Cossart and Piltant have removed the restrictions on the

characteristics.

Local uniformization has only been proved for all dimensions for certain well-behaved places, theAbhyankar places[Knaf-K 2006]. If one allows an extension of the function field

(alteration), then one has de Jongh’s Resolution by Alteration, a more precise local version [Knaf-K 2009],

and Temkin’s Inseparable Local Uniformization.

While all of these results explicitly or implicitly are instances of elimination of ramification, it has not been achieved in general.

(63)

## Partial results

Resolution of singularities has been proved up to dimension 3, for all but finitely many characteristics by Abhyankar. Recently, Cossart and Piltant have removed the restrictions on the

characteristics.

Local uniformization has only been proved for all dimensions for certain well-behaved places, theAbhyankar places[Knaf-K 2006]. If one allows an extension of the function field

(alteration), then one has de Jongh’s Resolution by Alteration, a more precise local version [Knaf-K 2009], and Temkin’s

Inseparable Local Uniformization.

While all of these results explicitly or implicitly are instances of elimination of ramification, it has not been achieved in general.

Franz-Viktor Kuhlmann Elimination of Ramification

(64)

## Partial results

Resolution of singularities has been proved up to dimension 3, for all but finitely many characteristics by Abhyankar. Recently, Cossart and Piltant have removed the restrictions on the

characteristics.

Local uniformization has only been proved for all dimensions for certain well-behaved places, theAbhyankar places[Knaf-K 2006]. If one allows an extension of the function field

(alteration), then one has de Jongh’s Resolution by Alteration, a more precise local version [Knaf-K 2009], and Temkin’s

Inseparable Local Uniformization.

While all of these results explicitly or implicitly are instances of elimination of ramification, it has not been achieved in general.

(65)

## Abhyankar’s Lemma

A basic result for the elimination of tame ramification is Abhyankar’s Lemma.

Here is a version a little more general than the original:

Theorem

Assume that(L, v)and(F, v)are two extensions of(L0, v), both contained in a common extension(M, v). If vL⊆vF and(L|L0, v) has only tame ramification (i.e., L⊆Lr0), then v(L.F) =vF.

Franz-Viktor Kuhlmann Elimination of Ramification

(66)

## Abhyankar’s Lemma

A basic result for the elimination of tame ramification is Abhyankar’s Lemma. Here is a version a little more general than the original:

Theorem

Assume that(L, v)and(F, v)are two extensions of(L0, v), both contained in a common extension(M, v). If vL⊆vF and(L|L0, v) has only tame ramification (i.e., L⊆Lr0), then v(L.F) =vF.

(67)

## Abhyankar’s Lemma

A basic result for the elimination of tame ramification is Abhyankar’s Lemma. Here is a version a little more general than the original:

Theorem

Assume that(L, v)and(F, v)are two extensions of(L0, v),

both contained in a common extension(M, v). If vL⊆vF and(L|L0, v) has only tame ramification (i.e., L⊆Lr0), then v(L.F) =vF.

Franz-Viktor Kuhlmann Elimination of Ramification

(68)

## Abhyankar’s Lemma

A basic result for the elimination of tame ramification is Abhyankar’s Lemma. Here is a version a little more general than the original:

Theorem

Assume that(L, v)and(F, v)are two extensions of(L0, v), both contained in a common extension(M, v).

If vL⊆vF and(L|L0, v) has only tame ramification (i.e., L⊆Lr0), then v(L.F) =vF.

(69)

## Abhyankar’s Lemma

A basic result for the elimination of tame ramification is Abhyankar’s Lemma. Here is a version a little more general than the original:

Theorem

Assume that(L, v)and(F, v)are two extensions of(L0, v), both contained in a common extension(M, v). If vL⊆vF and(L|L0, v) has only tame ramification (i.e., L⊆Lr0),

then v(L.F) =vF.

