Transcript
Page 1: Some Aspects of Modular Invariant Theory of Finite Groups ·

Some Aspects of ModularInvariant Theoryof Finite Groups

Peter Fleischmann

Institute of Mathematics, Statistics and Actuarial Science

University of Kent

Auckland, February 2005

Some Aspects of ModularInvariant Theoryof Finite Groups – p.1/35

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Geometric Background

affine algebra over ;

, corresponding affine variety;

group of automorphisms of .

categorical quotient defined by the

When is affine ?

Necessary condition:finitely generated over .

Some Aspects of ModularInvariant Theoryof Finite Groups – p.2/35

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Geometric Background

��� � affine algebra over

� � �

;

� � � � �� � � �

, corresponding affine variety;

group of automorphisms of

.

categorical quotient defined by the

When is affine ?

Necessary condition:finitely generated over .

Some Aspects of ModularInvariant Theoryof Finite Groups – p.2/35

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Geometric Background

��� � affine algebra over

� � �

;

� � � � �� � � �

, corresponding affine variety;

group of automorphisms of

.

� � � �� � � �� � � � �

categorical quotient defined by the

� ��� � � � ��� � � � � � � � � � ��� � � � � �� � � � � � � � ��

When is affine ?

Necessary condition:finitely generated over .

Some Aspects of ModularInvariant Theoryof Finite Groups – p.2/35

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Geometric Background

��� � affine algebra over

� � �

;

� � � � �� � � �

, corresponding affine variety;

group of automorphisms of

.

� � � �� � � �� � � � �

categorical quotient defined by the

� ��� � � � ��� � � � � � � � � � ��� � � � � �� � � � � � � � ��

When is

� � � �

affine ?

Necessary condition:finitely generated over .

Some Aspects of ModularInvariant Theoryof Finite Groups – p.2/35

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Geometric Background

��� � affine algebra over

� � �

;

� � � � �� � � �

, corresponding affine variety;

group of automorphisms of

.

� � � �� � � �� � � � �

categorical quotient defined by the

� ��� � � � ��� � � � � � � � � � ��� � � � � �� � � � � � � � ��

When is

� � � �

affine ?

Necessary condition:

� �finitely generated over

.

Some Aspects of ModularInvariant Theoryof Finite Groups – p.2/35

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Hilbert’s 14’ th problem

In case

� � �

and

� � ����

� � �

this wasposed by Hilbert (1’st It’l Congress, Paris 1900).

Answer no in general:(1958)Nagata (counter example ).

Answer yes in important special cases- ‘linear reductive groups’ e.g. Cayley,(Cayley, Sylvester, Gordan, Hilbert, Weyl.)

- finite groups, arbitrary (Emmy Noether (1926))

Theorem:

If is finite, then is finitely generated and the categoricalquotient is in bijection with the orbit space.

Some Aspects of ModularInvariant Theoryof Finite Groups – p.3/35

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Hilbert’s 14’ th problem

In case

� � �

and

� � ����

� � �

this wasposed by Hilbert (1’st It’l Congress, Paris 1900).

Answer no in general:(1958)Nagata (counter example

� � � � � � �).

Answer yes in important special cases- ‘linear reductive groups’ e.g. Cayley,(Cayley, Sylvester, Gordan, Hilbert, Weyl.)

- finite groups, arbitrary (Emmy Noether (1926))

Theorem:

If is finite, then is finitely generated and the categoricalquotient is in bijection with the orbit space.

Some Aspects of ModularInvariant Theoryof Finite Groups – p.3/35

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Hilbert’s 14’ th problem

In case

� � �

and

� � ����

� � �

this wasposed by Hilbert (1’st It’l Congress, Paris 1900).

Answer no in general:(1958)Nagata (counter example

� � � � � � �).

Answer yes in important special cases- ‘linear reductive groups’ e.g.

���

� � �� �� � � � �� � � Cayley,

(Cayley, Sylvester, Gordan, Hilbert, Weyl.)

- finite groups,

arbitrary (Emmy Noether (1926))

Theorem:

If is finite, then is finitely generated and the categoricalquotient is in bijection with the orbit space.

Some Aspects of ModularInvariant Theoryof Finite Groups – p.3/35

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Hilbert’s 14’ th problem

In case

� � �

and

� � ����

� � �

this wasposed by Hilbert (1’st It’l Congress, Paris 1900).

Answer no in general:(1958)Nagata (counter example

� � � � � � �).

Answer yes in important special cases- ‘linear reductive groups’ e.g.

���

� � �� �� � � � �� � � Cayley,

(Cayley, Sylvester, Gordan, Hilbert, Weyl.)

- finite groups,

arbitrary (Emmy Noether (1926))

Theorem:

If

is finite, then� �

is finitely generated and the categoricalquotient is in bijection with the orbit space.

� � � �� � � � ��

Some Aspects of ModularInvariant Theoryof Finite Groups – p.3/35

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Notation

From now on always:

finite group

a finite dimensional

� �

- module,

��

dual module with basis ��� � � � � � �� ,

� � ��� � � � � � �� � �� � � � � � ��

, polynomial ring;

� � � � �� � � � � �� � � � � � � � � ��

�� � � � � � � �

ring of polynomial invariants.

Example: Symmetric polynomials:, permuting the variables ,

polynomial ringgenerated by elementary symmetric functions indegrees .

Some Aspects of ModularInvariant Theoryof Finite Groups – p.4/35

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Notation

From now on always:

finite group

a finite dimensional

� �

- module,

��

dual module with basis ��� � � � � � �� ,

� � ��� � � � � � �� � �� � � � � � ��

, polynomial ring;

� � � � �� � � � � �� � � � � � � � � ��

�� � � � � � � �

ring of polynomial invariants.

