CONSTRUCTING ENDOTRIVIAL MODULES

JON F. CARLSON^{∗}

Dedicated to Eric Friedlander for his sixtieth birthday

Abstract. A classification of endotrivial modules over p-groups was recently completed by the author and Jacques Thévenaz relying on generators for the group of endotrivial modules created by Alperin, Dade and others. In this paper we present a completely new construction of a set of generators using methods of group cohomology, some of which were developed in the classification. The new method has the advantage that all endotrivial modules can be constructed by the homological technique. In addition, the method applies to endotrivial modules over groups that are not p-groups.

1. Introduction

The aim of this paper is to present a completely new approach to the construction of endotrivial modules. Let G be a finite group and k a field of characteristic p > 0. An endotrivial kG-module M has the property that Hom_{k}(M, M) ≅ k ⊕ P where P is a projective module. This is an isomorphism as kG-modules. Thus, a kG-module M is endotrivial if, in the stable category of kG-modules modulo projective modules, the endomorphism ring of M is a trivial module. There are at least two reasons for an interest in endotrivial modules. First, we should notice that Hom_{k}(M, M) ≅ M^{∗} ⊗ M, and hence the tensor product with an endotrivial module always induces a self-equivalence of the stable category of kG-modules. It is an element in the stable Picard group of all equivalences of the stable category of all kGmodules (e.g. see [6]). In addition, the endotrivial modules are the building blocks for the endopermutation modules, which for many groups, including p-nilpotent groups [13] and p-solvable groups [19] (see also [23]), are the sources of the irreducible modules.

A complete classification of the endotrivial modules for a p-group is now finally complete. The modules were defined by Dade [13] who proved that for an abelian p-group all endotrivial modules are syzygies of the trivial module. In [20], Puig proved that the group T (G) of endotrivial kG-modules, which is abelian, is finitely generated. The torsion part of that group was characterized recently by the author and J. Thévenaz [8]. Meanwhile, the rank of the torsion free part of the group had been found by Alperin [1] and somewhat later and independently by Bouc and Thévenaz [5]. Both found subgroups of finite index in T (G), but neither was able to show that the subgroups were all of T (G). This problem was finally solved in [9] where it was shown that the classes of the modules defined by Alperin actually generate the entire group of endotrivial modules. The last proof borrows heavily from methods developed in [8]. The characterization plays an essential role in the recently announced characterization by Bouc of the Dade group of endopermutation modules.

Date: January 17, 2005.

1991 Mathematics Subject Classification. 20C20.

* Supported in part by a grant from NSF.

Chronologically, this paper sits somewhere between [8] and [9]. Although it followed [1] and [5] by a few years, the method presented here was first to give a certifiably full set of generators for T (G) in the case that G is a p-group. The connections between the different sets of generators is still somewhat mysterious. We know that modules constructed by the techniques of this paper are isomorphic to certain relative syzygies, but the existence of the isomorphisms is only established using the full classification of endotrivial modules. Some discussion of this problem is given in Section 8. The construction given here has some advantages. For one thing we show that generators for the entire group of endotrivial modules, including the torsion part, can be constructed by the same method. Indeed, all endotrivial module can be found by carving up the known endotrivial modules, namely the syzygies Ω^{n}(k) of the trivial module k. In addition, the method can be used to establish the torsion free rank of T (G) even in the case that G is not a p-group. The the proof again relies on techniques developed in [8], and uses the theory of support varieties for modules in highly nontrivial ways.

Following this introduction, we recall some preliminaries of support varieties in Section 2 and endotrivial modules in Section 3. The main methods are proved in Section 4, though a couple of particularly messy technicalities are put off in the appendix (Section 9). The construction of the generators is laid out in the next three sections. First the torsion generators, then the generators for T (G) when G is extraspecial or almost extraspecial and then the general case of T (G) for G any p-group is proved in Section 7. The actual proof in that section is relatively easy. It is primarily a matter of showing that T (G) is determined by looking at the maximal extraspecial and almost extraspecial subgroups of G. Section 8 consists of some remarks on the development and some comments on remaining problems.