Franz-Viktor Kuhlmann Elimination of Ramification

(70)

## Abhyankar’s Lemma

A basic result for the elimination of tame ramification is Abhyankar’s Lemma. Here is a version a little more general than the original:

Theorem

Assume that(L, v)and(F, v)are two extensions of(L0, v), both contained in a common extension(M, v). If vL⊆vF and(L|L0, v) has only tame ramification (i.e., L⊆Lr0), then v(L.F) =vF.

(71)

## Epp’s Theorem

In 1973, Helmut Epp published a paper entitled Elimination of wild ramification in Inventiones 19.

His theorems deal with elimination of wild ramification in extensions of discrete valuation rings (DVRs). One version reads as follows. Take complete DVRs R⊂S0⊂S with S|S0finite. Denote by K, L0, L the quotient fields of R, S0, S, respectively.

Theorem

Assume that vL0 =vK. If char Kv>0, then assume in addition that the largest perfect subfield of Lv is Kv. Then there is a finite extension R0of R such that S.R0isweakly unramifiedover R0, i.e., their

quotient fields L0and K0 have the same value group.

Franz-Viktor Kuhlmann Elimination of Ramification

(72)

## Epp’s Theorem

In 1973, Helmut Epp published a paper entitled Elimination of wild ramification in Inventiones 19. His theorems deal with elimination of wild ramification in extensions of discrete valuation rings (DVRs).

One version reads as follows. Take complete DVRs R⊂S0⊂S with S|S0finite. Denote by K, L0, L the quotient fields of R, S0, S, respectively.

Theorem

Assume that vL0 =vK. If char Kv>0, then assume in addition that the largest perfect subfield of Lv is Kv. Then there is a finite extension R0of R such that S.R0isweakly unramifiedover R0, i.e., their

quotient fields L0and K0 have the same value group.

(73)

## Epp’s Theorem

In 1973, Helmut Epp published a paper entitled Elimination of wild ramification in Inventiones 19. His theorems deal with elimination of wild ramification in extensions of discrete valuation rings (DVRs). One version reads as follows.

Take complete DVRs R⊂S0⊂S with S|S0finite. Denote by K, L0, L the quotient fields of R, S0, S, respectively.

Theorem

Assume that vL0 =vK. If char Kv>0, then assume in addition that the largest perfect subfield of Lv is Kv. Then there is a finite extension R0of R such that S.R0isweakly unramifiedover R0, i.e., their

quotient fields L0and K0 have the same value group.

Franz-Viktor Kuhlmann Elimination of Ramification

(74)

## Epp’s Theorem

In 1973, Helmut Epp published a paper entitled Elimination of wild ramification in Inventiones 19. His theorems deal with elimination of wild ramification in extensions of discrete valuation rings (DVRs). One version reads as follows.

Take complete DVRs R⊂S0⊂S with S|S0finite. Denote by K, L0, L the quotient fields of R, S0, S, respectively.

Theorem

Assume that vL0 =vK. If char Kv>0, then assume in addition that the largest perfect subfield of Lv is Kv. Then there is a finite extension R0of R such that S.R0isweakly unramifiedover R0, i.e., their

quotient fields L0and K0 have the same value group.

(75)

## Epp’s Theorem

In 1973, Helmut Epp published a paper entitled Elimination of wild ramification in Inventiones 19. His theorems deal with elimination of wild ramification in extensions of discrete valuation rings (DVRs). One version reads as follows.

Take complete DVRs R⊂S0⊂S with S|S0finite. Denote by K, L0, L the quotient fields of R, S0, S, respectively.

Theorem

Assume that vL0 =vK.

If char Kv>0, then assume in addition that the largest perfect subfield of Lv is Kv. Then there is a finite extension R0of R such that S.R0isweakly unramifiedover R0, i.e., their

quotient fields L0and K0 have the same value group.

Franz-Viktor Kuhlmann Elimination of Ramification

(76)

## Epp’s Theorem

In 1973, Helmut Epp published a paper entitled Elimination of wild ramification in Inventiones 19. His theorems deal with elimination of wild ramification in extensions of discrete valuation rings (DVRs). One version reads as follows.