Example: Symmetric polynomials:�� � �� , permuting the variables � � � � � � � �� ,

� � � � �� � � � � � � � �

� polynomial ringgenerated by elementary symmetric functions � � � � � � � � � indegrees

�� �� � � � � �.

Some Aspects of ModularInvariant Theoryof Finite Groups – p.4/35

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Research Problem

I. Constructive Complexity of

� �

fundamental systems of invariantsdegree bounds for generators

II. Structural complexity of

distance of from being a polynomial ringcohomological co - dimension:

As in representation theory:

;

.

Some Aspects of ModularInvariant Theoryof Finite Groups – p.5/35

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Research Problem

I. Constructive Complexity of

� �

fundamental systems of invariantsdegree bounds for generators

II. Structural complexity of

� �

distance of

� �

from being a polynomial ringcohomological co - dimension:

� �� � � � � � �

As in representation theory:

;

.

Some Aspects of ModularInvariant Theoryof Finite Groups – p.5/35

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Research Problem

I. Constructive Complexity of

� �

fundamental systems of invariantsdegree bounds for generators

II. Structural complexity of

� �

distance of

� �

from being a polynomial ringcohomological co - dimension:

� �� � � � � � �

As in representation theory:

� �� � �� � � � �� � � � � � � ��� � �� � �;

� �� � � � � � �� � � � �� � �� � �.

Some Aspects of ModularInvariant Theoryof Finite Groups – p.5/35

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Constructive Aspects

Definition: (Degree bounds, Noether - number)

� � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � �

Theorem (Emmy Noether (1916)): If , then

- the Noether bound does not hold if divides .

- Noether’s proofs do not work for in general.

- Generalization to

(Fl., Fogarty/Benson, ).

Some Aspects of ModularInvariant Theoryof Finite Groups – p.6/35

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Constructive Aspects

Definition: (Degree bounds, Noether - number)

� � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � �

Theorem (Emmy Noether (1916)): If � �� � � �

, then

� � � � � � � � ��� � � � � � � � � � �� � � �

- the Noether bound does not hold if divides .

- Noether’s proofs do not work for in general.

- Generalization to

(Fl., Fogarty/Benson, ).

Some Aspects of ModularInvariant Theoryof Finite Groups – p.6/35

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Constructive Aspects

Definition: (Degree bounds, Noether - number)

� � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � �

Theorem (Emmy Noether (1916)): If � �� � � �

, then

� � � � � � � � ��� � � � � � � � � � �� � � �

- the Noether bound does not hold if � �� �

divides

� � �

.

- Noether’s proofs do not work for � �� � �� � � �

in general.

- Generalization to � �� � �� � � �

(Fl., Fogarty/Benson,

� � � � � � � �

).

Some Aspects of ModularInvariant Theoryof Finite Groups – p.6/35

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Constructive Aspects

� �� �

appears in two places:

surjectivity of transfer map

� � � � � � �� � �

� �� �

combinatorics to reduce degrees eg.:

(needed for ).

Some Aspects of ModularInvariant Theoryof Finite Groups – p.7/35

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Constructive Aspects

� �� �

appears in two places:

surjectivity of transfer map

� � � � � � �� � �

� �� �

combinatorics to reduce degrees eg.:

�� � � � �� �

� � � ��� �� ����� � � � � �

� � � � � � �

� � �

(needed for �� �� � � � �� � �

).

Some Aspects of ModularInvariant Theoryof Finite Groups – p.7/35

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Relative Noether Bound

Theorem (Fl., F Knop, M Sezer, (indep.)):

If

� � �

,

� � � � ��

, or

�� �

with

�� � � ��, then

� � � � � � � � � � � � �� � �

Some Aspects of ModularInvariant Theoryof Finite Groups – p.8/35

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Relative Noether Bound

Theorem (Fl., F Knop, M Sezer, (indep.)):

If

� � �

,

� � � � ��

, or

�� �

with

�� � � ��, then

� � � � � � � � � � � � �� � �

Sketch of proof:

�� � �� �� �� � �

;

� � �� � � � � � � � � � � � � � � � � � � �� � �

� ��

� �� ��

Some Aspects of ModularInvariant Theoryof Finite Groups – p.9/35

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Relative Noether Bound

Theorem (Fl., F Knop, M Sezer, (indep.)):

If

� � �

,

� � � � ��

, or

�� �

with

�� � � ��, then

� � � � � � � � � � � � �� � �

Sketch of proof:

�� � �� �� �� � �

;

� � �� � � � � � � � � � � � � � � � � � � �� � �

� ��

� �� ��

Let

�� � ��� � � � � � �� � � �

. For fixed i:

�� ��

� �� � ��

� � � � ��

� � � ��

Some Aspects of ModularInvariant Theoryof Finite Groups – p.10/35

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Relative Noether Bound

Theorem (Fl., F Knop, M Sezer, (indep.)):

If

� � �

,

� � � � ��

, or

�� �

with

�� � � ��, then

� � � � � � � � � � � � �� � �

Sketch of proof:Expansion and summation over

� � �� � � � � � gives:

� � � � � �� � � � ��

� �

� � �� � �� � � �� � �� � �� � � ��

� �� ���

� ��

� � � � � � � �� � �

��

��

a reduction in Hilbert - ideal

� �� � �

.

Some Aspects of ModularInvariant Theoryof Finite Groups – p.11/35

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Relative Noether Bound

Theorem (Fl., F Knop, M Sezer, (indep.)):

If

� � �

,

� � � � ��

, or

�� �

with

�� � � ��, then

� � � � � � � � � � � � �� � �

Sketch of proof:Expansion and summation over

� � �� � � � � � gives:

� � � � � �� � � � ��

� �

� � �� � �� � � �� � �� � �� � � ��

� �� ���

� ��

� � � � � � � �� � �

��

��

a reduction in Hilbert - ideal

� �� � �

.

Application of� �� or

� � � yields decomposition in

� �

.