The author would like to thank the Alexander von Humboldt Foundation for support and the Max Planck Institute for Mathematics in Bonn for their hospitality during an extended visit in the fall of 2003 when some of this paper was written. We are grateful to the University of Lausanne and the École Polytechnique in Lausanne for support during several visit. Thanks are also due to Jacques Thévenaz, Serge Bouc and Nadia Mazza for several stimulating and interesting conversations.

2. Support varieties and partial inflations

In this section we recall the definitions and properties of support varieties for modules. Our hypothesis throughout the section is that G is a finite group and k is an algebraically closed field of characteristic p > 0. The assumption of algebraic closure is not really necessary, but it makes the geometry easier and has no ill effects on the results that we will discuss. We refer the reader to the books by Benson [2], Evens [15] or the author and his collaborators [10] for references on cohomology rings and support varieties. All kG-modules are assumed to be finitely generated unless otherwise noted.

First we recall some notation. If M is a kG-module, then let P ⟶^{φ} M be its projective cover, and let Ω(M) be the kernel of φ. Iterating we let Ω^{n}(M) = Ω(Ω^{n−1}(M)) for n > 1. Similarly, let Ω^{−1}(M) be the cokernel of the injective hull M ⟶ Q, and Ω^{−n}(M) = Ω^{−1}(Ω^{−n+1}(M)). If we let Ω^{0}(M) be the sum of the nonprojective components of M, then we have that Ω^{n}(Ω^{m}(M)) ≅ Ω^{n+m}(M) for any integers n and m. Moreover, because the tensor as well as the dual of any projective module is projective, we have that Ω^{m}(M) ⊗ Ω^{n}(N) ≅ Ω^{m+n}(M ⊗ N) ⊕ (proj) and (Ω^{n}(M))^{∗} ≅ Ω^{−n}(M^{∗}). Here ⊗ means ⊗_{k}, and by ⊕ (proj) we mean the direct sum with some projective module.

The cohomology ring H^{∗}(G, k) is a finitely generated, graded commutative kalgebra and has a maximal ideal spectrum V_{G}(k) which is a homogeneous affine variety. One of the most valuable tools in the theory of module varieties is the Quillen’s Theorem, given below. See [21] or any of the standard references. If H is a subgroup of G, let res_{G,H} : H^{∗}(G, k) ⟶ H^{∗}(H, k) denote the restriction map, and let res_{G,H}^{∗} : V_{H}(k) ⟶ V_{G}(k) be the corresponding map on maximal ideal spectra.

Theorem 2.1. [21]

V_{G}(k) = ⋃_{E∈EA} res_{G,E}^{∗}(V_{E}(k))

where EA is the collection of all elementary abelian p-subgroups of G.

It is important to notice that the union could just as well have been taken over the maximal elementary abelian subgroups of G.

If M is a finitely generated kG-module, then its cohomology ring Ext_{kG}^{∗}(M, M) is a finitely generated module over H^{∗}(G, k), and we let J(M) denote its annihilator in H^{∗}(G, k). The support variety of M is the set V_{G}(M) = V_{G}(J(M)) ⊆ V_{G}(k) of all maximal ideal the contain J(M). Hence, V_{G}(M) is a closed homogeneous subvariety of V_{G}(k). The support varieties have some important properties. The properties were developed by many people including Alperin, Avrunin, Evens and Scott. Proofs and history can be found in the standard references [2, 15, 10].

Theorem 2.2. Suppose that L, M and N are kG-modules.

(1) The module M is projective if and only if V_{G}(M) = {0}.

(2) V_{G}(M ⊕ N) = V_{G}(M) ∪ V_{G}(N).

(3) V_{G}(M ⊗ N) = V_{G}(M) ∩ V_{G}(N).

(4) If the sequence 0 ⟶ L ⟶ M ⟶ N ⟶ 0 is exact, then V_{G}(M) ⊆ V_{G}(L) ∪ V_{G}(N).

