Symmetric Algebras and the Depth of Invariant Rings

A thesis submitted for the degree of Doctor of Philosophy

Jonathan Elmer

Department of Mathematics, Statistics and Actuarial Science University of Kent Canterbury, Kent

CT2 7NF

7 June 2007

Abstract

Let G be a finite group acting linearly on a finite-dimensional vector space V over a field k. We assume that the characteristic of k is a prime number p which divides |G|, so that V is a modular representation of G, and define a ring R := S(V^{∗}) on which G acts by graded algebra homomorphisms. Let P ∈ Syl_{p}(G). Following Fleischmann, Kemper and Shank we say that V is flat if depth(R^{G}) = min{dim(V), dim(V^{P}) + m + 1} where m := min{i > 0 : H^{i}(G, R) ≠ 0}. This is in fact the lower bound for depth(R^{G}).

In chapter one we set up the necessary machinery for the study of the depth of invariant rings. In chapter two we justify and define the notion of strongly flat kG-modules, and find necessary and sufficient conditions for a modular representation of a p-group to be strongly flat. The third chapter is devoted to finding strongly flat representations of p-groups. In particular we prove that every nonprojective indecomposable modular representation of the group C_{2} × C_{2} is flat, and using general facts about the symmetric algebra find a more general result for decomposable representations of this group. We also apply our results to the dihedral 2-groups.

In the fourth and fifth chapters we turn our attentions to groups which are not p-groups, viewing them as extensions of a particular Sylow-p-subgroup P < G. It follows easily from the definitions that for V a kG-module, V_{|P} is strongly flat if V is. In this chapter we give a number of examples of groups where the converse is also true. In particular we are able to classify the strongly flat representations of the dihedral groups D_{4n} for n odd. Using a different approach we classify the strongly flat representations of the alternating groups Alt_{4} and Alt_{5}, and show that the converse does not hold for these groups.

In the fifth chapter, we investigate a spectral sequence used by Fleischmann, Kemper and Shank to prove a key result in [21]. We obtain a slightly more general form of their result and use this to suggest a technique for calculating the depth of some invariant rings which are not flat.

Contents

Abstract i

List of Tables iii

Acknowledgements iv

List of Invariant Rings Described v

1 Introduction 1 1.1 Invariant Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Depth and Regular Sequences . . . . . . . . . . . . . . . . . . . . . 8 1.3 Symmetric Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.4 History of the Depth Problem . . . . . . . . . . . . . . . . . . . . . 17 1.5 Cohomology and Derivations . . . . . . . . . . . . . . . . . . . . . 19 1.6 The Relative Transfer Ideal . . . . . . . . . . . . . . . . . . . . . . 31

2 Flatness and Strong Flatness 36 2.1 A Lower Bound for Depth . . . . . . . . . . . . . . . . . . . . . . . 36 2.2 Necessary and Sufficient Conditions for Flatness . . . . . . . . . . 37 2.3 Strong Flatness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.4 The Rôle of Cohomological Connectivity . . . . . . . . . . . . . . . 41

3 Flatness for p-groups 44 3.1 The p-group Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.2 A Sufficient Condition . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.3 Groups of the Form C_{p} × C_{p} . . . . . . . . . . . . . . . . . . . . . . 48 3.4 Decomposable Representations . . . . . . . . . . . . . . . . . . . . 53 3.5 Dihedral 2-Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4 Extensions of p-Groups 74 4.1 Stable Cohomology and Strong Flatness . . . . . . . . . . . . . . . 74 4.2 Trivial Intersection Groups . . . . . . . . . . . . . . . . . . . . . . 76 4.3 p-Nilpotent Groups with Cyclic Sylow-p-subgroup . . . . . . . . . . 77 4.4 Direct Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5 Some groups with Sylow-2-subgroup P ≅ C_{2} × C_{2} 84 5.1 Dihedral Groups G := D_{4n} (n odd) . . . . . . . . . . . . . . . . . . 85 5.2 The Alternating Group Alt_{4} . . . . . . . . . . . . . . . . . . . . . . 89

5.2.1 Decomposable Representations . . . . . . . . . . . . . . . . 114

6 A Spectral Sequence 121 6.1 Setup and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 121 6.2 Almost Flat Representations . . . . . . . . . . . . . . . . . . . . . 127

7 Appendix: Classification of Representations of Alt_{4} 129

Bibliography 140

Index of Symbols 142

List of Tables

3.1 Decomposing Tensor Products of kP -modules . . . . . . . . . . . . 62 3.2 Flatness Data for String Modules . . . . . . . . . . . . . . . . . . . 71 3.3 Flatness Data for String Modules (cont.) . . . . . . . . . . . . . . . 72 3.4 Flatness Data for Band Modules . . . . . . . . . . . . . . . . . . . 73

5.1 Decomposing Tensor Products of kG-modules . . . . . . . . . . . . 116

Acknowledgements

The author would like to thank: Peter Fleischmann, my supervisor, for myriad corrections and for never being too busy to see me; Stuart Martin, my former director of studies at Cambridge, for getting me interested in algebra in the first place, others in the mathematical community inside and outside Kent, including, but not limited to, Jim Shank and Karin Erdmann, for various helpful comments and suggestions; friends too numerous to mention for keeping me sufficiently distracted so as to retain my sanity; and lastly, my parents, for their unending love and support.

List of Invariant Rings Described

Over the course of this work we will describe the structure (in terms of primary and secondary invariants) of several rings of invariants. Since these calculations are likely to be of independent interest, we provide a list of them here. Let V be a left kP -module, where P is the Klein four-group C_{2} × C_{2} generated by X and Y, and k is any perfect field of characteristic two. Let R := S(V^{∗}). Then R^{P} is described for the following modules V:

• V ≅ \overline{V}_{4}. This is the projective indecomposable module for P. R can be viewed as the ring k[x_{1}, x_{2}, x_{3}, x_{4}] on which the the generators X and Y of P act as the permutations (12)(34) and (13)(24) respectively. See page 54.

• V ≅ V_{3}. This is the representation in which X and Y act as the matrices

( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{matrix} ) , ( \begin{matrix}1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} )

respectively. See page 57.

• V ≅ V_{−3}. This is the representation in which X and Y act as the matrices

( \begin{matrix}1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) , ( \begin{matrix}1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} )

respectively. See page 58.

• V_{2,0}⊕V_{2,∞}⊕V_{2,1}. In this case R can be viewed as the ring k[x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}] on which X acts as the permutation (12)(56) and Y acts as (34)(56). See page 108.

Chapter 1

Introduction

In this first chapter, we aim to give some of the necessary background work needed to understand later chapters. We will explain each part of the thesis title in turn.

1.1 Invariant Rings

Invariant theory is usually thought of as the study of group actions on polynomial rings. Let k be any field and define R := k[x_{1}, x_{2} …, x_{n}]. R is endowed with a natural grading defined by the usual notion of monomial degree, and elements of R which are sums of monomials of R of the same degree are called homogenous elements. Let G be a group which acts on R so as to preserve multiplication and addition. Then the we define the ring of invariants R^{G} as follows:

Definition 1.1.

R^{G} := {r ∈ R : rg = r ∀g ∈ G}

It is easy to see that R^{G} is again a ring. Given a finite group G and a finite left kG-module V, we associate with it a ring as follows: let {x_{1}, x_{2}, … x_{n}} be a basis for the dual module V^{∗}, and let R := k[x_{1}, x_{2} …, x_{n}]. Then R has a natural right kG-module structure given by extending the action of G on V^{∗} by algebra automorphisms, and in fact R = S(V^{∗}), the symmetric algebra of V^{∗}, which we define formally in section 1.3.

Throughout this work we adopt the convention that V is a left kG-module and V^{∗} a right kG-module, instead of the alternative construction where V^{∗} becomes a left kG-module with action (g ·ϕ)(v) := ϕ(g^{−1}v). It is worth remarking that there is no reason we could not define R := S(V), and study the natural left action of G on R. Indeed, introducing the dual module seems an unnecessary extra complication. To understand the reasons for adopting this construction, consider the more general case where V is an algebraic variety over k. If k is algebraically closed, then the ring of functions O(V) is isomorphic to R := S(V^{∗}). If G acts on V, then we may define the quotient variety V/G, and its ring of functions O(V/G) is the invariant ring R^{G}. This is one motivation for studying rings of invariants.

The goal of invariant theory is to say something about the structure of R^{G} as a ring. We begin with an example which should be familiar to all readers. Let G be the symmetric group Σ_{n} and let V be the natural ℂG-module. Then R := ℂ[x_{1}, x_{2}, …, x_{n}], and the action of G on V^{∗} is given by x_{i}σ = x_{σ(i)}, with the action of G on R given in the usual way by algebraic extension. A polynomial in R which is invariant under this action is usually called a symmetric polynomial. Define (for i ≤ n) e_{i}, the ith elementary symmetric polynomial, as follows:

e_{i} := ∑_{1≤r_{1}<r_{2}<...<r_{i}≤n} ∏_{j=1}^{i} x_{r_{i}}

It is well known (see [32]) that every symmetric polynomial may be written as a polynomial in the elementary symmetric polynomials, and moreover this expression is unique. The uniqueness is equivalent to saying that there is no polynomial p with p(e_{1}, e_{2}, … e_{n}) = 0. We may express this by saying that

R^{G} = ℂ[e_{1}, e_{2}, …, e_{n}]

i.e. the ring of symmetric polynomials is a polynomial ring over ℂ generated by the elementary symmetric polynomials.

Not all rings of invariants are polynomial rings, as is shown by the following simple example: let G ≅ C_{2} := ⟨g|g^{2} = 1⟩ and let R := ℂ[x_{1}, x_{2}] with the action of G given by

x_{1}g = −x_{1}, x_{2}g = −x_{2}

Then R^{G} is generated by the invariants x_{1}^{2}, x_{2}^{2} and x_{1}x_{2}. However, R^{G} is not a polynomial ring, because x_{1}^{2}x_{2}^{2} − (x_{1}x_{2})^{2} = 0.

The linear groups whose invariants are polynomial have been classified. The classification in characteristic zero is a well-known result, due to Chevalley [9], Sheppard and Todd [38], and states that R^{G} is a polynomial ring if and only if G ≤ GL_{n}(k) is generated by pseudo-reflections. More recent work of Nakajima [34] tells us that a similar result holds for representations of p-groups over finite fields of order p, and a result due to Kemper and Malle [30] tells us that a similar result again holds for irreducible representations in finite characteristic.

Although the classification of rings of invariants which are polynomial rings is an unsolved problem, we do have necessary and sufficient conditions for a ring to be polynomial, due to Kemper (see [13]):

Theorem 1.1.1 ([29], Proposition 16). Let V be a kG-module of dimension n and R := S(V^{∗}). Let a_{1}, a_{2}, …, a_{n} ∈ R^{G} be a set of homogenous invariants with d_{i} := deg(a_{i}). Then the following are equivalent:

1. R^{G} = k[a_{1}, a_{2}, …, a_{n}];

2. The a_{i} are algebraically independent over k and ∏_{i=1}^{n} d_{i} = |G|.

Now that we have established that not every ring of invariants is polynomial, it is natural to ask the following question: how “close” is R^{G} to being a polynomial ring? It is easy to see that R^{G} must contain a polynomial ring: firstly, R^{G} is not zero, since if x ∉ R^{G}, 0 ≠ N(x) := ∏_{g∈G} x · g ∈ R^{G}; further, given any r ∈ R^{G}, k[r] ≤ R^{G}. Suppose A is a maximal polynomial subring of R^{G}. We may then view R^{G} as a module over A, but R^{G} may not be finitely generated. However, the following theorem tells us that we can always find an appropriate A such that R^{G} is a finitely generated module over A.

Theorem 1.1.2 (Noether’s Normalisation Theorem, graded form). Let A be a Noetherian graded ring, and let M be a finite A-module of (Krull) dimension m. Then there is a set of homogenous elements of positive degree a_{1}, a_{2} …, a_{m} ∈ A with the following properties:

1. dim(M/(a_{1}, a_{2}, …, a_{m})M) = 0 and the set {a_{1}, a_{2} …, a_{m}} is minimal with this property;

2. M is finitely generated as a module over A := k[a_{1}, … a_{m}].

Now suppose that A := R^{G}, where G acts linearly on the polynomial ring R := k[x_{1}, x_{2}, …, x_{n}], and that M = R^{G} is the regular R^{G}-module. Then the above applies with m = n.

Proof. The proof is by induction on dim(M) = m. Suppose first that m=0. Then (1) is vacuous, and M is finitely generated over k because M is finite. So suppose m > 0. Define

S = {P ∈ Ass_{A}(M) : dim(A/P) = m}.

By the prime avoidance theorem, (see for example [16], Theorem 3.1), there is a nonzero homogenous a_{1} ∈ A\ ∪_{P ∈S} P of positive degree. Let Q ∈ Ass_{A}(M/a_{1}M). Then I claim dim(A/Q) < m. Otherwise since a_{1} ∈ Q, Q ∉ S, but dim(A/Q) = m so Q ∉ Ass_{A} M. Since Q ∈ Ass_{A} M/a_{1}M there exists z ∈ M such that

Q = {r ∈ A : rz ∈ a_{1}M}.

Let Q′ := Ann_{A}(z). Then Q′ < Q and hence dim(A/Q′) > dim(A/Q). Since dim(M) is an upper bound for the dimensions of the associated primes of M, m > dim(A/Q) as per claim.

It follows that dim(M/a_{1}M) = max{dim(P) : P ∈ Ass(M/a_{1}M)} := l < m, and by induction there is a minimal set {a_{2}, a_{3} …, a_{l+1}} with 0 = dim((M/a_{1}M)/(a_{2}, a_{3}, …, a_{l+1})(M/a_{1}M)) = dim(M/(a_{1}, a_{2}, …, a_{l+1})M).

To see that this set is minimal with the stated property, consider the set {a_{1}, a_{2}, … â_{i}, …, a_{l+1}}, where we omit a_{i}. Suppose first that i ≠ 1. Then if

0 = dim(M/(a_{1}, a_{2}, … â_{i}, …, a_{l+1})M)

we have

0 = dim(M/a_{1})/((a_{2}, … â_{i}, …, a_{l+1})(M/a_{1}))

contradicting the minimality of the set {a_{2}, a_{3} …, a_{l+1}}. Now let i = 1. If

0 = dim(M/(a_{2}, a_{3}, …, a_{l+1})M)

then we must have dim(M) = l < m, contrary to our assumptions. Thus the set {a_{1}, a_{2}, …, a_{l+1}} is minimal as required and we have in fact m = l + 1. Furthermore, by induction M/a_{1}M is finitely generated over k[a_{2}, …, a_{m}], and I claim that M is finitely generated over A := k[a_{1}, a_{2}, …, a_{m}]. Let {b_{1}, b_{2}, …, b_{s}} be module generators for M/a_{1}M over k[a_{2}, …, a_{m}] and let x ∈ M be homogenous of minimal degree such that x ∉ A(b_{1}, b_{2}, …, b_{s}). Then there are homogenous polynomials {f_{1}, f_{2}, …, f_{s}} in k[a_{2}, … a_{m}] such that

x − ∑_{i=1}^{s} b_{i}f_{i} = a_{1}y

for some y ∈ M, and deg(y) < deg(x). By the minimality of deg(x) we may find polynomials {g_{1}, g_{2}, …, g_{s}} ∈ A such that

y = ∑_{i=1}^{s} b_{i}g_{i},

and so

x − ∑_{i=1}^{s} b_{i}f_{i} = a_{1} ∑_{i=1}^{s} b_{i}g_{i}

x = ∑_{i=1}^{s} b_{i}(f_{i} + a_{1}g_{i}) ∈ A(b_{1}, b_{2}, …, b_{s}).

This contradiction completes the proof of (2).

Finally suppose that G acts linearly on the polynomial ring R := k[x_{1}, x_{2}, …, x_{n}]. Let y be an indeterminate and consider the monic polynomial

ϕ_{x}(y) = ∏_{g∈G}(y − xg) ∈ R^{G}[y].

For each x ∈ R, x is a root of ϕ_{x}, and so R is an integral extension of R^{G}. In particular, we can find invariants {b_{i,1}, b_{i,2}, …, b_{i,m_{i}}} such that

x_{i}^{r_{i}} + b_{i,1}x_{i}^{r_{i}−1} + … + b_{i,r_{i}−1}x_{i} + b_{i,r_{i}}.

Let B be the subalgebra of R^{G} generated by the set {b_{i,j} : 1 ≤ i ≤ n, 1 ≤ j ≤ r_{i}}. Then R is integral over B, so R is generated as a B-module by finitely many monomials. Since R is Noetherian it follows that R^{G} is also a finitely generated B-module, and since B is a subalgebra of R^{G}, R^{G} is a Noetherian ring. So by part (1) of the theorem, there is a set

{a_{1}, a_{2}, …, a_{m}}

such that R^{G} is finitely generated over the subalgebra A := k[a_{1}, a_{2}, …, a_{m}], where m = dim(R^{G}) ≤ n. But R is finitely generated over B and hence over R^{G}, so R is also finitely generated over A. Again by part (1) of the theorem, we must have m = n.

The set {a_{1}, a_{2}, …, a_{n}} above is usually called a homogenous system of parameters, or hsop for short. When M := R^{G} these elements may be called primary invariants, and the generators of R^{G} over A secondary invariants. Often we will write

R^{G} = k[a_{1}, a_{2}, …, a_{n}](b_{1}, …, b_{r})

to mean {a_{1}, a_{2}, …, a_{n}} is an hsop for R^{G} and {b_{1}, …, b_{r}} a set of secondary invariants. This is a construction of great practical importance as it allows us, in principal, to describe invariant rings in terms of generators and syzygies. The projective dimension of R^{G} over A can be considered as a measure of the complexity of R^{G}, or indeed, as a measure of how far R^{G} is from being polynomial.

A ring which is free (of finite rank) over an hsop is called a Cohen-Macaulay ring. In section 1.2 we will define the depth of a ring, and show that R^{G} is Cohen-Macaulay if and only if depth(R^{G}) = dim(R^{G}). From this result it follows that if R^{G} is free over some hsop, it is free over any hsop, and so the notion of a Cohen-Macaulay ring is well defined. Like polynomial rings, the classification of rings of invariants which are Cohen-Macaulay is an unsolved problem, although we do have necessary and sufficient conditions for R^{G} to be Cohen-Macaulay, formulated by Kemper [28] in terms of primary and secondary invariants as follows:

Theorem 1.1.3 ([13], Theorem 3.7.1). Let R^{G} be a ring of invariants with hsop {a_{1}, a_{2}, …, a_{n}}. Define d_{i} := deg(a_{i}), and let the minimal size of a set of secondary invariants be r. Then

r ≥ ∏_{i=1}^{n} d_{i}/|G|

with equality if and only if R^{G} is Cohen-Macaulay.

We will use this result extensively in later chapters. We conclude this section with a result which completely answers our question for a certain special class of invariant rings:

Theorem 1.1.4. Let G be a finite group acting linearly on a polynomial ring R := k[x_{1}, x_{2} …, x_{n}]. Suppose that N < G and R^{N} is a Cohen-Macaulay ring. Suppose also that the characteristic of k does not divide |G|/|N|. Then R^{G} is a Cohen-Macaulay ring.

Proof. We define the Reynolds operator E : R^{N} → R^{G} by

E(r) := (|G|/|N|)^{−1} ∑_{g∈S} rg,

where S is a right-transversal of N in G. The restriction of E to R^{G} is then trivial, and so we have a splitting

R^{N} = R^{G} ⊕ ker(E).

Let A be a maximal polynomial subalgebra of R^{G} generated by an hsop. Now we know that R^{N} is a Cohen-Macaulay ring, and so R^{N} is finitely generated and free over A. Therefore the direct summand R^{G} is projective as a module over A. But A is a graded ring, and so every graded projective module over A is free (see [5], Lemma 4.1.1). Hence R^{G} is free over A and so is Cohen-Macaulay.

Note that applying the above with N = 1 shows us that R^{G} is Cohen-Macaulay when |G| is not divisible by the characteristic of k, an original result of Eagon and Hochster [15]. R^{G} is called a “modular” ring of invariants if the characteristic of k divides |G| and “non-modular” otherwise. So we have shown that every nonmodular ring of invariants is Cohen-Macaulay. We will see in later sections that the modular case is far less clear cut.

1.2 Depth and Regular Sequences

In this section we give several equivalent definitions of depth, and explain why we are interested in calculating the depth of invariant rings. Throughout this section R will denote a graded connected Noetherian ring over a field k, that is, a positively graded Noetherian ring ⨁_{i} R_{i} with R_{0} ≅ k, and unique maximal homogenous ideal R_{+} generated by homogenous elements of positive degree. Although all of our results are defined in this generality, we have in mind the setting where R := S(V^{∗}) is a polynomial ring.

Definition 1.2. Let R be a graded connected Noetherian ring, and let M be an R-module. Let I be an ideal of R with IM ≠ M. A sequence {y_{1}, y_{2}, …, y_{m}} of elements of I is called a M-Regular Sequence if the following conditions hold:

1. The map M → M given by multiplication by y_{1} is injective;

2. For every r ≤ m, the map

M/(y_{1}, y_{2} …, y_{r−1})M → M/(y_{1}, y_{2} …, y_{r−1})M

induced by multiplication by y_{r} is injective.

We call a regular sequence maximal if it cannot be extended to a longer regular sequence. Under the assumption that R is Noetherian, every maximal regular sequence has the same length. This allows us to make the following definitions:

Definition 1.3. The grade of I on M, denoted grade(I, M), is the length of a maximal M-regular sequence in I.

We extend this definition by saying that grade(I, M) = ∞ if IM = M.

Definition 1.4. Let R be a graded connected Noetherian ring. Then for any R-module M

depth(M) := grade(R_{+}, M).

There are a number of equivalent formulations for grade and depth.

Theorem 1.2.1 (Rees, [8], Theorem 1.2.5, and [7]). . Let R be a graded connected Noetherian ring, I an ideal of R and let M be an R-module such that IM ≠ M. Then the following integers are equal:

1. The grade of I on M;

2. min{i : Ext_{R}^{i}(R/I, M) ≠ 0}; 3. min{i : H_{I}^{i}(M) ≠ 0}.

The functor H_{I}^{i}(−) used in condition (3) is called a local cohomology functor. We include this result for historical reasons only — this is the formulation of depth used in the spectral sequence studied by Ellingsrud and Skjelbred in [17]. Since we shall make no further use of it, we shall not attempt to prove here its equivalence to (1) and (2). The definition of local cohomology, and the proof of this result, may be found in [7].

Proof. We prove the equivalence of (1) and (2) by induction on the grade of I on M. Suppose grade(I, M) = 0. Then I consists entirely of zero-divisors of M. The prime avoidance theorem (see, for example [8]) tells us that I is contained in an associated prime of M, that is,

I ⊂ Ann_{R}(z) ⊂ R

for some non-zero z ∈ M. The map ψ_{z} : R → M defined by

ψ_{z}(r) := rz

is a nonzero element of Hom_{R}(R, M) with I in the kernel, and therefore

Ext_{R}^{0}(R/I, M) = Hom_{R}(R/I, M) ≠ 0

as required.

Now suppose that grade(I, M) = m and {y_{1}, y_{2} …, y_{m}} is a maximal Mregular sequence in I. Then y_{1} is not a zero-divisor, and by definition {y_{2}, y_{3} …, y_{m}} is a maximal M/y_{1}M-sequence in I. Therefore grade(I, M/y_{1}M) = m − 1 and by induction

Ext_{R}^{i}(R/I, M/y_{1}M) =  \{ \begin{matrix}= 0 & i < m − 1, \\ ≠ 0 & i = m − 1. \end{matrix}

Now there is a short exact sequence

0 → M →^{y_{1}} M → M/y_{1}M → 0

which induces a long exact sequence

… → Ext_{R}^{i−1}(R/I, M/y_{1}M) → Ext_{R}^{i}(R/I, M) →^{y_{1}^{∗}} Ext_{R}^{i}(R/I, M) → …

The map y_{1}^{∗} is the map induced on Ext_{R}^{i}(R/I, M) by multiplication by y_{1}. But y_{1} ∈ I ⊆ Ann_{R}(R/I), and so this is the zero map. It follows that, for each i ≤ m,

Ext_{R}^{i}(R/I, M) ≅ Ext_{R}^{i−1}(R/I, M/y_{1}M)

Therefore min{i : Ext_{R}^{i}(R/I, M) ≠ 0} = min{i : Ext_{R}^{i−1}(R/I, M/y_{1}M) ≠ 0} = m by induction, since grade(I, M/y_{1}M) = m − 1. This gives us the required result.

Corollary 1.2.2. Let R be a graded connected Noetherian ring on and M a finitely-generated R-module. Then

depth(M) = min{i : Ext_{R}^{i}(k, M) ≠ 0}.

Our next result explains the connection between the depth and structural complexity of invariant rings.

Theorem 1.2.3 (Auslander-Buchsbaum, ([8], Theorem 1.3.3)). Let R be a graded connected Noetherian ring and let M be a finitely generated R-module with finite projective dimension. Then

depth(M) + projdim(M) = depth(R).

Proof. We prove this by induction on r := projdim(M). If r = 0, then M is a finitely generated projective R-module, and since R is a local ring, M is free (see, for example [37], Proposition 11.23 - alternatively this follows from [5], Lemma 4.1.1 which we used before), say M ≅ ∑_{j=1}^{m} R_{j} where each R_{j} ≅ R. Therefore

Ext_{R}^{i}(k, M) = ⨁_{j=1}^{m} Ext_{R}^{i}(k, R_{j})

which implies depth(M) = depth(R) as required.

Now suppose r > 0. So M is neither projective nor free, but we can find a resolution

0 → K → F → M → 0

where F is a finitely generated free R-module, and projdim(K) = r − 1. This induces a long exact sequence

… Ext_{R}^{i−1}(k, F) → Ext_{R}^{i−1}(k, M) → Ext_{R}^{i}(k, K) → Ext_{R}^{i}(k, F) …

Since F is free we can write in a similar way to before F ≅ ⊕_{j=1}^{l}R_{j} where each R_{j} ≅ R, for some l. Then Ext_{R}^{i}(k, F) ≅ ⊕_{j=1}^{l} Ext_{R}^{i}(k, R), and so depth(F) = depth(R). It follows that for each 0 ≤ i < depth(R) we have

Ext_{R}^{i}(k, F) = 0.

Therefore for all 0 < i < depth(R)

Ext_{R}^{i−1}(k, M) ≅ Ext_{R}^{i}(k, K).

Now since projdim(K) = r − 1 we have by induction

depth(K) = depth(R) − r + 1,

so by 1.2.1

0 = Ext_{R}^{i}(k, K) = Ext_{R}^{i−1}(k, M)

for each 0 < i < depth(R) − r + 1, with

0 ≠ Ext_{R}^{depth(R)−r+1}(k, K) = Ext_{R}^{depth(R−r)}(k, M).

Therefore depth(M) = depth(R) − r as required.

It should now be clear why we are interested in calculating the depth of invariant rings. Let R := k[x_{1}, x_{2} …, x_{n}] be a polynomial ring on which G acts. Clearly the depth of a polynomial ring is equal to its number of generators. Then Theorem 1.1.2 tells us that R^{G} is a finitely generated module over a maximal polynomial subring A spanned by a homogenous system of parameters, and by Theorem 1.2.3 the projective dimension of R^{G} over A is equal to n − depth(R^{G}). So depth is an effective measure of the structural complexity of R^{G}. This has applications in computer algebra too - algorithms such as those used in MAGMA for computing invariant rings tend to be quite efficient when R^{G} is a Cohen-Macaulay ring, but can become less so when the depth of R^{G} is especially small.

In practise, calculating the depth of invariant rings is an extremely difficult problem. Although there exist routines which calculate depth directly by finding maximal regular sequences, these tend to take a lot of computing power, especially when the dimension n of R is large. For this reason, we prefer to look for general principles which allow us to find the depth immediately given certain grouptheoretic or representation-theoretic properties of G.

We conclude this section with a result which we will find extremely useful in later sections:

Lemma 1.2.4 ([13], Lemma 3.7.2). Let M be an R-module for some graded connected algebra R over a field k, and let l be an integral extension of k. Define \overline{M} := M ⊗_{k} l). and S := R ⊗_{k} l. Then

depth_{R}(M) = depth_{S}(\overline{M}).

Proof. Clearly S is integral over R, and

grade(R_{+}, M) ≤ grade(S_{+}, \overline{M}),

so it is enough to prove the reverse inequality. Let {y_{1}, …, y_{m}} ∈ R_{+} be a maximal M-regular sequence. Then every element of R_{+} is a zero divisor of

M/(y_{1}, …, y_{m})M

and hence R_{+} is contained in the union of associated primes of the above. It follows that the ideal R_{+}S of S lies in some associated prime P of

\overline{M}/(y_{1}, …, y_{m})\overline{M}.

But since S is integral over R, only one prime ideal contains R_{+}S, namely S_{+}. Hence P = S_{+} and \{y_{1}, …, y_{m}\} is a maximal \overline{M}-sequence in S_{+}.

Later on we will be interested in calculating depth(R^{G}) where the group G is fixed and R := S(V^{∗}), where V is any indecomposable representation of G. The above lemma effectively allows us to assume that k is algebraically closed and V is absolutely indecomposable.

1.3 Symmetric Algebras

Let V be a kG-module. Then there is a natural action of G on the rth tensor power T^{r}(V) of V given by

g(v_{1} ⊗ v_{2} ⊗ … ⊗ v_{r}) = (gv_{1} ⊗ gv_{2} ⊗ … ⊗ gv_{r})

for each set {v_{1}, v_{2}, …, v_{r}} ∈ V^{r} and by extending linearly. Define T (V) := ∑_{r≥0} T^{r}(V). There is a natural multiplication T^{r}(V) × T^{s}(V) → T^{r+s}(V) on T (V) defined by

(v_{1} ⊗ v_{2} ⊗ … ⊗ v_{r}) · (w_{1} ⊗ w_{2} ⊗ … ⊗ w_{s}) = v_{1} ⊗ v_{2} ⊗ … ⊗ v_{r} ⊗ w_{1} ⊗ w_{2} ⊗ … ⊗ w_{s}

and by extending linearly. This gives T (V) the structure of a graded algebra. Define U(V) to be the ideal of T (V) generated by all elements of the form v ⊗ w − w ⊗ v. We may define a grading on U(V) by setting U^{r}(V) := U(V) ∩ T^{r}(V).

Definition 1.5. The rth Symmetric Power of V is defined to be the quotient module

S^{r}(V) := T^{r}(V)/U^{r}(V).

The Symmetric Algebra on V is defined to be the direct sum

S(V) := ⨁_{r≥0} S^{r}(V).

The natural multiplication ⊗ : T^{r}(V)×T^{s}(V) → T^{r+s}(V) induces a multiplication on S(V) which gives S(V) the structure of a graded commutative algebra.

Suppose {v_{1}, v_{2}, …, v_{n}} is a basis for V. Then S(V) can be identified with the polynomial ring k[v_{1}, v_{2}, …, v_{n}].

Remark: In characteristic zero, one often defines the symmetric powers of V to be the submodules of T^{r}(V) generated by the symmetric tensors

∑_{σ∈Sym(n)} v_{σ(1)} ⊗ v_{σ(2)} ⊗ … ⊗ v_{σ(r)}

It can be shown that in characteristic zero this construction leads to the same notion of symmetric algebra as that above. In prime characteristic this is no longer true, and this construction leads us to a different object, called the Algebra of Divided Powers.

Our main tool for investigation will be the cohomology modules H^{1}(G, R), (with coefficients in the symmetric algebra). It is therefore in our interests to know as much as possible about the module structure of the symmetric algebra.

Perhaps the most significant result of recent times in this area is the following:

Theorem 1.3.1 (Karagueuzian and Symonds [26], Theorem 1.2). Let G be a finite group of order divisible by p and V a kG-module of dimension n, where k is a finite field of characteristic p. Then the number of isomorphism classes of indecomposable modules which are direct summands of S(V) is finite. Moreover, if G is a p-group, every isomorphism class which occurs may be found in degree less than or equal to

q^{n} + q − 1

where q = |k|.

Let W be a direct summand of S(V), and let u ∈ R^{G}. Then uW is certainly a submodule of S(V) of the same isomorphism type as W, and it may or may not be a direct summand. When it is, we say that the invariant u propagates the summand W. The principle behind this result is that every summand of S(V) of sufficiently large degree is just the propagation of a direct summand of smaller degree. This particular result has far reaching consequences - for example, one can use it to find an upper bound for the degree of generators of R^{G}. We shall make particular use of it when we classify strongly flat representations of certain groups - see, for example, Theorem 3.4.1.

Somewhat in the same spirit, albeit much easier to prove, is the following result:

Lemma 1.3.2. Let V be a kP module for a p-group P and field k of characteristic p. Then the symmetric algebra R := S(V^{∗}) splits as

R = uR ⊕ B

where the invariant u and the submodule B are described below.

It is not entirely clear to whom this lemma should be attributed. An argument similar to the one below appears in [25], but this result seems to have been known for some time.

Proof. Since P is a p-group, the only irreducible kP -module is trivial. Consequently, P has an upper triangular representation on V and R = k[x_{1}, …, x_{n}] may be viewed as k[x_{2}, …, x_{n}][x_{1}] because k[x_{2}, …, x_{n}] is a kG-module. Let B be the kG-submodule of R consisting of polynomials whose degree (as polynomials in x_{1}) is less than |P |. Consider the invariant u := ∏_{g∈P} x_{1}.g. In k[x_{2}, …, x_{n}][x_{1}], this is a monic polynomial of degree |P |. Given any r ∈ R with deg_{x_{1}}(r) ≥ |P | we may perform long division by u, giving us a unique expression r = qu + b for some q ∈ R, b ∈ B. Clearly uR ∩ B = 0 and therefore R = uR ⊕ B.

Corollary 1.3.3. If W is a direct summand of S^{l}(V^{∗}) then uW is an isomorphic direct summand of S^{l+|P |}(V^{∗}).

We will find this lemma especially useful when we come to consider direct products of groups in chapter 3. In fact we can say a little more than this. Notice how we do not specify in this lemma that V need be indecomposable. Suppose then that V ≅ V_{1} ⊕ V_{2} ⊕ … ⊕ V_{m}, and the basis x_{1}^{(1)}, x_{2}^{(1)}, x_{n(1)}^{(1)}, x_{1}^{(2)}, … x_{n(m)}^{(m)} is arranged so that the representation of each summand V_{i} is upper triangular. Then each invariant u_{i} := ∏_{g∈P} x_{1}^{(i)}g propagates every direct summand in R.

1.4 History of the Depth Problem

It follows from Theorem 1.2.3 that n := dim(R) is an upper bound for depth(R^{G}), and we have equality if and only if R^{G} is Cohen-Macaulay. It is in our interests to find a lower bound for depth(R^{G}), and ideally find necessary and sufficient conditions for its attainment. Here we list some known cases where purely grouptheoretic properties enable the calculation of depth(R^{G}) for all representations V. There are also many examples in the literature of depth calculations for natural representations of certain classical groups.

Theorem 1.4.1 (Ellingsrud and Skjelbred, 1980 [17]). For any group G with Sylow-p-subgroup P and representation V over a field of characteristic p, we have

depth(R^{G}) ≥ min{dim(V^{P}) + 2, dim(V)}.

