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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Relative Noether Bound
Theorem (Fl., F Knop, M Sezer, (indep.)):
If
� � �
,
� � � � ��
, or
�� �
with
�� � � ��, then
� � � � � � � � � � � � �� � �
Some Aspects of ModularInvariant Theoryof Finite Groups – p.8/35
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
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
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
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
Known modular degree bounds
� � � � � � � � �
;
� � � � � � � � � � �� � �� � �� � � � �� � ��
, (Derksen - Kemper)
� � � �� � � � � �� � � �
, (Karaguezian - Symonds),
here � � � � ��
For permutation representations
� � �� :
� � � � � � �� �
� � � � � � � � � � �� ��
Some Aspects of ModularInvariant Theoryof Finite Groups – p.13/35
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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