(5) V_{G}(M) = ∪_{E∈EA} res_{G,E}^{∗}(V_{E}(M)).

(6) Suppose that V_{G}(M) = V_{1} ∪ V_{2} where V_{1} and V_{2} are closed sets such that V_{1} ∩ V_{2} = {0}. Then M ≅ M_{1} ⊕ M_{2} where V_{G}(M_{1}) = V_{1} and V_{G}(M_{2}) = V_{2}.

(7) Suppose that ζ ∈ H^{n}(G, k), and let ζ̂: Ω^{n}(k) ⟶ k be a cocycle representing ζ. Let L_{ζ} denote the kernel of ζ̂. Then V_{G}(L_{ζ}) = V_{G}(ζ).

For the remainder of the section assume that the group G has a normal subgroup Z = ⟨z⟩ of order p. Let \overline{G} = G/Z. For M a kG-module, let \overline{M} = M/M_{0} where M_{0} = {m ∈ M|(z − 1)^{p−1}m = 0}. Note that M_{0} is a kG-submodule of M. With this notation, we have the following theorem.

Theorem 2.3. [3] For any n ≥ 0, the support variety of the k\overline{G}-module \overline{Ω^{2p^{n}}(k)} is

V\overline{G}(\overline{Ω^{2p^{n}}(k)}) = V\overline{G}(I_{n})

where I_{n} is the kernel of the partial inflation inf_{n} : H^{∗}(\overline{G}, k) ⟶ E_{2p^{n}+1}^{∗,0} of the cohomology ring of \overline{G} onto the bottom row of the E_{2p^{n}+1} page of the Lyndon-Hochschild-Serre spectral sequence

E_{2}^{r,s} = H^{r}(\overline{G}, H^{s}(Z, k)) ⇒ H^{r+s}(G, k).

Notice that at the E_{2} page of the spectral sequence, the bottom row is the ring

E_{2}^{∗,0} = H^{∗}(\overline{G}, H^{0}(Z, k)) = H^{∗}(\overline{G}, k).

The image of the d_{2} differential is the ideal I_{0} and E_{3}^{∗,0} ≅ H^{∗}(\overline{G}, k)/I_{0}. Likewise, the bottom row of the E_{2p^{n}+1} page of the spectral sequence is the ring H^{∗}(\overline{G}, k)/I_{n}. To illustrate how this works, we present one example which is useful to us when we consider the semi-dihedral 2-groups in a later section.

Example 2.4. Suppose that G is a semi-dihedral group of order 2^{n} for n ≥ 4. Then G = ⟨g, h|g^{2^{n−1}} = 1 = h^{2}, hgh = g^{2^{n−2}−1}⟩. The element z is g^{2^{n−2}} and \overline{G} = G/Z is a dihedral group of order 2^{n−1}. The cohomology ring of \overline{G} is well known and has the form H^{∗}(\overline{G}, k) ≅ k[z, y, x]/(zy) where the degrees of z, y and x are 1, 1 and 2, respectively. Because \overline{G} acts trivially on Z, we have that E_{2}^{∗,∗} ≅ H^{∗}(\overline{G}, k)⊗H^{∗}(Z, k) as a ring. The cohomology of Z is a polynomial ring H^{∗}(Z, k) ≅ k[u]. Hence, E_{2}^{0,1} is spanned by u (or 1 ⊗ u), while E_{2}^{2,0} is spanned by z^{2}, y^{2}, x.

The cohomology ring H^{∗}(G, k) was computed by Evens and Priddy [16], and we could use their analysis to figure out what happens. However, the fact that we need also follows from some general principles. To be precise, the element x in the cohomology ring of \overline{G} is the generator that restricts nontrivially to the center of \overline{G}. Now the center of \overline{G} is cyclic of order 2 and in G it is extended to a cyclic group of order 4. So the point is that this element can not survive to the E_{3} page. Actually some minor changes of variables are possible here (e. g. replacing x by z^{2} + x), but it all comes to the same thing. The image d_{2}(u) must be an element that restricts nontrivially to the center of \overline{G}, and we can assume that d_{2}(u) = x.