For any cyclic p-group and representation V over a field of characteristic p, we have equality in the above.

Theorem 1.4.2 (Campbell et al, 2000 [10]). For any Abelian group G with cyclic Sylow-p-subgroup P and representation V over a field of characteristic p, we have

depth(R^{G}) = min{dim(V^{P}) + 2, dim(V)}.

Theorem 1.4.3 (Kemper, 2001 [27]). For any G with Sylow-p-subgroup P such that p^{2} does not divide |G| (the so-called “mildly modular” case) and representation V over a field of characteristic p, we have

depth(R^{G}) = min{dim(V^{P}) + 2, dim(V)}.

Theorem 1.4.4 (Fleischmann, Kemper, Shank, 2004 [21] ). For any group G with Sylow-p-subgroup P and representation V over a field of characteristic p, we have

depth(R^{G}) ≥ min{dim(V^{P}) + cc_{G}(R) + 1, dim(V)}.

If P is cyclic and G is p-nilpotent, then

depth(R^{G}) = min{dim(V^{P}) + 2, dim(V)}.

i.e. the inequality is an equality and cc_{G}(R) = 1.

In the final item, cc_{G}(R) , the cohomological connectivity, has the definition min{i > 0 : H^{i}(G, R) ≠ 0}. One should note that whilst we are mainly concerned with the right-action of G on V^{∗}, the V^{P} in the formula refers to the fixed points of P in the left-action of G on V.

In summary then, whilst all non-modular invariant rings are Cohen-Macaulay (depth as large as possible), most modular invariant rings are not. In fact, if the group structure is sufficiently simple, the depth of a modular invariant ring is very often as small as possible! For the rest of this paper we assume V is a modular representation. Representations (G, V) for which equality holds in the final item are called flat. Note that this is considered a property of V, and not of V^{∗} or R. An immediate corollary is that if cc_{G}(R) + dim(V^{P}) + 1 > dim(V), then we have both that V is flat and R^{G} is Cohen-Macaulay. Such representations will be called trivially flat.

1.5 Cohomology and Derivations

The methods we will use to find flat representations rely heavily on calculations of the first cohomology group H^{1}(G, R), and so we need a convenient way to calculate these. We use the following result:

Lemma 1.5.1.

H^{1}(G, R) ≅ \frac{Der(G, R)}{PDer(G, R)}.

Here Der(G, R) denotes the set of derivations from G to R, and PDer(G, R) the principal derivations. Derivations from G to R are maps ϕ satisfying ϕ(gh) = ϕ(h) + ϕ(g).h ^{1}, and the principal derivations are those of the form ϕ_{v} : G → R,

ϕ_{v}(X) = vX − v

for some v ∈ R.

Derivations may be specified by giving the images of the generators of G, and the fact that ϕ is a derivation will give additional restrictions on the images in R. For a proof of Lemma 1.5.1 from first principles, one could consult [40], but for the purposes of this paper we will take this as the definition of H^{1}(G, R). Let G be generated by {g_{1}, …, g_{r}}. We define [v_{1}, …, v_{r}] (with appropriate conditions on the v_{i}’s) to be the equivalence class in H^{1}(G, R) of the derivation ϕ : g_{i} ↦ v_{i}. With these definitions it is clear that H^{1}(G, R) becomes a R^{G}-module via the assignment x · [v_{1}, … v_{r}] := [xv_{1}, … xv_{r}].

1R is a right kG-module, so we make into a bimodule by giving a trivial left-action.

The purpose of this section is to derive some of the standard results of cohomology theory which we will use in later chapters, using this definition of H^{1}(G, R). Many of these results have analogues in higher cohomology groups - we may use these analogues implicitly when talking about the earlier results of others, but calculations in H^{1}(G, R) will be sufficient to prove all the new results in chapters 2–5.

Lemma 1.5.2. Let N ≤ G, α ∈ Der(N, R), and g ∈ G, X ∈ g^{−1}Ng. Define

α^{g}(X) = α(gXg^{−1})g.

Then the assignment α ↦ α^{g} induces a map H^{1}(N, R) → H^{1}(g^{−1}Ng, R). In particular if N is normal in G then we get a right action of G on H^{1}(N, R). Moreover, N acts trivially on H^{1}(N, R), so H^{1}(N, R) may be regarded as a G/Nmodule.

Proof. It is clear that α^{g} : g^{−1}Ng → R. Let X, Y ∈ g^{−1}Ng. Then

α^{g}(XY) = α(gXY g^{−1})g = α(gXg^{−1}gY g^{−1})g,

and since α ∈ Der(N, R),

α(gXg^{−1}gY g^{−1})g = α(gXg^{−1})gY g^{−1}g + α(gY g^{−1})g = α^{g}(X)Y + α^{g}(Y)

which proves that α^{g} ∈ Der(g^{−1}Ng, R). If α ∈ PDer(N, R), say α(X) = vX − v for some fixed v ∈ R, then

α^{g}(X) = α(gXg^{−1})g = (vgXg^{−1} − v)g = (vg)X − vg

so α^{g} ∈ PDer(g^{−1}Ng, R). So α → α^{g} induces a well-defined map H^{1}(N, R) → H^{1}(g^{−1}Ng). If g, h ∈ G then

(α^{g})^{h}(X) = (α^{g}(hXh^{−1}))h = α(ghXh^{−1}g^{−1})gh = α^{gh}(X)

which shows that we have a right action of G on Der(N, R) if N is normal and hence we have a right action of G on H^{1}(N, R) if N is normal. Finally if g ∈ N then

α^{g}(X) = α(gXg^{−1})g = α(g)X + α(Xg^{−1})g = α(g)X + α(X) + α(g^{−1})g,

and since 0 = α(1) = α(g) + α(g^{−1})g,

α^{g}(X) = α(X) + α(g)X − α(g)

which differs from α(X) by a principal derivation. So N acts trivially on H^{1}(N, R).

Definition 1.6. Let N ≤ G and let i : N → G be the usual inclusion. Then the restriction map is the map

res_{N}^{G} : H^{1}(G, R) → H^{1}(N, R)

induced by i and the module restriction R → R_{|N}.

Definition 1.7. Let N be a normal subgroup of G and let j : G → G/N be the usual projection. Then the inflation map is the map

inf_{N}^{G} : H^{1}(G/N, R^{N}) → H^{1}(G, R)

induced by j and the inclusion R^{N} ↪ R.

Definition 1.8. Let S be a left-transversal of N in G (where N < G is not necessarily a normal subgroup), so that G = ⊔_{g∈S}gN. There is a map tr_{N}^{G} :

R^{N} → R^{G} defined by

tr_{N}^{G}(x) := ∑_{g∈S}(xg)

called the relative transfer (just transfer if N is trivial). It is easy to see that this map is independent of the choice of transversal.

We will need to show that this induces a map

Tr_{N}^{G} : H^{1}(N, R) → H^{1}(G, R)

which we will also call the (relative) transfer. This is not immediate from the definition, since if α is a derivation, the map X ↦ α(X)g usually is not. We define the transfer of a derivation as follows: let {x_{i} : i ∈ I} be a right transversal of N in G, so that we can write G = ⊔_{i∈I}Nx_{i}. Then for each g ∈ G we can find a set {h_{i} : i ∈ I} of elements of N and a permutation σ ∈ Sym(I) such that x_{i}g = h_{i}x_{σ(i)}. We then define

Tr_{N}^{G}(α)(g) = ∑_{i∈I} α(h_{i})x_{σ(i)}.

Lemma 1.5.3. The transfer of a derivation α ∈ Der(N, R) as defined above is a derivation in Der(G, R). Moreover if α is principal, so is Tr_{N}^{G}(α), so that the transfer induces a well-defined map T r_{N}^{G} : H^{1}(N, R) → H^{1}(G, R).

Proof. Let g, g′ ∈ G. We can find {h_{i} : i ∈ I} and σ as above, and similarly a set {h′_{i} : i ∈ I} of elements of N and a permutation τ ∈ Sym(I) such that x_{i}g′ = h′_{i}x_{τ(i)}. We then have

x_{i}gg′ = h_{i}x_{σ(i)}g′ = h_{i}h′_{σ(i)}x_{τ(σ(i))}

and so

Tr_{N}^{G}(α)(gg′) = ∑_{i∈I} α(h_{i}h′_{σ(i)})x_{τ(σ(i))},

and since α is a derivation, this is

∑_{i∈I}(α(h_{i})h′_{σ(i)}x_{τ(σ(i))} + α(h′_{σ(i)})x_{τ(σ(i))})

= ∑_{i∈I}(α(h_{i})x_{σ(i)}g′ + ∑_{i∈I} α(h′_{σ(i)})x_{τ(σ(i))} = Tr_{N}^{G}(α)(g)g′ + Tr_{N}^{G}(α)(g′)

as required. Finally let ϕ_{v} denote the principal derivation in Der(N, R) defined by ϕ_{v}(h) := vh − v for some v ∈ R. Then

Tr_{N}^{G}(ϕ_{v})(g) = ∑_{i∈I} ϕ_{v}(h_{i})x_{σ(i)} = ∑_{i∈I} vh_{i}x_{σ(i)} − ∑_{i∈I} vx_{σ(i)}

= (∑_{i∈I} vx_{i})g − (∑_{i∈I} vx_{i})

so that Tr_{N}^{G}(ϕ_{v}) = ϕ_{w} ∈ Der(G, R) where w = tr_{N}^{G}(v).

We will also need the following results which tell us how these maps interact:

Lemma 1.5.4. If N ≤ G then we have

Tr_{N}^{G} res_{N}^{G} = [G : N] Id_{H^{1}(G,R)}

Proof. Let {x_{i} : i ∈ I} be a right transversal of N in G and let α ∈ Der(G, R), g ∈ N. Let {h_{i} : i ∈ I} and σ ∈ Sym(I) be such that x_{i}g = h_{i}x_{σ(i)}. We have

Tr_{N}^{G}(res_{N}^{G}(α))(g) = ∑_{i∈I} α(h_{i})x_{σ(i)}.

Now since α ∈ Der(G, R), we may evaluate

α(x_{i}g) = α(h_{i}x_{σ(i)}) = α(h_{i})x_{σ(i)} + α(x_{σ(i)})

so that

α(h_{i})x_{σ(i)} = α(x_{i})g + α(g) − α(x_{σ(i)}).

Therefore

Tr_{N}^{G}(res_{N}^{G}(α))(g) = ∑_{i∈I}(α(x_{i})g + α(g) − α(x_{σ(i)})) = ∑_{i∈I} α(x_{i})(g − 1) + [G : H]α(g)

which differs from [G : H]α(g) by a principal derivation. It follows that Tr_{N}^{G} res_{N}^{G}(α) = [G : N]α as required.

Lemma 1.5.5. [The Cartan-Eilenberg Double Coset Formula, ([1], Theorem 6.2)] Let K, N be subgroups of G and let {x_{i} : i ∈ I} be a set of double coset representatives, so that G = ⊔_{i∈I}Kx_{i}N. Then for any α ∈ Der(K, R),

res_{N}^{G}(Tr_{K}^{G})(α) = ∑_{i∈I} Tr_{N∩x_{i}^{−1}Kx_{i}}^{N}(res_{N∩x_{i}^{−1}Kx_{i}}^{x_{i}^{−1}Kx_{i}}(α^{x_{i}}))

It follows immediately that the above formula also holds for cohomology classes α ∈ H^{1}(K, R).

Proof. Let W_{i} := N ∩ x_{i}^{−1}Kx_{i} and let {z_{ij} : j ∈ J} be a set of right coset representatives of W_{i} ∈ N, so that N = ⊔_{j∈J}W_{i}z_{ij} for any i ∈ I. Then

x_{i}N = ⊔_{j∈J}x_{i}W_{i}z_{ij} = ⊔_{j∈J}(x_{i}N ∩ Kx_{i})z_{ij} = ⊔_{j∈J}Kx_{i}z_{ij}.

Therefore

G = ⊔_{i∈N}Kx_{i}N = ⊔_{i∈I,j∈J}Kx_{i}z_{ij}

i.e. {x_{i}z_{ij} : i ∈ I, j ∈ J} is a right transversal for K in G. Now let h ∈ N. Then since each z_{ij} is ∈ N, we can find a set {h_{i} : i ∈ I} where h_{i} ∈ W_{i} and a set of permutations {σ_{i} : i ∈ I} ⊂ Sym(J) such that

z_{ij}h = h_{i}z_{iσ_{i}(j)}

and so

x_{i}z_{ij}h = x_{i}h_{i}x_{i}^{−1}x_{i}z_{iσ_{i}(j)}.

Note that x_{i}h_{i}x_{i}^{−1} ∈ K for every i ∈ I. Therefore

Tr_{K}^{G}(α)(h) = ∑_{i∈I,j∈J} α(x_{i}h_{i}x_{i}^{−1})x_{i}z_{iσ_{i}(j)} = ∑_{i∈I,j∈J} α^{x_{i}}(h_{i})z_{iσ_{i}(j)}

= ∑_{i∈I}(∑_{j∈J} α^{x_{i}}(h_{i})z_{iσ_{i}(j)}) = ∑_{i∈I} Tr_{x_{i}^{−1}Kx_{i}}^{N}(α^{x_{i}})(h).

Lemma 1.5.6. [See also Lorenz-Pathak ([31], Theorem 1.3)] Elements in the kernel of the restriction res_{N}^{G} are annihilated by the images of the relative transfer tr_{N}^{G}.

Proof. Let α ∈ Der(G, R) represent the cohomology class τ ∈ H^{1}(G, R). Then res_{N}^{G}(τ) is represented by res(α_{N}^{G}). Let s ∈ R^{N}. We will show that (*)

Tr_{N}^{G}(s. res_{N}^{G}(τ)) = tr_{N}^{G}(s).τ

where “.” denotes the usual product in R. We may write G as a union of right cosets G = ⊔_{i∈I}Nx_{i} for some right transversal {x_{i} : i ∈ I}. So for each g ∈ G we can find {h_{i}} ⊂ N and a permutation σ ∈ Sym(I) such that

x_{i}g = h_{i}x_{σ(i)}

and since α is a derivation in Der(G, R) we may evaluate for each i ∈ I,

α(x_{i})g + α(g) = α(h_{i})x_{σ(i)} + α(x_{σ(i)}).

Now by definition

Tr_{N}^{G}(s. res_{N}^{G}(α))(g) = ∑_{i∈I}(s.α(h_{i}))x_{σ(i)}

= ∑_{i∈I}(sx_{σ(i)}.[α(x_{i})g+α(g)−α(x_{σ(i)})]) = ∑_{i∈I}(sx_{σ(i)}.α(g)+sx_{σ(i)}.[α(x_{i})g−α(x_{σ(i)})])

= tr_{N}^{G}(s).α + ∑_{i∈I}(sx_{σ(i)}.[α(x_{i})g − α(x_{σ(i)})]).

So since s ∈ R^{N},

Tr_{N}^{G}(s. res_{N}^{G}(α))(g) − tr_{H}^{G}(s).α(g) = ∑_{i∈I}(sh_{i}x_{σ(i)}.α(x_{i})g) − ∑_{i∈I} sx_{σ(i)}.(α(x_{σ(i)}))

= ∑_{i∈I}(sx_{i}g.α(x_{i})g) − ∑_{i∈I} sx_{σ(i)}.(α(x_{σ(i)})) = (∑_{i∈I} sx_{i}.α(x_{i}))(g − 1)

which is a principal derivation. So we have proved (*).

Suppose τ ∈ ker(res_{N}^{G}). Then for any s ∈ R^{N} we have

tr_{N}^{G}(s).τ = Tr_{N}^{G}(s. res_{N}^{G}(τ)) = 0

and we have proved that ker(res_{N}^{G}) is annihilated by the image of the relative transfer tr_{N}^{G}.

Lemma 1.5.7. Let N be a normal subgroup of G. The inflation inf_{N}^{G} is an injective map, and its image is the kernel of the restriction res_{N}^{G}.

This is a consequence of a much more general result - in fact the exact sequence

0 → H^{1}(G/N, R^{N}) →^{inf_{N}^{G}} H^{1}(G, R) →^{res_{N}^{G}} H^{1}(N, R)^{G/N} → ...(∗)

can be derived from the Leray-Hochschild-Serre spectral sequence - (see [18], Section 7.2).

With these definitions in mind, we show how to calculate the first cohomology group of a cyclic group with non-trivial coefficients.

Lemma 1.5.8. Let P be a cyclic group of order p which acts non-trivially on a kP -module W. Let X be a generator of P. Then we have

H^{1}(P, W) ≅ \frac{ker(tr_{1}^{P})}{\{z(X − 1) : z ∈ W \}}.

Proof. We calculate Der(P, W) and PDer(P, W) explicitly. Let ϕ ∈ Der(P, W). ϕ is specified by choosing the image of X. Let v := ϕ(X). Then for every i ∈ {1, …, p} we have

ϕ(X^{i}) = ϕ(X^{i−1}) + vX^{i−1}

and by induction

ϕ(X^{i}) = v(1 + X + X^{2} + … + X^{i−1}).

In particular we have

0 = ϕ(X^{p}) = tr_{1}^{P} (v)

and so v ∈ ker(tr_{1}^{P}). Evidently ϕ is principal if and only if ϕ(X^{i}) = zX^{i} − z for some z ∈ W. In particular we must have

ϕ(X) = v = zX − z.

Moreover if this condition holds then we have

ϕ(X^{i}) = z(X − 1)(1 + X + X^{2} + … + X^{i−1}) = z(X^{i} − 1)

so this condition is both necessary and sufficient for ϕ to be a principal derivation. By Lemma 1.5.1, the first cohomology group is as claimed.

Lemma 1.5.9 ([21], Lemma 6.2). Let M and N be maximal subgroups of a pgroup P. Suppose that v ∈ R^{N} such that the derivation [v] ∈ Der(P/N, R^{N}) is not principal, and thus [v] represents a non-zero element of H^{1}(P/N, R^{N}). Then

res_{M}^{P} inf_{N}^{P}([v]) = 0

if and only if [v] is a principal derivation (and so represents zero) in H^{1}(M/(M ∩ N), R^{M∩N}).

Proof. The inflation map H^{1}(P/N, R^{N}) → H^{1}(P, R) is induced by the canonical group homomorphism P → P/N and the module homomorphism R^{N} ↪ R. The group homomorphism factors as P → P/(M ∩ N) → P/N whereas the module homomorphism factors as R^{N} → R^{M∩N} → R. So the inflation map factors as a map

H^{1}(P/N, R^{N}) → H^{1}(P/(M ∩ N), R^{M∩N}) → H^{1}(P, R).

In a similar way the composition

H^{1}(P/(M ∩ N), R^{M∩N}) →^{inf_{M∩N}^{P}} H^{1}(P, R) →^{res_{M}^{P}} H^{1}(M, R)

is induced by the canonical group homomorphism M ↪ P → P/(N ∩M) and the module homomorphism R^{M∩N} → R. The former can alternatively be factored as M → M/(M ∩ N) ↪ P/(M ∩ N). Thus we have a commutative diagram

H^{1}(P/N, R^{N})  ⟶^{inf_{N}^{P}}  H^{1}(P, R)  ⟶^{res_{M}^{P}}  H^{1}(M, R)
↓  ^{inf}_{M∩N}^{P} ↑  ^{inf}_{M∩N}^{M} ↑
H^{1}(P/(M ∩ N), R^{M∩N})  H^{1}(P/(M ∩ N), R^{M∩N}) ⟶^{res_{M/(M∩N)}^{P/(M∩N)}} H^{1}(M/(M ∩ N), R^{M∩N})

All maps labeled by “inf” are of course injective. It follows that the kernel of the map H^{1}(P/N, R^{N}) → H^{1}(M, R) formed by going along the top of the diagram coincides with the kernel of the map H^{1}(P/N, R^{N}) → H^{1}(M/(M ∩ N), R^{M∩N}) formed by following the outside of the diagram anticlockwise coincide. This latter map takes [v] viewed as an element of H^{1}(P/N, R) to [v] viewed as an element of H^{1}(M/(M ∩ N), R^{M∩N}).

This result is extremely useful since it is usually easier to see whether [v] is non-zero in H^{1}(M/(M ∩ N), R^{M∩N}), because M/(M ∩ N) is a cyclic group and its cohomology can be easily evaluated using Lemma 1.5.8.

In order to see these principles in action, let us consider an example to which we shall return later. Consider the group P := C_{2} × C_{2}. The generators are labeled X and Y, and we have 3 subgroups

M := ⟨X⟩
N := ⟨Y ⟩
L := ⟨XY ⟩

Then we have:

Lemma 1.5.10.

Der(P, R) ≅ {(v, w) ∈ R^{2}|v ∈ R^{M}, w ∈ R^{N}, v + w ∈ R^{L}}

and

P Der(P, R) ≅ {(v, w) ∈ R^{2}|∃z ∈ R : v = z(X + 1), w = z(Y + 1)}.

Proof. Let ϕ(X) := v, ϕ(Y) := w. Then 0 = ϕ(X.X) = v + vX, and 0 = ϕ(Y.Y) = w + wY, so v ∈ R^{M} and w ∈ R^{N}. Finally ϕ(X.Y) = ϕ(Y.X) ⇒ w + vY = v + wX ⇒ v + w = vXY + wXY so v + w ∈ R^{L} as required. . The principal derivations are those of the form ϕ_{z}(g) = zg + z, so ϕ = ϕ_{z} if we can write v = ϕ_{z}(X) = z(X + 1), w = ϕ_{z}(Y) = z(Y + 1).

Using the definitions given in chapter two, we can determine the form of the inflation and restriction maps for each subgroup of P. The results are summarised in the diagram overleaf:

\frac{R^{G}}{R^{M}(Y + 1)} ↘^{inf_{M}^{G}}  ↗^{res_{M}^{G}} \{\frac{R^{M}}{R(X + 1)}\}^{G/M}

\frac{R^{G}}{R^{N}(X + 1)} →^{inf_{N}^{G}} \frac{\{(v, w) ∈ R^{2}|v ∈ R^{M}, w ∈ R^{N}, v + w ∈ R^{L}\}}{\{(u(X + 1), u(Y + 1)) ∈ R^{2}|u ∈ R\}} →^{res_{N}^{G}} \{\frac{R^{N}}{R(Y + 1)}\}^{G/N}

\frac{R^{G}}{R^{L}(Y + 1)} ↗^{inf_{L}^{G}}  ↘^{res_{L}^{G}} \{\frac{R^{L}}{R(XY + 1)}\}^{G/L}

With the cohomology groups as presented above, the mappings are described as follows:

• inf_{M}^{G} : [a] → [0, a]

• inf_{N}^{G} : [a] → [a, 0]

• inf_{L}^{G} : [a] → [a, a]

• res_{M}^{G} : [a, b] → [a]

• res_{N}^{G} : [a, b] → [b]

• res_{L}^{G} : [a, b] → [a + b] All these results follow immediately from the definitions of inflation and re-

striction on derivations, except for the final one. We should have

res_{L}^{G}[a, b] = [ϕ(XY)] = [ϕ(X)Y + ϕ(Y)] = [aY + b].

But [aY + b] = [aXY − a + a + b] = [a + b] since a ∈ R^{M} and ϕ_{a}(XY) := aXY − a is a principal derivation.

1.6 The Relative Transfer Ideal

Let N be a proper subgroup of P. Recall from section 1.5 the Relative Transfer Maps tr_{N}^{G} : R^{N} → R^{G}

tr_{N}^{G} : x → ∑_{g∈G/N} xg

The image of each of these maps is an ideal I_{N}^{G} in R^{G}. Of course we can equally well define transfer maps for left ideals, using instead a system of right coset representatives for N in G.

Definition 1.9. Define the relative transfer ideal for G

I_{<P}^{G} := ∑_{N<P} I_{N}^{G}.

Let \sqrt{I_{<P}^{G}} denote the radical of I_{<P}^{G}. Whilst I_{<P}^{G} depends explicitly on the choice of Sylow-p-subgroup, we will show that its radical does not. Relative transfer ideals and their radicals play a central rôle in modular invariant theory, and have been studied extensively, see [19] and [39]. For example, Fleischmann [19] has shown that R^{G}/\sqrt{I_{<P}^{G}} is always a Cohen-Macaulay ring, which is a substantial generalisation of our earlier result of Eagon and Hochster (Theorem 1.1.4). More recently it has been shown that R^{G}/\sqrt{I_{<P}^{G}} is in fact a polynomial ring if k is algebraically closed [11]. So there is a sense in which the relative transfer ideal contains information about the extra complexity in R^{G} when R^{G} is not a polynomial ring. We now discuss a few of the properties of relative transfer ideals of which we shall make use.

Theorem 1.6.1. Let 𝔦_{P} := ℑ(V^{P}) and let 𝔦_{G} := 𝔦_{P} ∩ R^{G} where ℑ(V^{P}) is the ideal of S(V^{∗}) generated by the set of linear forms which vanish on V^{P}. Then the following expressions are equal:

1. The variety 𝔙(I)

{v ∈ V : p|[G_{v} : G_{v} ∩ H] ∀H Where H < G is a p-group};

3.

⋃_{Q} V^{Q}

where Q runs through all Sylow-p-subgroups of G;

4. The variety 𝔙(𝔦_{G}).

Proof. (1)=(2): We first claim that, for each H < P,

𝔙(I_{H}^{G}) = {v ∈ V : p|[G_{v} : G_{v} ∩ H^{g}] ∀g ∈ G}

where G_{v} is the stabiliser of v in G. To see this, let v ∈ V and x ∈ R^{H}. For any subgroup U of G we have the formula

tr_{H}^{G}(x) = ∑_{g∈S} tr_{U∩H^{g}}^{U}(xg)

where S is a set of U–H double coset representatives, see [19]. Suppose that the index [G_{v} : G_{v} ∩ H^{g}] is divisible by p for every g ∈ G. Then we have

tr_{H}^{G}(x)v = ∑_{g∈S} tr_{G_{v}∩H^{g}}^{G_{v}}(xg)v = ∑_{g∈S}( ∑_{z∈T_{g}}(xg)(zv))

where T_{g} is a left transversal of G_{v} ∩ H^{g} in G_{v}. But clearly v ∈ V^{G_{v}} and so

tr_{H}^{G}(x)v = ∑_{g∈S}[G_{v} : G_{v} ∩ H^{g}]xg(v) = 0.

To prove the converse of the claim we will need the following technical lemma: Lemma 1.6.2. [Fleischmann [19], Theorem 5.8(ii); also Lorenz-Pathak [31], part of Lemma 1.2] Let v ∈ V and let Q be a Sylow-p-subgroup of the stabiliser G_{v}. Then there exists x ∈ R^{Q} with tr_{Q}^{G}(x)v ≠ 0.

Proof. Recall we express R as k[x_{1}, … x_{n}]. For each w ∈ V \{v} define

i(w) = min{i : x_{i}(w − v) ≠ 0}.

Now set

f := ∏_{w∈Gv}(x_{i(w)} − x_{i(w)}w)

where Gv denotes the orbit of v under G. Thus f(v) ≠ 0 and f(w) = 0 for all w ∈ Gv\{v}. Let

x := ∏_{q∈Q} fq

. Then x(v) = f(v)^{|Q|} ≠ 0 and x(w) = 0 for all w ∈ Gv\{v}. Now

tr_{Q}^{G}(x)v = tr_{G_{v}}^{G} tr_{Q}^{G_{v}}(xv) = [G_{v} : Q] ∑_{z∈T} x(zv)

where T is a left transversal of G_{v} in G. For each z ∈ T \G_{v}, zv ∈ Gv so x(zv) = 0, and there is exactly one element of T in G_{v}. So we have

tr_{Q}^{G}(x)v = [G_{v} : Q]f(v)^{|Q|} ≠ 0.

So now suppose that there is a g ∈ G with p |̸[G_{v} : G_{v} ∩ H^{g}]. Then there is a Sylow-p-subgroup Q of G_{v} with H^{g} < Q. By Lemma 1.6.2 there exists x ∈ R^{Q} with tr_{Q}^{G}(x)v ≠ 0. This completes the proof of the claim.

Now since I = ∑_{H<P} I_{H}^{G} we have

𝔙(I) = ⋂_{H<P} 𝔙(I_{H}^{G}) = {v ∈ V : p|[G_{v} : G_{v} ∩ H^{g}] ∀H < P, g ∈ G}

= {v ∈ V : p|[G_{v} : G_{v} ∩ H] ∀H Where H < G is a p-group}

It is worth noticing that this expression is independent of our earlier choice P of Sylow-p-subgroup.

(2) ⊆ (3): Suppose v ∉ V^{Q} for any Sylow-p-subgroup Q of G. Then no Sylow-p-subgroup satisfies Q < G_{v}. Let Q′ be a Sylow-p-subgroup of G_{v}. Then p |̸[G_{v} : Q′]. Q′ cannot be a Sylow-p-subgroup of G since v ∈ V^{Q′}, but there is a Sylow-p-subgroup Q of G with Q′ < Q since Q′ is still a p-group. This shows that

𝔙(I) ⊆ ⋃_{Q} V^{Q}

where Q varies through all Sylow-p subgroups of G.

To show (3) ⊆ (4): Let v ∈ V^{Q}. Then Q = P^{g} for some g ∈ G, and so gv ∈ V^{P}. If r ∈ 𝔦 then

rv = (rg)v = r(gv) = 0.

Finally we show that (4) ⊆ (1): Clearly this is equivalent to I ⊆ 𝔦_{G} := 𝔦_{P} ∩R^{G}. We know that I ⊆ R^{G}, and I = ∑_{H<P} I_{H}^{G} so it is enough to show that I_{H}^{G} ⊆ 𝔦_{P} for each H < P. Accordingly, let x ∈ R^{H} for some H < P, and let v ∈ V^{P}. Then we have

tr_{H}^{G}(x)v = ∑_{g∈S} tr_{P ∩H^{g}}^{P}(xg)v = ∑_{g∈S}( ∑_{z∈T_{g}}(xg)(zv))

where S is a set of P –H double coset representatives, and T_{g} is a left transversal of H^{g} ∩ P in P. But v ∈ V^{P} and so

tr_{H}^{G}(x)v = ∑_{g∈S}[P : H^{g} ∩ P]xgv = 0

because p|[P : H^{g} ∩ P] for each g ∈ S, since P is a p-group.

We conclude that (1) = (2) ⊆ (3) ⊆ (4) ⊆ (1), and hence that (1) = (2) = (3) = (4).

For any prime ideal J of R define

G_{J} := {g ∈ G : rg − r ∈ J ∀r ∈ R}

which is called the inertia group of J. Then we have

Corollary 1.6.3. The radical \sqrt{I} can be described as ℑ(V^{P}) ∩ R^{G} where P is any Sylow-p-subgroup of G. In keeping with the notation of the previous theorem, we set 𝔦_{G} := \sqrt{I} and note that this is independent of the choice of P. Then in addition, 𝔦_{G} is a prime ideal of R, with height dim(V) − dim(V^{P}). Moreover, the index [G : G_{𝔦_{G}}] is never divisible by p, and 𝔦_{G} is the minimal prime ideal ⊂ R^{G} of R with this property.

Proof. Suppose r ∈ 𝔦_{Q} ∩ R^{G} for some Sylow-p-subgroup Q ≠ P. Let v ∈ V^{P}. There exists g ∈ G with Q^{g} = P, and rv = rgv = 0 because gv ∈ V^{Q}. So 𝔦_{P} ∩ R^{G} is independent of our choice of Sylow-p-subgroup P.

By the previous theorem 𝔙(I) = 𝔙(𝔦_{G}). So by Hilbert’s Nullstellensatz (see [37]), we have

\sqrt{I} = \sqrt{𝔦_{G}}.

It is clear that ℑ(V^{P}) is a prime ideal of R with height dim(V) − dim(V^{P}). Since R^{P} is integral over R^{G} this is also true of 𝔦_{G} = ℑ(V^{P}) ∩ R^{G}, proving the second statement. Since 𝔦_{G} is prime we also have

𝔦_{G} = \sqrt{𝔦_{G}} = \sqrt{I}

which proves the first.

To prove the third statement, let r ∈ R and g ∈ P. Then for each v ∈ V^{P} we have

(rg − r)v = rv − rv = 0.

So when g ∈ P and r ∈ R, rg − r ∈ 𝔦_{P} This shows that P ≤ G_{𝔦_{P}} ≤ G_{𝔦_{G}} and so p does not divide [G : G_{𝔦_{G}}]. Now suppose 𝔧 ⊂ R^{G} is a prime ideal of R with p |̸[G : G_{𝔧}]. Then Q ≤ G_{𝔧} for some Q ∈ Syl_{p}(G) and so 𝔧 ⊆ 𝔦_{Q} ∩ R^{G} = 𝔦_{G}, since 𝔦_{G} is independent of our choice of Sylow-p-subgroup. This shows that 𝔦_{G} is minimal with the stated property.

Flatness and Strong Flatness

In this chapter we briefly discuss some of the results found in [21] which we will use to calculate the depth of invariant rings in later chapters. In the first section we will sketch a proof of the inequality in Theorem 1.4.4, and in the second section we will find necessary and sufficient conditions for flatness. In the third section we define the notion of “strong flatness”, and show how this can be used to find flat representations.

2.1 A Lower Bound for Depth

In [22], the following result is proven:

Theorem 2.1.1.

depth(R^{G}) = dim(R/J) + grade(J, R^{G})

where J is any ideal of R^{G} with the property that p does not divide the index [G : G_{J}], where G_{J} is the inertia group of J. Recall that the radical of the relative transfer ideal, 𝔦_{G}, has this property and is minimal among all prime ideals with this property (Theorem 1.6.3), so this ideal is a natural object to which to apply this result. We find

depth(R^{G}) = dim(V^{P}) + grade(𝔦_{G}, R^{G})

and the minimality of 𝔦_{G} tells us that this is the strongest inequality which can be obtained in this way. Thus the calculation of depth(R^{G}) is reduced to the (slightly!) simpler problem of calculating grade(𝔦_{G}, R^{G}). Now it follows from Kemper, ([27], Theorem 1.4)

grade(J, R^{P}) ≥ min(cc_{G}(R) + 1, height(J))

for any proper ideal J of R^{G}. Combining these two results, we see immediately that

depth(R^{G}) ≥ min(dim(V^{P} + cc_{G}(R) + 1, dim(V))).

Clearly dim(V) is an upper bound for depth(R^{G}), and we now see that, provided that dim(V) ≥ dim(V^{P}) + cc_{G}(R) + 1, (i.e. provided V is not trivially flat), dim(V^{P}) + cc_{G}(R) + 1 is a lower bound. Moreover, it follows from the above that this lower bound is attained under these conditions precisely when grade(𝔦_{G}, R^{G}) = cc_{G}(R) + 1. This will be our primary tool for identifying flat representations.

2.2 Necessary and Sufficient Conditions for Flatness

It is still not an easy task to spot occasions when grade(𝔦_{G}, R^{G}) = cc_{G}(R) + 1. For this reason, the following result is most useful:

Theorem 2.2.1. Let V be a kG-module, R := S(V^{∗}), and m := cc_{G}(R) where we assume m < codim(V^{P}) + 1. Then the following are equivalent:

1. 𝔦_{G} ≤ Ann_{R^{G}}(α) for some 0 ≠ α ∈ H^{m}(G, R);

2. grade(𝔦_{G}, H^{m}(G, R)) = 0; 3. grade(𝔦_{G}, R^{G}) = m+1;

4. V is flat, i.e depth(R^{G}) = dim(V^{P}) + m + 1.

Note that H^{m}(G, R) is an R^{G}-module via the cup product since H^{0}(G, R) = R^{G}. Also we showed directly that H^{1}(G, R) is an R^{G}-module in section 1.5.