Take complete DVRs R⊂S0⊂S with S|S0finite. Denote by K, L0, L the quotient fields of R, S0, S, respectively.

Theorem

Assume that vL0 =vK. If char Kv>0, then assume in addition that the largest perfect subfield of Lv is Kv.

Then there is a finite extension R0of R such that S.R0isweakly unramifiedover R0, i.e., their

quotient fields L0and K0 have the same value group.

(77)

## Epp’s Theorem

In 1973, Helmut Epp published a paper entitled Elimination of wild ramification in Inventiones 19. His theorems deal with elimination of wild ramification in extensions of discrete valuation rings (DVRs). One version reads as follows.

Take complete DVRs R⊂S0⊂S with S|S0finite. Denote by K, L0, L the quotient fields of R, S0, S, respectively.

Theorem

Assume that vL0 =vK. If char Kv>0, then assume in addition that the largest perfect subfield of Lv is Kv. Then there is a finite extension R0of R such that S.R0isweakly unramifiedover R0, i.e., their

quotient fields L0and K0 have the same value group.

Franz-Viktor Kuhlmann Elimination of Ramification

(78)

## Elements of the proof

It is important to note that the subgroup Grof the absolute Galois group Gal L is a p-group.

This means that every finite separable extension of Lris a tower of Galois extensions of degree p. If char L=p, then these areArtin-Schreier extensions: L|L0is an Artin-Schreier extension if L=L0(z)with zp−z∈L0. If d∈L0, then L=L0(z−d)and

(z−d)p− (z−d) = zp−z−dp+d .

That allows us to replace any summand in zp−z that is of the form dpby its p-th root d. This fact has been used by several authors, including Abhyankar, Epp and myself, to find normal forms for such Artin-Schreier extensions that fit our purposes.

(79)

## Elements of the proof

It is important to note that the subgroup Grof the absolute Galois group Gal L is a p-group. This means that every finite separable extension of Lris a tower of Galois extensions of degree p.

If char L=p, then these areArtin-Schreier extensions: L|L0is an Artin-Schreier extension if L=L0(z)with zp−z∈L0. If d∈L0, then L=L0(z−d)and

(z−d)p− (z−d) = zp−z−dp+d .

That allows us to replace any summand in zp−z that is of the form dpby its p-th root d. This fact has been used by several authors, including Abhyankar, Epp and myself, to find normal forms for such Artin-Schreier extensions that fit our purposes.

Franz-Viktor Kuhlmann Elimination of Ramification

(80)

## Elements of the proof

It is important to note that the subgroup Grof the absolute Galois group Gal L is a p-group. This means that every finite separable extension of Lris a tower of Galois extensions of degree p. If char L=p, then these areArtin-Schreier extensions:

L|L0is an Artin-Schreier extension if L=L0(z)with zp−z∈L0. If d∈L0, then L=L0(z−d)and

(z−d)p− (z−d) = zp−z−dp+d .

That allows us to replace any summand in zp−z that is of the form dpby its p-th root d. This fact has been used by several authors, including Abhyankar, Epp and myself, to find normal forms for such Artin-Schreier extensions that fit our purposes.

(81)

## Elements of the proof

It is important to note that the subgroup Grof the absolute Galois group Gal L is a p-group. This means that every finite separable extension of Lris a tower of Galois extensions of degree p. If char L=p, then these areArtin-Schreier extensions:

L|L0is an Artin-Schreier extension if L=L0(z)with zp−z∈L0.

If d∈L0, then L=L0(z−d)and

(z−d)p− (z−d) = zp−z−dp+d .

That allows us to replace any summand in zp−z that is of the form dpby its p-th root d. This fact has been used by several authors, including Abhyankar, Epp and myself, to find normal forms for such Artin-Schreier extensions that fit our purposes.