Some Aspects of ModularInvariant Theoryof Finite Groups – p.12/35

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Known modular degree bounds

� � � � � � � � �

;

� � � � � � � � � � �� � �� � �� � � � �� � ��

, (Derksen - Kemper)

� � � �� � � � � �� � � �

, (Karaguezian - Symonds),

here � � � � ��

For permutation representations

� � �� :

� � � � � � �� �

� � � � � � � � � � �� ��

Some Aspects of ModularInvariant Theoryof Finite Groups – p.13/35

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Degree bound conjectures

1 If �

�� �� �

, then

� � � � � � � � � � � � �� � (i.e.

�� �

not needed).

2 Noether - bound for Hilbert ideal (Derksen/Kemper):

3 General modular degree bound:

Conjecture 1 true, if .Conjectures 2. and 3. have been proved for - permutationrepresentations (Fl. 2000, Fl. - Lempken 1997).

Some Aspects of ModularInvariant Theoryof Finite Groups – p.14/35

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Degree bound conjectures

1 If �

�� �� �

, then

� � � � � � � � � � � � �� � (i.e.

�� �

not needed).

2 Noether - bound for Hilbert ideal (Derksen/Kemper):

� � � �� � � � � � � � ��

3 General modular degree bound:

Conjecture 1 true, if .Conjectures 2. and 3. have been proved for - permutationrepresentations (Fl. 2000, Fl. - Lempken 1997).

Some Aspects of ModularInvariant Theoryof Finite Groups – p.14/35

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Degree bound conjectures

1 If �

�� �� �

, then

� � � � � � � � � � � � �� � (i.e.

�� �

not needed).

2 Noether - bound for Hilbert ideal (Derksen/Kemper):

� � � �� � � � � � � � ��

3 General modular degree bound:

� � � � � � � �� � � � � � � � � � � � � � � � � � � ��

Conjecture 1 true, if .Conjectures 2. and 3. have been proved for - permutationrepresentations (Fl. 2000, Fl. - Lempken 1997).

Some Aspects of ModularInvariant Theoryof Finite Groups – p.14/35

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Degree bound conjectures

1 If �

�� �� �

, then

� � � � � � � � � � � � �� � (i.e.

�� �

not needed).

2 Noether - bound for Hilbert ideal (Derksen/Kemper):

� � � �� � � � � � � � ��

3 General modular degree bound:

� � � � � � � �� � � � � � � � � � � � � � � � � � � ��

Conjecture 1 true, if

�� � � � ��

.Conjectures 2. and 3. have been proved for � - permutationrepresentations (Fl. 2000, Fl. - Lempken 1997).

Some Aspects of ModularInvariant Theoryof Finite Groups – p.14/35

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The relative - transfer ideal

relative transfer map: for

� � �

:

� � � � � � � � �� � �

� � � � �� �� ��

image

� � � � � � � � � �

is ideal in

� �

.

For a Sylow - group ,

sum of all relative transfers from subgroups .

Some Aspects of ModularInvariant Theoryof Finite Groups – p.15/35

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The relative - transfer ideal

relative transfer map: for

� � �

:

� � � � � � � � �� � �

� � � � �� �� ��

image

� � � � � � � � � �

is ideal in

� �

.

For a Sylow � - group

,

� �� � � �

� � �� �� � �� �

� � �� � � � �� � � � ��

� �� � � � sum of all relative transfers from subgroups

� � �

.

Some Aspects of ModularInvariant Theoryof Finite Groups – p.15/35

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The relative - transfer ideal

Consider factor rings

� � � � � � � � �� �� � � � � � � � � �� �� � � � � � � � � � ��

By Brauer homomorphism + Mackey - formula:

Since , can use coprime Noether bound:

conjecture:

(again confirmed for - permutation representations).

Some Aspects of ModularInvariant Theoryof Finite Groups – p.16/35

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The relative - transfer ideal

Consider factor rings

� � � � � � � � �� �� � � � � � � � � �� �� � � � � � � � � � ��

By Brauer homomorphism + Mackey - formula:

� � � � �� � � � � �� � � � � ��

Since , can use coprime Noether bound:

conjecture:

(again confirmed for - permutation representations).

Some Aspects of ModularInvariant Theoryof Finite Groups – p.16/35

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The relative - transfer ideal

Consider factor rings

� � � � � � � � �� �� � � � � � � � � �� �� � � � � � � � � � ��

By Brauer homomorphism + Mackey - formula:

� � � � �� � � � � �� � � � � ��

Since �

�� � � �

, can use coprime Noether bound:

� � � � � � � �� � � � � �� � �

� � � � � � � � � � ��

conjecture:

(again confirmed for - permutation representations).

Some Aspects of ModularInvariant Theoryof Finite Groups – p.16/35

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The relative - transfer ideal

Consider factor rings

� � � � � � � � �� �� � � � � � � � � �� �� � � � � � � � � � ��

By Brauer homomorphism + Mackey - formula:

� � � � �� � � � � �� � � � � ��

Since �

�� � � �

, can use coprime Noether bound:

� � � � � � � �� � � � � �� � �

� � � � � � � � � � ��

conjecture:

� � � � � � � � �

� � � �� � � � � �� � � � � � � � � � � � � � � � � � � ��

(again confirmed for � - permutation representations).

Some Aspects of ModularInvariant Theoryof Finite Groups – p.16/35

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Localisations

with G Kemper (Munich) and C F Woodcock (Kent)

Noether bound for invariant field in arbitrary characteristic i.e.

For arbitrary , in degreesuch that

If , then .

Some Aspects of ModularInvariant Theoryof Finite Groups – p.17/35

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Localisations

with G Kemper (Munich) and C F Woodcock (Kent)

Noether bound for invariant field in arbitrary characteristic i.e.

� � � � � � � � � � � � � � � �� � � �

For arbitrary , in degreesuch that

If , then .