Proof. The equivalence of (3) and (4) was discussed in section 2.1. The equivalence of (2) and (3) follows from a spectral sequence argument which we give in chapter 6 (Theorem 6.1.3). So we require only to prove the equivalence of (1) and (2).

Suppose first that grade(𝔦_{G}, H^{m}(G, R)) = 0. Then no element of 𝔦_{G} acts regularly on H^{m}(G, R), so 𝔦_{G} consists of zero-divisors and hence lies in some associated prime 𝔭 := Ann_{R^{G}}(α) for some α ∈ H^{1}(G, R). Conversely suppose 𝔦_{G} ≤ \sqrt{Ann_{R^{G}}(α)}. Then for any r ∈ 𝔦_{G}, we have r · α = 0, and so r cannot act regularly on H^{1}(G, R).

We would like to use condition (1) in the above result to identify flat representations. Note that by [21], Corollary 2.2, we have Ann_{R^{G}}(α) ≥ 𝔦_{G} for all α ∈ H^{∗}(G, R). So we may replace the ≤ in condition (1) with an equals sign. Further one may define the variety associated with α ∈ H^{∗}(G, R) as

𝔙(α) = {v ∈ V : f(v) = 0 ∀f ∈ Ann_{R^{G}}(α)}.

Clearly then Ann_{R^{G}}(α) = 𝔦_{G} implies 𝔙(α) = 𝔙(𝔦_{G}), and from the Nullstellensatz we get the alternative equivalent condition

𝔦_{G} = \sqrt{Ann_{R^{G}}(α)}

since 𝔦_{G} is a prime ideal.

2.3 Strong Flatness

Theorem 2.2.1 says, in effect, that a representation is flat, and not trivially so, if and only if there is a non-zero cohomology class α ∈ H^{1}(G, R) which is annihilated by every element of the relative transfer ideal. In practise it is still quite difficult to find such a cohomology class. For this reason we introduce a slightly narrower class of representations.

Definition 2.1. Let V be a kG-module, P ∈ Syl_{p}(G) and let R := S(V^{∗}). Define V to be strongly flat if there exists a non-zero τ ∈ H^{1}(G, R) which restricts to zero in H^{1}(N, R) for each maximal subgroup N < P. If this τ is found in the degree one summand, H^{1}(G, V^{∗}), V is said to be linearly flat.

Note that we require 0 ≠ τ ∈ H^{1}(G, R), so the assumption cc_{G}(R) = 1 is implicit in the definition. If G is a p-group then we always have cc_{G}(R) = 1 (see Lemma 3.1.1), but this is not the case in general.

Lemma 2.3.1. Strongly flat representations are flat.

Proof. By Lemma 1.5.6, elements in the image of the relative transfer tr_{N}^{G}(R^{N}) are annihilated by the kernel of the corresponding restrictions res_{N}^{G}. So if τ ∈ ⋂_{N<P} ker(res_{N}^{G}) it is annihilated by ∑(tr_{N}^{P}(R^{N})). This result still holds if R is replaced by any graded connected Noetherian ring on which G acts linearly.

Note that the converse of this lemma is not true - for an example we refer the reader to section 3.4. However, we will see that strong flatness is a sufficiently general notion as to produce a wealth of new examples of flat representations. Strong flatness also has the following desirable properties, the first two of which are not shared by mere flatness:

Lemma 2.3.2. Strong flatness has the following properties:

1. A representation V is strongly flat if and only if there is a direct summand W^{∗} of S(V^{∗}) for which W is linearly flat.

2. If U, V are kG-modules and U is strongly flat, then so is U ⊕ V.

3. Suppose k ≤ 𝕂 is a field extension. Then V is strongly flat if and only if Ṽ := V ⊗_{k} 𝕂 is strongly flat.

Proof. Suppose α ∈ H^{1}(G, R) satisfies res_{N}^{G}(α) = 0. The splitting R = ∑_{i≥0} S^{i}(V^{∗}) induces a splitting

H^{1}(G, R) = ∑_{i≥0} H^{1}(G, S^{i}(V^{∗})).

So we may write α = ∑_{i≥0} α_{i} where α_{i} ∈ H^{1}(G, S^{i}(V^{∗})). Moreover the splitting above is compatible with the restriction maps in the sense that res_{N}^{G}(G, S^{i}(V^{∗})) ⊂ H^{1}(N, S^{i}(V^{∗}))^{G/N}, so res_{N}^{G}(α_{i}) = 0 for each i. Choose j with α_{j} ≠ 0. Then 0 ≠ α_{j} ∈ H^{1}(G, S^{j}(V^{∗})). Since S^{j}(V^{∗}) ≅ (S^{j}(V))^{∗}, it follows that S^{j}(V) is linearly flat.

To prove the second property, use the formula S(U ⊕V) = S(U)⊗S(V). Now k = S^{0}(V) is a direct summand of S(V), and so S(U) ⊗ k ≅ S(U) is a direct summand of S(U ⊕ V). The result now follows from the first property.

For the third, we use the fact that 𝕂 is a faithfully flat (in the sense of commutative algebra) k-module. This means that a sequence of k-modules

… → L → M → N → …

is exact if and only if the sequence of 𝕂-modules

… → L ⊗_{k} 𝕂 → M ⊗_{k} 𝕂 → N ⊗_{k} 𝕂 → …

is exact. In particular, for any finite group G and module W, H^{1}(G, W) ⊗_{k} 𝕂 = H^{1}(G, W ⊗_{k} 𝕂). Consider the sequence

0 → H^{1}(P, R) →^{f} ∏_{N<_{max}P} H^{1}(N, R)^{P/N} → …

where f(α) := (res_{N}^{P}(α) : N <_{max} P), and the next map is the product of corresponding maps in the exact sequence found in Lemma 1.5.7. V is strongly flat precisely when this sequence is not exact. Applying the functor ⊗_{k}𝕂 gives us the sequence

0 → H^{1}(P, R̃) →^{f⊗_{k}𝕂} ∏_{N<P} H^{1}(N, R̃)^{P/N} → …

where R̃ := R ⊗_{k} 𝕂. This sequence is exact precisely when Ṽ is not strongly flat. Since 𝕂 is faithfully flat over k, the two sequences are either both exact or both not exact, and the result follows.

Part (1) of the above lemma tells us that strong flatness depends only upon which direct summands appear in the symmetric algebra S(V^{∗}). Recall from Theorem 1.3.1 that only a finite number of isomorphism classes of modules can appear, and so this seems to be a tractable approach to finding flat representations. In particular, strong flatness is much easier to exhibit than flatness. Part (1) of this lemma also tells us that if the trivial kG-module k is linearly flat for some group G, then every kG-module V is strongly flat, since S(V^{∗}) has a direct summand S^{0}(V^{∗}) ≅ k. From this result and Lemma 2.3.1 a special case of the result of Ellingsrud and Skjelbred (theorem 1.4.1) follows immediately: let P be a cyclic group of order p and k a field of characterstic p. Then by Lemma 1.5.8, H^{1}(P, k) ≠ 0. So there is a non-zero cohomology class α ∈ H^{1}(P, k). The condition that α restrict to zero for any maximal subgroup of P is vacuous because P has a no non-trivial proper subgroups. Therefore k is strongly flat, and so every kP -module is flat. In fact, we may derive each of the remaining results on flatness listed in section 1.4 by considering only the direct summand H^{1}(G, k) of H^{1}(G, R). Contrast this with some of our calculations in chapter 3, in which we identify flat representations of certain groups by looking at their cohomology with non-trivial coefficients.

2.4 The Rôle of Cohomological Connectivity

We mentioned in the previous section that the assumption cc_{G}(R) = 1 is implicit in the definition of strong flatness. In this section we explain why we insist this is part of the definition. The following lemma is a natural generalisation of Lemma 2.3.1. Lemma 2.4.1. Suppose V is a representation of G over a field of characteristic p, and P is a (non-trivial) Sylow-p-subgroup of G. Let R := S(V^{∗}) and let m := cc_{G}(R). Suppose also that there is a non-zero cohomology class α ∈ H^{m}(G, R) satisfying (*)

res_{N}^{G}(α) = 0

for every maximal subgroup N of P. Then V is flat.

Proof. By Lorenz and Pathak ([31], Lemma 1.3), elements in the kernel of the restriction res_{N}^{G} : H^{∗}(G, R) → H^{∗}(N, R) are annihilated by the image of the relative transfer tr_{N}^{G} (note that this reduces to Theorem 1.5.6 when m = 1). It follows then that α is annihilated by every element of the relative transfer ideal 𝔦_{G}. By condition (1) of Theorem 2.2.1, V is flat.

Now consider the following situation: let V be a representation of G which satisfies the hypotheses of the theorem, with cohomological connectivity m_{V} > 1. Let W be another representation of G which is not flat and has cohomological connectivity m_{W} := 1. Now consider the representation V ⊕ W. The symmetric algebra S(V^{∗} ⊕ W^{∗}) contains a direct summand isomorphic to S(W^{∗}). Therefore

0 ≠ H^{1}(G, S(V^{∗})) | H^{1}(G, S(V^{∗} ⊕ W^{∗}))

and consequently the cohomological connectivity m_{V ⊕W} of V ⊕ W is 1. Furthermore, S(V^{∗} ⊕ W^{∗}) also contains a direct summand isomorphic to S(V^{∗}), and hence we have a non-zero α ∈ H^{m_{V}} (G, S(V^{∗} ⊕ W^{∗})) with res_{N}^{G}(α) = 0 for all maximal subgroups N of P. However, since 1 = m_{V ⊕W} < m_{V}, this does not guarantee that V ⊕ W is flat.

So if we were to define representations satisfying the hypotheses of Lemma 2.4.1 as strongly flat, we would lose the “desirable property” that V ⊕ W is strongly flat whenever V or W is so. Since this property underpins most of our work on decomposable representations, we insist that strongly flat representations have this property, and so we force our definition of strongly flat kG-modules to include the condition cc_{G}(R) = 1. This definition also justifies the fact that we have proved all our facts about cohomology only in the case of H^{1}.

43

Flatness for p-groups

In this chapter we address the problem of calculating depth(R^{P}) where P is a p-group and R := S(V^{∗}) where V is a modular representation of P. We do this primarily by exhibiting examples of flat representations.

In the first section we show that every modular representation V of a p-group P has cohomological connectivity one, and dim(V^{P}) ≥ 1.

In the second section we obtain necessary and sufficient conditions for strong flatness for p-groups (Theorem 2.2.1).

In the third section we apply Theorem 2.2.1 to some representations of elementary abelian p-groups of rank two. We prove our first major result in Corollary 3.3.2, which states that every non-projective indecomposable modular representation of the group P := C_{2} × C_{2} is flat. We also obtain a complete list of indecomposable kP -modules whose rings of invariants are Cohen-Macaulay.

In the fourth section we turn our attention to decomposable representations, and use the results of the previous section to derive some interesting consequences concerning the structure of symmetric algebras. We are also able to classify the strongly flat kP -modules. As a side result we give complete descriptions of a few rings of invariants for this group.

The final section is more experimental, and aims to apply Theorem 2.2.1 to representations of the dihedral 2-groups. We obtain plenty of new examples of flat representations, but unfortunately no general results in the manner of Theorem 3.3.2.

3.1 The p-group Case

Our specialisation to p-groups results in two major simplifications:

Lemma 3.1.1. Let V be a (left) kP -module, where k is a field of characteristic p, and let R := S(V^{∗}). Then we have:

1. dim(V^{P}) ≥ 1;

2. cc_{P} (R) = 1.

Proof. (1): V may be viewed as a abelian p-group under addition, which is infinite if and only if the field k is infinite. Let k′ be a finite subfield of k, and let 0 ≠ v ∈ V. Define V′ := vk′P, the subspace generated by v and the group algebra k′P, so that V′ may be viewed as a finite abelian p-group. Now we may define the semidirect product Q := P ⋉ V′ in which V′ is a normal subgroup, and the action of P on V by conjugation is the action of P on V′ as a vector space. Since Q is a finite p-group, the intersection of the centre of Q with any normal subgroup is non-empty and consequently there is an element of V′ on which P acts trivially. This is an element of V which is fixed by P, and so dim(V^{P}) ≥ 1 as required.

(2): A derivation from P to k is a linear map ϕ : P → k which satisfies

ϕ(XY) = ϕ(Y) + ϕ(X)Y

for X, Y ∈ P. But the action of P on k is trivial, so the definition becomes

ϕ(XY) = ϕ(Y) + ϕ(X)

i.e. ϕ is a homomorphism from P to the additive group of k. Moreover such a derivation is principal if and only if it takes the form ϕ = ϕ_{v} : X ↦ vX − v = v − v = 0 since the action is trivial, that is, the only principal derivation is the zero map. So it suffices to show there is a non-zero homomorphism from P to k. Since P is a p-group, it has a normal subgroup, Q, of index p (which is of course trivial if P ≅ C_{p}). The quotient map P → P/Q may be viewed as a map from P to the a subgroup of the additive group of k. Therefore this can be seen as a non-zero, non-principal derivation P → k and thus H^{1}(P, k) ≠ 0.

3.2 A Sufficient Condition

In this section we shall find a sufficient condition for any modular representation of any p-group to be strongly flat. Let P be a p-group and let M be a maximal subgroup of P. Let W be a (right) kP -module. Let g, g′ ∈ P \M, then g′ = hg^{i} for some h ∈ M and 1 ≤ i < p. Since M is normal in P, P acts on W^{M} and W^{M}(g′ − 1) = W^{M}(hg^{i} − 1) =

W^{M}(g^{i} − 1) = W^{M}(1 + g + ⋯ + g^{i−1})(g − 1) ⊆ W^{M}(g − 1),

hence W^{M}(g′ − 1) = W^{M}(g − 1) by symmetry. So we can define unambiguously

X_{M} := W^{M}(g − 1).

Define X := N ∩ M, which is a maximal subgroup of M and of N. Let u, u′ ∈ N\M, then N/X = ⟨\overline{u}⟩ = ⟨\overline{u}′⟩ and u′ = xu^{i} for some x ∈ X and 1 ≤ i < p. In the same way as above we see W^{X}(u′ − 1) = W^{X}(u − 1). So for any maximal subgroup N ≠ M we can pick a u_{N} ∈ N\M and define

Y_{M} := ⋂_{{N<maxP \atop N≠M}} W^{N∩M}(u_{N} − 1).

Theorem 3.2.1. For a non-cyclic p - group P the following are equivalent:

1. 0 ≠ ⋂_{M<_{max}P} ker(res_{M}^{P}(H^{1}(P, W))); 2. For some M ◁ P maximal, X_{M} < Y_{M} ∩ W^{M};

3. For all M ◁ P maximal, X_{M} < Y_{M} ∩ W^{M}.

Proof. (3) ⇒ (2) is clear.

Suppose (2) holds, and let v ∈ Y_{M} ∩ W^{M}\X_{M}. Pick g ∈ P \M, so that \overline{g} ∈ P/M generates P/M ≅ C_{p}. Since P is not cyclic there exists a maximal subgroup N ⊴ P with g ∈ N and we define X := N ∩ M. Then clearly N ≠ M and, by assumption, v = w′(u_{N} − 1) with w′ ∈ W^{X}. As before we see that W^{X}(u_{N} − 1) = W^{X}(g − 1) so v = w(g − 1) with some w ∈ W^{X}. Hence

Tr_{1}^{P/M}(v) = v(\overline{g} − 1)^{p−1} = w(g − 1)^{p} = w(g^{p} − 1) = 0

since g^{p} ∈ X. So v ∈ ker(Tr_{1}^{P/M}). Since v is not in W^{M}(g − 1), and P/M is a cyclic group generated by \overline{g}

0 ≠ [v] ∈ H^{1}(P/M, W^{M}).

Set τ := inf_{M}^{P}([v]) ∈ H^{1}(P, W). Then 0 ≠ τ ∈ ker(res_{M}^{P}) by [18], Corollary 7.2.3. By Lemma 1.5.9, if we can show that

0 = [v] ∈ H^{1}(M′/M ∩ M′, W^{M∩M′})

for any maximal subgroup M′ < P, M ≠ M′, then

τ ∈ ⋂_{N<P} ker res_{N}^{P}(H^{1}(P, W)).

Let M′ ≠ M be such a maximal subgroup of P and set Y := M′∩M, then there is u_{M′} ∈ M′\M and, by assumption v ∈ W^{Y} (u_{M′} − 1). Since \overline{u}_{M′} generates M′/Y, we have 0 = [v] ∈ H^{1}(M′/Y, W^{Y}). Thus we have proved (2) ⇒ (1).

Finally suppose that 0 ≠ τ ∈ ⋂_{N<_{max}P} ker res_{N}^{P}, and let M ◁ P be a maximal subgroup. Then by [18], Corollary 7.2.3

τ = inf_{M}^{P}([v])

for some 0 ≠ [v] ∈ H^{1}(P/M, W^{M}). This implies that

v ∈ W^{M}\W^{M}(g − 1)

for each g ∈ P \M. Let N < P be a maximal subgroup with N ≠ M, set X := N ∩M and let u_{N} ∈ N\M, then N/X = ⟨\overline{u}_{N}⟩. By assumption res_{N}^{P}(τ) = 0, so by Lemma 1.5.9,

0 = [v] ∈ H^{1}(N/X, W^{X})

i.e. v ∈ W^{X}(u_{N} − 1). So we have shown that for each maximal subgroup M of P we can a find v with

v ∈ ( ⋂_{M≠N<_{max}P}(W^{N∩M}(u_{N} − 1))) ∩ W^{M}\W^{M}(g − 1)

i.e. there exists v ∈ Y_{M} ∩ W^{M}\X_{M}.

3.3 Groups of the Form C_{p} × C_{p}

We now apply our results in the simplest case in which the depth of invariant rings is unknown. Let P ≅ C_{p} × C_{p} be generated by X and Y, and define maximal subgroups L := ⟨XY ⟩, M := ⟨X⟩, N := ⟨Y ⟩. Let W ≅ V^{∗} be a right P -module over a field k of characteristic p. Every maximal subgroup H ≠ N of P is of the form ⟨XY^{i}⟩ for 0 ≤ i < p. Hence, by condition (2) of Theorem 2.2.1, V is linearly flat if and only if

W^{N}(X − 1) < ⋂_{l=0}^{p−1} W (XY^{l} − 1) ∩ W^{N}.

When p = 2, the condition takes the particularly simple form

W^{N}(X − 1) < W (X − 1) ∩ W (XY − 1) ∩ W^{N},

although by part (iii) of Theorem 2.2.1, we may interchange the roles of maximal subgroups, and when p = 2 we will frequently use the alternative condition

W^{L}(X − 1) < W (X − 1) ∩ W (Y − 1) ∩ W^{L}.

There are four infinite families of kP -Modules which are described below. These can be taken to be left or right modules, depending on which side the matrices act. We will take them to be left modules with notation as shown, with an asterisk denoting the dual module in which the same matrices act on the right. Heller and Reiner [23] have shown that if p = 2 then these together with the projective indecomposable kP -module form a complete set of isomorphism classes of indecomposable kP -modules.^{1} In the following, I_{n} denotes the n × n - identity matrix and for λ ∈ k, J_{λ} denotes an n × n matrix in indecomposable (upper triangular) rational canonical form with eigenvalue λ, i.e. an upper triangular matrix with λ on the diagonal, 1 on the super-diagonal and zero elsewhere.

• For every even dimension 2n there are representations V_{2n,λ}

X ∼ ( \begin{matrix}I & I \\ 0 & I \end{matrix} )

Y ∼ ( \begin{matrix}I & J_{λ} \\ 0 & I \end{matrix} )

The dimension of the fixed point space of the corresponding left-module, denoted dim(V^{P}) is n.

• For every even dimension 2n there is a representation V_{2n,∞}

1The classification of modules for C_{2} ×C_{2} over a field of characteristic two is an original result of Kronecker; the version we present here is due to Heller and Reiner.

X ∼ ( \begin{matrix}I & J_{0} \\ 0 & I \end{matrix} )

Y ∼ ( \begin{matrix}I & I \\ 0 & I \end{matrix} )

In this case again dim(V^{P}) = n.

• For every odd dimension 2n + 1 there is a representation V_{−(2n+1)}

X ∼ ( \begin{matrix}I_{n} & 0 ⋮ 0 & I_{n} \\ 0 & & I_{n+1} \end{matrix} )

Y ∼ ( \begin{matrix}I_{n} & I_{n} & 0 ⋮ 0 \\ 0 & I_{n+1} \end{matrix} )

These have dim(V^{P}) = n.

• For every odd dimension 2n + 1 there is a representation V_{2n+1}

X ∼ ( \begin{matrix}I_{n+1} & 0...0 \\ & I_{n} \\ 0 & I_{n} \end{matrix} )

Y ∼ ( \begin{matrix}I_{n+1} & I_{n} \\ & 0...0 \\ 0 & I_{n} \end{matrix} )

For these modules dim(V^{P}) = n + 1.

Theorem 3.3.1. Each even-dimensional module V_{2n,λ} with n ≥ 2 and λ ∈ k or λ = ∞ is linearly flat. Each odd-dimensional module of the form V_{2n+1} with n ≥ 2 is linearly flat, while the modules of the form V_{−(2n+1)} are linearly flat provided n ≥ p. Corollary 3.3.2. Every non-projective indecomposable modular representation of the group P := C_{2} × C_{2} is flat. Proof. To prove the corollary observe that when p = 2, the theorem states that every non-projective indecomposable kP -module of dimension ≥ 4 is linearly flat. But it is also easily seen that each indecomposable kP -module of dimension < 4 in our classification above is trivially flat. Note that there is only one projective indecomposable module, namely the regular module.

Proof. We now show in each of the three cases that there is an element v ∈ ⋂_{l=0}^{p−1} W (XY^{l} −1)∩W^{N}\W^{N}(X −1). This will show that the left-module V with V^{∗} ≅ W is linearly flat, by Theorem 2.2.1.

Consider the representation V := V_{2n,λ}, where for the time being λ ≠ ∞. Then we have

W^{N} = \{ \begin{matrix}⟨x_{n+1}, …, x_{2n}⟩ & λ ≠ 0 \\ ⟨x_{n}, …, x_{2n}⟩ & λ = 0 \end{matrix}

so that

W^{N}(X − 1) = \{ \begin{matrix}0 & λ ≠ 0 \\ ⟨x_{2n}⟩ & λ = 0 \end{matrix}

Also for each 0 ≤ l ≤ p − 1

W (XY^{l} − 1) = ⟨(1 + lλ)x_{n+1} + lx_{n+2}, (1 + lλ)x_{n+2} + lx_{n+3}, …, (1 + lλ)x_{2n}⟩

W (XY^{l} − 1) = \{ \begin{matrix}⟨x_{n+1}, …, x_{2n}⟩ & λl ≠ −1 \\ ⟨x_{n+2}, …, x_{2n}⟩ & λl = −1 \end{matrix}

Now k is a field of characteristic p, so it contains a copy of the field 𝔽_{p} of order p. If λ ∈ 𝔽_{p}\{0}, there is precisely one l ∈ {1, … p − 1} for which λl = −1. If λ = 0 or if λ ∈ k\𝔽_{p}, then λl ≠ −1 for every l. It follows that

⋂_{l=0}^{p−1} W (XY^{l} − 1) ∩ W^{N} = \{ \begin{matrix}⟨x_{n+1}, …, x_{2n}⟩ & λ = 0 or λ ∈ k\𝔽_{p} \\ ⟨x_{n+2}, …, x_{2n}⟩ & λ ∈ 𝔽_{p}\\{0\} \end{matrix}

So we have linear flatness for all λ if n ≥ 2, and for n = 1 when λ ∉ 𝔽_{p}. V_{2n,∞} is obtained from V_{2n,0} under the automorphism of P which swaps X and Y, and so the same results hold here.

Now consider the representation V := V_{2n+1}. This time, we have

W^{N} = ⟨x_{n+1}, …, x_{2n+1}⟩

so that W^{N}(X − 1) = ⟨x_{2n+1}⟩. Further we have

W (XY^{l} − 1) = ⟨lx_{n+2}, x_{n+2} + lx_{n+3}, x_{n+3} + lx_{n+4}, …, x_{2n} + lx_{2n+1}, x_{2n+1}⟩

W (XY^{l} − 1) = ⟨x_{n+2}, x_{n+3}, …, x_{2n+1}⟩

for all l ∈ {0, …, p − 1}. Therefore

⋂_{l=0}^{p−1} W (XY^{l} − 1) ∩ W^{N}\W^{N}(X − 1) = ⟨x_{n+2}, x_{n+3}, …, x_{2n}⟩

and we have linear flatness provided n ≥ 2.

Finally consider the representation V := V_{−(2n+1)}. In this case

W^{N} = ⟨x_{n+1}, …, x_{2n+1}⟩

so W^{N}(X − 1) = 0, and

W (XY^{l} − 1) = ⟨lx_{n+1} + x_{n+2}, lx_{n+2} + x_{n+3}, …, lx_{2n} + x_{2n+1}⟩

Let us denote each space W (XY^{l} − 1) ∩ W^{N} by the shorthand W_{l}. It is enough to show that ⋂_{l=0}^{p−1} W_{l} ≠ 0. It is easy to see that each W_{l} is a hyperplane in the n + 1-dimensional space W^{N}. It then follows that

dim ⋂_{l=0}^{p−1} W_{l} ≥ n + 1 − p,

and so V_{−(2n+1)} is linearly flat when n ≥ p.

3.4 Decomposable Representations

We have proved that all non-projective indecomposable representations of C_{2} × C_{2} are flat. We now go on to classify completely representations of C_{2} × C_{2} which are strongly flat by investigating which direct summands may appear in their symmetric algebras. Before we do, we restate and extend Corollary 3.3.2 specifically for the case p = 2.

Theorem 3.4.1. Let V be an indecomposable representation of P := C_{2} × C_{2} over a field of characteristic 2. Then:

1. If V ≅ V_{2,λ} for (λ ≠ 0, 1, ∞), V_{4,λ}, or V ≅ an indecomposable of dimension ≥ 5, then V is linearly flat.

2. If dim(V) ≤ 4 and V is not projective, then V is trivially flat.

3. If V is isomorphic to the projective indecomposable of dimension 4, then V is not flat.

4. The representations V_{2,0}, V_{2,1}, V_{2,∞}, V_{3} and V_{−3}, though flat, are not strongly so.

Recall that since this is a p-group in characteristic p, there is only one projective indecomposable, namely the regular representation. From now on this is abbreviated to \overline{V}_{4} .

Proof. : (1) and (2) are clear. The remaining statements will be proven in the forthcoming lemmas.

\overline{V}_{4} is equivalent to a permutation representation. We can think of X and Y acting on V^{∗} as the permutations (12)(34) and (13)(24) respectively, by which we mean that

x_{1}X = x_{2}, x_{2}X = x_{1}, x_{3}X = x_{4}, x_{4}X = x_{3}

and the action of Y calculated similarly. It’s clear that we have dim(V^{P}) = 1, therefore either V is flat or R^{P} is Cohen-Macaulay. In this case we can find explicit generators for R^{P} in order to show that the latter holds, and V is not flat.

Remark: The ring of invariants for C_{2} × C_{2} and \overline{V}_{4} has been determined in [1], where k is assumed to be the field of order 2. The version we present here for convenience and completeness, assumes k to be any perfect field of characteristic 2. We use the following lemma:

Lemma 3.4.2. Let R := S(V^{∗}) = k[x_{1}, x_{2}, x_{3}, x_{4}] with V ≅ \overline{V}_{4}. Then the ring of invariants R^{M} may be described as

S := k[a, b, c, d](1, β_{1})

where a := x_{1} + x_{2}, b := x_{3} + x_{4}, c := x_{1}x_{2}, d := x_{3}x_{4} form an hsop and β_{1} := x_{1}x_{3} + x_{2}x_{4}, with β_{1}^{2} = β_{1}ab + a^{2}d + b^{2}c. Proof. It is clear that a, b, c, d and β_{1} are indeed invariants, and also the relation given for β_{1} is easily checked. Since x_{i}^{2} + ax_{i} + c = 0 = x_{j}^{2} + x_{j}b + d for i = 1, 2 and j = 3, 4, the ring R is integral over B := k[a, b, c, d], and a, b, c, d form a homogenous system of parameters. For any subalgebra T ⊆ R let Frac(T) denote its quotient field, then [Frac(R) : Frac(B)] = 4 and Frac(R) = Frac(B)(β_{1}, x_{1}), because

x_{1} ∈ Frac(B)(β_{1}, x_{1})

x_{2} = c/x_{1} ∈ Frac(B)(β_{1}, x_{1})

x_{3} = \frac{β_{1}x_{1} − cb}{x_{1}^{2} − c},

which tells us that β_{1} ∉ Frac(B). Hence [Frac(R) : Frac(S)] = 2. But from Galois Theory we know that [Frac(R) : Frac(R^{M})] = |M| = 2 and together with S ⊆ R^{M} we conclude that Frac(R^{M}) = Frac(S). In order to show that R^{M} = S it now suffices to prove that S is normal, i.e. integrally closed in its quotient field, because then we get

R^{M} = IntCls_{Frac(R^{M})}(B) ⊆ IntCls_{Frac(S)}(S) = S ⊆ R^{M}.

To show that S is normal we use Serre’s normality criterion (see [8], Theorem 2.2.22), stating that S is normal if and only if the following two conditions are satisfied:

• “Regular in Codimension 1”: For each prime ideal 𝔭 of S with height 1, the localisation S_{𝔭} is regular.

• Serre’s Condition S_{2}: For each prime ideal 𝔮 of S, we have depth(S_{𝔮}) ≥

min{2, dim(S_{𝔮})}. Since S is a hypersurface and hence Cohen-Macaulay, the second condition (S_{2}) is automatically satisfied. Suppose that the first condition fails to hold. Then there is a prime ideal 𝔭 of S with height 1, such that the localisation S_{𝔭} is not regular. We view S as the quotient of the polynomial ring S = k[â, b̂, ĉ, d̂, β̂_{1}] by the principal ideal generated by f := β̂_{1}^{2} + β̂_{1}âb̂ + â^{2}d̂+ b̂^{2}ĉ. Then, by [16], Theorem 16.20, 𝔭 must properly contain the ideal J(f) := ({ \atop J(f)} + (f))/(f) in S, where { \atop J(f)} is generated by the 1 × 1 - minors, i.e. the entries, of the Jacobian corresponding to f. Hence J(f) = (β_{1}b, β_{1}a, b^{2}, a^{2}, ab), \sqrt{J(f)} = (a, b, β_{1}) ⊆ 𝔭 and S/\sqrt{J(f)} ≅ k[c, d]. We conclude

2 = Dim(S) − Dim(S/\sqrt{J(f)}) < Dim(S) − Dim(S/𝔭) = height(𝔭) = 1.

This contradiction finishes the proof.

To find R^{P}, we apply the lemma twice and obtain

Lemma 3.4.3. We keep the notation from 3.4.2. Then the ring of invariants R^{P} is Cohen-Macaulay and can be described as

R^{P} = k[a + b, c + d, ab, cd](1, β_{1})(1, ac + bd)

= k[x_{1} + x_{2} + x_{3} + x_{4}, x_{1}x_{2} + x_{3}x_{4}, (x_{1} + x_{2})(x_{3} + x_{4}), x_{1}x_{2}x_{3}x_{4}](1, β_{1}, β_{2}, β_{1}β_{2})

where β_{2} = x_{1}^{2}x_{2} + x_{1}x_{2}^{2} + x_{3}^{2}x_{4} + x_{3}x_{4}^{2}. In particular R^{P} is not flat.

Proof. The quotient P/M acts on R^{M} = k[a, b, c, d](1, β_{1}) by swapping a with b, swapping c with d, and by fixing β_{1}. Hence 3.4.2 gives the first equality, with primary invariants of degrees 1,2,2 and 4. Since the product of these degrees is 16 and the minimal set of secondary invariants has 4 = 16/|P | elements, we conclude by Theorem 1.1.3, that R^{P} is a Cohen-Macaulay ring. Since dim(R^{P}) = 4 and dim(\overline{V} _{4}^{P}) = 1, the ring R^{P} cannot be flat.

The proof of (4) will again be broken down into lemmas. Recall that in order to prove strong flatness we require to find an element of R(X + 1) ∩ R(Y + 1) ∩ R^{L}\R^{L}(X +1). If V ≅ V_{2,0} then R(Y +1) = 0, if V ≅ V_{2,∞} then R(X +1) = 0 and if V ≅ V_{2,1} then R^{L}(X + 1) = R(X + 1). So none of these modules are strongly flat as P -modules. Of course their vector invariants are flat, being representations of the cyclic group C_{2}. The point is that we cannot say that representations of P containing one of these modules as a direct summand automatically are flat.

Lemma 3.4.4. Let R := S(V^{∗}) = k[x_{1}, x_{2}, x_{3}] with V ≅ V_{3}. Then the ring of invariants R^{P} is polynomial of the form B := k[u, v, w] u := x_{3}, v := x_{1}(x_{1} + x_{3}) and w := x_{2}(x_{2} + x_{3}). The representation V_{3} is flat but not strongly flat.

Proof. Note that P acts as follows

X : x_{1} ↦ x_{1}, x_{2} ↦ x_{2} + x_{3}, x_{3} ↦ x_{3}
Y : x_{1} ↦ x_{1} + x_{3}, x_{2} ↦ x_{2}, x_{3} ↦ x_{3}
XY : x_{1} ↦ x_{1} + x_{3}, x_{2} ↦ x_{2} + x_{3}, x_{3} ↦ x_{3}

hence u, v, w ∈ R^{P}. Now since x_{3} ∈ B, it’s clear that R = B[x_{1}, x_{2}]. Further since

x_{1}^{2} = x_{1}x_{3} + x_{1}(x_{1} + x_{3}) = ux_{1} + v ∈ B(1, x_{1})
x_{2}^{2} = x_{2}x_{3} + x_{2}(x_{2} + x_{3}) = ux_{2} + w ∈ B(1, x_{2})

we have R = B(1, x_{1}, x_{2}, x_{1}x_{2}) as a B-module. The relations above also tell us that R is integral over B; moreover the product of the degrees of u, v and w is 4 = |P |, hence by Theorem 1.1.1, R^{P} = k[u, v, w] as required. Since [Frac(R) : Frac(B)] = 4, we see that R = B ⊕ Bx_{1} ⊕ Bx_{2} ⊕ Bx_{1}x_{2}. Let r ∈ R. We may then write r = b_{0} + b_{1}x_{1} + b_{2}x_{2} + b_{3}x_{1}x_{2}, with each b_{i} ∈ B = R^{P}. If r ∈ R^{M}, then r = rX, that is

b_{0} + b_{1}x_{1} + b_{2}x_{2} + b_{3}x_{1}x_{2} = b_{0} + b_{1}x_{1} + b_{2}(x_{2} + x_{3}) + b_{3}x_{1}(x_{2} + x_{3})

so we must have b_{2} = b_{3} = 0 and R^{M} = B(1, x_{1}). In a similar way, R^{N} = B(1, x_{2}) and R^{L} = B(1, x_{1} + x_{2}). Now we obtain immediately

R^{L}(X + 1) = Bx_{3}
R(X + 1) = R^{M}(x_{3}) = B(x_{3}, x_{1}x_{3})
R(Y + 1) = R^{N}(x_{3}) = B(x_{3}, x_{2}x_{3}).