We conclude from the Theorem 2.3 that the variety

V\overline{G}(\overline{Ω^{2}(k_{G})}) = V\overline{G}((x))

is the maximal ideal spectrum of the ring H^{∗}(\overline{G}, k)/(x) ≅ k[z, y]/(zy). Notice that this variety is the union of two lines through the origin. Hence \overline{Ω^{2}(k)} is the direct sum of two submodules corresponding to the two lines, by Theorem 2.2(6).

We should be a little careful in the above analysis. The usual thing is that the element x in the cohomology of \overline{G} is chosen so that Sq^{1}(x) = (z + y)x, where Sq^{1} is the Steenrod operator of degree 1. This choice really specifies the restriction of x to the elementary abelian subgroups of \overline{G}. With this choice, we should have that d_{2}(u) = z^{2} + x. Then d_{3}(v) = (z + y)x where v is the class of u^{2} on the E_{3} page.

This is because Sq^{1}(u) = u^{2}, and hence its image under d_{3} is Sq^{1} of the image of u under d_{2}. With this notation we get the cohomology ring H^{∗}(G, k) in the form

H^{∗}(G, k) ≅ k[ẑ, ŷ, x̂, ŵ]/(ẑ^{3}, ẑŷ, ẑx̂, x̂^{2} − ŷ^{2}ŵ),

where the elements ẑ, ŷ, x̂, ŵ correspond to the classes of z, y, u^{2}y and u^{4} in the spectral sequence. This is precisely the formulation of [16]. With this notation, we have also that z^{3} = 0 and hence E_{4}^{∗,0} = k[z, y]/(z^{3}, zy). So \overline{Ω^{4}(k)} has an irreducible variety.

3. Preliminaries on endotrivial modules

In this section we remind the reader of some basic facts about endotrivial modules. Throughout, we assume that G is a finite group and that k is a field of characteristic p > 0. All kG-modules are assumed to be finitely generated. First the definition: A kG-module M is endotrivial provided Hom_{k}(M, M) ≅ k ⊕ P where P is a projective kG-module and the isomorphism is an isomorphism of kG-modules. Recall that Hom_{k}(M, N) ≅ M^{∗} ⊗_{k} N, as kG-modules, where M^{∗} is the k-dual of M. Consequently, M is an endotrivial module if and only if M^{∗} ⊗ M ≅ k ⊕ (proj), where ⊕ (proj) means direct sum with some projective module and ⊗ means ⊗_{k}.

The main example is that for every n, Ω^{n}(k) is an endotrivial modules. In fact, the calculus is that

Ω^{m}(k) ⊗ Ω^{n}(k) ≅ Ω^{m+n}(k) ⊕ (proj), and (Ω^{n}(k))^{∗} ≅ Ω^{−n}(k).

Therefore, Ω^{n}(k) ⊗ Ω^{−n}(k) ≅ k ⊕ (proj). Note also that if M is endotrivial, then so is M ⊕ P for any projective module P. Two facts that we need are the following. The first is an easy consequence of Chouinard’s Theorem (see Theorem 8.2.12 of [10]) that a module is projective if and only if its restriction to every elementary abelian p-subgroup is projective. A proof of the proposition can be found in [7], among other places.

Proposition 3.1. A kG-module is endotrivial if and only if its restriction to every elementary abelian p-subgroup is endotrivial.

The second is a deep theorem of Dade [13].

Theorem 3.2. Suppose that G is an abelian p-group and that M is an endotrivial module. Then M ≅ Ω^{n}(k) ⊕ (proj) for some integer n.