Some Aspects of ModularInvariant Theoryof Finite Groups – p.17/35

Page 39: Some Aspects of Modular Invariant Theory of Finite Groups ·

Localisations

with G Kemper (Munich) and C F Woodcock (Kent)

Noether bound for invariant field in arbitrary characteristic i.e.

� � � � � � � � � � � � � � � �� � � �

For arbitrary

,

� � � � �� � � �� � � �in degree

� � � �

such that

� � � ��

� � � � � � ��

If

� � � � � � �

, then� � � �

�� � � � �

.

Some Aspects of ModularInvariant Theoryof Finite Groups – p.17/35

Page 40: Some Aspects of Modular Invariant Theory of Finite Groups ·

Method to construct

Pick "easy" constructible subalgebra

� � � �

, such that

� � � � � � �

with � � � � �

.

let h. s. o. p.

note: is free with submodules .

can be computed using“Groebner bases for modules over polynomial rings"(e.g. Adams / Loustaunau AMS)

If follows

Some Aspects of ModularInvariant Theoryof Finite Groups – p.18/35

Page 41: Some Aspects of Modular Invariant Theory of Finite Groups ·

Method to construct

Pick "easy" constructible subalgebra

� � � �

, such that

� � � � � � �

with � � � � �

.

let

� �� � � � � � � � � � � � �

h. s. o. p.

note: � � �� ��

is free with submodules � �� �

.

can be computed using“Groebner bases for modules over polynomial rings"(e.g. Adams / Loustaunau AMS)

If follows

Some Aspects of ModularInvariant Theoryof Finite Groups – p.18/35

Page 42: Some Aspects of Modular Invariant Theory of Finite Groups ·

Method to construct

Pick "easy" constructible subalgebra

� � � �

, such that

� � � � � � �

with � � � � �

.

let

� �� � � � � � � � � � � � �

h. s. o. p.

note: � � �� ��

is free with submodules � �� �

.

� � � � � � � � � � � � � � � � �� � � � � � ���

� �

can be computed using“Groebner bases for modules over polynomial rings"(e.g. Adams / Loustaunau AMS)

If follows

Some Aspects of ModularInvariant Theoryof Finite Groups – p.18/35

Page 43: Some Aspects of Modular Invariant Theory of Finite Groups ·

Method to construct

Pick "easy" constructible subalgebra

� � � �

, such that

� � � � � � �

with � � � � �

.

let

� �� � � � � � � � � � � � �

h. s. o. p.

note: � � �� ��

is free with submodules � �� �

.

� � � � � � � � � � � � � � � � �� � � � � � ���

� �

can be computed using“Groebner bases for modules over polynomial rings"(e.g. Adams / Loustaunau AMS)

If follows

Some Aspects of ModularInvariant Theoryof Finite Groups – p.18/35

Page 44: Some Aspects of Modular Invariant Theory of Finite Groups ·

Method to construct

Pick "easy" constructible subalgebra

� � � �

, such that

� � � � � � �

with � � � � �

.

let

� �� � � � � � � � � � � � �

h. s. o. p.

note: � � �� ��

is free with submodules � �� �

.

� � � � � � � � � � � � � � � � �� � � � � � ���

� �

can be computed using“Groebner bases for modules over polynomial rings"(e.g. Adams / Loustaunau AMS)

If follows

� � � � �� � �� � � � � ���

� � �

Some Aspects of ModularInvariant Theoryof Finite Groups – p.18/35

Page 45: Some Aspects of Modular Invariant Theory of Finite Groups ·

Vector Invariants of �

Let

be arbitrary commutative ring

with - action by

ring of ( - fold) “vector invariants".

Define Galois - resolvent

Some Aspects of ModularInvariant Theoryof Finite Groups – p.19/35

Page 46: Some Aspects of Modular Invariant Theory of Finite Groups ·

Vector Invariants of �

Let

be arbitrary commutative ring

� �� � � � � � � �� � � � � � � ��� � � � � � � �� � � � � � � �� �

with

�� - action by

� � � � � � � � � � � � �

ring of ( - fold) “vector invariants".

Define Galois - resolvent

Some Aspects of ModularInvariant Theoryof Finite Groups – p.19/35

Page 47: Some Aspects of Modular Invariant Theory of Finite Groups ·

Vector Invariants of �

Let

be arbitrary commutative ring

� �� � � � � � � �� � � � � � � ��� � � � � � � �� � � � � � � �� �

with

�� - action by

� � � � � � � � � � � � �

� �� � � � � � � ring of (

- fold) “vector invariants".

Define Galois - resolvent

Some Aspects of ModularInvariant Theoryof Finite Groups – p.19/35

Page 48: Some Aspects of Modular Invariant Theory of Finite Groups ·

Vector Invariants of �

Let

be arbitrary commutative ring

� �� � � � � � � �� � � � � � � ��� � � � � � � �� � � � � � � �� �

with

�� - action by

� � � � � � � � � � � � �

� �� � � � � � � ring of (

- fold) “vector invariants".

Define Galois - resolvent� � �� � � � � � �� � � � � �

�� ��

� � ��

��� � �� � � � �� � � � � � � � � � � � � ��

Some Aspects of ModularInvariant Theoryof Finite Groups – p.19/35

Page 49: Some Aspects of Modular Invariant Theory of Finite Groups ·

Vector Invariants of �

Theorem(H Weyl, D Richman)If � � � ��

, then

� �� � � � � � is generated by the coefficients of� � �� � � � � � �� � � �

. In particular

� � � �� � � � � � ��

This is false for or fields with .

Theorem (Fl. 1997)

with equality if and or .