Therefore R(X + 1) ∩ R(Y + 1) = Bx_{3} = R^{L}(X + 1), and so there cannot be an element in R(X + 1) ∩ R(Y + 1) ∩ R^{L}\R^{L}(X + 1).

Lemma 3.4.5. Let R := S(V^{∗}) = k[x_{1}, x_{2}, x_{3}] with V ≅ V_{−3}. Then the ring of invariants R^{P} is polynomial of the form B := k[x_{2}, x_{3}, z] with z := N_{P} (x_{1}) := ∏_{g∈P} (x_{1}g) = x_{1}^{4} + x_{1}^{2}x_{2}^{2} + x_{1}^{2}x_{2}x_{3} + x_{1}^{2}x_{3}^{2} + x_{1}x_{2}^{2}x_{3} + x_{1}x_{2}x_{3}^{2}. The representation V_{−3} is flat but not strongly flat.

Proof. Clearly B ⊆ R^{P} and the equation

x_{1}^{4} = x_{1}^{2}(x_{2}^{2} + x_{3}^{2} + x_{2}x_{3}) + x_{1}(x_{2}^{2}x_{3} + x_{2}x_{3}^{2}) + z

shows that R is integral over B. Since the degree product of x_{2}, x_{3} and z is 4 = |P |, again by Theorem 1.1.1, we have B = R^{P} with a decomposition

R = B[x_{1}] = B ⊕ Bx_{1} ⊕ Bx_{1}^{2} ⊕ Bx_{1}^{3}.

Suppose r ∈ R^{M}. We may write r = b_{0} + b_{1}x_{1} + b_{2}x_{1}^{2} + b_{3}x_{1}^{3}, and then r = rX tells us

b_{0} + b_{1}x_{1} + b_{2}x_{1}^{2} + b_{3}x_{1}^{3} = b_{0} + b_{1}(x_{1} + x_{3}) + b_{2}(x_{1}^{+}x_{3}^{2}) + b_{3}(x_{1}^{3} + x_{1}^{2}x_{3} + x_{1}x_{3}^{2} + x_{3}^{3})

= b_{0} + b_{1}x_{3} + b_{2}x_{3}^{2} + b_{3}x_{3}^{3} + (b_{1} + b_{3}x_{3}^{2})x_{1} + (b_{2} + b_{3}x_{3})x_{1}^{2} + b_{3}x_{1}^{3}

By equating coefficients of x_{1}, we find b_{3} = 0 and b_{1} = b_{2}x_{3}. Thus r = b_{0} + b_{2}x_{3}x_{1} + b_{2}x_{1}^{2} = b_{0} + b_{2}(x_{1}^{2} + x_{1}x_{3}) and thus

R^{M} = B(1, x_{1}^{2} + x_{1}x_{3})

In a similar way,

R^{N} = B(1, x_{1}^{2} + x_{1}x_{2})
R^{L} = B(1, x_{1}^{2} + x_{1}(x_{2} + x_{3}))

So we have

R(X + 1) = R^{M}(x_{3}) = B(x_{3}, x_{1}^{2}x_{3} + x_{1}x_{3}^{2})
R(Y + 1) = R^{N}(x_{2}) = B(x_{2}, x_{1}^{2}x_{2} + x_{1}x_{2}^{2})
R^{L}(X + 1) = B(1, x_{1}^{2} + x_{1}(x_{2} + x_{3}))(X + 1) = B(x_{2}x_{3})

Suppose s ∈ R(X + 1) ∩ R(Y + 1) ∩ R^{L}. Then there are expressions

s = b_{0}x_{2} + b_{1}(x_{1}^{2} + x_{1}x_{2}) = c_{0}x_{3} + c_{1}(x_{1}^{2} + x_{1}x_{3})

Where b_{0}, b_{1}, c_{0}, c_{1} ∈ B. Applying XY to both sides

s = b_{0}x_{2} + b_{1}(x_{1}^{2} + x_{1}x_{2} + x_{2}x_{3} + x_{3}^{2}) = c_{0}x_{3} + c_{1}(x_{1}^{2} + x_{1}x_{3} + x_{2}^{2} + x_{2}x_{3}) Therefore b_{1} = c_{1} = 0 and s = b_{0}x_{2} = c_{0}x_{3} ∈ B(x_{2}x_{3}) = R^{L}(X + 1), and so V_{−3} cannot be strongly flat as R^{L}(X + 1) ⊃ R(X + 1) ∩ R(Y + 1) ∩ R^{L}.

This last lemma also terminates the proof of 3.4.1. Corollary 3.4.6. A complete list of indecomposable kP -modules for P := C_{2}×C_{2} whose polynomial invariants are Cohen-Macaulay is given by

\{V_{1}, V_{2,λ}, V_{−3}, V_{3}, V_{4,λ}, \overline{V}_{4}, V_{5}\}.

Proof. We have stated already that the modules V_{1}, V_{2,λ}, V_{−3}, V_{3} are trivially flat.

Observe that dim(V_{4,λ}^{P}) = 2 and dim(V_{5}^{P}) = 3, so both of these are trivially flat. That \overline{V}_{4} gives rise to a Cohen-Macaulay invariant ring is proved in Theorem 3.4.1. Finally we check the the statement of Corollary 3.3.2, to see that every other indecomposable representation is flat, but not trivially so.

Corollary 3.4.7. The symmetric algebras of the representations V_{3},V_{−3} and \overline{V}_{4} contain only direct summands of the form V_{1}, V_{2,1}, V_{2,0}, V_{2,∞}, V_{3},V_{−3} and \overline{V}_{4}.^{2}

Proof. We refer to the set \{V_{1}, V_{2,1}, V_{2,0}, V_{2,∞}, V_{3},V_{−3}, \overline{V}_{4}\}, here and elsewhere in this section, as NSF. Let V ∈ NSF. All indecomposable modules not in NSF are linearly flat. If S(V^{∗}) contains a direct summand W^{∗} for which W is linearly flat, then V is strongly flat. So S(V^{∗}) must decompose as a direct sum only of modules of isomorphism types in the set

NSF^{∗} = \{V_{1}^{∗}, V_{2,1}^{∗}, V_{2,0}^{∗}, V_{2,∞}^{∗}, V_{3}^{∗}, V_{−3}^{∗}, \overline{V} _{4}^{∗}\}.

, and therefore S(V) has as direct summands only modules in the set NSF.

Now if we can determine precisely which summands appear in the symmetric algebras of modules in NSF, and how the tensor products decompose into indecomposable summands, we will then be able to list all possible kP -modules which are strongly flat.

Theorem 3.4.8. Let add(NSF) denote the set of modules which are direct sums of modules in NSF. Let W be any kP -module. Then W is strongly flat unless W ∈ add(NSF) and neither V_{3} ⊕ V_{3} nor V_{−3} ⊕ V_{−3} is a direct summand of W.

Proof. If W ∉ add(NSF), then W must contain a direct summand which is strongly flat. Consequently, W is strongly flat by property (2) in Lemma 2.3.2. So we assume W ∈ add(NSF). We write

W = aV_{1} ⊕ b_{0}V_{2,0} ⊕ b_{1}V_{2,1} ⊕ b_{∞}V_{2,∞} ⊕ c_{+}V_{3} ⊕ c_{−}V_{−3} ⊕ d\overline{V}_{4},

2V_{1} denotes the trivial module

so that

W^{∗} ≅ aV_{1}^{∗} ⊕ b_{0}V_{2,0}^{∗} ⊕ b_{1}V_{2,1}^{∗} ⊕ b_{∞}V_{2,∞}^{∗} ⊕ c_{+}V_{3}^{∗} ⊕ c_{−}V_{−3}^{∗} ⊕ d\overline{V} _{4}^{∗}

and

S(W^{∗}) ≅ S(V_{1}^{∗})^{⊗a}⊗S(V_{2,0}^{∗})^{⊗b_{0}}⊗S(V_{2,1}^{∗})^{⊗b_{1}}⊗S(V_{2,∞}^{∗})^{b_{∞}}⊗S(V_{3}^{∗})^{⊗c_{+}}⊗S(V_{−3}^{∗})^{⊗c_{−}}⊗S(\overline{V} _{4}^{∗})^{⊗d}.

In order to determine which isomorphism classes of modules may appear as summands in the above expression, we need to know which summands appear in the symmetric algebras of the modules in NSF, and also how their tensor products decompose. Clearly the symmetric algebras of the non-faithful modules V_{2,0}, V_{2,1} and V_{2,∞} contain as direct summands only themselves and the trivial module, as any direct summands in the symmetric algebra must also be nonfaithful. Still more obviously, the symmetric algebra of V_{1} contains only copies of the trivial module. In order to ascertain which modules appear in the symmetric algebras of V_{3}, V_{−3} and \overline{V}_{4}, we use Theorem 1.3.1. This states that all summands which do appear as summands of the symmetric algebra can be found in degrees ≤ 2^{n} − n − 1, where n is the dimension of the representation. This allows us to compute, using the meataxe in MAGMA:

• S(V_{−3}) contains summands isomorphic to V_{1}, V_{2,1}, V_{2,∞}, V_{2,0}, V_{−3} and \overline{V}_{4}.

• S(V_{3}) contains summands isomorphic to V_{1}, V_{3} and \overline{V}_{4}.

• S(\overline{V}_{4}) contains summands isomorphic to V_{1}, V_{2,1}, V_{2,∞}, V_{2,0} and \overline{V}_{4}.

The first two results require a decomposition of the first four symmetric powers, while the third requires a decomposition of the first 11. This takes considerable computation time. However there is a far easier way to obtain this result - \overline{V}_{4} is a permutation module of dimension 4, so each summand in its symmetric algebra has dimension dividing 4. This tells us that S(\overline{V}_{4}) cannot contain summands isomorphic to V_{3} or V_{−3} and the result now follows from Corollary 3.4.7.

The following table lists the decompositions of tensor products in NSF. We omit from the table results concerning V_{1}, understanding that W ⊗ V_{1} = W for any kP -module W.

Table 3.1: Decomposing Tensor Products of kP -modules

⊗  V_{2,1} V_{2,∞} V_{2,0}  V_{3}  V_{−3}  \overline{V}_{4}
V_{2,1} 2V_{2,1}  \overline{V}_{4}  \overline{V}_{4} V_{2,1} ⊕ \overline{V}_{4} V_{2,1} ⊕ \overline{V}_{4} 2\overline{V}_{4}
V_{2,∞}  2V_{2,∞}  \overline{V}_{4} V_{2,∞} ⊕ \overline{V}_{4} V_{2,∞} ⊕ \overline{V}_{4} 2\overline{V}_{4}
V_{2,0}  2V_{2,0} V_{2,0} ⊕ \overline{V}_{4} V_{2,0} ⊕ \overline{V}_{4} 2\overline{V}_{4}
V_{3}  \overline{V}_{4} ⊕ V_{5} V_{1} ⊕ 2\overline{V}_{4} 3\overline{V}_{4}
V_{−3}  \overline{V}_{4} ⊕ V_{−5} 3\overline{V}_{4}
\overline{V}_{4}  4\overline{V}_{4}

We conclude that the modules in the set add\{V_{1}, V_{2,1}, V_{2,∞}, V_{2,0}, \overline{V}_{4}\} form a closed system under taking symmetric algebras. Moreover all the modules in the decomposition of S(W^{∗}) belong to NSF unless either c_{+} or c_{−} is at least two. If c_{+} ≥ 2, then S(W^{∗}) contains a direct summand isomorphic to V_{5}^{∗}. We know from the proof of Lemma 3.3.2 that V_{5} is linearlt flat and so W is strongly flat. Similarly if c_{−} ≥ 2, then S(W^{∗}) contains a direct summand isomorphic to V_{−5}^{∗} and again W is strongly flat.

It must be stressed that we have determined the depth of R^{P} for any indecomposable representation of P := C_{2} × C_{2}, (and so classified its flat indecomposable representations), and also classified all the strongly flat representations of this group. We have not, however, classified all the flat representations of this group. Consider, for example, the kP -module W := V_{−3} ⊕ V_{2,1} ⊕ V_{2,1}. Then W ∈ add(NSF) and neither V_{3} ⊕ V_{3} nor V_{−3} ⊕ V_{−3} is a direct summand of W, so by Theorem 3.4.8, W is not strongly flat. A calculation in MAGMA establishes that the depth of the corresponding invariant ring R^{G} is 5, and so W is in fact flat. We may easily verify that W is not trivially flat. So then, it must be the case (by Theorem 2.2.1) that there is a cohomology class 0 ≠ α ∈ H^{1}(P, R) which is annihilated by each element of the relative transfer ideal. At the same time, the intersection of the kernels of the restrictions to each maximal subgroup is zero. Finding such an α directly is, at least for the moment, beyond the scope of the methods presented here.

3.5 Dihedral 2-Groups

Now let P be generated by a pair of involutions X and Y, subject to the condition that (XY)^{2^{n}} = 1. Then for n ≥ 2, P is isomorphic to the dihedral group D_{2^{n+1}} of order 2^{n+1}, which is a non-abelian 2-group. When n = 1, P is isomorphic to the Klein 4-group C_{2} × C_{2} which we have considered in detail in the previous section. For the purposes of this section we regard this group as a special case of a dihedral 2-group. Let V be a left kP -module, where k is any field of characteristic 2, and again let R := S(V^{∗}). Define (normal) subgroups of P

M := ⟨XY ⟩, N := ⟨(XY)^{2}⟩.

P then has three maximal subgroups, which are the cyclic group M and the smaller dihedral groups

H_{X} := ⟨X, (XY)^{2}⟩, H_{Y} := ⟨Y, (XY)^{2}⟩

with H_{X} ∩ M = H_{Y} ∩ M = N. Then by Theorem 2.2.1, V is strongly flat if and only if

R^{M}(X − 1) < R^{M} ∩ ⋂_{H<_{max}P} ⋂_{g∈H\M} R^{H∩M}(g − 1)

R^{M}(X − 1) < R^{M} ∩ ( ⋂_{g∈H_{X}\M} R^{N}(g − 1)) ∩ ( ⋂_{g∈H_{Y} \M} R^{N}(g − 1))

i.e.

R^{M}(X − 1) < R^{M} ∩ ⋂_{g∈P \M} R^{N}(g − 1).

Our aim is to use this theorem to construct examples of flat kP -modules. Somewhat surprisingly, we have the following result:

Lemma 3.5.1. . Let V be a left kP -module, where P ≅ D_{2^{n}}. Then, for any n ≥ 2, V is strongly flat if and only if

R^{M}(X − 1) < R^{M} ∩ R^{N}(X − 1) ∩ R^{N}(Y − 1).

Proof. We need only to show that ⋂_{g∈P \M} R^{N}(g − 1) = R^{N}(X − 1) ∩ R^{N}(Y − 1). So suppose g ∈ P \M. Elements of P can be written uniquely as words in the letters X, Y subject to the following conditions:

1. X and Y follow alternately (XX and Y Y are not allowed);

2. The greatest power of XY which can appear is (XY)^{2^{n−1}−1}.

So g can be written uniquely as either (XY)^{m}X or (Y X)^{m}Y for some m ≤ 2^{n−1} − 1 (pure powers of XY are precisely the elements of M). Suppose first that g = (XY)^{m}X and let v ∈ R^{N}(g − 1). Then for some w ∈ R^{N} we have

v = w(g − 1) = w((XY)^{m}X − 1) = \{ \begin{matrix}w(X − 1) & m even \\ w(XY X − 1) & m odd \end{matrix}

So clearly R^{N}(g − 1) = R^{N}(X − 1) if m is even, while if m is odd

w(g − 1) = w(XY X − 1) = w(XY XY − Y)Y = w(1 − Y)Y = w(Y − 1).

By a similar argument, if g := (Y X)^{m}Y then R^{N}(g − 1) = R^{N}(Y − 1) when m is even, and R^{N}(g − 1) = R^{N}(X − 1) when m is odd. So for every g ∈ P \M we have either R^{N}(g − 1) = R^{N}(X − 1) or R^{N}(Y − 1).

The fact that this lemma holds independently of the exponent n is particularly interesting. This tells us that there is a well-defined notion of strong flatness which is the same for each member of the family of dihedral 2-groups. Thus a (faithful) representation of D_{2^{n}} which is strongly flat remains so when regarded as a representation of D_{2^{n+1}} with non-trivial kernel. By way of contrast, consider once more the kP -module V := V_{2,0} where P ≅ C_{2} × C_{2}. Then V is a kP -module with nontrivial kernel and V is not strongly flat, and we can construct faithful kP -modules containing V as a direct summand which are not flat. However, considered as a faithful module for the cyclic quotient P/⟨X⟩, V is strongly flat, and any modular representation of P/⟨X⟩ containing V will also be flat.

The above lemma tells us that provided X, Y and XY each act non-trivially, a similar situation will not arise for dihedral 2-groups. It also gives us a means of constructing faithful representations of P ≅ D_{2^{n}} which are strongly flat; namely, take any faithful representation of P and take its direct sum with any strongly flat module for the quotient group C_{2} × C_{2}. This is, of course, a rather artificial construction, and it is perhaps more interesting to know that there are faithful indecomposable kP -modules which are strongly flat in their own right.

Example 3.1. Let P ≅ D_{8} be generated by the matrices

X := ( \begin{matrix}1 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix} )

Y := ( \begin{matrix}1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 1 \end{matrix} )

Let V be the natural left kP -module and let W denote the dual module, which is the degree one part of R. Then we have

XY := ( \begin{matrix}1 & 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 & 1 \end{matrix} )

and so W^{XY} = ⟨x_{3}, x_{5}⟩, and W^{XY} (X − 1) = 0. Then

(XY)^{2} := ( \begin{matrix}1 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix} )

and so W^{(XY)^{2}} = ⟨x_{2}, x_{3}, x_{4}, x_{5}⟩, then W^{(XY)^{2}}(X − 1) = ⟨x_{3}⟩ and W^{(XY)^{2}}(Y − 1) = ⟨x_{3}, x_{5}⟩. It is easy to see that

x_{3} ∈ R^{M} ∩ R^{N}(X − 1) ∩ R^{N}(Y − 1)\R^{M}(X − 1).

By Lemma 3.5.1, V is strongly flat.

There is, of course, a complete classification of indecomposable modules for dihedral 2-groups in characteristic 2. We present a short summary here, but for further details, see [35]. The modules are divided into two separate classes, called string modules and band modules. We concentrate first on the first kind. These are in one-one correspondence with a certain subclass of words made from the letters a and b and their inverses. Let w(a, b) := w_{1}w_{2}w_{3}...w_{n−1} be a word such that the a or a^{−1} is always followed by b or b^{−1}, so we have no powers of 2 or higher and do not need to worry about “cancelation”. Then we construct the module V:= M(w(a, b)) as follows: write V := kx_{1} ⊕ ... ⊕ kx_{n} as a direct sum of vector spaces, and draw a diagram like so:

kx_{1} ⟶^{w_{1}} kx_{2} ⟶^{w_{2}} kx_{3} ⟶^{w_{3}} … ⟶^{w_{n−1}} kx_{n} Where w_{i} = a^{−1} or b^{−1},

we replace the arrow pointing right with one pointing left and labeled by a or b. So for example, the word aba^{−1}b corresponds to the diagram

kx_{1} ⟶^{a} kx_{2} ⟶^{b} kx_{3} ⟵^{a} kx_{4} ⟶^{b} kx_{5}

The idea now is that X − 1 acts as the identity along edges labeled a and as zero elsewhere, with Y − 1 acting as the identity along edges labeled b and as zero elsewhere. Thus, the diagram in our example gives rise to the module

X := ( \begin{matrix}1 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix} )

Y := ( \begin{matrix}1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 1 \end{matrix} )

Remarks: Define the formal inverse of w to be the word w_{n−1}^{−1}w_{n−2}^{−1} … w_{1}^{−1}. Then the construction gives a natural isomorphism M(w) ≅ M(w^{−1}). One should also notice that (XY)^{2^{n}} is in the kernel of M(w) if w does not contain the subword (ab)^{2^{n−1}}.

The second type of module is defined as follows: we take a word w(a, b) in a, b as before, but with the additional condition that w is not an exact power of a smaller word. This time w must be a word of even length. Let w := w_{1}w_{2} … w_{n}, and let {U_{1}, U_{2}, … U_{n}} be a set of k-vector spaces of equal dimension r. Now construct V := ⨁_{i=1}^{n} U_{i} and let Ψ be an indecomposable homomorphism U_{n} → U_{1}. Now we construct a diagram as follows:

U_{2} ⟶^{w_{2}} U_{3} ⟶^{w_{3}} … ⟶^{w_{n−2}} U_{n−1}
w_{1}↑  ↓^{w_{n−1}}
U_{1}  ⟵_{w_{n}}^{Ψ}  U_{n}

Once more, where w_{i} = a^{−1} or b^{−1}, we replace the arrow pointing right with one pointing left and labeled by a or b. So for example, the word w := aba^{−1}b, corresponds to the diagram

U_{2} ⟶^{b} U_{3}
a↑  ^{a}↑
U_{1} ⟵_{b}^{Ψ} U_{n}

Again X − 1 acts as the identity along edges labeled a and as zero elsewhere, with Y − 1 acting as the identity along edges labeled b and as zero elsewhere. Thus, the diagram in our example gives rise to the module

X := ( \begin{matrix}I_{r} & I_{r} & 0 & 0 \\ 0 & I_{r} & 0 & 0 \\ 0 & 0 & I_{r} & 0 \\ 0 & 0 & I_{r} & I_{r} \end{matrix} )

Y := ( \begin{matrix}I_{r} & 0 & 0 & Ψ \\ 0 & I_{r} & I_{r} & 0 \\ 0 & 0 & I_{r} & 0 \\ 0 & 0 & 0 & I_{r} \end{matrix} )

It is clear that there is an isomorphism M(w, Ψ) ≅ M(w^{−1}, Ψ), and also that M(w, Ψ) ≅ M(\overline{w}, Ψ) where \overline{w} is obtained from w under cyclic permutation. In a similar way to before, notice that (XY)^{2^{n}} is in the kernel of M(w, Ψ) if w does not contain the subword (ab)^{2^{n−1}}, with one exception: the groups which act on the modules M((ab)^{2^{n}}(ab)^{−2^{n}}, 1) have order 2^{n+2}, not 2^{n+3} as expected. These are the projective indecomposable modules for the groups D_{2^{n+2}}.

Using the invariant theory package in MAGMA, it is possible to write a routine which tests explicitly for the non-emptiness or otherwise of Y_{M}\X_{M}, where

X_{M} = R^{M}(X − 1)

and

Y_{M} = R^{M} ∩ ⋂_{g∈P \M} R^{N}(g − 1).

In some cases this can give a value for depth(R^{G}) when the library routine “depth” uses too much memory. Furthermore, testing for linear flatness is a simple exercise in linear algebra, and we can get results for this in dimensions well beyond those where the depth function crashes.

Table 3.5 gives the results of such a program for string modules of dimension ≤ 7, while table 3.4 gives the corresponding results for band modules. Note that we can obtain additional modules simply by exchanging the rôles of X and Y, and these will be associated to the schema obtained by exchanging a and b. These modules are not usually isomorphic to our original modules, but its easy to see that the data in the table will be equivalent. Allowing for this, we have results for all indecomposable modules of dimension ≤ 7. The column ”depth” is the simple calculation dim(V^{P}) + 2 if our module is strongly flat, and the result returned by the library function ”depth” in all other cases. “?” denotes that the insufficient memory was available to perform the calculation. In the second table, a single λ in the second column means that Ψ is multiplication by any non-zero λ ∈ k, unless some range is specified. J_{n}(λ) is the n × n Jordan block matrix with single eigenvalue λ, and where this appears in the table we mean that Ψ is the homomorphism associated to this matrix.

What can we conclude from these results? My hope was that, in a manner analogous to our investigations into the group C_{2}×C_{2}, we would find that all modules of dimension greater than some integer m are linearly flat. This would mean that in order to classify strongly flat modules, we would need only to investigate the composition of symmetric algebras of small dimensional modules. However, the range of dimensions in which we have investigated holds no evidence of this, and it may be the case that there are arbitrarily large indecomposable modules which are not strongly flat. I have also tested for linear flatness in all modules of dimension ≤ 10, with no discernable patterns emerging.

Perhaps the best I can hope for is to find necessary and sufficient conditions on the schemas for the corresponding modules to be linearly flat. Although it might be possible to find the strongly flat modules in a more direct manner. Look carefully at the (string) modules of dimension 7, and observe that each of these is either strongly flat, or causes the computer to run out of memory. Running out of memory could in this case be regarded as evidence of flatness - the Gröbner basis calculations take longer when the projective dimension of a module over a homogeneous system of parameters is large. But this is hardly rigourous!

Table 3.2: Flatness Data for String Modules

Schema  |P | dim(V^{P}) lin. flat? str. flat? Depth(R)  Notes
ab  8  1  No  No  3  Trivially flat
ab^{−1}  4  2  No  No  3  Trivially flat
a^{−1}b  4  1  No  No  3  Trivially flat
aba  8  1  No  No  4  Cohen-Macaulay
aba^{−1}  8  2  No  No  4  Trivially flat
ab^{−1}a  4  2  Yes  Yes  4  ≅ V_{4,∞}
a^{−1}ba  8  1  No  No  4  Cohen-Macaulay
abab  16  1  No  No  5  Cohen-Macaulay
abab^{−1}  8  2  No  No  5  Cohen-Macaulay
aba^{−1}b  8  2  Yes  Yes  4  x_{3} ∈ Y_{M}\X_{M}
aba^{−1}b^{−1}  8  2  No  No  5  Cohen-Macaulay
ab^{−1}ab  8  2  No  Yes  4  x_{2}^{2} ∈ Y_{M}\X_{M}
ab^{−1}ab^{−1}  4  3  Yes  Yes  5  ≅ V_{−5}
a^{−1}bab  8  1  No  No  4
a^{−1}ba^{−1}b  4  2  Yes  Yes  4  ≅ V_{5}
a^{−1}b^{−1}ab  8  1  No  No  4
ababa  16  1  No  No  5
ababa^{−1}  16  2  No  No  5
abab^{−1}a  8  2  Yes  Yes  4  x_{4} ∈ Y_{M}\X_{M}
abab^{−1}a^{−1}  8  2  No  No  5
aba^{−1}ba  8  2  No  No  5
aba^{−1}ba^{−1}  8  3  Yes  Yes  5  x_{3}, x_{5} ∈ Y_{M}\X_{M}
aba^{−1}b^{−1}a  8  2  No  No  5
ab^{−1}aba  8  2  No  Yes  4  x_{2}^{3} ∈ Y_{M}\X_{M}
ab^{−1}aba^{−1}  8  3  No  Yes  5  x_{2}^{2} ∈ Y_{M}\X_{M}
ab^{−1}ab^{−1}a  4  3  Yes  Yes  5  ≅ V_{6,∞}
ab^{−1}a^{−1}ba  8  2  No  No  5
ab^{−1}a^{−1}b^{−1}a 8  2  No  No  5
a^{−1}baba  16  1  No  No  ?
a^{−1}bab^{−1}a  8  2  Yes  Yes  4  x_{4} ∈ Y_{M}\X_{M}
a^{−1}ba^{−1}ba  8  2  No  Yes  4  x_{3}^{2} ∈ Y_{M}\X_{M}
a^{−1}b^{−1}aba  8  1  No  No  ?
ababab  16  1  No  ?  ?
ababab^{−1}  16  2  No  ?  ?
ababa^{−1}b  16  2  Yes  Yes  4  x_{5} ∈ Y_{M}\X_{M}
ababa^{−1}b^{−1} 16  2  No  ?  ?
abab^{−1}ab  8  2  No  ?  ?
abab^{−1}ab^{−1}  8  3  Yes  Yes  5  x_{4}, x_{6} ∈ Y_{M}\X_{M}
abab^{−1}a^{−1}b  8  2  No  ?  ?

Table 3.3: Flatness Data for String Modules (cont.)

Schema  |P | dim(V^{P}) lin. flat? str. flat? Depth  Notes
abab^{−1}a^{−1}b^{−1}  8  2  No  ?  ?
aba^{−1}bab^{−1}  8  3  No  Yes  5  x_{3}^{2}x_{7} = x_{2}x_{4}x_{6} ∈ Y_{M}\X_{M}
aba^{−1}ba^{−1}b  8  3  Yes  Yes  5  x_{3}, x_{5} ∈ Y_{M}\X_{M}
aba^{−1}b^{−1}ab  8  2  No  ?  ?
aba^{−1}b^{−1}ab^{−1}  8  3  No  Yes  5  x_{6}^{2} ∈ Y_{M}\X_{M}
aba^{−1}b^{−1}a^{−1}b  8  2  No  ?  ?
ab^{−1}abab  16  2  No  ?  ?
ab^{−1}abab^{−1}  8  3  No  Yes  5  x_{1}x_{6} + x_{2}^{2} + x_{2}x_{5} ∈ Y_{M}\X_{M}
ab^{−1}aba^{−1}b  8  3  Yes  Yes  5  x_{2}, x_{5} ∈ Y_{M}\X_{M}
ab^{−1}ab^{−1}ab  8  3  Yes  Yes  5  x_{2}, x_{4} ∈ Y_{M}\X_{M}
ab^{−1}ab^{−1}ab^{−1}  4  4  Yes  Yes  6  ≅ V_{−7}
ab^{−1}ab^{−1}a^{−1}b  8  3  Yes  Yes  5  x_{2}, x_{4} ∈ Y_{M}\X_{M}
ab^{−1}a^{−1}bab  8  2  No  ?  ?
ab^{−1}a^{−1}bab^{−1}  8  3  No  Yes  5  x_{1}x_{6} + x_{2}x_{5} ∈ Y_{M}\X_{M}
ab^{−1}a^{−1}ba^{−1}b  8  3  No  Yes  5  x_{2}x_{3}x_{7} + x_{3}^{2}x_{7} ∈ Y_{M}\X_{M}
ab^{−1}a^{−1}b^{−1}ab  8  2  No  ?  ?
a^{−1}babab  16  1  No  ?  ?
a^{−1}babab^{−1}  16  2  No  ?  ?
a^{−1}baba^{−1}b  8  2  Yes  Yes  4  x_{5} ∈ Y_{M}\X_{M}
a^{−1}bab^{−1}ab  8  2  No  ?  ?
a^{−1}bab^{−1}a^{−1}b  8  2  No  ?  ?
a^{−1}ba^{−1}bab  8  2  No  ?  ?
a^{−1}ba^{−1}ba^{−1}b  4  3  Yes  Yes  5  ≅ V_{7}
a^{−1}ba^{−1}b^{−1}ab  8  2  Yes  Yes  4  x_{3} ∈ Y_{M}\X_{M}
a^{−1}ba^{−1}b^{−1}a^{−1}b 8  3  Yes  Yes  5  x_{3} ∈ Y_{M}\X_{M}
a^{−1}b^{−1}abab  16  1  No  ?  ?
a^{−1}b^{−1}ab^{−1}ab  8  2  No  ?  ?
a^{−1}b^{−1}a^{−1}bab  8  1  No  ?  ?

Table 3.4: Flatness Data for Band Modules

Schema  Ψ  |P | dim(V^{P}) lin. flat? str. flat? Depth  Notes
ab^{−1}  1  2  1  no  no  2  Trivially flat
ab^{−1}  λ ≠ 1 4  1  yes  yes  2  ≅ V_{2,λ}
ab^{−1}  J_{2n}(λ) 4  1  yes  yes  2  ≅ V_{2n,λ}
abab^{−1}  λ  8  1  no  no  4  Cohen-Macaulay
aba^{−1}b^{−1}  1  4  1  no  no  4  ≅ \overline{V}_{4}
aba^{−1}b^{−1}  λ ≠ 1 8  1  no  no  4  Cohen-Macaulay
ababab^{−1}  λ  16  1  no  ?  ?
ababa^{−1}b^{−1}  λ  16  2  no  ?  ?
abab^{−1}ab^{−1}  λ  8  3  yes  yes  5  x_{4} ∈ Y_{M}\X_{M}
aba^{−1}bab^{−1}  λ  8  2  no  yes  4  x_{6}^{2} ∈ Y_{M}\X_{M}
abab^{−1}a^{−1}b^{−1}  λ  8  1  no  ?  ?
aba^{−1}ba^{−1}b^{−1}  λ  8  2  yes  yes  4  x_{3} ∈ Y_{M}\X_{M}

Extensions of p-Groups

In this chapter we attempt to find general conditions for flatness or strong flatness of representations of certain groups which are not p-groups. In the first section we explain how the cohomology modules H^{1}(P, R) and H^{1}(G, R), where P ∈ Syl_{p}(G), are related. In the second we identify a certain class of groups (the trivial intersection groups) for which this relationship is particularly transparent. In the third section we use some of these ideas to give a proof of the second part of Theorem 1.4.4.

In the final section we turn our attention to groups G of the form P ×Q where P is a p-group and Q an abelian p′-group. We show that a representation V of G is strongly flat if and only if V_{|P} is so, where V_{|P} is the restriction of V to P.

4.1 Stable Cohomology and Strong Flatness

Let us examine the definition of strongly flat a little more closely. V is strongly flat if and only if

∃α ∈ H^{1}(G, R)\{0} with res_{N}^{G}(α) = 0

for every maximal N < P. Each restriction map res_{N}^{G} factors as

res_{N}^{G} = res_{N}^{P} res_{P}^{G}

so we could equally well formulate this condition as

∃α ∈ res_{P}^{G}(H^{1}(G, R))\{0} with res_{N}^{P}(α) = 0

for every maximal N < P. Since Tr_{P}^{G} res_{P}^{G}(α) = [G : P]α ≠ 0 (Lemma 1.5.4), restriction provides an injective homomorphism H^{1}(G, R) → H^{1}(P, R), and H^{1}(G, R) may be viewed as a direct summand of H^{1}(P, R). In this section we investigate more closely the image of res_{P}^{G}.

Lemma 4.1.1. α ∈ res_{P}^{G}(H^{1}(G, R)) if and only if

res_{P ∩g^{−1}P g}^{P}(α) = res_{P ∩g^{−1}P g}^{g^{−1}P g}(α^{g}) ∀g ∈ G.

Elements α ∈ H^{1}(P, R) which satisfy this condition are called G-stable elements. Note that, since the Sylow-p-subgroups are all conjugate, the definition of G-stability does not depend on the choice of P.

Proof. Let K and N be subgroups of a finite group G, and let D be a set of double coset representatives, so that G = ⊔_{x∈D}KgN. Then the Cartan-Eilenberg formula (Lemma 1.5.5) tells us that, for any cohomology class α ∈ H^{1}(N, R),

res_{K}^{G}(Tr_{N}^{G})(α) = ∑_{x∈D} Tr_{K∩xNx^{−1}}^{K}(res_{K∩xNx^{−1}}^{xNx^{−1}}(α^{x})).