Franz-Viktor Kuhlmann Elimination of Ramification

(82)

## Elements of the proof

It is important to note that the subgroup Grof the absolute Galois group Gal L is a p-group. This means that every finite separable extension of Lris a tower of Galois extensions of degree p. If char L=p, then these areArtin-Schreier extensions:

L|L0is an Artin-Schreier extension if L=L0(z)with zp−z∈L0. If d∈L0, then L=L0(z−d)and

(z−d)p− (z−d) = zp−z−dp+d .

That allows us to replace any summand in zp−z that is of the form dpby its p-th root d. This fact has been used by several authors, including Abhyankar, Epp and myself, to find normal forms for such Artin-Schreier extensions that fit our purposes.

(83)

## Elements of the proof

It is important to note that the subgroup Grof the absolute Galois group Gal L is a p-group. This means that every finite separable extension of Lris a tower of Galois extensions of degree p. If char L=p, then these areArtin-Schreier extensions:

L|L0is an Artin-Schreier extension if L=L0(z)with zp−z∈L0. If d∈L0, then L=L0(z−d)and

(z−d)p− (z−d) = zp−z−dp+d .

That allows us to replace any summand in zp−z that is of the form dpby its p-th root d.

This fact has been used by several authors, including Abhyankar, Epp and myself, to find normal forms for such Artin-Schreier extensions that fit our purposes.

Franz-Viktor Kuhlmann Elimination of Ramification

(84)

## Elements of the proof

It is important to note that the subgroup Grof the absolute Galois group Gal L is a p-group. This means that every finite separable extension of Lris a tower of Galois extensions of degree p. If char L=p, then these areArtin-Schreier extensions:

L|L0is an Artin-Schreier extension if L=L0(z)with zp−z∈L0. If d∈L0, then L=L0(z−d)and

(z−d)p− (z−d) = zp−z−dp+d .

That allows us to replace any summand in zp−z that is of the form dpby its p-th root d. This fact has been used by several authors,

including Abhyankar, Epp and myself, to find normal forms for such Artin-Schreier extensions that fit our purposes.

(85)

## Elements of the proof

It is important to note that the subgroup Grof the absolute Galois group Gal L is a p-group. This means that every finite separable extension of Lris a tower of Galois extensions of degree p. If char L=p, then these areArtin-Schreier extensions:

L|L0is an Artin-Schreier extension if L=L0(z)with zp−z∈L0. If d∈L0, then L=L0(z−d)and

(z−d)p− (z−d) = zp−z−dp+d .

That allows us to replace any summand in zp−z that is of the form dpby its p-th root d. This fact has been used by several authors, including Abhyankar, Epp and myself,

to find normal forms for such Artin-Schreier extensions that fit our purposes.

Franz-Viktor Kuhlmann Elimination of Ramification

(86)

## Elements of the proof

It is important to note that the subgroup Grof the absolute Galois group Gal L is a p-group. This means that every finite separable extension of Lris a tower of Galois extensions of degree p. If char L=p, then these areArtin-Schreier extensions:

L|L0is an Artin-Schreier extension if L=L0(z)with zp−z∈L0. If d∈L0, then L=L0(z−d)and

(z−d)p− (z−d) = zp−z−dp+d .

That allows us to replace any summand in zp−z that is of the form dpby its p-th root d. This fact has been used by several authors, including Abhyankar, Epp and myself, to find normal

(87)

## Elements of the proof

Let us discuss the case of char L=p,

and let us assume that we have reduced the proof to the case where L|L0is an

Artin-Schreier extension. By the assumptions of the theorem, we can write

R = (Kv)[[π]] and S0 = (L0v)[[π]]. So we have that

zp−z = anπn+an1πn1+. . .

with n∈Z and ai∈L0v. If n ≥0, then it is easy to show that L⊂Li0and there is no wild ramification. So let us assume that n<0.

Franz-Viktor Kuhlmann Elimination of Ramification

(88)

## Elements of the proof

Let us discuss the case of char L=p, and let us assume that we have reduced the proof to the case where L|L0is an

Artin-Schreier extension.