Theorem (Fl. Kemper, Woodcock)

Some Aspects of ModularInvariant Theoryof Finite Groups – p.20/35

Page 50: Some Aspects of Modular Invariant Theory of Finite Groups ·

Vector Invariants of �

Theorem(H Weyl, D Richman)If � � � ��

, then

� �� � � � � � is generated by the coefficients of� � �� � � � � � �� � � �

. In particular

� � � �� � � � � � ��This is false for

� � �

or fields with � � � ��� � �

.

Theorem (Fl. 1997)

� � � �� � � � � � � � � �� � �� � � � � � � �

with equality if � � � � and � �� � � � or

� � �

.

Theorem (Fl. Kemper, Woodcock)

� � � � � � � � �� � � � � � � � ��

Some Aspects of ModularInvariant Theoryof Finite Groups – p.20/35

Page 51: Some Aspects of Modular Invariant Theory of Finite Groups ·

Noether - Homomorphism

For

� � �

,

� � � � �� � � � � � ��

� �

:

Noether - homomorphism

� � � �� � � � � � � � � �

- equivariant homomorphism of�

- algebras.

� � �� � �� � � � � � � � � �

(Noether - image)

If � � �� � � ��

, � � �and

� � � � � � � �� � �� � � � � � � � � � � � ��

If � � � ��

, Weyl’s theorem applies and

� � � � � � � � � � � � ��

Some Aspects of ModularInvariant Theoryof Finite Groups – p.21/35

Page 52: Some Aspects of Modular Invariant Theory of Finite Groups ·

in the modular case

Theorem(Fl.+Kemper+Woodcock)

is purely inseparable overof index , the maximal - power in .

, hence

:

Some Aspects of ModularInvariant Theoryof Finite Groups – p.22/35

Page 53: Some Aspects of Modular Invariant Theory of Finite Groups ·

in the modular case

Theorem(Fl.+Kemper+Woodcock)

� � � � � � � � � � � � � � � � � � � � � � � � � � ��

is purely inseparable overof index , the maximal - power in .

, hence

:

Some Aspects of ModularInvariant Theoryof Finite Groups – p.22/35

Page 54: Some Aspects of Modular Invariant Theory of Finite Groups ·

in the modular case

Theorem(Fl.+Kemper+Woodcock)

� � � � � � � � � � � � � � � � � � � � � � � � � � ��

� � � ��

is purely inseparable over �of index � �

, the maximal � - power in� � �

.

, hence

:

Some Aspects of ModularInvariant Theoryof Finite Groups – p.22/35

Page 55: Some Aspects of Modular Invariant Theory of Finite Groups ·

in the modular case

Theorem(Fl.+Kemper+Woodcock)

� � � � � � � � � � � � � � � � � � � � � � � � � � ��

� � � ��

is purely inseparable over �of index � �

, the maximal � - power in� � �

.

� � � � �� � � � �

, hence

� � � � � � � � � � � � � � ��

:

Some Aspects of ModularInvariant Theoryof Finite Groups – p.22/35

Page 56: Some Aspects of Modular Invariant Theory of Finite Groups ·

in the modular case

Theorem(Fl.+Kemper+Woodcock)

� � � � � � � � � � � � � � � � � � � � � � � � � � ��

� � � ��

is purely inseparable over �of index � �

, the maximal � - power in� � �

.

� � � � �� � � � �

, hence

� � � � � � � � � � � � � � ��

� � � � � � � �� � � �

:� � � � � � � � � � �

Some Aspects of ModularInvariant Theoryof Finite Groups – p.22/35

Page 57: Some Aspects of Modular Invariant Theory of Finite Groups ·

Structural Aspects

Have seen is polynomial ring.

However : acting on byand .

non - unique factorization with irreducibles and in.

Hence is not an UFD.

Some Aspects of ModularInvariant Theoryof Finite Groups – p.23/35

Page 58: Some Aspects of Modular Invariant Theory of Finite Groups ·

Structural Aspects

Have seen

� �� � � � � � �� � � �� � � � � � � � � � �

is polynomial ring.

However : acting on byand .

non - unique factorization with irreducibles and in.

Hence is not an UFD.

Some Aspects of ModularInvariant Theoryof Finite Groups – p.23/35

Page 59: Some Aspects of Modular Invariant Theory of Finite Groups ·

Structural Aspects

Have seen

� �� � � � � � �� � � �� � � � � � � � � � �

is polynomial ring.

However :

� � � � � �� �� acting on

��� � � �� � by

� � � � � �

and � � � � � � .

� �� � � � � � �

� � � �

non - unique factorization with irreducibles and in.

Hence is not an UFD.

Some Aspects of ModularInvariant Theoryof Finite Groups – p.23/35

Page 60: Some Aspects of Modular Invariant Theory of Finite Groups ·

Structural Aspects

Have seen

� �� � � � � � �� � � �� � � � � � � � � � �

is polynomial ring.

However :

� � � � � �� �� acting on

��� � � �� � by

� � � � � �

and � � � � � � .

� �� � � � � � �

� � � �

� � � � � � � � � �

non - unique factorization with irreducibles

� � � � �

and

� �

in

� �� � �

.Hence

� �

is not an UFD.

Some Aspects of ModularInvariant Theoryof Finite Groups – p.23/35

Page 61: Some Aspects of Modular Invariant Theory of Finite Groups ·

Structural Aspects

How far away is

� �

from being a polynomial ring in general?

Noether normalisation: polynomial subalgebrasuch that finite as - module.

Cohen - Macaulay (CM) free.

Theorem: (Eagon - Hochster):If , then is Cohen - Macaulay ring.

Proof: is free over ;

(transfer map)

for graded - modules: “projective" = “free".

free, so is CM.

False in the modular case

Example: , .

Some Aspects of ModularInvariant Theoryof Finite Groups – p.24/35

Page 62: Some Aspects of Modular Invariant Theory of Finite Groups ·

Structural Aspects

How far away is

� �

from being a polynomial ring in general?

Noether normalisation:

polynomial subalgebra� � � �

such that

� �

finite as

- module.