Suppose α ∈ res_{P}^{G}(H^{1}(G, R)), say α = res_{P}^{G}(β) for some β ∈ H^{1}(G, R). Then we have

α^{x} = (res_{P}^{G}(β))^{x} = res_{xP x^{−1}}^{G}(β).

Restricting both sides to P ∩ xP x^{−1} gives

res_{P ∩xP x^{−1}}^{xP x^{−1}}(α^{x}) = res_{P ∩xP x^{−1}}^{G}(β) = res_{P ∩xP x^{−1}}^{P}(res_{P}^{G}(β)) = res_{P ∩xP x^{−1}}^{P}(α)

as required. Conversely suppose that α is stable. Then we have

Tr_{P ∩xP x^{−1}}^{P}(res_{P ∩xP x^{−1}}^{xP x^{−1}}(α^{x})) = Tr_{P ∩xP x^{−1}}^{P}(res_{P ∩xP x^{−1}}^{P}(α)) = [P : P ∩ xP x^{−1}]α

and using Lemma 1.5.5 we get

res_{P}^{G}(Tr(_{P}^{G})(α)) = ∑_{x∈D} Tr_{P ∩xP x^{−1}}^{P}(res_{P ∩xP x^{−1}}^{xP x^{−1}} α^{x}) = ∑_{x∈D}[P : P ∩xP x^{−1}]α = [G : P]α,

and since [G : P] is invertible mod P, α ∈ res_{P}^{G}(H^{1}(G, R)).

Let V be a kG-module and R := S(V^{∗}). Let P be a Sylow-p-subgroup of G. Then it follows from the definitions that V is strongly flat if and only if there is some α ∈ H^{1}(P, R) which is both G-stable and in the kernel of each restriction map res_{N}^{P} for N < P a maximal subgroup. In particular V_{|P} must be strongly flat.

4.2 Trivial Intersection Groups

Definition 4.1. Define G to be a trivial intersection (T.I.) group if, given any Sylow-p-subgroup P, the following holds:

gP g^{−1} ∩ P = \{ \begin{matrix}P & g ∈ N_{G}(P); \\ 1 & otherwise. \end{matrix}

Lemma 4.2.1. Suppose G is a trivial intersection group. Then

H^{1}(G, R) ≅ H^{1}(N_{G}(P), R) ≅ H^{1}(P, R)^{N_{G}(P)/P}

where the action of N_{G}(P)/P is described in Lemma 1.5.2.

Proof. By Lemma 4.1.1, α ∈ H^{1}(P, R) is G-stable if and only if

res_{P ∩g^{−1}P g}^{P}(α) = res_{P ∩g^{−1}P g}^{g^{−1}P g}(α^{g})∀g ∈ G.

If G is a trivial intersection group, then for g ∉ N_{G}(P) we find res_{P ∩g^{−1}P g}^{P} = res_{P ∩g^{−1}P g}^{g^{−1}P g} = 0, and so to prove that α is G-stable it is enough to show that α is N_{G}(P)-stable. This proves the first equality. For each g ∈ N_{G}(P) we have res_{P ∩g^{−1}P g}^{P} = res_{P ∩g^{−1}P g}^{g^{−1}P g} = Id_{H^{1}(P,R)}, and the stability condition reduces to

α^{g} = α ∀g ∈ N_{G}(P)

Since P acts trivially on H^{1}(P, R) by Lemma 1.5.2, the action may be regarded as an action of the quotient N_{G}(P)/P.

4.3 p-Nilpotent Groups with Cyclic Sylow-p-subgroup

Examples of trivial intersection groups include any group in which the Sylow-p-subgroups are cyclic of prime order, as well as any group whose Sylow-p-subgroups are normal in G. Suppose G is a finite group whose Sylow-p-subgroups are cyclic of prime order. Then G must be a trivial intersection group, since P ∩g^{−1}P g ≤ P and P has no non-trivial proper subgroups. If in addition P = N_{G}(P), (P is self-normalising) then for any kG-module W we have H^{1}(G, W) = H^{1}(P, W). So in this case it is obvious that a kG-module V is strongly flat if and only if V_{|P} is so. Since every representation of a cyclic group of prime order is strongly flat, we conclude that every kG-module is strongly flat when G is of the form described above.

In fact we can do a little better than this. Let P be a cyclic p-group, but not necessarily of prime order. Each kP -module is strongly flat, and the “reason” for this, is that the trivial module k is linearly flat and a direct summand of R := S(V^{∗}) for every choice of V. Exploiting the triviality of this action allows us to prove the following:

Theorem 4.3.1. [[21], Theorem 5.3 and this paper, Theorem 1.4.4(ii)] Let G be a p-nilpotent group with cyclic Sylow-p-subgroup P. Then every kG-module is strongly flat.

It is worth noting that if P has prime order, then the hypotheses of this theorem are precisely those discussed in the preceding paragraph.

Proof. If G is p-nilpotent, then there is a proper normal subgroup H of G such that G = P H. Therefore

N_{G}(P) = P N_{H}(P) = P C_{H}(P) = C_{G}(P)

that is to say, every pair of elements in P which are conjugate in G are in fact conjugate in P. It is enough to show that the trivial kG-module is strongly flat. Let α be a derivation representing an element of H^{1}(P, k). We will show that α is G-stable. For every g ∈ N_{G}(P) and every X ∈ P we have

α^{g}(X) := α(gXg^{−1})g.

Since g ∈ N_{G}(P) = C_{G}(P), we have gXg^{−1} = X, and since the action of G on k is trivial, α(X)g = α(X). So we have shown α^{g}(X) = α(X) for all g ∈ N_{G}(P) and X ∈ P, i.e. α is G-stable. So we can find 0 ≠ β ∈ H^{1}(G, k) with res_{P}^{G}(β) = α. P has a unique maximal subgroup, Q, and by Theorem 1.5.7

ker(res_{Q}^{P}(H^{1}(G, k))) ≠ 0.

Now let 0 ≠ α ∈ ker(res_{Q}^{P}(H^{1}(G, k))). Then we may find a corresponding β as above which satisfies

res_{Q}^{G}(β) = res_{Q}^{P} res_{P}^{G}(β) = res_{Q}^{P}(α) = 0.

Therefore k is strongly flat.

4.4 Direct Products

Direct products are in some sense the simplest form of group extension. Throughout this section let G := P × Q, where P is a p-group and Q a p′ group. Let k be a field of characteristic p and V a finite-dimensional kG-module. We aim to investigate under what circumstances flatness of V is governed entirely by the action of P on V, i.e. for which representations V do we have

V is flat ⇔ V_{|P} is flat?

Note that when V is trivially flat, the above is a triviality. For this reason we shall always assume that codim(V^{P}) > 2. Our intuition about non-modular invariant theory tells us that the depth of R^{G} ought to depend on P only, and we can find no counterexample to this statement. However, we shall see in this section just how difficult it can be to verify this in even the simplest of cases.

Let W be any right kG-module. It is clear that G is a trivial intersection group with P a normal subgroup and G/P ≅ Q, so we have immediately that H^{1}(G, W) ≅ H^{1}(P, R)^{Q}. Suppose that α is a non-zero cohomology class in H^{1}(G, R). We may regard α as an equivalence class of derivations. Now

(\frac{Der(P, W)}{PDer(P, W)})^{Q} ≅ \begin{matrix}Der(P, W) & Q \\ PDer(P, W) & Q \end{matrix}

because Q is a p′-group. So we may choose a derivation ϕ ∈ Der(P, W)^{Q} representing α. Let g ∈ Q, and X ∈ P. Then

ϕ(X) = ϕ^{g}(X) = ϕ(gXg^{−1})g = ϕ(X)g

i.e. ϕ ∈ Der(P, R^{Q}). It follows that α ∈ H^{1}(P, W^{Q}). Conversely if α′ ∈ H^{1}(P, W^{Q}), α can be represented by a non-principal derivation ϕ′ ∈ Der(P, W^{Q}). For X ∈ P and g ∈ Q this satisfies

ϕ′(X) = ϕ′(X)g = ϕ′(gXg^{−1})g = ϕ′^{g}(X)

and so α ∈ H^{1}(P, W)^{Q}. This establishes the key fact that

H^{1}(P, W^{Q}) ≅ H^{1}(G, W) ≅ H^{1}(P, W)^{Q}.

Note that the assumption that P be a p-group and Q a p′-group is not necessary to reach this conclusion.

Let G ≅ P × Q, V a finite dimensional kG - module with codim(V^{P}) > 2 and R := S(V^{∗}). Since R^{Q} ≠ 0 we may apply the above with W = R to show that the cohomological connectivities of V and of V_{|P} are both equal to one. Define as before 𝔦_{P} := ℑ(V^{P}) ∈ Spec(R) with 𝔦_{H} := 𝔦_{P} ∩ R^{H} for any H ≤ G. Then, using theorem 2.2.1, we can compare the flatness conditions for V_{|P} and V:

• V_{|P} is flat if and only if there is 0 ≠ α ∈ H^{1}(P, R) with 𝔦_{P} = \sqrt{ann_{R^{P}} (α)};

• V is flat if and only if there is 0 ≠ β ∈ H^{1}(P, R^{Q}) with 𝔦_{G} = \sqrt{ann_{R^{G}}(β)}.

Since Q is a p′ - group it is clear that H^{1}(P, R)^{Q} = tr_{1}^{Q}(H^{1}(P, R)) and therefore

H^{1}(P, R) ≅ H^{1}(P, R)^{Q} ⊕ K,

where K denotes the kernel of the transfer map tr_{1}^{Q}. Note that Q acts on H^{1}(P, R), and so by tr_{1}^{Q} we mean the transfer for this action, i.e. tr_{1}^{Q}(α) := ∑_{g∈Q} α^{g}.

Corollary 4.4.1. Let G ≅ P × Q with p′ - group Q and V a finite dimensional kG-module. Then the following are equivalent:

1. V is flat;

2. V_{|P} is flat and moreover ∃ 0 ≠ α ∈ H^{1}(P, R^{Q}) with 𝔦_{P} = \sqrt{ann_{R^{P}} (α)}.

Condition (2) is satisfied, if there is 0 ≠ γ ∈ H^{1}(P, R) with 𝔦_{P} = \sqrt{ann_{R^{P}} (γ)} and tr_{1}^{Q}(γ) ≠ 0.

Proof. “1. ⇒ 2.”: Let V be flat, then there is 0 ≠ α ∈ H^{1}(P, R^{Q}) with

𝔦_{G} = \sqrt{ann_{R^{G}}(α)}.

(Note that R^{G} ≤ R^{Q} has a natural action on H^{1}(P, R^{Q}). ) It follows that 𝔦_{G} ≤ \sqrt{ann_{R^{P}} (α)} ≤ 𝔦_{P}, where the last inclusion follows for example from [21], Corollary 2.2. But 𝔦_{P} is the only prime ideal of R^{P} lying over 𝔦_{G} (any of these is G-conjugate to 𝔦_{P}, hence equal to 𝔦_{P}), so it follows that

𝔦_{P} = \sqrt{𝔦_{G}R^{P}} ≤ \sqrt{ann_{R^{P}} (α)} ≤ 𝔦_{P} .

In particular V_{|P} is flat. (This fact alone also follows from the inequality depth(R^{G}) ≥ depth(R^{P}) for p′ - extensions of P.)

“2. ⇒ 1.”: Let 0 ≠ α be as described, then by Lemma 4.1.1 α = res_{P}^{G}(β) for some 0 ≠ β ∈ H^{1}(G, R). Since 𝔦_{G} = 𝔦_{P} ∩ R^{G}, we have for some N ∈ ℕ:

0 = 𝔦_{G}^{N}α = res_{P}^{G}(𝔦_{G}^{N}β),

and by the injectivity of res_{P}^{G},

𝔦_{G} ≤ \sqrt{ann_{R^{G}}(R^{G}β)} ≤ 𝔦_{G}.

For the last remark observe that, since P ⊴ G, the group Q stabilizes 𝔦_{P}, hence for every x ∈ 𝔦_{P} and suitable N ∈ ℕ we get (x^{N}q) · γ = 0 ∀q ∈ Q. Therefore, we have

x^{N} · (tr_{1}^{Q}γ) = ∑_{g∈Q} x^{N} · γ^{g} = ∑_{g∈Q}(x^{N}g^{−1} · γ)^{g} = 0.

From this the last remark follows immediately.

In practice it can be difficult to verify the criterion (2) or to prove that tr_{1}^{Q}(γ) ≠ 0 for a given 0 ≠ γ ∈ H^{1}(P, R), even if Q is abelian. In this case however, there is a more transparent connection between the properties of V and V_{|P} to be strongly flat:

Theorem 4.4.2. Let G be a group of the form P × Q with P a p-group and Q an abelian p′-group. Then V ∈ kG-mod is strongly flat if and only if V_{|P} ∈ kP -mod is so.

Proof. Let k ≤ 𝕂 be a field extension, then by Lemma 2.3.2(3), V is strongly flat if and only if V ⊗_{k} 𝕂 is so, by . So we may assume that k is algebraically closed. Now let V be strongly flat and 0 ≠ α ∈ H^{1}(G, R) with res_{N}^{G}(α) = 0 for all N < P. Then 0 ≠ γ = res_{P}^{G}(α) ∈ H^{1}(P, R^{Q}) satisfies res_{N}^{P}(γ) = 0, so V_{|P} is strongly flat. Now assume that V_{|P} is strongly flat and let 0 ≠ α ∈ H^{1}(P, R) with res_{N}^{P}(α) = 0 for all N < P. Then V^{∗} = ⊕_{j=1}^{k}W_{j} with W_{j} ≅ U_{j} ⊗ k_{ϵ_{j}}, where each U_{j} is an indecomposable kP - module, and k_{ϵ_{j}} is a one-dimensional Q-module with character ϵ_{j} ∈ Hom(Q, k^{×}) and diagonal action (u ⊗ λ)(p, q) := up ⊗ λϵ_{j}(q). Hence we have a decomposition of the G-module

R = ⊗_{j}S(W_{j}) = ⊕_{s=0}^{∞} ⊕_{{\underline{ℓ}∈ℕ^{k}, \atop |\underline{ℓ}|=s}} X\underline{ℓ}

with X\underline{ℓ} := X_{ℓ_{1}} ⊗ ⋯ ⊗ X_{ℓ_{k}} and X_{ℓ_{i}} := S(W_{i})_{ℓ_{i}}. Note that Q acts on each S(W_{i})_{ℓ_{i}} by the linear character ϵ_{i}^{ℓ_{i}} and on X\underline{ℓ} by the character ∏_{j=1}^{k} ϵ_{j}^{ℓ_{j}}. For every N < P we have decompositions of k-spaces

H^{1}(P, R) = ⊕\underline{ℓ}H^{1}(P, X\underline{ℓ})
H^{1}(N, R) = ⊕\underline{ℓ}H^{1}(N, X\underline{ℓ})

which are preserved by the corresponding restriction map. It follows that α = ⊕\underline{ℓ}α\underline{ℓ} with α\underline{ℓ} ∈ H^{1}(P, X\underline{ℓ}) and res_{N}^{P}(α\underline{ℓ}) = 0 for every N < P and \underline{ℓ}. Hence we can assume that 0 ≠ α ∈ H^{1}(P, X\underline{ℓ}). For every j = 1, ⋯, k, the space X_{ℓ_{j}} is a kG - direct summand of S(W_{j}) and by Theorem 1.3.2 there is a suitable homogeneous invariant u_{j} ∈ S(W_{j})^{P} of degree |P |, propagating direct summands of S(W_{j}). Choosing a_{j}, b_{j} ∈ ℕ such that b_{j}|Q| − a_{j}|P | = ℓ_{j}, we see that u_{j}^{a_{j}} · X_{ℓ_{j}} is a direct summand of S(W_{j}) and also a submodule of

S^{ℓ_{j}+|P |a_{j}}(W_{j}) = S^{b_{j}|Q|}(W_{j}) ≤ S(W_{j})^{Q}

Let u := ⊗_{j=1}^{k}u_{j}^{a_{j}} ∈ R^{P}, then X\underline{ℓ} and u · X\underline{ℓ} = ⊗_{j=1}^{k}u_{j}^{a_{j}} · X_{ℓ_{j}} are isomorphic kP - modules and the latter one is a direct summand of R^{Q}. It follows that there is

0 ≠ α̃ ∈ H^{1}(P, uX\underline{ℓ}) | H^{1}(P, R^{Q})

satisfying res_{N}^{P}(α̃) = 0 for all N < P. Therefore α̃ = res_{P}^{G}(β) with β ∈ H^{1}(G, R) and

res_{N}^{G}(β) = res_{N}^{P}(res_{P}^{G}(β)) = res_{N}^{P}(α̃) = 0

for all N < P. This shows that V is strongly flat as kG - module.

Some groups with

Sylow-2-subgroup P ≅ C_{2} × C_{2}

In this chapter we apply some of the ideas from chapter 4 to groups G whose Sylowp-subgroups are isomorphic to C_{2} × C_{2}. We focus on these groups in particular because we have a complete classification of strongly flat representations for P. So we want to know which representations of V of G, if any, have the property (*)

V Strongly flat ⇔ V_{|P} Strongly flat.

Note that we have proved (theorem 4.4.2) that every representation V of a group G of the form P × Q with P a p-group and Q an abelian p′-group.

In the first section we concentrate on dihedral groups D of order 4n where n is odd. These are 2-nilpotent groups with Sylow-2-subgroups isomorphic to C_{2} ×C_{2}. We prove that V ∈ kD-mod is strongly flat if and only if V_{|P} is so, where P is any Sylow-2-subgroup. In conjunction with Theorem 3.4.8, this gives us a classification of strongly flat kG-modules. We also prove a corresponding result for groups of the form D × Q where Q is an abelian group of odd order. In the second section we turn our attentions to the alternating group G := Alt_{4}, which is a trivial intersection group with normal Sylow-2-subgroup P ≅ C_{2} × C_{2}. We use Conlon’s [12] classification of kG-modules to classify strongly flat kG-modules, and show by example that V ∈ kG need not be flat even if V_{|P} is strongly flat. In this section we also show how to calculate the cohomology groups H^{1}(G, W) for any kG-module W, and prove that the cohomology group H^{1}(G, R) is always non-zero when R = S(V^{∗}) and V is a faithful kG-module. Aside from this, we also give a complete list of indecomposable kG-modules whose rings of invariants are Cohen-Macaulay, and prove some interesting results regarding the structure of symmetric algebras for kG-modules.

5.1 Dihedral Groups G := D_{4n} (n odd)

The dihedral groups D_{4n} with n odd have Sylow-2-subgroups isomorphic to the group C_{2} × C_{2}. They are not trivial intersection groups, but we shall see that they are sufficiently close to being so that we can find stable cohomology classes with relative ease. We begin this section with a result which is applicable more generally:

Lemma 5.1.1. Suppose G is a finite group and P ∈ Syl_{p}(G), which satisfies

P ∩ g^{−1}P g = \{ \begin{matrix}P & g ∈ P; \\ N & otherwise. \end{matrix}

Here N is some fixed subgroup of Z(G) ∩ P. We could call such a group a central intersection group. Let W be a right kG-module. Then ker(res_{N}^{P}) is contained in the set of G-stable elements of H^{1}(P, W_{|P}).

Proof. Let 0 ≠ α ∈ H^{1}(P, W_{|P}) satisfy res_{N}^{P}(α) = 0. We must show that α is G-stable, i.e.

res_{g^{−1}P g∩P}^{P} (α) = res_{g^{−1}P g∩P}^{g^{−1}P g} (α^{g})

for all g ∈ G. If g ∈ P this is automatically satisfied, since g^{−1}P g ∩ P = P = g^{−1}P g and P acts trivially on H^{1}(P, W_{|P}), by Lemma 1.5.2. So its enough to show that res_{N}^{P}(α) = res_{N}^{g^{−1}P g}(α^{g}) for all g ∈ G\P.

Let z ∈ N, and let ϕ ∈ Der(P, W) be a derivation representing the cohomology class α. Then since α ∈ ker(res_{N}^{P}), ϕ acts as a principle derivation on N. Therefore there is an element r ∈ W such that ϕ(z) = rz − r. Now ϕ^{g} ∈ Der(g^{−1}P g, W) is a derivation representing α^{g}, and for any z′ ∈ N we have

ϕ^{g}(z′) := ϕ(gz′g^{−1})g = ϕ(z′)g = (rz′ − r)g = rgz′ − rg

and so ϕ^{g} acts as a principal derivation on N. This shows that res_{N}^{g^{−1}P g}(α^{g}) = res_{N}^{P}(α) = 0 as required.

Now let G be the group

⟨X, B|X^{2} = B^{2n} = 1, XBX = B^{−1}⟩

where n is some odd number. G is a dihedral group of order 4n. Let P be the subgroup generated by X and B^{n}. Then P is a Sylow-2-subgroup, and each Sylow-2-subgroup takes the form B^{i}P B^{−i} for some i ∈ {0, ...n − 1}. Let N be the subgroup of P generated by B^{n} (This coincides with our earlier definition of N as a subgroup of P). Then N is the centre of G. Theorem 5.1.2. Let G be isomorphic to a dihedral group of order 4n, where n is an odd number, and let P be a fixed Sylow-p-subgroup. Let W be a right G-module, and suppose 0 ≠ α ∈ H^{1}(P, W_{|P}) satisfies res_{H}^{P}(α) = 0 for each maximal subgroup H < P. Then there is a non-zero element β ∈ H^{1}(G, W) with res_{H}^{G}(β) = 0 for every maximal subgroup H < P. Proof. We show first that G satisfies the conditions of Lemma 5.1.1. Let g ∈ G, then we can write g := B^{i}X^{j}, where 0 ≤ i ≤ 2n − 1 and 0 ≤ j ≤ 1. Then g^{−1}B^{n}g = B^{n} and g^{−1}Xg = X^{j}B^{−i}XB^{i}X^{j}

= X^{j}B^{−2i}X^{j+1} = \{ \begin{matrix}XB^{2i} & j = 0; \\ XB^{−2i} & j = 1. \end{matrix}

In particular P = g^{−1}P g if and only if g ∈ P and otherwise P ∩ g^{−1}P g = N as required.

Now let 0 ≠ α ∈ H^{1}(P, W_{|P}) be a cohomology class which satisfies res_{H}^{P}(α) = 0 for each maximal subgroup H < P. Then in particular res_{N}^{P}(α) = 0, and by Lemma 5.1.1 α is a G-stable cohomology class. So α = res_{P}^{G}(β) for some nonzero cohomology class β ∈ H^{1}(G, W), and for each maximal subgroup H < P we have

res_{H}^{G}(β) = res_{H}^{P}(res_{P}^{G}(β)) = res_{H}^{P}(α) = 0.

Corollary 5.1.3. Let G and P be as above. Then V ∈ kG-mod is linearly flat if and only if V_{|P} is so. In particular, V is strongly flat if and only if V_{|P} is so.

Proof. Suppose V is a kG-module for which V_{|P} is linearly flat. Then there exists a non-zero cohomology class α ∈ H^{1}(P, V^{∗}) which satisfies res_{H}^{P}(α) = 0 for each maximal H < P. In particular, res_{N}^{P}(α) = 0. By Theorem 5.1.2, α is Gstable. This means that α = res_{P}^{G}(β) for some 0 ≠ β ∈ H^{1}(G, V^{∗}) which satisfies res_{H}^{G}(β) = res_{H}^{P}(α) = 0 for each maximal H < P. Consequently, V is linearly flat

To prove the second part, suppose that V is a kG-module for which V_{|P} is strongly flat. Then there is a direct summand W^{∗} of S(V_{|P}^{∗}) for which W is linearly flat. Then there is a non-zero cohomology class α ∈ H^{1}(P, W^{∗})|H^{1}(P, S(V^{∗})) which satisfies res_{H}^{P}(α) = 0 for each maximal H < P. The first part of the corollary tells us that α is G-stable, and so α = res_{P}^{G}(β) for some non-zero cohomology class β ∈ H^{1}(G, S(V^{∗})) which satisfies res_{H}^{G}(β) = res_{H}^{P}(α) = 0 for each maximal H < P. Therefore V is strongly flat.

Remarks: Clearly the conclusion of Theorem 5.1.2 gives a stronger property than (*) for these dihedral groups. In fact, a result of this form cannot possibly hold for the groups of the form P × Q, since there are no faithful linearly flat representations of these groups - we require that Q act trivially for linear flatness to hold.

Corollary 5.1.4. Suppose G is a group of the form D × Q, where D is a dihedral group of order 4n, n odd, and Q is an abelian group of odd order. Let V be a representation of G. Then V is strongly flat if and only if V_{|P} is so.

Proof. Note first that by Corollary 5.1.3, it is enough to show that V is strongly flat if and only if V_{|D} is so. Let k ≤ 𝕂 be a field extension, then by Lemma 2.3.2, V is strongly flat if and only if V ⊗_{k} 𝕂 is so. So we may assume that k is algebraically closed. Now let V be strongly flat and 0 ≠ α ∈ H^{1}(G, R) with res_{H}^{G}(α) = 0 for all H < P. Then 0 ≠ γ = res_{D}^{G}(α) ∈ H^{1}(D, R^{Q}) satisfies res_{H}^{D}(γ) = 0 for all H < P, so V_{|D} is strongly flat.

To prove the converse, we argue in a similar fashion to Theorem 4.4.2. Let V be a kG-module for which V_{|P} is strongly flat. We write

V_{|P ×Q} ⊕_{j=1}^{k} W_{j}

with W_{j} ≅ U_{j} ⊗k_{ϵ_{j}}, where each U_{j} is an indecomposable kD - module, k_{ϵ_{j}} is a one - dimensional Q - module with character ϵ_{j} ∈ Hom(Q, k^{×}) and diagonal action (u ⊗ λ)(p, q) := up ⊗ λϵ_{j}(q). Hence we have a decomposition of the G-module

R = ⊗_{j}S(W_{j}) = ⊕_{s=0}^{∞} ⊕_{{\underline{ℓ}∈ℕ^{k}, \atop |\underline{ℓ}|=s}} X\underline{ℓ}

with X\underline{ℓ} := X_{ℓ_{1}} ⊗ ⋯ ⊗ X_{ℓ_{k}} and X_{ℓ_{i}} := S(W_{i})_{ℓ_{i}}. Using the same argument as before, we have a homogenous invariant u ∈ R^{P} with X\underline{ℓ} and uX\underline{ℓ} isomorphic kP -modules and the latter a direct summand of R^{Q}, and a nonzero cohomology class

α ∈ H^{1}(P, uX\underline{ℓ}) | H^{1}(P, R^{Q})

satisfying res_{N}^{P}(α) = 0 for each maximal N < P. By Theorem 5.1.2, α = res_{P}^{D}(β) for some nonzero β ∈ H^{1}(D, R^{Q}). But the argument preceding Theorem 4.4.2 tells us that

H^{1}(D, R^{Q}) ≅ H^{1}(G, R) ≅ H^{1}(D, R)^{Q}

since G = D × Q. Therefore β = res_{D}^{G}(γ) for some 0 ≠ γ ∈ H^{1}(G, R) which satisfies in turn

res_{N}^{G}(γ) = res_{N}^{D}(res_{D}^{G}(γ)) = res_{N}^{D}(β) = res_{N}^{P}(res_{P}^{D}(β)) = res_{N}^{P}(α) = 0

for each maximal subgroup N < P as required.

Since we know which representations of P are strongly flat (Theorem 3.4.8), we can use this and the preceding results to classify strongly flat representations of D and of D × Q.

It is perhaps worth noting that these dihedral groups (and the direct products D×Q) are 2-nilpotent groups - since we may write D = P H where H is the normal subgroup ⟨B^{2}⟩. So the results in this section could be viewed as an extension of Theorem 4.3.1. Also note that although we can classify kD-modules over a field k of characteristic two, we have not used any classification here. Contrast this with the approach we take for the alternating group in the next section.

5.2 The Alternating Group Alt_{4}

In this section we turn our attentions to the group G ≅ Alt_{4} (in characteristic two). This is a trivial intersection group with normal Sylow-2-subgroup P ≅ C_{2} × C_{2}. Define generators as follows:

G := ⟨X, Y, T |X^{2} = Y^{2} = (XY)^{2} = T^{3} = 1, T^{−1}XT = Y, T^{−1}Y T = XY ⟩.

Of course we only need to pick T and one involution as generators, but we stick to this notation so that P := ⟨X, Y ⟩ as in chapter two. The subgroups L, M and N also retain their original definitions, that is to say, L := ⟨XY ⟩, M := ⟨X⟩ and N := ⟨Y ⟩. It is clear that if V is a strongly flat kG-module, then V_{|P} is a strongly flat kP -module. In this section we ask to what extent the converse holds. We do this by finding a classification of kG-modules, and then determining which of these are strongly flat. Note that the arguments of section 4.2 apply to show that, if V is a kF -module for F a T.I. group with Sylow-2-subgroups P ≅ C_{2} × C_{2} and N_{F} (P) := G ≅ Alt_{4}, then V is strongly flat if and only if V_{|G} is so since in this case H^{1}(F, R) ≅ H^{1}(G, R). So if we can classify the strongly flat kG-modules, we will also know which kF -modules are strongly flat. In particular this applies to the simple group SL_{2}(4) ≅ Alt_{5}. We begin by describing the group cohomology. Define Θ_{T,Y} ∈ kG as follows:

Θ_{T,Y} := T^{2} + Y T + 1.

Lemma 5.2.1. Let W be a right-kG-module. Then

H^{1}(G, W) = \frac{ker(Θ_{T,Y}) ∩ W^{M}}{\{z(X + 1) ∈ W : zT + z ∈ W^{N}\} ∩ ker(Θ_{T,Y}) ∩ W^{M}} .

Consequently H^{1}(G, W) ≠ 0 if and only if we can find an element

v ∈ ker(Θ_{T,Y}) ∩ W^{M}

with v ≠ z(X + 1) for any z with zT + z ∈ W^{N}. Moreover there is a nonzero element α ∈ H^{1}(G, W) with res_{H}^{G}(α) = 0 for each H < P if and only if we can find an element

v ∈ ker(Θ_{T,Y}) ∩ W (X + 1)

with v ≠ z(X + 1) for any z with zT + z ∈ W^{N}. Proof. Since P is a normal subgroup of G, we have

H^{1}(P, W) ≅ H^{1}(P, W_{|P})^{G/P} := (\frac{Der(P, W_{|P})}{PDer(P, W_{|P})})^{G/P} ≅ (\frac{Der(P, W_{|P})^{G/P}}{PDer(P, W_{|P})^{G/P}} )

where the last isomorphism arises because G/P is a group of order 3 (generated by the image of T under the canonical map G → G/P) and k a field of characteristic 2. So let 0 ≠ α ∈ H^{1}(G, W). Since res_{P}^{G} is injective, τ := res_{P}^{G}(α) ∈ H^{1}(P, W_{|P})^{G/P} is nonzero, and τ may be represented by a non-principal derivation ϕ ∈ Der(P, W) with ϕ^{T} = ϕ.

Now let v := ϕ(X) and define w := ϕ(Y). By Lemma 1.5.10 we have v ∈ W^{M}, w ∈ W^{N} and v + w ∈ W^{L}, such that there is no z ∈ W with z(X + 1) = v, z(Y + 1) = w. Since ϕ(Y) = ϕ^{T} (Y), we have w = ϕ(T Y T^{−1})T = vT and

v = ϕ(X) = ϕ^{T} (X) = ϕ(T XT^{−1})T = ϕ(XY)T = ϕ(X)Y T +ϕ(Y)T = vY T +vT^{2}

so that v ∈ ker(Θ_{T,Y}). Furthermore if v = 0 then w = vT = 0 and ϕ is the zero derivation, a contradiction. So we have found a non-zero v ∈ ker(Θ_{T,Y} ∩ W^{M}) as required. Now suppose v = z(X + 1) for some z with zT + z ∈ W^{N}. Then vT = z(X +1)T = zT (Y +1) = z(Y +1) and ϕ is principal, another contradiction. For the second part, suppose α ∈ H^{1}(G, W) satisfies res_{H}^{G}(α) = 0 for each H < P. Since res_{P}^{G} is injective, τ := res_{P}^{G}(α) ∈ H^{1}(P, W_{|P})^{G/P} is nonzero and satisfies res_{H}^{P}(τ) = 0 for each H < P. Since res_{M}^{P}(τ) = 0, we have ϕ_{|M} is principal, i.e. v := ϕ(X) ∈ W (X + 1), as required.

Conversely, let 0 ≠ v ∈ ker(Θ_{T,Y}) ∩ W^{M} be such that v ≠ z(X + 1) for any z with zT + z ∈ W^{N} and define w := vT ∈ W^{N}. Then we have

v + w = v + vT = vT^{2} + vY T + vT = vT^{2} + vT (XY + 1) ∈ W^{L}.

Further if v = z(X +1) and w = z(Y +1) then we have vT = zT (Y +1) = z(Y +1) so that zT + z ∈ W^{N}, a contradiction. Therefore by Lemma 1.5.10, τ := [v, w] is a non-zero element of H^{1}(P, W_{|P}), and there is a non-principal derivation ϕ ∈ Der P, W_{|P} representing τ with ϕ(X) = v, ϕ(Y) = w. Furthermore we have

ϕ^{T} (Y) = ϕ(T Y T^{−1})T = ϕ(X)T = vT = w = ϕ(Y)

and

ϕ^{T} (X) = ϕ(T XT^{1})T = ϕ(XY)T = vY T + wT = vY T + vT^{2} = v

so that ϕ = ϕ^{T}. Since ϕ^{T} is a derivation representing τ^{T}, we have τ^{T} = τ and so 0 ≠ τ ∈ H^{1}(P, W_{|P})^{G/P}. Then by Lemma 1.5.5, τ = res_{P}^{G}(α), for some nonzero α ∈ H^{1}(G, W), as required. For the second part, suppose there is a z′ ∈ W with v = z′(X +1). By the first part of our argument, τ := [v, vT] is a nonzero element of H^{1}(P, W_{|P}) in the image of the restriction map res_{P}^{G}. The exact form of the restriction maps to subgroups of P is given in Lemma 1.5.10 - this states that if γ := [a, b], then res_{M}^{P}(γ) = [a], res_{N}^{P}(γ) = [b] and res_{L}^{P}(γ) = [a + b] as elements of H^{1}(M, W), H^{1}(N, W) and H^{1}(L, W) respectively. Therefore we have

res_{M}^{P}(τ) = [v] = 0

because v := z′(X + 1) ∈ W (X + 1),

res_{N}^{P}(τ) = [vT] = 0

because vT = z′(X + 1)T = z′T (Y + 1) ∈ W (Y + 1) and finally

res_{L}^{P}(τ) = [v + vT] = 0

because

v + vT = (vT^{2} + v)T = vY T^{2} = z′Y (X + 1)T^{2} = z′Y T^{2}(XY + 1) ∈ W (XY + 1).

Therefore res_{H}^{P}(τ) = 0 for every proper subgroup H < P, and so res_{H}^{G}(α) = res_{H}^{P}(res_{P}^{G}(α)) = res_{H}^{P}(τ) = 0 for every H < P as required.

Corollary 5.2.2. Let V be a kG-module. Then V is strongly flat if and only if we can find an element

v ∈ ker(Θ_{T,Y}) ∩ R(X + 1)

with v ≠ z(X + 1) for any z with zT + z ∈ R^{N}.

