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Some Aspects of Modular Invariant Theory of 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

Some Aspects of Modular Invariant Theory of Finite Groups ·

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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

Page 2: Some Aspects of Modular Invariant Theory of Finite Groups ·

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

Page 3: Some Aspects of Modular Invariant Theory of Finite Groups ·

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

Page 4: Some Aspects of Modular Invariant Theory of Finite Groups ·

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

Page 5: Some Aspects of Modular Invariant Theory of Finite Groups ·

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

Page 6: Some Aspects of Modular Invariant Theory of Finite Groups ·

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

Page 7: Some Aspects of Modular Invariant Theory of Finite Groups ·

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

Page 8: Some Aspects of Modular Invariant Theory of Finite Groups ·

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

Page 9: Some Aspects of Modular Invariant Theory of Finite Groups ·

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

Page 10: Some Aspects of Modular Invariant Theory of Finite Groups ·

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

Page 11: Some Aspects of Modular Invariant Theory of Finite Groups ·

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

Page 12: Some Aspects of Modular Invariant Theory of Finite Groups ·

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

Page 13: Some Aspects of Modular Invariant Theory of Finite Groups ·

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

Page 14: Some Aspects of Modular Invariant Theory of Finite Groups ·

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

Page 15: Some Aspects of Modular Invariant Theory of Finite Groups ·

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

Page 16: Some Aspects of Modular Invariant Theory of Finite Groups ·

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

Page 17: Some Aspects of Modular Invariant Theory of Finite Groups ·

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

Page 18: Some Aspects of Modular Invariant Theory of Finite Groups ·

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

Page 19: Some Aspects of Modular Invariant Theory of Finite Groups ·

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

Page 20: Some Aspects of Modular Invariant Theory of Finite Groups ·

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

Page 21: Some Aspects of Modular Invariant Theory of Finite Groups ·

Relative Noether Bound

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

If

� � �

,

� � � � ��

, or

�� �

with

�� � � ��, then

� � � � � � � � � � � � �� � �

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

Page 22: Some Aspects of Modular Invariant Theory of Finite Groups ·

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

Page 23: Some Aspects of Modular Invariant Theory of Finite Groups ·

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

Page 24: Some Aspects of Modular Invariant Theory of Finite Groups ·

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

Page 25: Some Aspects of Modular Invariant Theory of Finite Groups ·

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

Page 26: Some Aspects of Modular Invariant Theory of Finite Groups ·

Known modular degree bounds

� � � � � � � � �

;

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

, (Derksen - Kemper)

� � � �� � � � � �� � � �

, (Karaguezian - Symonds),

here � � � � ��

For permutation representations

� � �� :

� � � � � � �� �

� � � � � � � � � � �� ��

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

Page 27: Some Aspects of Modular Invariant Theory of Finite Groups ·

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

Page 28: Some Aspects of Modular Invariant Theory of Finite Groups ·

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

Page 29: Some Aspects of Modular Invariant Theory of Finite Groups ·

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

Page 30: Some Aspects of Modular Invariant Theory of Finite Groups ·

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

Page 31: Some Aspects of Modular Invariant Theory of Finite Groups ·

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

Page 32: Some Aspects of Modular Invariant Theory of Finite Groups ·

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

Page 33: Some Aspects of Modular Invariant Theory of Finite Groups ·

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

Page 34: Some Aspects of Modular Invariant Theory of Finite Groups ·

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

Page 35: Some Aspects of Modular Invariant Theory of Finite Groups ·

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

Page 36: Some Aspects of Modular Invariant Theory of Finite Groups ·

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

Page 37: 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 degreesuch that

If , then .

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

Page 38: 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 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