We can form the group T (G) of endotrivial kG-modules as follows. An element of the group is a class of an endotrivial module M. Two endotrivial modules M and N are in the same class provided there exist projective modules P and Q such that M ⊕P ≅ N ⊕Q. The operation in the group is given by the rule [M]+[N] = [M ⊗N]. By Dade’s Theorem, we have that T (G) = {0} if G ≅ C_{2}, a cyclic group of order 2, while T (G) = ℤ/(2) if G ≅ C_{p^{n}} for either p > 2 or n > 1. For any abelian p-group of rank at least 2, we have that T (G) ≅ ℤ.

For nonabelian p-groups we have the following.

Theorem 3.3. [8] Suppose that G is a nonabelian p-group and that G is neither a quaternion 2-group nor a semi-dihedral 2-group. Then the group T (G) is torsion free. Moreover, if EA the collection of maximal elementary abelian p-subgroups of G, then the restriction homomorphism

( ∏_{E∈EA} res_{G,E}) : T (G) ⟶ ∏_{E∈EA} T (E)

is injective.

Thus, we see that the isomorphism class of an indecomposable endotrivial module can be detected from its restrictions to the maximal elementary abelian subgroups. We should be clear that if G is a quaternion or semi-dihedral 2-group, then T (G) is known by reasonably elementary means. See [7] for details. We show later that the generators for the torsion part of T (G) for these groups can be also constructed as modules using the technique of the next section.

In his calculation of the torsion free rank of T (G), Alperin [1] noted the basics of the proposition that follows. Notice that the second statement of the proposition requires also the fact that T (G) is torsion free by the previous theorem.

Proposition 3.4. Suppose that M is an endotrivial module and that E_{1} and E_{2} are two elementary abelian p-subgroups of G, both having rank at least 3. If M_{E_{i}} ≅ Ω^{n_{i}}(k) ⊕ (proj) for i = 1, 2, then n_{1} = n_{2}. In particular, if G is not a quaternion or semi-dihedral 2-group, then the product of the restriction maps

( ∏_{E∈EA_{2}} res_{G,E}) : T (G) ⟶ ∏_{E∈EA_{2}} T (E),

where EA_{2} is the set of all elementary abelian p-subgroups of rank 2, is injective.

From these results we can deduce that if G is not cyclic, quaternion or semidihedral and if G has no maximal elementary abelian subgroup of rank 2 or has only one conjugacy class of maximal elementary abelian p-subgroups then T (G) ≅ ℤ. Hence, the cases of interest are those in which G has at least two conjugacy classes of maximal elementary abelian subgroups and has at least one class of maximal elementary abelian subgroups of rank 2. In [9] (Lemma 2.2), the following is demonstrated. It is proved in slightly different form in [1] and [17].

Proposition 3.5. Suppose that G is a p-group that is not cyclic, quaternion, dihedral or semi-dihedral. Assume also that G has at least two conjugacy classes of maximal elementary abelian subgroups and that at least one of these classes has an element of rank 2. Then there exists a collection E_{1}, E_{2}, …, E_{n} such that the following hold.

(1) Every E_{i} is elementary abelian of rank 2.

(2) E_{n} is normal in G.

(3) E_{1}, …, E_{n−1} are maximal elementary abelian subgroups of G.

(4) If E is a maximal elementary abelian subgroup of rank 2 in G, then E is conjugate to E_{i} for exactly one i.

(5) If the p-rank of G is three or larger then E_{n} is not a maximal elementary abelian p-subgroup.

Note that if the rank of G is 2, then E_{1}, …, E_{n} are simply representatives of the conjugacy classes of maximal elementary abelian subgroups. On the other hand, if the rank of G is larger than 2, then E_{1}, …, E_{n−1} are representatives of the classes of maximal elementary abelian subgroups of rank 2, and E_{n} is not maximal. The normality of E_{n} is very useful to us in the later sections. The main consequence of all of this is the following.

Corollary 3.6. Assume the hypothesis and notation of the proposition. Then the map

(∏_{i=1}^{n} res_{G,E_{i}}) : T (G) ⟶ ∏_{i=1}^{n} T (E_{i})

is injective.

Alperin [1] shows further that the image of the map in the corollary has finite index in T (G), and hence that the rank of T (G) is n. This statement is also a consequence of the construction in this paper.