We now use these results to classify strongly flat representations of the alternating group Alt_{4} over a field of characteristic two, which we may assume to be algebraically closed, by Lemma 2.3.2. In contrast to the case of dihedral groups, we will need a complete classification of modules for Alt_{4}. A classification is partially described in Benson’s book ([4], Remark 4.3.3). Using this as a guideline, we are able to prove the following result, which was proven originally by S.B. Conlon:

Theorem 5.2.3 (Conlon [12]). The absolutely indecomposable (left) kG-Modules fall into the families listed below. ^{1}

• For every even dimension 2n there are representations W_{2n}(ω^{j})

X ∼ ( \begin{matrix}I_{n} & ωω^{2}0...0 \\ & 0ωω^{2}...0 \\ ... & ... \\ & 0....ωω^{2} \\ & 0......ω \\ 0 & I_{n} \end{matrix} )

Y ∼ ( \begin{matrix}I_{n} & ω^{2}ω0...0 \\ & 0ω^{2}ω...0 \\ ... & ... \\ & 0....ω^{2}ω \\ & 0......ω^{2} \\ 0 & I_{n} \end{matrix} )

T ∼ ω^{j} diag(1, ω, ...ω^{n−1}, ω, ω^{2}, ...ω^{n}) These modules have W_{2n}(ω^{j})_{|P} ≅ V_{2n,ω^{j}} and dim(V^{P}) = n. There are also the “conjugate” representations W′_{2n}(ω^{j}), obtained by taking the complex conjugates of the matrices above. These are equivalent to the representations obtained from W_{2n}(ω^{j}) under the automorphism Ψ of G defined by

Ψ(X) = Y, Ψ(Y) = X, Ψ(T) = T^{−1}.

1Here ω ∈ k denotes a primitive cube root of unity.

• For every odd dimension 2n + 1 there are representations W_{2n+1}(ω^{j})

X ∼ ( \begin{matrix}I_{n+1} & ω^{2}0...0 \\ & ωω^{2}...0 \\ ... & ... \\ & 0....ωω^{2} \\ & 0......ω \\ 0 & I_{n} \end{matrix} )

Y ∼ ( \begin{matrix}I_{n+1} & ω0...0 \\ & ω^{2}ω...0 \\ ... & ... \\ & 0.....ω^{2}ω \\ & 0.......ω^{2} \\ 0 & I_{n} \end{matrix} )

T ∼ ω^{j} diag(1, ω, ...ω^{n}, ω^{2}, ω^{3}, ...ω^{n+1}) These modules have W_{2n+1}(ω^{j})_{|P} ≅ V_{2n+1} and dim(V^{P}) = n + 1.

• For every odd dimension 2n + 1 there are representations W_{−(2n+1)}(ω^{j})

X ∼ ( \begin{matrix}I_{n} & ωω^{2}0...0 \\ & 0ωω^{2}...0 \\ ... & ... \\ & 0.....ωω^{2} \\ 0 & I_{n+1} \end{matrix} )

Y ∼ ( \begin{matrix}I_{n} & ω^{2}ω0...0 \\ & 0ω^{2}ω...0 \\ ... & ... \\ & 0.....ω^{2}ω \\ 0 & I_{n+1} \end{matrix} )

T ∼ ω^{j} diag(1, ω, ...ω^{n−1}, ω, ω^{2}, ...ω^{n+1}) These modules have W_{−(2n+1)}(ω^{j})_{|P} ≅ V_{−(2n+1)} and dim(V^{P}) = n. There is an obvious isomorphism W_{1}(ω^{j}) ≅ W_{−1}(ω^{j}). These modules, which on which P acts trivially, will be denoted W_{1}(ω^{j}) by preference.

• For each dimension 6n and each λ ≠ ω, ω^{2} there are representations \overline{W}_{6n}(λ)

X ∼ ( \begin{matrix}I & I & 0 & 0 & 0 & 0 \\ 0 & I & 0 & 0 & 0 & 0 \\ 0 & 0 & I & J(λ) & 0 & 0 \\ 0 & 0 & 0 & I & 0 & 0 \\ 0 & 0 & 0 & 0 & I & J(λ + 1) \\ 0 & 0 & 0 & 0 & 0 & I \end{matrix} )

Y ∼ ( \begin{matrix}I & J(λ) & 0 & 0 & 0 & 0 \\ 0 & I & 0 & 0 & 0 & 0 \\ 0 & 0 & I & J(λ + 1) & 0 & 0 \\ 0 & 0 & 0 & I & 0 & 0 \\ 0 & 0 & 0 & 0 & I & I \\ 0 & 0 & 0 & 0 & 0 & I \end{matrix} )

T ∼ ( \begin{matrix}0 & 0 & I & 0 & 0 & 0 \\ 0 & 0 & 0 & I & 0 & 0 \\ 0 & 0 & 0 & 0 & I & 0 \\ 0 & 0 & 0 & 0 & 0 & I \\ I & 0 & 0 & 0 & 0 & 0 \\ 0 & I & 0 & 0 & 0 & 0 \end{matrix} )

In this family there are equivalences \overline{W}_{6n}(λ) ≅ \overline{W}_{6n}(1/(1 − λ)) ≅ \overline{W}_{6n}(λ −

1/λ). These modules have \overline{W}_{6n}(λ)_{|P} ≅ V_{2n,1/(1−λ)} ⊕ V_{2n,(λ−1)/λ} ⊕ V_{2n,λ} and

dim(V^{P}) = 3n.

• There are 3 projective modules \overline{W}_{4}(ω^{j})

X ∼ ( \begin{matrix}1 & ω & ω^{2} & 1 \\ 0 & 1 & 0 & ω^{2} \\ 0 & 0 & 1 & ω \\ 0 & 0 & 0 & 1 \end{matrix} )

Y ∼ ( \begin{matrix}1 & ω^{2} & ω & 1 \\ 0 & 1 & 0 & ω \\ 0 & 0 & 1 & ω^{2} \\ 0 & 0 & 0 & 1 \end{matrix} )

T ∼ ω^{j} ( \begin{matrix}ω^{j} & 0 & 0 & 0 \\ 0 & ω^{j+1} & 0 & 0 \\ 0 & 0 & ω^{j+2} & 0 \\ 0 & 0 & 0 & ω^{j} \end{matrix} )

These modules have \overline{W}_{4}(ω^{j})_{|P} ≅ \overline{V}_{4} and dim(V^{P}) = 1. The full proof of this result requires the use of some machinery not used

elsewhere in this work. For this reason, we defer the proof to the Appendix. We use this result to classify the strongly flat representations of G. Theorem 5.2.4. An indecomposable modular representation V of Alt_{4} over an algebraically closed field of characteristic 2 is strongly flat if and only if it is isomorphic to one of the following:

1. Any indecomposable module of dimension ≥ 7;

2. Any of the six-dimensional modules W_{6}(ω^{i}) (or W′_{6}(ω^{i}) );

3. Any of the five-dimensional modules W_{5}(ω^{i});

4. Any of the five-dimensional modules W_{−5}(ω^{i});

5. Any none-projective module of dimension 4;

6. Any of the two-dimensional modules W_{2}(ω), W_{2}(ω^{2}), W′_{2}(ω) and W′_{2}(ω^{2});

7. A six-dimensional module \overline{W}_{6}(λ) for some λ ∈ k where λ ≠ 0, 1, ∞ or any cube root of unity..

The proof of this result is divided into a number of lemmas. The modules not listed above are those which are not strongly flat. We will refer to the set of these as NSF (G). This set is

{W_{1}(1), W_{1}(ω), W_{1}(ω^{2}), W_{2}(1), W′_{2}(1), W_{3}(1), W_{3}(ω), W_{3}(ω^{2}), W_{−3}(1), W_{−3}(ω),

W_{−3}(ω^{2}), \overline{W}_{4}(1), \overline{W}_{4}(ω), \overline{W}_{4}(ω^{2}), \overline{W}_{6}(1)\}.

With the exception of W_{2}(1) and W′_{2}(1), it is clear that no module in NSF (G) is strongly flat, since their restrictions to P are not. This module provides us with our first example of a modules V for which V not strongly flat but V_{|P} is so.

The following observation simplifies our problem considerably: Lemma 5.2.5. Let M be an indecomposable direct summand of S(V^{∗}) where V is a kG-module. Then there is an invariant u ∈ R^{P} (which is also a relative invariant with respect to G) of degree four such that for any integer d, u^{d}M is an indecomposable direct summand of S(V^{∗}) of degree (deg(M) + 4d). Moreover if V ≅ W_{m}(ω^{j}), (where m may be odd and positive or negative, or positive and even, and n := dim(V^{P})) then there is a set of such invariants u_{1}, u_{2}, …, u_{n} ∈ R^{P} for which u_{i}^{d}M is a direct summand of S(V^{∗}) for each i = 1, 2, … n and every d, with

u_{r}^{d}W_{m}^{∗}(ω^{j}) ≅ W_{m}^{∗}(ω^{j+d(j+r−1)})

. Proof. Choose a basis {x_{1}, x_{2}, …, x_{m}} of V^{∗} such that the matrices representing the indecomposable summands of V^{∗} are in the form given by the classification in Theorem 5.2.3 - in particular, so that T acts diagonally on V^{∗}, with eigenvalues cube roots of unity. Then we define, for each r ≤ n, u_{r} := ∏_{g∈P} (x_{i}g). Now M_{|P} is a direct summand of S(V_{|P}^{∗}). We may write R = A[x_{r}] with A := k[x_{1}, x_{2}, … x_{r−1}, x_{r+1}, …, x_{m}] which is a kG-module, and by Lemma 1.3.2, u_{r}^{d}M_{|P} is a direct summand of S(V_{|P}^{∗}) of degree (deg(M) + |P |d) which is isomorphic as a kP -module to M_{|P}. Note that M_{|P} is not necessarily indecomposable even if M is, but every indecomposable summand of M_{|P} is still propagated by u_{r}. Since u_{r} ∈ R^{P} for each r, the actions of X and Y on u_{r}M are the same as those on M. If {y_{1}, y_{2}, … y_{l}} is a basis for M, then {u_{r}y_{1}, u_{r}y_{2}, … u_{r}y_{l}} is a basis for u_{r}M and the action of T on basis elements of u_{r}M is given by

(u_{r}y_{i})T = (u_{r}T)(y_{i}T) = (∏_{g∈P} x_{r}g)T (y_{i}T) = ( ∏_{g′∈P} x_{r}T g′)(y_{i}T) = ϵ_{r}^{4}u_{r}(y_{i}T)

= ϵ_{r}u_{r}(y_{i}T) ∈ u_{r}M

where ϵ_{r} is the cube root of unity such that x_{r}T = ϵ_{r}x_{r}.

This shows that u_{r}M, and by induction u_{r}^{d}M, is an indecomposable kGsubmodule of S(V^{∗}), which is also a kP -direct summand of S(V^{∗}). Moreover, since P is the Sylow-2-subgroup of G, every kG module is projective relative to P, and this tells us that u_{r}^{d}M is an indecomposable kG-direct summand of S(V^{∗}). Now suppose that V ≅ W_{m}ω^{j}, for some m (which may be odd and positive or negative, or positive and even, with n := dim(V^{P})) and any j. Then ϵ_{r} = ω^{j+r−1}, and our formulae for the action of T tell us

u_{r}W_{m}^{∗}(ω^{j}) ≅ W_{m}^{∗}(ω^{j}) ⊗ W_{1}^{∗}(ω^{j+r−1}) = W_{m}^{∗}(ω^{2j+r−1})

and by induction u_{r}^{d}W_{m}^{∗}(ω^{j}) ≅ W_{m}^{∗}(ω^{j+d(j+r−1)}).

Corollary 5.2.6. Suppose W := V^{∗} where V ≅ W_{m}(ω) or W_{m}(ω^{2}). Then S(W) has direct summands isomorphic to W_{m}^{∗}(1), W_{m}^{∗}(ω) and W_{m}^{∗}(ω^{2}). If W ≅ W_{m}^{∗}(1), then S(W) always has direct summands isomorphic to W_{m}^{∗}(1), and has summands isomorphic to W_{m}^{∗}(ω) and W_{m}^{∗}(ω^{2}) provided dim(V^{P}) ≥ 2. Consequently if W_{m}(1) is linearly flat, then W_{m}(ω) and W_{m}(ω^{2}) are strongly flat, and if W_{m}(ω) is linearly flat, W_{m}(ω^{2}) is always strongly flat and W_{m}(1) is strongly flat provided dim(V^{P}) ≥ 2. Similarly if W_{m}(ω^{2}) is linearly flat, W_{m}(ω) is always strongly flat and W_{m}(1) is strongly flat provided dim(V^{P}) ≥ 2.

Remark: The condition dim(V^{P}) ≥ 2 excludes only the modules W_{1}(−), W′_{2}(−), W_{2}(−), W_{3}(−), and \overline{W}_{4}(−).

Proof. Let V ≅ W_{m}(ω). Then

u_{1}(V^{∗}) ≅ W_{m}^{∗}(ω^{2})|S^{5}(V^{∗}),
u_{1}^{2}(V^{∗}) ≅ W_{m}^{∗}(1)|S^{9}(V^{∗}).

Let V ≅ W_{m}(ω^{2}). Then

u_{1}(V^{∗}) ≅ W_{m}^{∗}(ω)|S^{5}(V^{∗}),
u_{1}^{2}(V^{∗}) ≅ W_{m}^{∗}(1)|S^{9}(V^{∗}).

Now V ≅ W_{m}(1), and suppose dim(V^{P}) ≥ 2. Then

u_{2}(V^{∗}) ≅ W_{m}^{∗}(ω)|S^{5}(V^{∗}),
u_{2}^{2}(V^{∗}) ≅ W_{m}^{∗}(ω^{2})|S^{9}(V^{∗}).

Lemma 5.2.7. Suppose V^{∗} := W where V ≅ W_{2n}(ω^{i}) or W′_{2n}(ω^{i}), for some i ∈ {0, 1, 2} and n ≥ 2. Then V is strongly flat.

Remark: This establishes parts (2) and (5) of Theorem 5.2.4, as well as contributing to part (1).

Proof. It is clear (by symmetry) that if V ≅ W_{2n}(ω^{i}), V is strongly flat if and only if it is so when V ≅ W′_{2n}(ω^{i}), so we consider only the modules W_{2n}(ω^{i}). By Corollary 5.2.6, it is sufficient to show that if V ≅ W_{2n}(1), V is linearly flat. So let V ≅ W_{2n}(1). Then

W (X + 1) = W (Y + 1) = W^{N} = W^{M} = ⟨x_{n+1}, x_{n+2}, … x_{2n}⟩

is a kG-submodule of W on which X and Y act trivially, and x_{n+j} ∈ ker(Θ_{T,Y}) ∩ W (X + 1) if and only if j ≠ 0 mod 3. Further, if z ∈ W, zT + z ∈ W^{N} if and only if z ∈ K ⊕ ⟨x_{n+1}, …, x_{2n}⟩, where

K := ker(T + 1) ∩ ⟨x_{1}, …, x_{n}⟩ = ⟨x_{i} : i = 1 mod 3, i ≤ n⟩.

Now if z ∈ ⟨x_{n+1}, …, x_{2n}⟩, z(X + 1) = 0, and so

{z(X+1) : zT +z ∈ W^{N}} = K(X+1) = ⟨ωx_{n+i}+ω^{2}x_{n+1+i} : i = 1 mod 3, i ≤ n⟩,

where we define x_{2n+1} = 0 so that ωx_{2n} ∈ K(X +1) if n = 1 mod 3. In particular for any value of n,

x_{n+1} ∈ W (X + 1) ∩ ker(Θ_{T,Y})\{z(X + 1) : zT + z ∈ W^{N}}.

By Lemma 5.2.2, V is linearly flat.

Lemma 5.2.8. Let V^{∗} := W. If V ≅ W_{2}(1), V is not strongly flat, but is trivially flat with cc_{G}(R) = 1. If V ≅ W_{2}(ω), V is linearly flat. If V ≅ W_{2}(ω^{2}), V is strongly flat, but not linearly so.

Remark: This establishes part (6) of Theorem 5.2.4, since the analogous results for W′_{2}(1), W′_{2}(ω) and W′_{2}(ω^{2}) hold by symmetry.

Proof. Let V ≅ W_{2}(ω^{i}) where i ∈ {0, 1, 2}. G acts on W by the matrices

X ≃ ( \begin{matrix}1 & ω \\ 0 & 1 \end{matrix} )

Y ≃ ( \begin{matrix}1 & ω^{2} \\ 0 & 1 \end{matrix} )

T ≃ ( \begin{matrix}ω^{i} & 0 \\ 0 & ω^{i+1} \end{matrix} ) .

In each case we have W^{M} = W (X + 1) = ⟨x_{2}⟩, so we must simply determine whether x_{2} ∈ ker Θ_{T,Y} and whether x_{2} ∈ {z(X + 1) : zT + z ∈ W^{N}}. The element x_{2}Θ_{T,Y} = (ω^{i+1} + ω^{−i−1}) + 1 = 0 unless i = 2. In particular this shows that cc_{G}(R) = 1 when i = 0, using the first part of Lemma 5.2.1. Further, x_{2} = ω^{2}x_{1}(X + 1), and x_{1}(T + 1) = x_{1}(ω^{i} + 1) is in W^{N} only when i = 0. So V is linearly flat if and only if i = 1. The remainder of the result follows easily on applying Corollary 5.2.6. We show that W_{2}(1) is not strongly flat in Theorem 5.2.14.

Lemma 5.2.9. Suppose V^{∗} := W and V is isomorphic to one of the representations W_{2n+1}(ω^{i}) for n ≥ 3. Then V is strongly flat.

Remark: This contributes to the proof of part (1) of Theorem 5.2.4.

Proof. It is sufficient to show that for i = 0, V is linearly flat, by Corollary 5.2.6. So let V ≅ W_{2n+1}(1) for some n ≥ 3. In this case W (X + 1) = ⟨x_{n+2}, ...x_{2n+1}⟩ whereas W^{N} = ⟨x_{n+2}, ...x_{2n+1}, y⟩ where y := ∑_{j=1}^{n+1} ω^{1−j}x_{j}. The coefficient of x_{1} in y is not zero, and for each element w ∈ W (T + 1) the coefficient of x_{1} in w is zero, therefore y ∉ W (T + 1). Let z ∈ W. If zT + z ∈ W^{N}, we must have

z ∈ K ⊕ ⟨x_{n+2}, ..., x_{2n+1}⟩

where

K := ker(T + 1) ∩ ⟨x_{1}, x_{2}, …, x_{n+1}⟩ = ⟨x_{i} : i = 1 mod 3, i ≤ n + 1⟩.

If z ∈ ⟨x_{n+2}, ...x_{2n+1}⟩, then z(X + 1) = 0, and therefore

= K(X + 1) = ⟨ω^{2}x_{n+2}⟩ ⊕ ⟨ω^{2}x_{n+i} + ωx_{n+1+i} : i = 1 mod 3, 1 < i ≤ n + 1⟩.

Now on the submodule W (X + 1), X and Y act trivially and so the action of (Θ_{Y,T}) is given by

x_{n+j}Θ_{T,Y} = (ω^{j} + ω^{−j} + 1)x_{n+j}

j ∈ {2, ...n + 1}. Therefore x_{n+j} ∈ ker(Θ_{T,Y}) j ≠ 0 mod 3. In particular, for each n ≥ 3, x_{n+4} ∈ ker(Θ_{T,Y}) ∩ W (X + 1)\{z(X + 1) : zT + z ∈ W^{N}}. So by Lemma 5.2.2 V is linearly flat.

Lemma 5.2.10. Let V^{∗} := W. If V ≅ W_{5}(1), V is strongly flat, but not linearly so. If V ≅ W_{5}(ω^{2}), V is linearly flat. If V ≅ W_{2}(ω), V is strongly flat, but not linearly so.

Remark: This establishes part (3) of Theorem 5.2.4.

Proof. Let V ≅ W_{5}(ω^{i}) where i ∈ {0, 1, 2}. G acts on W by the matrices

X ≃ ( \begin{matrix}1 & 0 & 0 & ω^{2} & 0 \\ 0 & 1 & 0 & ω & ω^{2} \\ 0 & 0 & 1 & 0 & ω \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix} )

Y ≃ ( \begin{matrix}1 & 0 & 0 & ω & 0 \\ 0 & 1 & 0 & ω^{2} & ω \\ 0 & 0 & 1 & 0 & ω^{2} \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix} )

T ≃ ( \begin{matrix}ω^{i} & 0 & 0 & 0 & 0 \\ 0 & ω^{i+1} & 0 & 0 & 0 \\ 0 & 0 & ω^{i+2} & 0 & 0 \\ 0 & 0 & 0 & ω^{i+2} & 0 \\ 0 & 0 & 0 & 0 & ω^{i} \end{matrix} )

We have W (X + 1) = ⟨x_{4}, x_{5}⟩ in all cases. If i = 0, then x_{5} ∉ ker(Θ_{T,Y}) and x_{4} = ωx_{1}(X + 1) with x_{1} ∈ ker(T + 1), so x_{4} ∈ {z(X + 1) : zT + z ∈ W^{N}}. If i = 1, then x_{4} ∉ ker(Θ_{T,Y}) and x_{5} = ω^{2}x_{3}(X + 1) with x_{3} ∈ ker(T + 1), so x_{5} ∈ {z(X + 1) : zT + z ∈ W^{N}}. So if i = 0 or 1, V is not linearly flat. If, however, i = 2, then

{z(X + 1) : zT + z ∈ W^{N}} = ⟨ωx_{4} + ω^{2}x_{5}⟩,

and x_{4} ∈ ker(Θ_{T,Y}). So V is linearly flat in this case. The fact that V is strongly flat when i = 1 now follows easily from Corollary 5.2.6, as does the fact that V is strongly flat when i = 0, since dim(V^{P}) = 3 ≥ 2.

Lemma 5.2.11. Suppose V^{∗} := W and V is isomorphic to one of the representations W_{−(2n+1)}(ω^{i}) for n ≥ 3. Then V is strongly flat.

Remark: This contributes to the proof of part (1) of Theorem 5.2.4

Proof. By Corollary 5.2.6, it is sufficient to prove that V is linearly flat when i = 0. In this case we have

W^{N} = W^{M} = ⟨x_{n+1}, …, x_{2n+1}⟩,

whereas W (X + 1) = ⟨α_{i} : i ≤ n⟩ where α_{i} := x_{i}(X + 1) = ωx_{n+i} + ω^{2}x_{n+1+i}. If zT + z ∈ W^{N}, we must have

z ∈ K ⊕ ⟨x_{n+1}, …, x_{2n+1}⟩

where

K := ker(T + 1) ∩ ⟨x_{1}, …, x_{n}⟩ = ⟨x_{i} : i = 1 mod 3, i ≤ n⟩.

If z ∈ W^{N} then z(X + 1) = 0, so

{z(X + 1) : zT + z ∈ W^{N}} = K(X + 1) = ⟨α_{i} : i = 1 mod 3, i ≤ n⟩

On the space ⟨x_{n+1}, ...x_{2n+1}⟩, the action of Θ_{T,Y} is given by x_{n+j}Θ_{T,Y} = (ω^{j} + ω^{2j} + 1)x_{n+j}, so

α_{j}Θ_{T,Y} = (ω^{j} + ω^{2j} + 1)ωx_{n+j} + (ω^{j+1} + ω^{2j+2} + 1)ω^{2}x_{n+j+1}

= \{ \begin{matrix}ωx_{n+j} & j = 0 mod 3 \\ 0 & j = 1 mod 3 \\ ω^{2}x_{n+1+j} & j = 2 mod 3 \end{matrix}

Let v := α_{2} + ωα_{3} = ωx_{n+2} + x_{n+4}. Then vΘ_{T,Y} = ω^{2}x_{n+3} + ω^{2}x_{n+3} = 0, so

0 ≠ v ∈ ker Θ_{T,Y} ∩ W (X + 1)\{z(X + 1) : zT + z ∈ W^{N}}.

By Lemma 5.2.2, V is linearly flat. Lemma 5.2.12. Let V^{∗} := W with V ≅ W_{−5}(ω^{i}). If i = 0, V is strongly, but not linearly flat. If i = 1, V is linearly flat. If i = 2, then V is strongly but not linearly flat.

Remark: This establishes part (4) of Theorem 5.2.4. Proof. Let V ≅ W_{−5}(ω^{i}) where i ∈ {0, 1, 2}. G acts on W by the matrices

X ≃ ( \begin{matrix}1 & 0 & ω & ω^{2} & 0 \\ 0 & 1 & 0 & ω & ω^{2} \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix} )

Y ≃ ( \begin{matrix}1 & 0 & ω^{2} & ω & 0 \\ 0 & 1 & 0 & ω^{2} & ω \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix} )

T ≃ ( \begin{matrix}ω^{i} & 0 & 0 & 0 & 0 \\ 0 & ω^{i+1} & 0 & 0 & 0 \\ 0 & 0 & ω^{i+1} & 0 & 0 \\ 0 & 0 & 0 & ω^{i+2} & 0 \\ 0 & 0 & 0 & 0 & ω^{i} \end{matrix} )

In each case we have W (X + 1) = ⟨α_{1}, α_{2}⟩. If i = 0 we have α_{1} = x_{1}(X + 1), and x_{1}(T + 1) = 0, so α_{1} ∈ {z(X + 1) : zT + z ∈ W^{N}}, and for each pair λ_{1}, λ_{2} ∈ k, (λ_{1}α_{1} + λ_{2}α_{2})Θ_{T,Y} = λ_{2}ω^{2}x_{5}, so (λ_{1}α_{1} + λ_{2}α_{2}) ∈ ker Θ_{T,Y} if and only if λ_{2} = 0. If i = 2, we have α_{2} = x_{2}(X + 1), and x_{2}(T + 1) = 0, so α_{2} ∈ {z(X + 1) : zT + z ∈ W^{N}}, and for each pair λ_{1}, λ_{2}, (λ_{1}α_{1} + λ_{2}α_{2})Θ_{T,Y} = λ_{1}ωx_{4}, so (λ_{1}α_{1} + λ_{2}α_{2}) ∈ ker Θ_{T,Y} if and only if λ_{1} = 0. If i = 1, {z(X + 1) : zT + z ∈ W^{N}} = 0, and α_{1} + ωα_{2} ∈ ker Θ_{T,Y}. So in this case V is linearly flat. The fact that V is strongly flat when i = 2 now follows easily from Lemma 5.2.6, as does the fact that V is strongly flat when i = 0, since dim(V^{P}) = 2.

Lemma 5.2.13. Suppose V^{∗} := W where V ≅ \overline{W}_{6n}(λ), where λ ≠ 0 if n = 1, and λ ≠ ω, ω^{2}. Then V is linearly flat.

Remark: This establishes part (7) of Theorem 5.2.4, and also completes part (1), thus completing the whole proof. This particular result is a special case of [21], Lemma 4.2.

Proof. Let A be a kP -module, with A^{∗} := U and A ≅ V_{2n}(λ). By Theorem 3.3.2, A is linearly flat, so there is a nonzero cohomology class τ ∈ H^{1}(P, U) with res_{H}^{P}(τ) = 0 for each H < P. Now U ↑_{P}^{G}≅ W, and W_{|P} has a direct summand isomorphic to U. So there is a nonzero cohomology class τ̃ ∈ H^{1}(P, W_{|P}) with res_{H}^{P}(τ̃) = 0 for each H < P. Define σ := Tr_{P}^{G}(τ̃). By the Eckman-Shapiro Lemma (see for example [18], Corollary 4.2.7), Tr_{P}^{G} : H^{1}(P, W)_{G−Stab} → H^{1}(G, W ↑_{P}^{G}) is an isomorphism, so σ is not zero. If H < P, then Lemma 1.5.5 tells us

res_{H}^{G}(σ) = res_{H}^{G}(Tr_{P}^{G}(τ̃)) = ∑_{g∈⟨T ⟩} res_{H}^{P}(τ^{g}) = res_{g^{−1}Hg}^{P}(τ̃) = 0.

This shows that V is linearly flat, and also completes the proof of Theorem 5.2.4.

Our main conclusion is that the situation for strong flatness of indecomposable kG-modules is very different to that for kP -modules. It is certainly not true that V ∈ kG-mod is strongly flat if and only if V_{|P} is so, in contrast to when G is a direct product P × Q - we show this by example in the next lemma. Also, we have examples of indecomposable modules (e.g. W_{2}(ω^{2})) which are strongly, but not linearly, flat.

We include the following lemma for completeness’ sake.

Lemma 5.2.14. 1. For every faithful representation V of G with R := S(V^{∗}), we have cc_{G}(R) = 1.

2. The representations V with V ≅ \overline{W}_{4}(ω^{j}), which are neither trivially nor strongly flat, are not flat.

3. The representations V with V ≅ W_{2}(1) or W′_{2}(1) are trivially flat, but not strongly flat. However, V_{|P} is linearly flat in both cases.

4. The representation V with V ≅ \overline{W}_{6}(0) is not flat

Proof. It is clear that a module V which is strongly flat has cc_{G}(R) = 1, so we need only to show (1) for the faithful modules in NSF (G), namely the modules in the set

{W_{2}(1), W′_{2}(1), W_{3}(1), W_{3}(ω), W_{3}(ω^{2}), W_{−3}(1), W_{−3}(ω), W_{−3}(ω^{2}),
\overline{W}_{4}(1), \overline{W}_{4}(ω), \overline{W}_{4}(ω^{2}), \overline{W}_{6}(1)\}.

We showed in Lemma 5.2.7 that cc_{G}(R) = 1 when W := V^{∗} ≅ W_{2}(1) (or W′_{2}(1)).

Let U be a right kG-module isomorphic to W_{3}^{∗}(1). Observe that we have U^{M} = ⟨ωx_{1} + ω^{2}x_{2}, x_{3}⟩, and the matrix which represents Θ_{T,Y} is

( \begin{matrix}1 & 0 & 1 \\ 0 & 0 & ω \\ 0 & 0 & 0 \end{matrix} )

In this case we have x_{3} ∈ ker(Θ_{T,Y})∩U^{M}, so by the first part of Lemma 5.2.1, H^{1}(G, U) ≠ 0. Now let V^{∗} := W with V ≅ W_{3}(ω^{i}). If i = 0 then S^{1}(V^{∗}) ≅ U and so H^{1}(G, R) has a direct summand isomorphic to H^{1}(G, U). If i = 1 or i = 2 then u_{1}^{2}W ≅ U|S^{9}(V^{∗}) by Lemma 5.2.5 and so again H^{1}(G, R) has a direct summand isomorphic to H^{1}(G, U). In each case, H^{1}(G, R) contains a nonzero direct summand and so cc_{G}(R) = 1.

Now let U be a kG-module isomorphic to W_{−3}^{∗}(1). Observe that U^{M} = ⟨x_{2}, x_{3}⟩, and the matrix which represents Θ_{T,Y} is

( \begin{matrix}1 & 1 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} ) .

So x_{3} ∈ ker(Θ_{T,Y}) ∩ U^{M}, and by the first part of Lemma 5.2.1, H^{1}(G, U) ≠ 0. Now let V^{∗} := W ≅ W_{−3}(ω^{i}). If i = 0 then S^{1}(V^{∗}) ≅ U and so H^{1}(G, R) contains a non-zero direct summand ≅ H^{1}(G, U). If i = 1 or i = 2 then u_{1}^{2}W ≅ U|S^{9}(V^{∗}) and again H^{1}(G, R) contains a non-zero direct summand ≅ H^{1}(G, U). So we have shown that cc_{G}(R) = 1 whenever V ≅ W_{−3}(ω^{i}).

Now if W is projective, then H^{1}(G, W) = 0. Therefore H^{1}(G, W) = 0 whenever W ≅ \overline{W}_{4}(ω^{i}). However, using the meataxe in MAGMA we can show that S^{2}(\overline{W}_{4}(ω^{i})) ≅ \overline{W}_{6}(0) for all i ∈ \{0, 1, 2\}. So we may complete the proof of (1) by showing that H^{1}(G, W) ≠ 0 when W ≅ \overline{W} _{6}^{∗}(0). In this case X and Y act as the matrices

( \begin{matrix}1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} )

and

( \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} )

and T acts as the matrix

( \begin{matrix}0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \end{matrix} ) .

So then, W^{M} = ⟨x_{2}, x_{4}, x_{5}, x_{6}⟩, while the matrix which represents Θ_{T,Y} is

( \begin{matrix}1 & 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 & 1 & 1 \\ 0 & 1 & 0 & 1 & 0 & 1 \\ 1 & 1 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 & 0 & 1 \end{matrix} ) .

Observe that x_{2} + x_{4} ∈ ker(Θ_{T,Y}) ∩ W^{M}. So H^{1}(G, W) ≠ 0. This completes the proof of part (1).

To prove (2), let V ≅ \overline{W}_{4}(ω^{j}). Then we have shown that cc_{G}(R) = 1 and we must prove that depth(R^{G}) > dim(V^{P}) + 2 = 3, i.e. depth(R^{G}) = 4. We know from Theorem 3.4.1 that depth(R^{P}) = 4, ie R^{P} is a Cohen-Macaulay ring. Now |G|/|P | = 3 ≠ 0 mod 2, so by Theorem 1.1.4 R^{G} is again Cohen-Macaulay.

There are two possible approaches to proving (3). Let V ≅ W_{2}(1) with V^{∗} := W, and let R := S(V^{∗}). Then by Lemma 5.2.2, V is not strongly flat if and only if we have

R(X + 1) ∩ ker(Θ_{T,Y}) ≤ {z(X + 1) : z ∈ R, zT + z ∈ R^{N}}.

We have little hope of showing this condition holds directly, because ker(Θ_{T,Y}) is not even an ideal of R. The second approach would be to find the isomorphism type of every summand appearing in S(V^{∗}) and show that these are not linearly flat. Note that by Theorem 1.3.1 (a result due to Karagueuzian and Symonds), the number of isomorphism types appearing in S(V^{∗}) is finite; however, since G is not a p-group, this theorem does not give us an upper bound for the degree below which all isomorphism types are found. However, since dim(V) = 2, G may be viewed as a subgroup of GL_{2}(k), with V the natural module, and we can find all isomorphism types using an older result due to Alperin and Kovacs (see [2]) which states:

Let V be the Natural Module for a subgroup G ≤ GL_{2}(𝔽_{q}), for q a prime power p^{m}. Then up to a largest projective summand, the sequence S^{r}(V)_{r∈ℕ} is q(q − 1)-periodic

First suppose that k = 𝔽_{4}, the smallest field over which W_{2}(1) is realizable. Then we can use MAGMA to find, up to a largest projective summand, the decomposition of the first 11 symmetric powers of V^{∗}. For kG-modules A and B, we define the symbol ≃ to mean A ≃ B ⇔ A ≅ B ⊕ Q where Q is projective. We find

S^{0}(W) ≃ W_{1}^{∗}(1), S^{1}(W) ≃ W_{2}^{∗}(1), S^{2}(W) ≃ W_{3}^{∗}(1), S^{3}(W) ≃ 0,
S^{4}(W) ≃ W_{1}^{∗}(1), S^{5}(W) ≃ W_{2}^{∗}(1), S^{6}(W) ≃ W_{3}^{∗}(1), S^{7}(W) ≃ 0,
S^{8}(W) ≃ W_{1}^{∗}(1), S^{9}(W) ≃ W_{2}^{∗}(1), S^{10}(W) ≃ W_{3}^{∗}(1), S^{11}(W) ≃ 0.