By the assumptions of the theorem, we can write

R = (Kv)[[π]] and S0 = (L0v)[[π]]. So we have that

zp−z = anπn+an1πn1+. . .

with n∈Z and ai∈L0v. If n ≥0, then it is easy to show that L⊂Li0and there is no wild ramification. So let us assume that n<0.

(89)

## Elements of the proof

Let us discuss the case of char L=p, and let us assume that we have reduced the proof to the case where L|L0is an

Artin-Schreier extension. By the assumptions of the theorem, we can write

R = (Kv)[[π]] and S0 = (L0v)[[π]].

So we have that

zp−z = anπn+an1πn1+. . .

with n∈Z and ai∈L0v. If n ≥0, then it is easy to show that L⊂Li0and there is no wild ramification. So let us assume that n<0.

Franz-Viktor Kuhlmann Elimination of Ramification

(90)

## Elements of the proof

Let us discuss the case of char L=p, and let us assume that we have reduced the proof to the case where L|L0is an

Artin-Schreier extension. By the assumptions of the theorem, we can write

R = (Kv)[[π]] and S0 = (L0v)[[π]]. So we have that

zp−z = anπn+an1πn1+. . . with n∈Z and ai∈L0v.

If n≥0, then it is easy to show that L⊂Li0and there is no wild ramification. So let us assume that n<0.

(91)

## Elements of the proof

Let us discuss the case of char L=p, and let us assume that we have reduced the proof to the case where L|L0is an

Artin-Schreier extension. By the assumptions of the theorem, we can write

R = (Kv)[[π]] and S0 = (L0v)[[π]]. So we have that

zp−z = anπn+an1πn1+. . .

with n∈Z and ai∈L0v. If n ≥0, then it is easy to show that L⊂Li0and there is no wild ramification.

So let us assume that n<0.

Franz-Viktor Kuhlmann Elimination of Ramification

(92)

## Elements of the proof

Let us discuss the case of char L=p, and let us assume that we have reduced the proof to the case where L|L0is an

Artin-Schreier extension. By the assumptions of the theorem, we can write

R = (Kv)[[π]] and S0 = (L0v)[[π]]. So we have that

zp−z = anπn+an1πn1+. . .

with n∈Z and ai∈L0v. If n ≥0, then it is easy to show that L⊂Li0and there is no wild ramification. So let us assume that

(93)

## Elements of the proof

If ai ∈Kv, then we can get rid of the term aiπi by putting some y with yp−y=aiπiin the extension R0of R

(this may change the value group, but that is accepted). We will be left with summands aiπiwhere ai ∈/Kv if i is negative.

By our assumption on Lv, for every ai ∈L0v\Kv, there is a maximal k such that its pk-th root is still in L0v.

The idea of Epp is now to replace aiπiby its pk-th root a1/pi kπi/pk, putting πi/pk into R0.

After doing this for all negative i, Epp states that in the above form for zp−z we have that anhas no p-th root in L0v. From this one easily deduces that for the new extension L0|K0we have that[L0v : K0v] =p and vL0 =vK0, so S.R0is weakly unramified over R0.

Franz-Viktor Kuhlmann Elimination of Ramification

(94)

## Elements of the proof

If ai ∈Kv, then we can get rid of the term aiπi by putting some y with yp−y=aiπiin the extension R0of R (this may change the value group, but that is accepted).

We will be left with summands aiπiwhere ai ∈/Kv if i is negative.

By our assumption on Lv, for every ai ∈L0v\Kv, there is a maximal k such that its pk-th root is still in L0v.

The idea of Epp is now to replace aiπiby its pk-th root a1/pi kπi/pk, putting πi/pk into R0.

After doing this for all negative i, Epp states that in the above form for zp−z we have that anhas no p-th root in L0v. From this one easily deduces that for the new extension L0|K0we have that[L0v : K0v] =p and vL0 =vK0, so S.R0is weakly unramified over R0.

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