Cohen - Macaulay (CM) free.

Theorem: (Eagon - Hochster):If , then is Cohen - Macaulay ring.

Proof: is free over ;

(transfer map)

for graded - modules: “projective" = “free".

free, so is CM.

False in the modular case

Example: , .

Some Aspects of ModularInvariant Theoryof Finite Groups – p.24/35

Page 63: Some Aspects of Modular Invariant Theory of Finite Groups ·

Structural Aspects

How far away is

� �

from being a polynomial ring in general?

Noether normalisation:

polynomial subalgebra� � � �

such that

� �

finite as

- module.

� �

Cohen - Macaulay (CM)� �� �

� �free.

Theorem: (Eagon - Hochster):If , then is Cohen - Macaulay ring.

Proof: is free over ;

(transfer map)

for graded - modules: “projective" = “free".

free, so is CM.

False in the modular case

Example: , .

Some Aspects of ModularInvariant Theoryof Finite Groups – p.24/35

Page 64: Some Aspects of Modular Invariant Theory of Finite Groups ·

Structural Aspects

How far away is

� �

from being a polynomial ring in general?

Noether normalisation:

polynomial subalgebra� � � �

such that

� �

finite as

- module.

� �

Cohen - Macaulay (CM)� �� �

� �free.

Theorem: (Eagon - Hochster):If

� � � � ��

, then

� �

is Cohen - Macaulay ring.

Proof: is free over ;

(transfer map)

for graded - modules: “projective" = “free".

free, so is CM.

False in the modular case

Example: , .

Some Aspects of ModularInvariant Theoryof Finite Groups – p.24/35

Page 65: Some Aspects of Modular Invariant Theory of Finite Groups ·

Structural Aspects

How far away is

� �

from being a polynomial ring in general?

Noether normalisation:

polynomial subalgebra� � � �

such that

� �

finite as

- module.

� �

Cohen - Macaulay (CM)� �� �

� �free.

Theorem: (Eagon - Hochster):If

� � � � ��

, then

� �

is Cohen - Macaulay ring.

Proof:

is free over

;

(transfer map)

for graded - modules: “projective" = “free".

free, so is CM.

False in the modular case

Example: , .

Some Aspects of ModularInvariant Theoryof Finite Groups – p.24/35

Page 66: Some Aspects of Modular Invariant Theory of Finite Groups ·

Structural Aspects

How far away is

� �

from being a polynomial ring in general?

Noether normalisation:

polynomial subalgebra� � � �

such that

� �

finite as

- module.

� �

Cohen - Macaulay (CM)� �� �

� �free.

Theorem: (Eagon - Hochster):If

� � � � ��

, then

� �

is Cohen - Macaulay ring.

Proof:

is free over

;

� �� �

� � � �

(transfer map)

for graded - modules: “projective" = “free".

free, so is CM.

False in the modular case

Example: , .

Some Aspects of ModularInvariant Theoryof Finite Groups – p.24/35

Page 67: Some Aspects of Modular Invariant Theory of Finite Groups ·

Structural Aspects

How far away is

� �

from being a polynomial ring in general?

Noether normalisation:

polynomial subalgebra� � � �

such that

� �

finite as

- module.

� �

Cohen - Macaulay (CM)� �� �

� �free.

Theorem: (Eagon - Hochster):If

� � � � ��

, then

� �

is Cohen - Macaulay ring.

Proof:

is free over

;

� �� �

� � � �

(transfer map)

for graded

- modules: “projective" = “free".

free, so is CM.

False in the modular case

Example: , .

Some Aspects of ModularInvariant Theoryof Finite Groups – p.24/35

Page 68: Some Aspects of Modular Invariant Theory of Finite Groups ·

Structural Aspects

How far away is

� �

from being a polynomial ring in general?

Noether normalisation:

polynomial subalgebra� � � �

such that

� �

finite as

- module.

� �

Cohen - Macaulay (CM)� �� �

� �free.

Theorem: (Eagon - Hochster):If

� � � � ��

, then

� �

is Cohen - Macaulay ring.

Proof:

is free over

;

� �� �

� � � �

(transfer map)

for graded

- modules: “projective" = “free".

� �

free, so� �

is CM.

False in the modular case

Example: , .

Some Aspects of ModularInvariant Theoryof Finite Groups – p.24/35

Page 69: Some Aspects of Modular Invariant Theory of Finite Groups ·

Structural Aspects

How far away is

� �

from being a polynomial ring in general?

Noether normalisation:

polynomial subalgebra� � � �

such that

� �

finite as

- module.

� �

Cohen - Macaulay (CM)� �� �

� �free.

Theorem: (Eagon - Hochster):If

� � � � ��

, then

� �

is Cohen - Macaulay ring.

Proof:

is free over

;

� �� �

� � � �

(transfer map)

for graded

- modules: “projective" = “free".

� �

free, so� �

is CM.

False in the modular case

Example: , .

Some Aspects of ModularInvariant Theoryof Finite Groups – p.24/35

Page 70: Some Aspects of Modular Invariant Theory of Finite Groups ·

Structural Aspects

How far away is

� �

from being a polynomial ring in general?

Noether normalisation:

polynomial subalgebra� � � �

such that

� �

finite as

- module.

� �

Cohen - Macaulay (CM)� �� �

� �free.

Theorem: (Eagon - Hochster):If

� � � � ��

, then

� �

is Cohen - Macaulay ring.

Proof:

is free over

;

� �� �

� � � �

(transfer map)

for graded

- modules: “projective" = “free".

� �

free, so� �

is CM.