So besides projective summands (which we know are not linearly flat because their restrictions to P are not so), S(W_{2}^{∗}(1)) contains only summands of the form W_{1}^{∗}(1) and W_{3}^{∗}(1). Since all these summands are absolutely indecomposable (in fact 𝔽_{4} is a splitting field for G), this decomposition remains true when we extend to an algebraically closed field k. Therefore, if V ≅ W_{2}(1), S(V^{∗}) contains no summands W^{∗} in its symmetric algebra such that W is linearly flat, and so V is not strongly flat. By a similar argument, up to a largest projective direct summand, W_{2}^{′∗}(1) contains in its symmetric algebra only direct summands of the form W_{1}^{′∗}(1), W_{2}^{′∗}(1) and W_{3}^{′∗}(1), where the prime denotes complex conjugation. Obviously W_{1}^{∗}(1) = W_{1}^{′∗}(1). W′_{3}(1) is an indecomposable kG-module of dimension 3, with dim(W′_{3}(1)^{P}) = 1, so by our classification theorem (Theorem 5.2.3) must be isomorphic to a representation of the form W_{3}(ω^{j}) for some j, none of which are linearly flat ^{2}. Therefore W′_{2}(1) is not strongly flat either.

To prove (4), we notice again that if \overline{W}_{6}(0) is not flat then Depth R^{G} = 6 so R^{G} is Cohen-Macaulay, and its enough to show that if V ≅ W_{6}(0)_{|P} = V_{2,0}⊕V_{2,1}⊕ V_{2,∞}, R^{G} is Cohen-Macaulay. Since this is likely to be of independent interest, we include it as a separate lemma.

Lemma 5.2.15. Let V be the left kP -module V_{2,0} ⊕ V_{2,1} ⊕ V_{2,∞}. By changing the basis, we can view V^{∗} as a permutation representation on the variables x_{1}, x_{2}, …, x_{6}, where X acts as the permutation (12)(34) and Y as the permutation (34)(56). Then the ring of invariants R^{P} with R := S(V^{∗}) is Cohen-Macaulay.

2In fact it is not hard to see that W_{3}(1) ≅ W′_{3}(1) - just consider the transformation x_{2} ↔ x_{3}

Proof. By Lemma 3.4.2, we may write

R^{M} = k[x_{1} + x_{2}, x_{3} + x_{4}, x_{5}, x_{6}, x_{1}x_{2}, x_{3}x_{4}](1, x_{1}x_{4} + x_{2}x_{3}).

Alternatively we may view R^{M} as isomorphic to the quotient

R^{M} = \frac{S}{β^{2} + βab + a^{2}d + b^{2}c}

where S := k[a, b, c, d, e, f, β] and a, b, c, d, e, f, β are free generators with degrees 1,1,2,2,1,1 and 2 respectively, having pre-images x_{1} +x_{2}, x_{3} +x_{4}, x_{1}x_{2}, x_{3}x_{4}, x_{5}, x_{6} and x_{1}x_{4} + x_{2}x_{3} respectively. Y now acts on this new ring S by fixing a,b,c,d, swapping e and f, and by sending β to β + ab. Applying the same lemma again, we find

S^{N} = k[a, b, c, d, e + f, ef, ab, β(β + ab)](1, e(β + ab) + fβ)

(of course the generator ab is not needed) and so

R^{P} = S^{N}/(β^{2} + βab + a^{2}d + b^{2}c) = \frac{k[a, b, c, d, e + f, ef](1, e(β + ab) + fβ)}{(β^{2} + βab + a^{2}d + b^{2}c)}.

(The generator β(β + ab) has disappeared, because in the quotient ring this is equivalent to a^{2}d + b^{2}c ∈ k[a, b, c, d]. Note also that we continue to view this as a subring of S := k[a, b, c, d, e, f, β] even though β is not Y -invariant.)

This in turn is isomorphic to the quotient

\frac{k[â, b̂, ĉ, d̂, ê, f̂, γ̂]}{(β̂^{2} + β̂âb̂ + â^{2}d̂+ b̂^{2}ĉ, γ̂^{2} + γ̂âb̂ê + ê^{2}(β̂^{2} + β̂^{2}âb̂) + f̂â^{2}b̂^{2})}

where â, b̂, ĉ, d̂, β̂ are free generators with pre-images a, b, c, d, β, ê is a free generator of degree 1 with pre-image e + f, f̂ is a free generator of degree 2 with image ef, and γ̂ is a free generator of degree 3 with pre-image e(β + ab) + fβ. We can use the first relation to eliminate the terms involving β̂ in the second thus

R^{P} = \frac{k[â, b̂, ĉ, d̂, ê, f̂, γ̂]}{(γ̂^{2} + γ̂âb̂ê + ê^{2}(â^{2}d̂+ b̂^{2}ĉ) + f̂â^{2}b̂^{2})}.

It is now easy to see that the pre-images of {â, b̂, ĉ, d̂, ê, f̂} form a set of primary invariants for R^{P}, and the pre-images of {1, γ̂} a minimal set of secondary invariants. So the degree product of the primary invariants is 1 · 1 · 1 · 2 · 2 · 2 = 8, and the minimal number of secondary invariants is 8/|P | = 2. Therefore by Theorem 1.1.3, R^{P} is a Cohen-Macaulay ring (in fact, a hyper-surface) as claimed.

Corollary 5.2.16. (Compare with 3.4.6) The following is a complete list of indecomposable kG-modules for G ≅ Alt_{4} whose corresponding rings of invariants are Cohen-Macaulay:

W_{1}(ω^{j}), W_{2}(ω^{j}), W′_{2}(ω^{j}), W_{3}(ω^{j}), W_{−3}(ω^{j}), W_{4}(ω^{j}), W′_{4}(ω^{j})\overline{W}_{4}(ω^{j}), W_{5}(ω^{j}), \overline{W}_{6}(0).

Corollary 5.2.17. (Compare with Corollary 3.4.7) The set NSF (G) is closed under taking symmetric algebras.

Proof. Let W ∈ NSF (G). If W ≅ W_{2}(1) (or W′_{2}(1)) then our result follows from part (3) of Theorem 5.2.14, and if W ≅ W_{1}(ω^{j}) its clear that S(W) has only summands on which P acts trivially. Note that A ≅ V_{3}|S(W_{|P}), if and only if S(W) contains a direct summand B ≅ W_{3}(ω^{i}) (for some i) which restricts to it, since these are the only absolutely indecomposable kG-modules with V_{3}|B_{|P}. For the same reason, S(W_{|P}) contains a direct summand ≅ V_{−3} if and only if S(W) contains a direct summand ≅ W_{−3}(ω^{i}) which restricts to it, S(W_{|P}) contains a direct summand ≅ \overline{V}_{4} if and only if S(W) contains a direct summand ≅ \overline{W}_{4}(ω^{i}) which restricts to it and S(W_{|P}) contains a direct summand ≅ V_{2,0}, V_{2,1} or V_{2,∞} if and only if S(W) contains a direct summand ≅ \overline{W}_{6}(0) which restricts to it. Using the calculations in Theorem 3.4.8, we discern that:

• S(V_{−3}) contains summands isomorphic to V_{1}, V_{2,1}, V_{2,∞}, V_{2,0}, V_{−3} and \overline{V}_{4}; • S(V_{3}) contains summands isomorphic to V_{1}, V_{3} and \overline{V}_{4};

• S(\overline{V}_{4}) contains summands isomorphic to V_{1}, V_{2,1}, V_{2,∞}, V_{2,0} and \overline{V}_{4};

• S(V_{2,0} ⊕ V_{2,1} ⊕ V_{2,∞}) contains summands isomorphic to V_{1}, V_{2,1}, V_{2,∞}, V_{2,0} and \overline{V}_{4}.

This last calculation comes from the facts that S(V_{2,0} ⊕ V_{2,1} ⊕ V_{2,∞}) ≅ S(V_{2,0}) ⊗ S(V_{2,1}) ⊗ S(V_{2,∞}) and the decompositions of tensor products in Table 3.4. From these calculations and the preceding discussion we deduce:

• S(W_{−3}(−)) may only contain summands isomorphic to W_{1}(−), \overline{W}_{6}(0), V_{−3}(−) and \overline{W}_{4}(−);

• S(W_{3}(−)) may only contain summands isomorphic to W_{1}(−), W_{3}(−) and \overline{W}_{4}(−);

• S(\overline{W}_{4}(−)) may only contain summands isomorphic to W_{1}(−), \overline{W}_{6}(0) and \overline{W}_{4};

• S(\overline{W}_{6}(0)) may only contain summands isomorphic to W_{1}(−), \overline{W}_{4} and \overline{W}_{6}(0). where each pair of empty brackets may contain any cube root of unity in k. This establishes the result.

As we mentioned earlier, G does not have the property that for all indecomposable kG-modules V, V is strongly flat if and only if V_{|P} is so. What is true is the following: Theorem 5.2.18. Let G ≅ Alt_{4}, let P = Syl_{2}(G) and let k be an algebraically closed field of characteristic 2. Let V ∈ kG-mod. Then exactly one of the following holds:

• V is strongly flat;

• V_{|P} is not strongly flat;

• V is isomorphic to W_{2}(1) or W′_{2}(1). Proof. This follows easily on comparing the sets NSF (P) and NSF (G).

5.2.1 Decomposable Representations

The aim of this subsection is to classify strongly flat representations of G, by decomposing tensor products in a similar way to Theorem 3.4.8. Theorem 5.2.19. Let add(NSF (G)) denote the set of modules which are direct sums of modules in NSF (G). Let V be any kG-module. Then V is strongly flat unless V ∈ add(NSF (G)). If V ∈ add(NSF (G)) then V is strongly flat if and only if it contains a direct summand of one the following isomorphism types:

• W_{3}(−) ⊕ W_{3}(−);

• W_{−3}(−) ⊕ W_{−3}(−);

• W_{2}(1) ⊕ W_{3}(−);

• W′_{2}(1) ⊕ W_{3}(−);

• W_{2}(1) ⊕ W_{2}(1);

• W′_{2}(1) ⊕ W′_{2}(1);

• W_{2}(1) ⊕ W′_{2}(1);

• W_{2}(1) ⊕ W_{1}(ω);

• W_{2}(1) ⊕ W_{1}(ω^{2});

• W_{2}(1) ⊕ W_{1}(ω);

• W_{2}(1) ⊕ W_{1}(ω^{2});

• W′_{2}(1) ⊕ W_{1}(ω);

• W′_{2}(1) ⊕ W_{1}(ω^{2}). Proof. If V ∉ add(NSF (G)), then W := V^{∗} has a direct summand A^{∗} for which A is strongly flat. Therefore V is strongly flat. So we may assume V ∈ add(NSF (G)). Table 5.2.1 describes how tensor products of modules in NSF (G) decompose up to a largest projective summand; we omit the columns for the trivial module W_{1}(1), understanding that V ⊗W_{1}(1) ≅ V for each V ∈ kGmod.

Table 5.1: Decomposing Tensor Products of kG-modules

⊗  W_{1}(ω) W_{1}(ω^{2}) W_{2}(1) W′_{2}(1) W_{3}(1) W_{3}(ω) W_{3}(ω^{2}) W_{−3}(1) W_{−3}(ω) W_{−3}(ω^{2})  \overline{W}_{6}(0)
W_{1}(ω) W_{1}(ω^{2}) W_{2}(ω) W′_{2}(ω) W_{1}(1) W_{3}(ω) W_{3}(ω^{2}) W_{3}(1) W_{−3}(ω) W_{−3}(ω^{2}) W_{−3}(1)  W_{6}(0)
W_{1}(ω^{2})  W_{1}(ω) W_{2}(ω^{2}) W′_{2}(ω^{2}) W_{3}(ω^{2}) W_{3}(1) W_{3}(ω) W_{−3}(ω^{2}) W_{−3}(1) W_{−3}(ω)  W_{6}(0)
W_{2}(1)  W_{4}(1)  0  W_{2}(ω) W_{2}(ω^{2}) W_{2}(1) W_{2}(ω) W_{2}(ω^{2}) W_{2}(1)  0
W′_{2}(1)  W′_{4}(1) W′_{2}(ω) W′_{2}(1) W′_{2}(ω^{2}) W′_{2}(ω)  W′_{2}(1) W′_{2}(ω^{2})  0
W_{3}(1)  W_{5}(1) W_{5}(ω) W_{5}(ω^{2}) W_{1}(ω^{2}) W_{1}(1)  W_{1}(ω)  \overline{W}_{6}(0)
W_{3}(ω)  W_{5}(ω^{2}) W_{5}(1) W_{1}(1)  W_{1}(ω) W_{1}(ω^{2})  \overline{W}_{6}(0)
W_{3}(ω^{2})  W_{5}(ω) W_{1}(ω) W_{1}(ω^{2}) W_{1}(1)  \overline{W}_{6}(0)
W_{−3}(1)  W_{−5}(ω) W_{−5}(ω^{2}) W_{−5}(1)  \overline{W}_{6}(0)
W_{−3}(ω)  W_{−5}(1) W_{−5}(ω)  \overline{W}_{6}(0)
W_{−3}(ω^{2})  W_{−5}(ω^{2})  \overline{W}_{6}(0)
\overline{W}_{6}(0)  \overline{W}_{6}(0) ⊕ \overline{W}_{6}(0)

Let us write

V = ⨁_{i=0}^{2} a_{i}W_{1}(ω^{i}) ⊕ bW_{2}(1) ⊕ b′W′_{2}(1) ⊕ ⨁_{i=0}^{2} c_{i}^{+}W_{3}(ω^{i}) ⊕ ⨁_{i=0}^{2} c_{i}^{−}W_{−3}(ω^{i})

⊕ ⨁_{i=0}^{2} d_{i}\overline{W}_{4}(ω^{i}) ⊕ e\overline{W}_{6},

so that

V^{∗} ≅ ⨁_{i=0}^{2} a_{i}W_{1}^{∗}(ω^{i}) ⊕ bW_{2}^{∗}(1) ⊕ b′W_{2}^{′∗}(1) ⊕ ⨁_{i=0}^{2} c_{i}^{−}W_{−3}^{∗}(ω^{i}) ⊕ ⨁_{i=0}^{2} c_{i}^{+}W_{3}^{∗}(ω^{i})

⊕ ⨁_{i=0}^{2} d_{i}\overline{W} _{4}^{∗}(ω^{i}) ⊕ e\overline{W}_{6}

and

S(V^{∗}) ≅ ⨂_{i=1}^{2} S(W_{1}^{∗}(ω^{i}))^{⊗a_{i}} ⊗ S(W_{2}^{∗}(1))^{⊗b} ⊗ S(W_{2}^{′∗}(1))^{⊗b′} ⊗ ⨂_{i=0}^{2} S(W_{−3}^{∗}(ω^{i}))^{⊗c_{i}^{−}}

⊗ ⨂_{i=0}^{2} S(W_{3}^{∗}(ω^{i}))^{⊗c_{i}^{+}} ⊗ ⨂_{i=0}^{2} S(\overline{W} _{4}^{∗}(ω^{i}))^{⊗d} ⊗ S(\overline{W} _{6}^{∗})^{⊗e}.

Using Table 5.2.1 and the list of isomorphism types which may appear in symmetric algebras from Corollary 5.2.17, it is easy to see that if b = b′ = c_{i}^{+} = c_{i}^{−} = 0 for all i then V is not strongly flat. So we may assume that S(V^{∗}) contains a direct summand of the form

⨂_{i=1}^{2} S(W_{1}^{∗}(ω^{i}))^{⊗a_{i}}⊗S(W_{2}^{∗}(1))^{⊗b}⊗S(W_{2}^{′∗}(1))^{⊗b′}⊗⨂_{i=0}^{2} S(W_{3}^{∗}(ω^{i}))^{⊗c_{i}^{−}}⊗⨂_{i=0}^{2} S(W_{−3}^{∗}(ω^{i}))^{⊗c_{i}^{+}}.

Suppose first that b ≥ 2. Then S(V^{∗}) contains a direct summand of the form S^{1}(W_{2}^{∗}(1)) ⊗ S^{1}(W_{2}^{∗}(1)) ≅ W_{4}^{∗}(1) and V is thus strongly flat because W_{4}(1) is linearly flat by Lemma 5.2.7. Similarly if b′ ≥ 2 then S(V^{∗}) contains a direct summand of the form S^{1}(W_{2}^{′∗}(1)) ⊗ S^{1}(W_{2}^{′∗}(1)) ≅ W_{4}^{′∗}(1) and V is again strongly flat. So we may assume b, b′ ≤ 1. Similarly again, if b = b′ = 1 then S(V^{∗}) contains a direct summand of the form S^{2}(W_{2}^{′∗}(1))⊗S^{2}(W_{2}^{′∗}(1)) ≅ W_{3}^{∗}(1)⊗W_{3}^{∗}(1) ≅ W_{5}^{∗}(1) and V is again strongly flat. So we may assume b+b′ ≤ 1. Suppose b = 1. If a_{1} ≥ 1, then S(V^{∗}) contains a direct summand of the form S^{1}(W_{2}^{∗}(1)) ⊗ S^{1}(W_{1}^{∗}(ω)) ≅ W_{2}^{∗}(ω) and V is thus strongly flat. If a_{2} ≥ 1, then S(V^{∗}) contains a direct summand of the form S^{1}(W_{2}^{∗}(1)) ⊗ S^{2}(W_{1}^{∗}(ω^{2})) ≅ W_{2}^{∗}(1) ⊗ W_{1}^{∗}(ω) ≅ W_{2}^{∗}(ω) and V is thus strongly flat. Similarly, suppose b′ = 1. If a_{1} ≥ 1, then S(V^{∗}) contains a direct summand of the form S^{1}(W_{2}^{′∗}(1)) ⊗ S^{1}(W_{1}^{∗}(ω)) ≅ W_{2}^{′∗}(ω) and V is thus strongly flat. If a_{2} ≥ 1, then S(V^{∗}) contains a direct summand of the form S^{1}(W_{2}^{′∗}(1)) ⊗ S^{2}(W_{1}^{∗}(ω^{2})) ≅ W_{2}^{′∗}(1) ⊗ W_{1}^{∗}(ω) ≅ W_{2}^{′∗}(ω) and V is thus strongly flat.

Now let c^{+} := ∑_{i=0}^{2} c_{i}^{+} > 1. Then V contains a direct summand of the form W_{3}(ω^{i}) ⊕ W_{3}(ω^{j}) for some i, j ∈ {0, 1, 2}, and V^{∗} contains a direct summand of the form W_{3}^{∗}(ω^{i}) ⊕ W_{3}^{∗}(ω^{j}). By Lemma 5.2.6, S(W_{3}^{∗}(−)) always contains a direct summand isomorphic to W_{3}^{∗}(1). So S(V^{∗}) contains a direct summand of the form W_{3}^{∗}(1) ⊗ W_{3}^{∗}(1) ≅ W_{5}^{∗}(ω), and so V is strongly flat, because W_{5}(ω) is linearly flat by Lemma 5.2.10. Similarly, if c^{−} := ∑_{i=0}^{2} c_{i}^{−} > 1. Then V contains a direct summand of the form W_{−3}(ω^{i}) ⊕ W_{−3}(ω^{j}) for some i, j ∈ {0, 1, 2}, and V^{∗} contains a direct summand of the form W_{−3}^{∗}(ω^{i}) ⊕ W_{−3}^{∗}(ω^{j}). By Lemma 5.2.6, S(W_{−3}(−)) always contains a direct summand isomorphic to W_{−3}(ω). So S(V^{∗}) contains a direct summand of the form W_{−3}^{∗}(ω) ⊗ W_{−3}^{∗}(ω) ≅ W_{−5}^{∗}(ω^{2}), and so V is strongly flat.

So we may now assume that c^{−} ≤ 1 and c^{+} ≤ 1. Suppose first that b = 1. If c_{0}^{+} > 0, then S(V^{∗}) contains a direct summand of the form S^{2}(W_{2}^{∗}(1)) ⊗ S^{9}(W_{3}^{∗}(1)) ≅ W_{3}^{∗}(1) ⊗ S^{9}(W_{3}^{∗}(1)). By Lemma 5.2.6, S^{9}(W_{3}^{∗}(1)) contains a direct summand of the form W_{3}^{∗}(ω^{2}), and so S(V^{∗}) contains a direct summand of the form W_{3}^{∗}(1) ⊗ W_{3}^{∗}(ω^{2}) ≅ W_{5}^{∗}(ω^{2}). Since W_{5}(ω^{2}) is linearly flat, V is strongly flat. If c_{1}^{+} > 0, then S(V^{∗}) contains a direct summand of the form S^{2}(W_{2}^{∗}(1)) ⊗ S^{5}(W_{3}^{∗}(ω)) ≅ W_{3}^{∗}(1)⊗S^{5}(W_{3}^{∗}(ω)). By Lemma 5.2.6, S^{9}(W_{3}^{∗}(ω)) contains a direct summand of the form W_{3}^{∗}(ω^{2}), and so S(V^{∗}) contains a direct summand of the form W_{3}^{∗}(1) ⊗ W_{3}^{∗}(ω^{2}) ≅ W_{5}^{∗}(ω^{2}). Since W_{5}(ω^{2}) is linearly flat, V is strongly flat. If c_{2}^{+} ≥ 0, then S(V^{∗}) contains a direct summand of the form S^{2}(W_{2}^{∗}(1)) ⊗ S^{1}(W_{3}^{∗}(ω^{2})) ≅ W_{3}^{∗}(1) ⊗ (W_{3}^{∗}(ω^{2})) ≅ W_{5}^{∗}(ω^{2}). Since W_{5}(ω^{2}) is linearly flat, V is strongly flat. In a similar way, suppose that b′ = 1. If c_{0}^{+} ≥ 0, then S(V^{∗}) contains a direct summand of the form S^{2}(W_{2}^{′∗}(1)) ⊗ S^{9}(W_{3}^{∗}(1)) ≅ W_{3}^{∗}(1) ⊗ S^{9}(W_{3}^{∗}(1)). By Lemma 5.2.6, S^{9}(W_{3}^{∗}(1)) contains a direct summand of the form W_{3}^{∗}(ω^{2}), and so S(V^{∗}) contains a direct summand of the form W_{3}^{∗}(1) ⊗ W_{3}^{∗}(ω^{2}) ≅ W_{5}^{∗}(ω^{2}). Since W_{5}(ω^{2}) is linearly flat, V is strongly flat. If c_{1}^{+} ≥ 0, then S(V^{∗}) contains a direct summand of the form S^{2}(W_{2}^{′∗}(1)) ⊗ S^{5}(W_{3}^{∗}(ω)) ≅ W_{3}(1) ⊗ S^{5}(W_{3}^{∗}(ω)). By Lemma 5.2.6, S^{5}(W_{3}^{∗}(ω)) contains a direct summand of the form W_{3}^{∗}(ω^{2}), and so S(V^{∗}) contains a direct summand of the form W_{3}^{∗}(1) ⊗ W_{3}^{∗}(ω^{2}) ≅ W_{5}^{∗}(ω^{2}). Since W_{5}(ω^{2}) is linearly flat, V is strongly flat. If c_{2}^{+} ≥ 0, then S(V^{∗}) contains a direct summand of the form S^{2}(W_{2}^{′∗}(1)) ⊗ S^{1}(W_{3}^{∗}(ω^{2})) ≅ W_{3}^{∗}(1) ⊗ (W_{3}^{∗}(ω^{2})) ≅ W_{5}^{∗}(ω^{2}). Since W_{5}(ω^{2}) is linearly flat, V is strongly flat.

Now suppose a_{1} = a_{2} = c^{+} = 0 and c^{−} = 1, with either b = 1 or b′ = 1. We must show that V is not strongly flat in this case. It is sufficient to show that if V := W_{2}(1) ⊕ W_{−3}(ω^{i}) or W′_{2}(1) ⊕ W_{3}(ω^{i}), V is not strongly flat - since we can show using Table 5.2.1 that add(\overline{W}_{4}(−), \overline{W}_{6}(0)) is an ideal (under ⊗ and ⊕) in add(NSF (G)). We have V^{∗} ≅ W_{2}^{∗}(1) ⊕ W_{−3}^{∗}(ω^{i}) or V^{∗} ≅ W_{2}^{′∗}(1) ⊕ W_{−3}^{∗}(ω^{i}), and

S(V^{∗}) = S(W_{2}^{∗}(1)) ⊗ S(W_{−3}^{∗}(ω^{i}))

or

S(V^{∗}) = S(W_{2}^{′∗}(1)) ⊗ S(W_{−3}^{∗}(ω^{i})).

Now we know which isomorphism classes of modules may appear in S(W_{2}^{∗}(1)) and S(W_{2}^{′∗}(1)) from part (3) of Theorem 5.2.14, and which may appear in S(W_{−3}^{∗}(ω^{i})) from our calculations in Corollary 5.2.17. S(V^{∗}) is a direct sum of modules of the form A ⊗ B, where A ∈ \{W_{1}^{∗}(1), W_{2}^{∗}(1), W_{3}^{∗}(1), \overline{W} _{4}^{∗}(−)\} (or A ∈ \{W_{1}^{∗}(1), W_{2}^{′∗}(1), W_{3}^{∗}(1), \overline{W} _{4}^{∗}(−)\}) and B ∈ \{W_{1}^{∗}(−), W_{−3}^{∗}(−), \overline{W} _{4}^{∗}(−), \overline{W} _{6}^{∗}(0)\}, where an empty bracket may contain any cube root of unity. Referring again to Table 5.2.1, we see that S(V^{∗}) cannot have any direct summands M^{∗} for which M is linearly flat (or even strongly flat) Therefore V is not strongly flat.

So we may now assume b = 0. To finish the proof, it is sufficient to show that V is not strongly flat when V ≅ M ⊕ W_{−3}(ω^{i}) ⊕ W_{3}(ω^{j}), where M is a module on which P acts trivially. Note that then P acts trivially on each summand in S(M) too. Now V^{∗} ≅ M^{∗} ⊕ W_{−3}^{∗}(ω^{i}) ⊕ W_{3}^{∗}(ω^{j}) and S(V^{∗}) ≅ S(M^{∗}) ⊗ S(W_{3}^{∗}(ω^{i})) ⊗ S(W_{−3}^{∗}(ω^{j})). S(V^{∗}) is a direct sum of modules of the form A ⊗ B ⊗ C, where A ∈ \{W_{1}^{∗}(−)\}, B ∈ \{W_{1}^{∗}(−), W_{−3}^{∗}(−), \overline{W} _{4}^{∗}(−), \overline{W} _{6}^{∗}(0), and C ∈ \{W_{1}^{∗}(−), W ∗_{3} (−), \overline{W} _{4}^{∗}(−)\} where an empty bracket may contain any cube root of unity. Referring again to Table 5.2.1, we see that V is not strongly flat. This completes the proof.

In conclusion, then, the depth question for kG-modules, G ≅ Alt_{4} is quite different to that for kP -modules, and it is certainly not true that V ∈ kG-mod is strongly flat if and only if V_{|P} is so, in contrast to the cases where G is dihedral of order 4n, n odd, or a direct product P × Q. A counterexample is provided by the module W_{2}(1), which is not strongly flat, whereas V_{|P} ≅ V_{2,ω} is strongly flat as a kP -module. Perhaps more interesting still are the modules V := W_{2}(1) ⊕ \overline{W}_{4}(ω^{i}) for any i. In this case V_{|P} ≅ V_{2,ω} ⊕\overline{V}_{4}, which is strongly flat by Theorem 3.4.8. So if R := S(V^{∗}) we have depth(R^{P}) = 4. We can show (using the invariant theory package in MAGMA) that depth(R^{G}) = 5, so V is not even flat in this case.

A Spectral Sequence

In this final chapter, we study in some detail the spectral sequence used by Fleischmann, Kemper and Shank in [21] to prove Theorem 2.2.1. We also describe a generalisation of their result which may possibly shed some light on the depth of invariant rings R^{G} for which the representation V is not flat.

6.1 Setup and Definitions

Let R be a commutative ring over a field k on which a finite group G acts linearly. Let M be a finite left R^{G}-module, and let

M^{∗} := … M^{2} → M^{1} → M^{0} → M → 0

be a projective resolution for M. Let k denote the trivial kG-module, and let

N^{∗} := … N^{2} → N^{1} → N^{0} → k → 0

be a free resolution for k as a kG-module. The maps in the first resolution are labeled δ^{i} : M^{i+1} → M^{i} and those in the second resolution are labeled ϵ^{i} := N^{i+1} → N^{i}. We may then define a first-quadrant double complex

C^{i,j} := Hom_{R^{G}}(M^{i}, Hom_{kG}(N^{j}, R))

and associated total complex

Tot(C)^{k} = ⨁_{i+j=k} C^{i,j}.

There are horizontal and vertical differentials on C

∂_{v}^{i,j} : C^{i,j} → C^{i,j+1}, ∂_{h}^{i,j} : C^{i,j} → C^{i+1,j}.

where ∂_{h}^{i,∗} is induced by δ^{i} and ∂_{v}^{∗,j} is induced by (−1)^{j}ϵ^{j} (We make the sign change so that ∂_{v}∂_{h} + ∂_{h}∂_{v} = 0). A differential on Tot(C) may then be defined by setting ∂_{t} := ∂_{h} + ∂_{v}.

We may filter the total complex in two different ways: define filtrations

F_{h}^{p}(Tot(C)^{k}) := ⨁_{r≥p} C^{r,k−r}, F_{v}^{q}(Tot(C)^{k}) := ⨁_{r≥q} C^{k−r,r}.

Associated to these filtrations are two spectral sequences _{v}E^{∗,∗} and _{h}E^{∗,∗} with E_{1} terms

hE_{1}^{p,q} = H^{p+q}(F_{h}^{p}(Tot(C)^{p+q})/F_{h}^{p+1}(Tot(C)^{p+q}), ∂_{t})
vE_{1}^{p,q} = H^{p+q}(F_{v}^{q}(Tot(C)^{p+q})/F_{v}^{q+1}(Tot(C)^{p+q}), ∂_{t}).

Since C^{i,j} = 0 when i < 0 or j < 0, the filtrations are bounded, and general theory tells us that both spectral sequences converge to the cohomology of the total complex - see, for example [33], Theorem 2.15.

Lemma 6.1.1.

vE_{1}^{p,q} = Ext_{R^{G}}^{q}(M, Hom_{kG}(N^{p}, R)) _{h}E_{1}^{p,q} = Hom_{R^{G}}(M^{p}, H^{q}(G, R)).

Proof. We have

∂_{h}(F_{h}^{p}(Tot(C)^{p+q})) ⊂ F_{h}^{p+1}(Tot(C)^{p+q+1})

and similarly

∂_{v}(F_{v}^{q}(Tot(C)^{p+q})) ⊂ F_{h}^{q+1}(Tot(C)^{p+q+1})

. Therefore

∂_{t}(F_{h}^{p}(Tot(C)^{p+q})/F_{h}^{p+1}(Tot(C)^{p+q})) = ∂_{v}(F_{h}^{p}(Tot(C)^{p+q})/F_{h}^{p+1}(Tot(C)^{p+q})) and ∂_{t}(F_{v}^{q}(Tot(C)^{p+q})/F_{v}^{q+1}(Tot(C)^{p+q})) = ∂_{h}(F_{v}^{q}(Tot(C)^{p+q})/F_{v}^{q+1}(Tot(C)^{p+q})). Since F_{v}^{q}(Tot(C)^{p+q})/F_{v}^{q+1}(Tot(C)^{p+q}) = (F_{h}^{p}(Tot(C)^{p+q})/F_{h}^{p+1}(Tot(C)^{p+q})) ≅ C^{p+q} we get

hE_{1}^{p,q} = H^{p,q}(C^{p,q}, ∂_{v}) _{v}E_{1}^{p,q} = H^{p,q}(C^{p,q}, ∂_{h})

and so

hE_{1}^{p,q} = H^{q}(Hom_{R^{G}}(M^{p}, Hom_{kG}(N^{∗}, R))) = Hom_{R^{G}}(M^{p}, H^{q}(G, R))

whereas vE_{1}^{p,q} = H^{q}(Hom R^{G}(M^{∗}, Hom_{kG}(N^{p}, R)) = Ext_{R^{G}}^{q}(M, Hom_{kG}(N^{p}, R).

Lemma 6.1.2. The E_{2} term in the first (horizontal) spectral sequence is

hE_{2}^{p,q} = Ext_{R^{G}}^{p}(M, H^{q}(G, R)).

Proof. We require to show that

H(_{h}E_{1}^{∗,∗}, ∂_{t}) = H(_{h}E_{1}^{∗,∗}, ∂_{h}).

Let α ∈ _{h}E^{p,q}. By Lemma 6.1.1 we can write α = \overline{x} where x ∈ ker(∂_{v}^{p,q}) and \overline{x} is the equivalence class of x in ker(∂_{v}^{p,q})/∂_{v}^{p,q+1}(C^{p,q+1}). Clearly then ∂_{t}(x) = ∂_{h}(x). Moreover if \overline{x} = \overline{y}, then we have

x − y ∈ ∂_{v}^{p,q+1}(z)

for some z ∈ C^{p,q+1},

∂_{t}(x − y) = ∂_{h}^{p,q}∂_{v}^{p,q+1}(z) + ∂_{v}^{p,q}∂_{v}^{p,q+1}(z)
= ∂_{v}^{p+1,q+1}∂_{h}^{p,q+1}(z) ∈ ∂_{v}^{p+1,q+1}C^{p+1,q+1}

and so the map induced on hE_{1} by ∂_{t} is the same as that induced by ∂_{h}.

We examine this spectral sequence in the special case where M = R^{G}/I for some ideal I of R^{G}. We may then identify

grade(I, R^{G}) = min{i : _{h}E_{2}^{i,0} ≠ 0}.

Also we have

vE_{1}^{p,q} = Ext_{R^{G}}^{q}(R^{G}/I, R^{s_{p}})

where s_{p} is the rank of the free kG-module N^{p}. Clearly we have _{v}E_{1}^{p,q} = 0 for every q < grade(I, R). Note that when R is a Cohen-Macaulay ring, grade(I, R) = height(I). It follows then that _{v}E_{∞} = 0 for every q < grade(I, R), and hence

F_{v}^{q}(H^{p+q}(Tot(C)))/F_{v}^{q+1}(H^{p+q}(Tot(C))) = 0

for every q < grade(I, R) and any choice of p. This tells us that, for each pair with p + q < grade(I, R)

F_{v}^{q+1}(H^{p+q}(Tot(C))) = F_{v}^{q}(H^{p+q}(Tot(C))) = F_{v}^{(q−1)+1}(H^{(p+1)+(q−1)}(Tot(C)))

= F_{v}^{q−1}(H^{p+q}(Tot(C))) = … = F_{v}^{0}(H^{p+q}(Tot(C))) = 0

and in particular we have

hE_{∞}^{p,q} = 0

whenever p + q < grade(I, R).

We now apply these results to prove a key result from chapter two. The first part of the following theorem is the “missing part” of Theorem 2.2.1. We will use the second part in the next section.