If M is an endotrivial kG-module, then we say that M has type

T ype(M) = a = (a_{1}, a_{2}, …, a_{n}) ∈ ℤ^{n}

provided for each i, M_{E_{i}} ≅ Ω^{a_{i}}(k_{E_{i}}) ⊕ (proj). Then the map in Corollary 3.6 can be regarded as taking M to T ype(M). The results of this section can be interpreted as implying the following.

Corollary 3.7. Assume the hypothesis and notation of Proposition 3.5. If M and N are indecomposable endotrivial kG-modules having the same type, then M ≅ N.

4. New endotrivial module from old

In this section we develop the basic technique that we use in the construction of endotrivial modules. The primary method (Theorem 4.2) is not really new. It was first proved in the paper [8] by the author and Jacques Thévenaz. The basic idea was used again in [9], though some modifications were necessary. Again, we need some minor modifications for this paper. For the sake of completeness, we present a thorough proof. We also prove the result without the assumption that the group G is a p-group or that the normal subgroup of order p is central. The generalization has applications in other works [11].

Our basic assumption throughout the section is that G is a finite group and that ⟨z⟩ is a normal subgroup of order p in G. Hence z is an element of order p. We follow the notation at the end of Section 2. If M is a kG-module then let \overline{M} = M/M_{0} where M_{0} = \{m ∈ M|(z − 1)^{p−1}m = 0\}. Then \overline{M} ≅ (z − 1)^{p−1}M where the isomorphism is induced by multiplication by (z − 1)^{p−1}. Let \overline{G} = G/⟨z⟩. We have an inflation map infG ; \overline{G} : H^{∗}(\overline{G}, k) ⟶ H^{∗}(G, k) which induces a map on varieties (infG ; \overline{G})^{∗} : V_{G}(k) ⟶ V\overline{G}(k). For the main theorem we require the following generalization of Lemma 5.5 of [8]. Assume also that k is algebraically closed.

Lemma 4.1. Suppose that N is a kG-module with the property that (z − 1)^{p−1}N = {0}. Hence, N is a module over the algebra R = k⟨z⟩/((z − 1)^{p−1}). Assume that N is a free R-module. Suppose that V\overline{G}((z − 1)^{p−2}N) = V_{1} ∪ V_{2} where V_{1} ∩ V_{2} = \{0\}. There exist L_{1} and L_{2} such that (z − 1)^{p−2}N = L_{1} ⊕ L_{2} where V\overline{G}(L_{1}) = V_{1} and V\overline{G}(L_{2}) = V_{2}. Then there must exist submodules N_{1} and N_{2} such that N = N_{1} ⊕ N_{2} and for i = 1, 2, we have that L_{i} = (z − 1)^{p−2}N_{i}.

Proof. Note first that for any i, (z − 1)^{i}N = (Rad kZ)^{i}N is a kG-submodule of N, since ⟨z⟩ = Z is a normal subgroup of G. In fact, for any i, the module

(z − 1)^{i}N/(z − 1)^{i+1}N ≅ (z − 1)^{p−2}N ⊗ T_{i}

for some kG-module T_{i} of dimension one. In particular, V\overline{G}((z−1)^{i}N/(z−1)^{i+1}N) = V\overline{G}((z − 1)^{p−2}N).

The statement concerning the existence of L_{1} and L_{2} is a direct consequence of Theorem 2.2. In the same way, we must have that (z−1)^{p−3}N/(z−1)^{p−2}N ≅ L′_{1}⊕L′_{2} where V\overline{G}(L′_{i}) = V_{i}. So there is another exact sequence

0 ⟶ W ⟶ (z − 1)^{p−3}N ⟶ L′_{2} ⟶ 0

where W contains (z − 1)^{p−2}N and W/(z − 1)^{p−2}N ≅ L′_{1}. So we have a sequence

0 ⟶ L_{2} ⟶ W/L_{1} ⟶ L′_{1} ⟶ 0

where L_{2} is the quotient (z − 1)^{p−2}N/L_{1}. Because V\overline{G}(L_{2}) ∩ V\overline{G}(L_{1}) = \{0\} the last sequence splits. Hence, there is a submodule U of W such that L_{1} ⊆ U, U/L_{1} ≅ L′_{1} and also U ∩ L_{2} = {0}. Therefore W ≅ U ⊕ L_{2}.