False in the modular case

Example:� � � � ���

�� � � � � � � � �� , � � �

.Some Aspects of ModularInvariant Theoryof Finite Groups – p.24/35

Page 71: Some Aspects of Modular Invariant Theory of Finite Groups ·

Structural Aspects

Obstruction: relative transfer ideal: let

� � �

,

� � � � ��

� � �

and

�� � � �� �;

Theorem (Fl. 1998)

is prime ideal of height .( space of - fixed points.)

is CM of Krull - dimension

Some Aspects of ModularInvariant Theoryof Finite Groups – p.25/35

Page 72: Some Aspects of Modular Invariant Theory of Finite Groups ·

Structural Aspects

Obstruction: relative transfer ideal: let

� � �

,

� � � � ��

� � �

and

�� � � �� �;

Theorem (Fl. 1998)

is prime ideal of height .( space of - fixed points.)

is CM of Krull - dimension

Some Aspects of ModularInvariant Theoryof Finite Groups – p.25/35

Page 73: Some Aspects of Modular Invariant Theory of Finite Groups ·

Structural Aspects

Obstruction: relative transfer ideal: let

� � �

,

� � � � ��

� � �

and

�� � � �� �;

Theorem (Fl. 1998)

� �� � � � � ��� � � � � � ��

�� � � � � ��

is prime ideal of height .( space of - fixed points.)

is CM of Krull - dimension

Some Aspects of ModularInvariant Theoryof Finite Groups – p.25/35

Page 74: Some Aspects of Modular Invariant Theory of Finite Groups ·

Structural Aspects

Obstruction: relative transfer ideal: let

� � �

,

� � � � ��

� � �

and

�� � � �� �;

Theorem (Fl. 1998)

� �� � � � � ��� � � � � � ��

�� � � � � ��

� � � � � � � � � �

is prime ideal of height � � � � � � � �

.(

� � � � space of

- fixed points.)

is CM of Krull - dimension

Some Aspects of ModularInvariant Theoryof Finite Groups – p.25/35

Page 75: Some Aspects of Modular Invariant Theory of Finite Groups ·

Structural Aspects

Obstruction: relative transfer ideal: let

� � �

,

� � � � ��

� � �

and

�� � � �� �;

Theorem (Fl. 1998)

� �� � � � � ��� � � � � � ��

�� � � � � ��

� � � � � � � � � �

is prime ideal of height � � � � � � � �

.(

� � � � space of

- fixed points.)

� � � �

is CM of Krull - dimension

� � ��� � ��

Some Aspects of ModularInvariant Theoryof Finite Groups – p.25/35

Page 76: Some Aspects of Modular Invariant Theory of Finite Groups ·

Structural Aspects

Obstruction: relative transfer ideal: let

� � �

,

� � � � ��

� � �

and

�� � � �� �;

Theorem (Fl. 1998)

� � � �

is CM of Krull - dimension� � ��� � �

Some Aspects of ModularInvariant Theoryof Finite Groups – p.26/35

Page 77: Some Aspects of Modular Invariant Theory of Finite Groups ·

Structural Aspects

Obstruction: relative transfer ideal: let

� � �

,

� � � � ��

� � �

and

�� � � �� �;

Theorem (Fl. 1998)

� � � �

is CM of Krull - dimension� � ��� � �

Simplified proof: let

� � � � � �� �

Some Aspects of ModularInvariant Theoryof Finite Groups – p.27/35

Page 78: Some Aspects of Modular Invariant Theory of Finite Groups ·

Structural Aspects

Obstruction: relative transfer ideal: let

� � �

,

� � � � ��

� � �

and

�� � � �� �;

Theorem (Fl. 1998)

� � � �

is CM of Krull - dimension� � ��� � �

Simplified proof: let

� � � � � �� �

� � � � � � � � � �� � � � �

with� � � � � � � � � �

;

Some Aspects of ModularInvariant Theoryof Finite Groups – p.28/35

Page 79: Some Aspects of Modular Invariant Theory of Finite Groups ·

Structural Aspects

Obstruction: relative transfer ideal: let

� � �

,

� � � � ��

� � �

and

�� � � �� �;

Theorem (Fl. 1998)

� � � �

is CM of Krull - dimension� � ��� � �

Simplified proof: let

� � � � � �� �

� � � � � � � � � �� � � � �

with� � � � � � � � � �

;

consider the projection� �� � � � � � � � � � � ��

Some Aspects of ModularInvariant Theoryof Finite Groups – p.29/35

Page 80: Some Aspects of Modular Invariant Theory of Finite Groups ·

Structural Aspects

Obstruction: relative transfer ideal: let

� � �

,

� � � � ��

� � �

and

�� � � �� �;

Theorem (Fl. 1998)

� � � �

is CM of Krull - dimension� � ��� � �

Simplified proof: let

� � � � � �� �

� � � � � � � � � �� � � � �

with� � � � � � � � � �

;

consider the projection� �� � � � � � � � � � � ��

now statement follows from J. Chuai, (Kingston, 2004):

� � � � �� �� � � ��� � � � � � � ��� ��

Some Aspects of ModularInvariant Theoryof Finite Groups – p.30/35

Page 81: Some Aspects of Modular Invariant Theory of Finite Groups ·

Cohomological - codimension

� �� � � � � � �

length of maximal regular sequence in ;

projective co dimension of as module overhomogeneous system of parameters.

Some Aspects of ModularInvariant Theoryof Finite Groups – p.31/35

Page 82: Some Aspects of Modular Invariant Theory of Finite Groups ·

Cohomological - codimension

� �� � � � � � �

length of maximal regular sequence in

� �

;

projective co dimension of as module overhomogeneous system of parameters.

Some Aspects of ModularInvariant Theoryof Finite Groups – p.31/35

Page 83: Some Aspects of Modular Invariant Theory of Finite Groups ·

Cohomological - codimension

� �� � � � � � �

length of maximal regular sequence in

� �

;

projective co dimension of

� �

as module overhomogeneous system of parameters.