Theorem 6.1.3. Let V be a left kG-module, R := S(V^{∗}), and m := cc_{G}(R). Let 𝔦_{G} denote the radical of the relative transfer ideal as defined in chapter 1. Suppose dim(V^{P}) + m + 1 < dim(V). Then the following are equivalent:

• grade(𝔦_{G}, H^{m}(G, R)) = 0;

• grade(𝔦_{G}, R^{G}) = m + 1.

When these conditions are satisfied, we have

depth(R^{G}) = dim(V^{P}) + cc_{G}(R) + 1.

Further, suppose that dim(V^{P})+m+2 < dim(V) and that grade(𝔦_{G}, H^{m}(G, R)) = 1. Then we have grade(𝔦_{G}, R^{G}) = m + 2 and

depth(R^{G}) = dim(V^{P}) + cc_{G}(R) + 2.

Proof. Let r = 1 or r = 2, and assume that dim(V^{P})+m+r < dim(V). Consider the spectral sequence _{h}E_{∗}^{p,q} described earlier, with R := S(V^{∗}) a polynomial ring and with I the relative transfer ideal defined in chapter one, (Definition 1.9). Recall that this is a prime ideal of R^{G} with height dim(V) − dim(V^{P}) (Corollary 1.6.3). Therefore the condition dim(V^{P}) + m + r < dim(V) ensures that

m + r < height(𝔦_{G}) = grade(𝔦_{G}, R).

Let d_{k} denote the induced differential on _{h}E_{k}, which has bidegree (k, 1−k). Since H^{q}(G, R) = 0 for 0 < q < m,_{h}E_{2}^{p,q} = 0 for 0 < q < m. So if grade(𝔦_{G}, H^{m}(G, R)) = r−1, we have a nonzero term α ∈ _{h}E_{2}^{r−1,m}. This term cannot survive the spectral sequence because _{h}E_{∞}^{r−1,m} = 0. We must therefore have

0 ≠ d_{m}(α) ∈ _{h}E_{2}^{m+r,0}.

Hence grade(𝔦_{G}, H^{m}(G, R)) = r − 1 ⇒ grade(𝔦_{G}, R^{G}) ≤ m + r.

At this point it is worth pointing out that a similar result for r > 2 cannot be obtained in this way - a nonzero α ∈ _{h}E_{2}^{r−1,m} in this case will also fail to survive the spectral sequence if α ∈ d_{r−1}(_{h}E_{r−1}^{0,m+r−2}).

Furthermore, if grade(𝔦_{G}, R^{G}) = l + 1, l < m then the nonzero β ∈ _{h}E_{2}^{l+1,0} must be the image under d_{l} of some nonzero α ∈ _{h}E_{2}^{0,l} or else survive the spectral sequence. Since neither of these are allowed, if r = 1 we must have grade(𝔦_{G}, R^{G}) = m + 1. If r = 2 then we conclude that either grade(𝔦_{G}, R^{G}) = m + 2 or that grade(𝔦_{G}, R^{G}) = m + 1.

Conversely suppose that r = 1 and grade(𝔦_{G}, R^{G}) = m + 1. Then there is a nonzero β ∈ E_{2}^{m+1,0} which survives the spectral sequence unless β = d_{m}(α) for some 0 ≠ α ∈ E_{2}^{0,m}. In particular we have grade(𝔦_{G}, H^{m}(G, R)) = 0. So we have now shown, provided that dim(V^{P}) + m + 1 < dim(V), that

grade(𝔦_{G}, H^{m}(G, R)) = 0 ⇔ grade(𝔦_{G}, R^{G}) = m + 1.

Now suppose r = 2, so dim(V^{P})+m+2 < dim(V), and grade(𝔦_{G}, H^{m}(G, R)) = 1. Then we have either grade(𝔦_{G}, R^{G}) = m + 2 or grade(𝔦_{G}, R^{G}) = m + 1. Since dim(V^{P})+m+2 < dim(V) we certainly have dim(V^{P})+m+1 < dim(V) and therefore if grade(𝔦_{G}, R^{G}) = m + 1 we must conclude that grade(𝔦_{G}, H^{m}(G, R)) = 0, which is a contradiction. Therefore grade(𝔦_{G}, H^{m}(G, R)) = 1 ⇒ grade(𝔦_{G}, R^{G}) = m+2. Both parts of the proof are now completed using the fact that grade(𝔦_{G}, R^{G})+ dim(V^{P}) = depth(R^{G}).

A representation satisfying depth(R^{G}) = dim(V^{P}) + m + 1 is called flat. We shall call a representation satisfying depth(R^{G}) = dim(V^{P}) + m + 2 almost flat.

6.2 Almost Flat Representations

We end this final chapter with a further application of the above spectral sequence, which I hope will eventually help us to say something about those representations which are not flat. Throughout this section, we have R := S(V^{∗}), 𝔦_{G} denotes the radical of the relative transfer ideal, and m := cc_{G}(R).

We have shown earlier that a representation is flat if either dim(V^{P}) + m + 1 ≥ dim(V) or grade(𝔦_{G}, H^{m}(G, R)) = 0. If this second condition fails to hold, there must be some nonzero element ρ ∈ 𝔦_{G} which acts regularly on H^{m}(G, R). Unfortunately we have no systematic means of locating such an element at this time. However, if we could find such an element, as the next result shows, we could make wide use of it.

Theorem 6.2.1. Suppose V is not flat and dim(V^{P}) + m + 2 < dim(V). Then there is a nonzero element ρ ∈ 𝔦_{G} which acts regularly on H^{m}(G, R). Suppose in addition that ρ acts regularly on H^{m+1}(G, R). S(V^{∗})/ρ is a decomposable kGmodule. If it has a direct summand W^{∗} for which W is linearly flat, then V is almost flat.

Lemma 6.2.2. There is an isomorphism

\frac{H^{1}(G, R)}{ρ^{∗}H^{1}(G, R)} ≅ H^{1}(G, R/ρR).

Proof. From the short exact sequence

0 → R →^{×ρ} R → R/ρR → 0

we get the long exact sequence (see [4], Theorem 2.5.3)

...0 → H^{m−1}(G, R/ρR) →^{∂_{m}} H^{(}G, R) →^{ρ^{∗}} H^{m}(G, R)

→^{α} H^{r}(G, R/ρR) →^{∂_{m+1}} H^{m+1}(G, R) →^{ρ^{∗}} H^{m+1}(G, R)...

Since the maps induced on by ρ on H^{m}(G, R) and H^{m+1}(G, R) are injective, the maps ∂_{m} and ∂_{m+1} are zero. So there is a short exact sequence

0 → H^{m}(G, R) →^{ρ^{∗}} H^{m}(G, R) →^{α} H^{m}(G, R/ρR) → 0

and α induces the required isomorphism. This completes the proof of Lemma 6.2.2.

Suppose that the decomposition of R/ρR contains a direct summand W^{∗} for which W is linearly flat. This means that there is a non-zero cohomology class α ∈ H^{m}(G, R/ρR) with res_{N}^{G}(α) = 0 for all N < P, and by Lemma 2.4.1, grade(𝔦_{G}, H^{m}(G, R/ρR)) = 0. Hence

grade(𝔦_{G}, H^{m}(G, R)/ρ^{∗}(H^{m}(G, R))) = 0

and therefore

grade(𝔦_{G}, H^{m}(G, R)) = 1.

So by Theorem 6.1.3, grade(𝔦_{G}, R^{G}) = m + 2 and V is almost flat.

Modulo finding elements acting regularly on H^{1}(G, R) and H^{2}(G, R), this gives us a process for finding the almost flat representations of a finite group G given a list of all the linearly flat representations.

Appendix: Classification of Representations of

G ≅ Alt_{4} over an Algebraically Closed Field of

Characteristic 2

Let V be a representation of G over a field k of characteristic 2, and let P ∈ Syl_{2}(G). For convenience we assume that k is algebraically closed, and in particular k contains the cubic roots of unity 1, ω, ω^{2}. Since P is a Sylow-2-subgroup of G, each kG-module V is projective relative to P (see for example [4], section 3.6), so by Higman’s criterion, each V is a direct summand of V_{|P} ↑_{P}^{G}. Therefore all indecomposable representations of G appear as direct summands of modules of the form W ↑_{P}^{G} where W is an indecomposable kP -module. This allows us to classify all absolutely indecomposable kG-modules.

Theorem 7.0.3 ([4], Remark 4.3.3 and [12]). The absolutely indecomposable kG-Modules fall into the following families

• For every even dimension 2n there are representations W_{2n}(ω^{j})

X ∼ ( \begin{matrix}I_{n} & ωω^{2}0...0 \\ & 0ωω^{2}...0 \\ ... & ... \\ & 0....ωω^{2} \\ & 0......ω \\ 0 & I_{n} \end{matrix} )

Y ∼ ( \begin{matrix}I_{n} & ω^{2}ω0...0 \\ & 0ω^{2}ω...0 \\ ... & ... \\ & 0....ω^{2}ω \\ & 0......ω^{2} \\ 0 & I_{n} \end{matrix} )

T ∼ ω^{j} diag(1, ω, ...ω^{n−1}, ω, ω^{2}, ...ω^{n}). There are also the “conjugate” representations W′_{2n}(ω^{j}), obtained by taking the complex conjugates of the matrices above. These are equivalent to the representations obtained from W_{2n}(ω^{j}) under the automorphism Ψ of G defined by

Ψ(X) = Y, Ψ(Y) = X, Ψ(T) = T^{−1}.

• For every odd dimension 2n + 1 there are representations W_{2n+1}(ω^{j})

X ∼ ( \begin{matrix}I_{n+1} & ω^{2}0...0 \\ & ωω^{2}...0 \\ ... & ... \\ & 0....ωω^{2} \\ & 0......ω \\ 0 & I_{n} \end{matrix} )

Y ∼ ( \begin{matrix}I_{n+1} & ω0...0 \\ & ω^{2}ω...0 \\ ... & ... \\ & 0.....ω^{2}ω \\ & 0.......ω^{2} \\ 0 & I_{n} \end{matrix} )

T ∼ ω^{j} diag(1, ω, ...ω^{n}, ω^{2}, ω^{3}, ...ω^{n+1}).

• For every odd dimension 2n + 1 there are representations W_{−(2n+1)}(ω^{j})

X ∼ ( \begin{matrix}I_{n} & ωω^{2}0...0 \\ & 0ωω^{2}...0 \\ ... & ... \\ & 0.....ωω^{2} \\ 0 & I_{n+1} \end{matrix} )

Y ∼ ( \begin{matrix}I_{n} & ω^{2}ω0...0 \\ & 0ω^{2}ω...0 \\ ... & ... \\ & 0.....ω^{2}ω \\ 0 & I_{n+1} \end{matrix} )

T ∼ ω^{j} diag(1, ω, ...ω^{n−1}, ω, ω^{2}, ...ω^{n+1}. There is (obviously) an isomorphism W_{1}(ω^{j}) ≅ W_{−1}(ω^{j}) for every j.

• For each dimension 6n and each λ ≠ ω, ω^{2} there are representations \overline{W}_{6n}(λ)

X ∼ ( \begin{matrix}I & I & 0 & 0 & 0 & 0 \\ 0 & I & 0 & 0 & 0 & 0 \\ 0 & 0 & I & J(λ) & 0 & 0 \\ 0 & 0 & 0 & I & 0 & 0 \\ 0 & 0 & 0 & 0 & I & J(λ + 1) \\ 0 & 0 & 0 & 0 & 0 & I \end{matrix} )

Y ∼ ( \begin{matrix}I & J(λ) & 0 & 0 & 0 & 0 \\ 0 & I & 0 & 0 & 0 & 0 \\ 0 & 0 & I & J(λ + 1) & 0 & 0 \\ 0 & 0 & 0 & I & 0 & 0 \\ 0 & 0 & 0 & 0 & I & I \\ 0 & 0 & 0 & 0 & 0 & I \end{matrix} )

T ∼ ( \begin{matrix}0 & 0 & I & 0 & 0 & 0 \\ 0 & 0 & 0 & I & 0 & 0 \\ 0 & 0 & 0 & 0 & I & 0 \\ 0 & 0 & 0 & 0 & 0 & I \\ I & 0 & 0 & 0 & 0 & 0 \\ 0 & I & 0 & 0 & 0 & 0 \end{matrix} )

In this family there are equivalences \overline{W}_{6n}(λ) ≅ \overline{W}_{6n}(1/(1 − λ)) ≅ \overline{W}_{6n}(λ −

1/λ).

• There are 3 projective modules \overline{W}_{4}(ω^{j})

X ∼ ( \begin{matrix}1 & ω & ω^{2} & 1 \\ 0 & 1 & 0 & ω^{2} \\ 0 & 0 & 1 & ω \\ 0 & 0 & 0 & 1 \end{matrix} )

Y ∼ ( \begin{matrix}1 & ω^{2} & ω & 1 \\ 0 & 1 & 0 & ω \\ 0 & 0 & 1 & ω^{2} \\ 0 & 0 & 0 & 1 \end{matrix} )

T ∼ ( \begin{matrix}ω^{j} & 0 & 0 & 0 \\ 0 & ω^{j+1} & 0 \\ 0 & 0 & ω^{j+2} & 0 \\ 0 & 0 & 0 & ω^{j} \end{matrix} )

Proof. Recall that a kG-module V is indecomposable if the endomorphism ring E := End_{kG}(V) contains no non-trivial idempotents, and is absolutely indecomposable if the endomorphism ring is local. We prove first that V := V_{2n,λ} ↑^{G} is indecomposable unless λ is a non-trivial cube root of unity. Temporarily, let X, Y and Z be the 2n by 2n matrices

X = ( \begin{matrix}I & I \\ 0 & I \end{matrix} ) ,  Y = ( \begin{matrix}I & J(λ) \\ 0 & I \end{matrix} ) ,  Z = ( \begin{matrix}I & J(λ + 1) \\ 0 & I \end{matrix} )

So now the pair X, Y give an absolutely indecomposable kP -module action of the form V_{2n,λ}. Writing in terms of 2n by 2n blocks, the matrices which give the action of G on V are now

( \begin{matrix}X & 0 & 0 \\ 0 & Y & 0 \\ 0 & 0 & Z \end{matrix} )

( \begin{matrix}Y & 0 & 0 \\ 0 & Z & 0 \\ 0 & 0 & X \end{matrix} )

( \begin{matrix}0 & I & 0 \\ 0 & 0 & I \\ I & 0 & 0 \end{matrix} )

E then consists of the k-endomorphisms of V which commute with these three matrices. If a k-endomorphism commutes with the last of these three, then it is represented by a matrix which looks like

M := ( \begin{matrix}A & B & C \\ C & A & B \\ B & C & A \end{matrix} )

This commutes with the first two matrices if the following relations are satisfied:

AX = XA,  AY = Y A,  AZ = ZA,
BX = ZB,  BY = XB,  BZ = Y B,
CX = Y C,  CY = ZC,  CZ = XC.

Furthermore, M represents an idempotent endomorphism if the following relations are satisfied:

A = A^{2} + BC + CB,
B = C^{2} + AB + BA,

C = B^{2} + AC + CA.

Suppose the only solutions to all these equations have BC = CB = C^{2} = B^{2} = 0. Then A is an idempotent which commutes with X and Y. Since the actions of X and Y form an indecomposable representation of P equivalent to V_{2n,λ}, we must have A = I_{2n} or A = 0. It then follows that C = B^{2} = 0 and B = C^{2} = 0 and M = I_{6n} or M = 0. Under these conditions then, V is indecomposable.

Let us consider the equations BY = XB and BX = ZB. Writing our matrices in n by n block form, we have

( \begin{matrix}B_{1} & B_{2} \\ B_{3} & B_{4} \end{matrix} ) ( \begin{matrix}I_{n} & J(λ) \\ 0 & I_{n} \end{matrix} ) = ( \begin{matrix}I_{n} & I_{n} \\ 0 & I_{n} \end{matrix} ) ( \begin{matrix}B_{1} & B_{2} \\ B_{3} & B_{4} \end{matrix} )

and therefore we find B_{3} = 0 and B_{1}J(λ) = B_{4}. In a similar way, the second equation gives us

( \begin{matrix}B_{1} & B_{2} \\ B_{3} & B_{4} \end{matrix} ) ( \begin{matrix}I_{n} & I_{n} \\ 0 & I_{n} \end{matrix} ) = ( \begin{matrix}I_{n} & J(λ + 1) \\ 0 & I_{n} \end{matrix} ) ( \begin{matrix}B_{1} & B_{2} \\ B_{3} & B_{4} \end{matrix} )

so that B_{3} = 0 and B_{1} = J(λ + 1)B_{4}. We can now substitute for B_{4}, finding that B_{1}J(λ)^{−1} = J(λ + 1)B_{1}. Now the Jordan Normal Form of J(λ)^{−1} is J(λ^{−1}), and therefore J(λ)^{−1} is conjugate to J(λ + 1) only when λ is a nontrivial cube root of unity. Otherwise we must have B_{1} = B_{4} = B_{3} = 0. By considering the equations for C in the same way, we find that C_{1} = C_{3} = C_{4} = 0. It follows that BC = CB = 0 and A^{2} = A as required. Therefore V ≅ \overline{W}_{6n}(λ) is indecomposable provided λ ≠ (ω, ω^{2}).

Now consider the three modules W_{2n}(1), W_{2n}(ω) and W_{2n}(ω^{2}), and their complex conjugate modules. Evidently W_{2n}(1), W_{2n}(ω) and W_{2n}(ω^{2}) are nonisomorphic kG-modules, each of which restricts to the same kP -module, V say. I claim that V is absolutely indecomposable. Each of these modules must be a direct summand of V ↑^{G}, and this module has dimension 6n = dim(W_{2n}(1) ⊕ W_{2n}(ω) ⊕ W_{2n}(ω^{2})). If V is absolutely indecomposable, then since dim(V) = 2n and dim(V^{P}) = n it follows from the classification of kP -modules (Section 3.3) that V ≅ V_{2n,λ} for some λ. But we’ve shown that V_{2n,λ} ↑^{G} is indecomposable unless λ is a non-trivial cube root of unity. So we must now have V_{2n,ω} ↑^{G}≅ W_{2n}(1)⊕W_{2n}(ω)⊕W_{2n}(ω^{2}), and similarly V_{2n,ω^{2}} ↑^{G}≅ W′_{2n}(1)⊕W′_{2n}(ω)⊕W′_{2n}(ω^{2}), since V_{2n,ω^{2}} is the complex conjugate of V_{2n,ω}^{1}.

The matrices which give the action of P on V are as follows:

X := ( \begin{matrix}I_{n} & Ω \\ 0 & I_{n} \end{matrix} )

Y := ( \begin{matrix}I_{n} & Ω′ \\ 0 & I_{n} \end{matrix} ) .

Here Ω is the n by n matrix with ω on the diagonal, ω^{2} on the superdiagonal and zeroes elsewhere, and the prime denotes complex conjugation. Let N denote the n by n matrix with ones on the superdiagonal and zero everywhere else. If

A := ( \begin{matrix}A_{1} & A_{2} \\ A_{3} & A_{4} \end{matrix} )

is a matrix which commutes with X and Y, then we find A_{3} = 0 and ΩA_{4} = A_{1}Ω and Ω′A_{4} = A_{1}Ω′. This can be written as

ωA_{4} + ω^{2}NA_{4} = ωA_{1} + ω^{2}A_{1}N,
ω^{2}A_{4} + ωNA_{4} = ω^{2}A_{1} + ωA_{1}N.

By taking appropriate linear combinations of the above equations we can reduce them to A_{1} = A_{4} and A_{1}N = NA_{1}. This means that A_{1} is an upper triangular matrix with all diagonal entries equal. In particular, this means that the endomorphism ring E := End_{kP} (V) consists of upper triangular matrices with only one eigenvalue. The Jacobson radical consists of those matrices with zeros on the main diagonal. Therefore E/ rad(E) ≅ k and E is local as required. This proves that V_{2n,ω} ↑^{G}≅ W_{2n}(1) ⊕ W_{2n}(ω) ⊕ W_{2n}(ω^{2}).

1We have not ruled out the alternative possibility that V_{2n,ω^{2}} ↑^{G}≅ W_{2n}(1) ⊕ W_{2n}(ω) ⊕ W_{2n}(ω^{2}), and V_{2n,ω} ↑^{G}≅ W′_{2n}(1) ⊕ W′_{2n}(ω) ⊕ W′_{2n}(ω^{2}), for some or all values of n. This does not matter, as the same classification arises.

Now consider the three modules W_{2n+1}(1), W_{2n+1}(ω), W_{2n+1}(ω^{2}). Again its clear that the restrictions to P of each of these three modules are the same module, V say. If we can show that V is absolutely indecomposable, it will follow from the classification theorem that V ≅ V_{2n+1}, as this is the only absolutely indecomposable kP -module with the correct dimensions for V and V^{P}. Relative projectivity will then tell us that V_{2n+1} ↑^{G}≅ W_{2n+1}(1) ⊕ W_{2n+1}(ω) ⊕ W_{2n+1}(ω^{2}).

So suppose

K := ( \begin{matrix}A & B \\ C & D \end{matrix} )

is a matrix which commutes with the action of P on V. In this case we have

X := ( \begin{matrix}I_{n+1} & Ω \\ 0 & I_{n} \end{matrix} )

Y := ( \begin{matrix}I_{n+1} & Ω′ \\ 0 & I_{n} \end{matrix} )

In the above, B is a matrix with n + 1 rows and n columns, and C is a matrix with n rows and n + 1 columns. In this case we define Ω to be the n by n + 1 matrix

( \begin{matrix}ω^{2} & 0 & … & 0 \\ ω & ω^{2} & 0 & … \\ 0 & ω & ω^{2} & … \\ ⋮ & ... & ... & ... \\ 0 & … & ω & ω^{2} \\ 0 & … & 0 & ω \end{matrix} )

Define also the n + 1 by n matrices

N := ( \begin{matrix}I_{n} \\ 0 \end{matrix} )

M := ( \begin{matrix}0 \\ I_{n} \end{matrix} )

Then the commutation relations are AΩ = ΩD, AΩ′ = Ω′D and C = 0, or alternatively by taking linear combinations AM = MD, AN = ND and C = 0. It follows that A and D are diagonal matrices with only one eigenvalue, so that K is in turn an upper triangular matrix with one eigenvalue. By the same argument as for the even dimensional case, E is local and hence V is absolutely indecomposable and V_{2n+1} ↑^{G}≅ W_{2n+1}(1) ⊕ W_{2n+1}(ω) ⊕ W_{2n+1}(ω^{2}).

Now consider the three modules W_{−(2n+1)}(1), W_{−(2n+1)}(ω), W_{−(2n+1)}(ω^{2}). Again its clear that the restrictions to P of each of these three modules are the same module, V say. If we can show that V is absolutely indecomposable, it will follow from the classification theorem that V ≅ V_{−(2n+1)}, as this is the only absolutely indecomposable kP -module with the correct dimensions for V and V^{P}. Relative projectivity will then tell us that V_{−(2n+1)} ↑^{G}≅ W_{−(2n+1)}(1) ⊕ W_{−(2n+1)}(ω) ⊕ W_{−(2n+1)}(ω^{2}). So suppose

K := ( \begin{matrix}A & B \\ C & D \end{matrix} )

is a matrix which commutes with the action of P on V. In this case we have

X := ( \begin{matrix}I_{n} & Ω \\ 0 & I_{n+1} \end{matrix} )

Y := ( \begin{matrix}I_{n} & Ω′ \\ 0 & I_{n+1} \end{matrix} )

Here B is a matrix with n rows and n + 1 columns, and C is a matrix with n + 1 rows and n columns. In this case Ω is the matrix

( \begin{matrix}ω & ω^{2} & 0 & … & 0 \\ 0 & ω & ω^{2} & … & 0 \\ ⋮ & ... & ... & ... & ⋮ \\ 0 & … & 0 & ω & ω^{2} \end{matrix} )

Define also the n by n + 1 matrices

N := ( I_{n} 0 )

M := ( 0 I_{n} )

Then the commutation relations are AΩ = ΩD, AΩ′ = Ω′D and C = 0, or alternatively by taking linear combinations AM = MD, AN = ND and C = 0. It follows that A and D are diagonal matrices with only one eigenvalue, so that K is in turn an upper triangular matrix with one eigenvalue. By the same argument as for the even dimensional case, E is local. Hence V is absolutely indecomposable and V_{−(2n+1)} ↑^{G}≅ W_{−(2n+1)}(1) ⊕ W_{−(2n+1)}(ω) ⊕ W_{−(2n+1)}(ω^{2}).

It is perhaps worth noting that, since the matrices representing the action of G on V_{2n+1} and V_{−(2n+1)} are real, we must have

V_{2n+1} ↑^{G}≅ V′_{2n+1} ↑^{G}≅ W′_{2n+1}(1) ⊕ W′_{2n+1}(ω) ⊕ W′_{2n+1}(ω^{2})

and similarly

V_{−(2n+1)} ↑^{G}≅ V′_{−(2n+1)} ↑^{G}≅ W′_{−(2n+1)}(1) ⊕ W′_{−(2n+1)}(ω) ⊕ W′_{−(2n+1)}(ω^{2})

where the prime (′) again denotes taking the complex conjugate of the matrices which describe the action of G. This means that W′_{2n+1}(ω^{i}) ≅ W_{2n+1}(ω^{j(i)}) for some j(i), and similarly that W′_{−(2n+1)}(ω^{i}) ≅ W_{−(2n+1)}(ω^{l(i)}) for some l(i), for each i and n. So we do not need to include the conjugate modules W′_{2n+1}(ω^{j}) and W′_{−(2n+1)}(ω^{j}) in our classification.

Finally consider the modules \overline{W}_{4}(1), \overline{W}_{4}(ω) and \overline{W}_{4}(ω^{2}). Each of these restricts to the same kP -module V, and we have dim(V) = 4, dim(V^{P}) = 1. Let \overline{k} be the algebraic closure of k. If V ⊗ \overline{k} is decomposable, say V ⊗ \overline{k} ≅ M_{1} ⊕ M_{2}, then we must have dim M_{i}^{P} = 0 for some i, which is impossible. Therefore V is absolutely indecomposable and by the classification theorem is isomorphic to the projective indecomposable module \overline{V}_{4}. Since \overline{W}_{4}(ω^{i}) is a direct summand of \overline{W}_{4}(ω^{i})_{|P} ↑^{G}= \overline{V}_{4} ↑^{G}, it follows that \overline{V}_{4} ↑^{G}≅ \overline{W}_{4}(1) ⊕ \overline{W}_{4}(ω) ⊕ \overline{W}_{4}(ω^{2}). For the same reasons as for the odd dimensional modules, we do not need to consider the complex conjugates of these modules. This can also be deduced from the 1-1 correspondence between projective indecomposable and irreducible modules, which tells us that there are precisely three non-isomorphic projective indecomposable modules.

We have now shown that V_{2n,λ} ↑_{P}^{G}= \overline{W}_{6n}(λ) if λ ∈ P^{1}(k), λ ≠ ω, ω^{2}. In these exceptional cases, we have V_{2n,ω} ↑_{P}^{G}= W_{2n}(1) ⊕ W_{2n}(ω) ⊕ W_{2n}(ω^{2}) and V_{2n,ω^{2}} ↑_{P}^{G}= W′_{2n}(1) ⊕ W′_{2n}(ω) ⊕ W′_{2n}(ω^{2}). Similarly V_{2n+1} ↑_{P}^{G}= W_{2n+1}(1) ⊕ W_{2n+1}(ω)⊕W_{2n+1}(ω^{2}), V_{−(2n+1)} ↑_{P}^{G}= W_{−(2n+1)}(1)⊕W_{−(2n+1)}(ω)⊕W_{−(2n+1)}(ω^{2}), and \overline{V}_{4} ↑_{P}^{G}= \overline{W}_{4}(1) ⊕ \overline{W}_{4}(ω) ⊕ \overline{W}_{4}(ω^{2}). This completes the proof.

Bibliography

[1] Alejandro Adem, R. James Milgram Cohomology of Finite Groups Springer-Verlag 309 (1994).

[2] J. Alperin and L.G. Kovacs, Periodicity of Weyl Modules for SL(2, q) Journal of Algebra 74 (1982) no.1, pp 52–55.

[3] M. Aschbacher Finite Group Theory Cambridge Studies in Advanced Mathematics 10 (2000).

[4] D.J.Benson, Representations and Cohomology 1, Cambridge Studies in Advanced Mathematics 30 (1995).

[5] D.J.Benson, Polynomial Invariants of Finite Groups, London Mathematical Society Lecture Notes 190 (1993).

[6] Wieb Bosma, John J. Cannon, Catherine Playoust The MAGMA algebra system 1: The user language, J. Symbolic Computing 24 (1997).

[7] M. P. Brodmann, R. Y. Sharp Local Cohomology: An algebraic introduction with geometric applications Cambridge Studies in Advanced Mathematics 60 (1998).

[8] Winfried Bruns, Jürgen Herzog Cohen-Macaulay Rings Cambridge Studies in Advanced Mathematics 39 (1997).

[9] C. Chevalley Invariants of finite groups generated by reflections Amer. J. Math 77 (1955) 778–782.

[10] H.E.A Campbell, I.P Hughes, G.Kemper, R.J Shank, D.L. Wehlau Depth of Modular Invariant Rings, Transformation Groups 5 (2000) pp 21–34.

[11] J. Chuai, On the polynomial property of quotients of invariant rings, preprint, Queen’s University, Kingston, 2004.

[12] S.B. Conlon Certain Representation Algebras J. Aust. Math. Soc. 5 pp. 83–89 (1965).

[13] H. Derksen, G. Kemper Computational Invariant Theory Invariant Theory and Algebraic Transformation Groups I, Encyclopaedia of Mathematical Sciences 130, Springer-Verlag.

[14] B.M. Puttaswamiah, John D. Dixon Modular Representations of Finite Groups New York Academic Press (1977).

[15] J. Eagon and M. Hochster Cohen-Macaulay rings, invariant theory, and the generic perfection of determinantal loci, Amer. J. Math. 93 (1971), pp 1020–1058.

[16] D. Eisenbud Commutative Algebra With A View Toward Algebraic Geometry, Graduate Texts in Mathematics 150, Springer-Verlag (1995).

[17] G. Ellingsrud and T.Skjelbred Profondeur d’anneaux d’invariants en caractéristique p, Compos. Math. 41 (1980) pp 233–244.

[18] Leonard Evens, The Cohomology of Groups, Oxford Science Publications (1991).

[19] Peter Fleischmann Relative Trace Ideals and Cohen-Macaulay Quotients of Modular Invariant Rings in Computational Methods for Representations of Groups and Algebras, Euroconference in Essen 1997 Progress in Mathematics 173 Birkhäuser, Basel (1999).

[20] J. A. Humphreys, Reflection Groups and Coxeter Groups Cambridge Studies in Advanced Mathematics 29 (1990).

[21] Peter Fleischmann, Gregor Kemper, R.J. Shank Depth and Cohomological Connectivity in Modular Invariant Theory Transactions of the American Mathematical Society 357 (2005) pp 3605-3621.

[22] Peter Fleischmann and R. James Shank The relative trace ideal and the depth of modular rings of invariants Arch. Math 80 (2003) pp 347–353.

[23] A. Heller and I. Reiner, Indecomposable Representations Illinois Journal of Mathematics 5 (1961) pp 314–323.

[24] B Huppert, N Blackburn Finite Groups II, Grundlehren der mathematischen Wissenschaften 242 (1982).

[25] Ian Hughes, Gregor Kemper Symmetric Powers of Modular Representations, Hilbert Series and Degree Bounds Communications in Algebra 28 (2000) pp 2059–2088.

[26] Dikran B. Karagueuzian, Peter Symonds The Module Structure of a Group Action on a Polynomial Ring: A Finiteness Theorem (2001) Unpublished.

[27] Gregor Kemper The Depth of Invariant Rings and Cohomology, (with an appendix by Kay Magaard) Journal of Algebra 245 (2001) pp 463–531.

[28] Gregor Kemper, On the Cohen-Macaulay property of Invariant Rings Journal of Algebra 215 (1999) pp 330–351.

[29] Gregor Kemper, Calculating Invariant Rings of Finite Groups over Arbitrary Fields Journal of Sympbolic Computing 21 (1996) pp 351–366.

[30] Gregor Kemper, Gunther Malle The finite Irreducible Groups with Polynomial Ring of Invariants Transformation Groups 2 (1997) pp 57–89.

[31] M. Lorenz, J. Pathak On Cohen-Macaulay Rings of Invariants Journal of Algebra 245 pp 247–264 (2001).

[32] Macdonald, I.G., Symmetric Functions and Hall Polynomials Oxford Mathematical Monographs (1979).

[33] John McCleary A User’s Guide to Spectral Sequences Cambridge Studies in Advanced Mathematics 58 (2001).

[34] H. Nakajima Invariants of finite groups generated by pseudoreflections Tsukba J. Math 3 (1979) 109–122.

[35] C. M. Ringel The indecomposable representations of the dihedral 2-groups Ann. Math. 214 (1975) pp 19–34.

[36] Derek J. S. Robinson A course in the theory of groups Springer Graduate Texts in Mathematics 80 (1995).

[37] Jospeh J.Rotman Advanced Modern Algebra Prentice-Hall (2002).

[38] G. C. Sheppard and J. A. Todd Finite unitary reflection groups Canada J. Math 6 (1954) pp 274–306.

[39] R. James Shank and David Wehlau The Transfer in Modular Invariant Theory Journal of Pure and Applied Algebra 142 (1999) pp 63–77.

[40] Charles A. Weibel, An introduction to homological algebra Cambridge Studies in Advanced Mathematics 38 (1994).

Index

C^{i,j}, 121 C_{n}, 3 D_{2^{n}}, 63 D_{4n}, 85 G, 1 G_{J}, 34 H_{I}^{i}(−), 10 H_{X}, 63 H_{Y}, 63 I_{<P}^{G}, 31 NSF, 60 NSF (G), 112 R^{G}, 1 R_{+}, 8 R_{i}, 8 S(V), 15 T, 89 V^{∗}, 1 V_{−(2n+1)}, 50 V_{2n+1}, 50 V_{2n,∞}, 49 V_{2n,λ}, 49 V_{|P}, 74 W_{(−2n+1)}(ω^{j}), 94 W_{2n+1}(ω^{j}), 94 W_{2n}(ω^{j}), 93 X, 48 Y, 48 [v_{1}, …, v_{r}], 20 Alt_{n}, 89 Der(G, R), 19 End, 132 Frac, 55 PDer(G, R), 19 Σ_{n}, 2 Θ_{T,Y}, 90 Tr_{N}^{G}, 22 add, 60 α^{g}, 20 depth, 9 grade, 9 inf_{N}^{G}, 21 IntCls, 55 A, 3 O(V), 2 X_{M}, 46 Y_{M}, 46 ℑ(V^{P}), 31 𝔙(I);, 31 𝔦_{G}, 31 ω, 93 \overline{V}_{4}, 54 \overline{W}_{4}(ω^{j}), 95 \overline{W}_{6n}(λ), 95 ∂_{h}^{i,j}, 122 ∂_{v}^{i,j}, 122 res_{N}^{G}, 21 Tot(C)^{k}, 122 tr_{N}^{G}, 22 hE^{∗,∗}, 122 vE^{∗,∗}, 122 cc_{G}(R), 18 k[a_{1}, a_{2}, …, a_{n}](b_{1}, …, b_{r}), 7 u_{r}, 97

W′_{2n}(ω^{j}), 93