Next we reverse the roles of L_{1} and L_{2} to find a submodule V of N such that we have an exact sequence

0 ⟶ L_{2} ⟶ V ⟶ L′_{2} ⟶ 0

and U ∩ V = {0}. Thus (z − 1)^{p−3}N = U ⊕ V. The next step of the proof is simply a repeat of arguments of this type, next showing a decomposition of (z − 1)^{p−4}N. Then repeat again as needed. □

Now we are ready for the main construction. Again let Z = ⟨z⟩.

Theorem 4.2. Suppose that M is an endotrivial kG-module such that M_{Z} ≅ k_{Z} ⊕ (proj). Assume that V\overline{G}(\overline{M}) = V_{1} ∪ V_{2} where V_{1} and V_{2} are closed subsets of V\overline{G}(k) such that V_{1}∩V_{2} = \{0\}. We have that \overline{M} ≅ N_{1}⊕N_{2} where V\overline{G}(N_{i}) = V_{i}, for i = 1, 2. Then there exists an endotrivial kG-module U such that \overline{U} ≅ N_{2}, and satisfying the following property. Let H be any p-subgroup that contains Z.

(1) If (infG ; \overline{G})^{∗} res_{G,H}^{∗}(V_{H}(k)) ∩ V_{2} = \{0\}, then U_{H} ≅ k_{H} ⊕ (proj).

(2) If (infG ; \overline{G})^{∗} res_{G,H}^{∗}(V_{H}(k)) ∩ V_{1} = \{0\}, then M_{H} ≅ U_{H} ⊕ (proj).

Proof. It is important for the proof that H be a p-subgroup. We use the fact that a module M over kH, for H a p-group, is projective if and only if the dimension of the set M/(Rad kH)M is precisely (1/|H|) Dim(M). We begin by constructing the module U, and then we show that it has the desired properties.

Because of the assumption that M_{⟨z⟩} ≅ k ⊕ (proj), we have that

\overline{M} ≅ M/M_{0} ≅ ((z − 1)^{i}M/(z − 1)^{i+1}M) ⊗ T_{i}

for some kG-module T_{i} of dimension one, for M_{0} = {m ∈ M|(z − 1)^{p−1}m = 0} and for all i = 1, …, p − 1. Because of the condition on the varieties, we have that \overline{M} = N_{1} ⊕ N_{2} where V\overline{G}(N_{1}) = V_{1} and V\overline{G}(N_{2}) = V_{2}. Then by Lemma 4.1, (z−1)M ≅ L_{1}⊕L_{2} where L_{1} and L_{2} are free as modules over the ring k⟨z⟩/(z−1)^{p−1} and (z − 1)^{p−2}L_{1} = N_{1}, (z − 1)^{p−2}L_{2} = N_{2}.

Let N′_{1} and N′_{2} be submodules of M/M_{0} such that the isomorphism M/M_{0} ⟶ (z − 1)^{p−1}M induced by multiplication by (z − 1)^{p−1} takes N′_{1} to N_{1} and N′_{2} to N_{2} precisely. Then clearly, M/M_{0} = N′_{1} ⊕ N′_{2}. For the purposes of the proof, we want to be even more exact. Choose elements a_{1}, …, a_{r} which are kG-generators for L_{1} and choose elements b_{1}, …, b_{s} that generate L_{2}. Then let N′_{1} be generated by classes a′_{1} + M_{0}, …, a′_{r} + M_{0} such that (z − 1)a′_{i} = a_{i} for i = 1, … r. Similarly, we let N′_{2} be generated by the classes of elements b′_{i} where (z − 1)b′_{i} = b_{i} for i = 1, …, s. In particular, if W is the submodule of M generated by b_{1}, … b_{s}, then (z − 1)W = L_{2}.