Some Aspects of ModularInvariant Theoryof Finite Groups – p.31/35

Page 84: Some Aspects of Modular Invariant Theory of Finite Groups ·

Cohomological - codimension

� �� � � � � � �

length of maximal regular sequence in

� �

;

projective co dimension of

� �

as module overhomogeneous system of parameters.

� �� � � � � � � � � � � � � � � �� � � � ��

Some Aspects of ModularInvariant Theoryof Finite Groups – p.31/35

Page 85: Some Aspects of Modular Invariant Theory of Finite Groups ·

Cohomological - codimension

For ideal

�� � �

: � � � � � �� � � � � �

length of maximal regular sequence inside

.

Theorem (Fl., Shank 2000)

For :

Some Aspects of ModularInvariant Theoryof Finite Groups – p.32/35

Page 86: Some Aspects of Modular Invariant Theory of Finite Groups ·

Cohomological - codimension

For ideal

�� � �

: � � � � � �� � � � � �

length of maximal regular sequence inside

.

Theorem (Fl., Shank 2000)

For

�� � � �� �:

� �� � � � �� � � � � � �� � � � � � � � � ��

Some Aspects of ModularInvariant Theoryof Finite Groups – p.32/35

Page 87: Some Aspects of Modular Invariant Theory of Finite Groups ·

Calculation of � � ��

(joint work with G Kemper and RJ Shank)

Using D Rees’ definition

� � � � � �� � � � � � � � �� � � � ��

� � � � � � �� � � � � � � � � � � � � � � � � �

one can approach computation of viaEllingsrud - Skjelbred spectral sequence (1980):

Some Aspects of ModularInvariant Theoryof Finite Groups – p.33/35

Page 88: Some Aspects of Modular Invariant Theory of Finite Groups ·

Calculation of � � ��

(joint work with G Kemper and RJ Shank)

Using D Rees’ definition

� � � � � �� � � � � � � � �� � � � ��

� � � � � � �� � � � � � � � � � � � � � � � � �

one can approach computation of � � � � � �� � � �

viaEllingsrud - Skjelbred spectral sequence (1980):

Some Aspects of ModularInvariant Theoryof Finite Groups – p.33/35

Page 89: Some Aspects of Modular Invariant Theory of Finite Groups ·

Calculation of � � ��

(joint work with G Kemper and RJ Shank)

Using D Rees’ definition

� � � � � �� � � � � � � � �� � � � ��

� � � � � � �� � � � � � � � � � � � � � � � � �

one can approach computation of � � � � � �� � � �

viaEllingsrud - Skjelbred spectral sequence (1980):

� �� �� � � � � �

� �� � � � �� � � � �� � � �

Some Aspects of ModularInvariant Theoryof Finite Groups – p.33/35

Page 90: Some Aspects of Modular Invariant Theory of Finite Groups ·

Calculation of � � ��

(joint work with G Kemper and RJ Shank)

Let

� � � height of

� � � � � � � �

and

�� � � � � �� � � � � � � �� � � � � � �

(cohomologicalconnectivity).

Theorem 1

equality, if for some

( call flat, if equality holds here.)

Some Aspects of ModularInvariant Theoryof Finite Groups – p.34/35

Page 91: Some Aspects of Modular Invariant Theory of Finite Groups ·

Calculation of � � ��

(joint work with G Kemper and RJ Shank)

Let

� � � height of

� � � � � � � �

and

�� � � � � �� � � � � � � �� � � � � � �

(cohomologicalconnectivity).

Theorem 1

equality, if for some

( call flat, if equality holds here.)

Some Aspects of ModularInvariant Theoryof Finite Groups – p.34/35

Page 92: Some Aspects of Modular Invariant Theory of Finite Groups ·

Calculation of � � ��

(joint work with G Kemper and RJ Shank)

Let

� � � height of

� � � � � � � �

and

�� � � � � �� � � � � � � �� � � � � � �

(cohomologicalconnectivity).

Theorem 1

� � � � � �� � � � � � � � � � � �� � ��

equality, if for some

( call flat, if equality holds here.)

Some Aspects of ModularInvariant Theoryof Finite Groups – p.34/35

Page 93: Some Aspects of Modular Invariant Theory of Finite Groups ·

Calculation of � � ��

(joint work with G Kemper and RJ Shank)

Let

� � � height of

� � � � � � � �

and

�� � � � � �� � � � � � � �� � � � � � �

(cohomologicalconnectivity).

Theorem 1

� � � � � �� � � � � � � � � � � �� � ��equality, if

� � � � �

for some� � � � � � � � �� � �

( call flat, if equality holds here.)

Some Aspects of ModularInvariant Theoryof Finite Groups – p.34/35

Page 94: Some Aspects of Modular Invariant Theory of Finite Groups ·

Calculation of � � ��

(joint work with G Kemper and RJ Shank)

Let

� � � height of

� � � � � � � �

and

�� � � � � �� � � � � � � �� � � � � � �

(cohomologicalconnectivity).

Theorem 1

� � � � � �� � � � � � � � � � � �� � ��equality, if

� � � � �

for some� � � � � � � � �� � �

� �� � � � � � � � � � � � � � � � � � � � �� � � � � ��

( call

� �

flat, if equality holds here.)

Some Aspects of ModularInvariant Theoryof Finite Groups – p.34/35

Page 95: Some Aspects of Modular Invariant Theory of Finite Groups ·

Calculation of � � ��

(joint work with G Kemper and RJ Shank)

Let

� � � height of

� � � � � � � �

and

�� � � � � �� � � � � � � �� � � � � � �

(cohomologicalconnectivity).

Theorem 2

If

is � - nilpotent with cyclic� � ��� ��

� � � � � �

flat.

� � � � � � � � � � �� � �Cohen - Macaulay + flat

If � � � � � � � � � �, then:

� � � � � �� � � � � � � � � � �� � � ��

Some Aspects of ModularInvariant Theoryof Finite Groups – p.35/35


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