Now let V be the pullback of the module N′_{2} under the quotient map φ in the sequence

0 ⟶ M_{0} ⟶ M ⟶^{φ} N′_{1} ⊕ N′_{2} ⟶ 0.

Thus, if b_{1}, …, b_{s} ∈ M is the collection of elements whose classes modulo M_{0} generate N′_{2} as above, then V is generated by the elements b_{1}, …, b_{s}, the submodule L_{1} and by any element u ∈ M_{0} that generates the direct summand of M_{⟨z⟩} that is isomorphic to the trivial module k. Now let U = V/L_{1} be the quotient. It remains to show that U has the desired properties. Note that from the construction, \overline{U} = N′_{2} as desired, and the dimension of U is p Dim N_{2} + 1

Let H be a p-subgroup of G, and suppose we have the first of the two cases, namely that (infG ; \overline{G})^{∗} res_{G,H}^{∗}(V_{H}(k)) ∩ V_{2} = \{0\}. Then (N_{2})_{↓\overline{H}} is a free k\overline{H}-module. Now suppose that c_{1}, …, c_{t} is a collection of elements in the submodule W generated by b_{1}, …, b_{s} such that c_{1}+M_{0}, …, c_{t}+M_{0} form a free k\overline{H}-basis for N′_{2}. In particular, Dim N′_{2} = t|\overline{H}|. We can see that the submodule (N_{2} + L_{1})/L_{1} ≅ N_{2} is generated as a k\overline{H}-module by the classes of the elements (z − 1)^{p−1}c_{1}, … (z − 1)^{p−1}c_{t}. Hence the kH-submodule of U generated by c_{1} + L_{1}, …, c_{t} + L_{1} has dimension t|H| = p Dim N_{2}. By dimension arguments, this must be a free submodule, and it must be that U_{H} ≅ k ⊕ (proj).

Now suppose the second case that (infG ; \overline{G})^{∗} res_{G,H}^{∗}(V_{H}(k)) ⊆ V_{2}. Then (N_{1})_{↓\overline{H}} is a free k\overline{H}-module. Let X be the kG-submodule of M generated by a_{1}, …, a_{r}. Then (z − 1)X = L_{1} and (X + M_{0})/M_{0} = N′_{1}. Choose elements d_{1}, …, d_{n} ∈ X so that the classes d_{i} + M_{0} form a free k\overline{H}-basis for N′_{1}. Then by a dimension argument, the kH-submodule X′ of M_{H} generated by d_{1}, …, d_{n} is a free kH-module and also

(z − 1)X′ = X′ ∩ V_{H} = X′ ∩ (M_{0})_{H} = {x ∈ X′|(z − 1)^{p−1}x = 0} = (L_{1})_{H}

as kH-modules. Here V is the submodule generated by M_{0} and b_{1}, …, b_{s} as above. We have then that

U_{H} ≅ V_{H}/(L_{1})_{H} ≅ V_{H}/(X′ ∩ V_{H}) ≅ (X′ + V_{H})/X′ ≅ M_{H}/X′.

But X′, being projective, is a direct summand of M_{H}. It follows that M_{H} ≅ U_{H} ⊕X′.

Notice finally that U is an endotrivial module because its restriction to every elementary abelian p-subgroup is an endotrivial module (see 3.1). That is, suppose that E is an elementary abelian p-subgroup of G. Without loss of generality we can also assume that Z ⊆ E. In the event that M_{E} ≅ k ⊕ (proj), then it must be the case that (infG ; \overline{G})^{∗} res_{G,E}^{∗}(V_{E}(k)) ∩ V_{i} = \{0\} for both i = 1 and i = 2. So then U_{E} ≅ k ⊕(proj). Otherwise, M_{E} ≅ Ω^{r}(k_{E})⊕(proj) for some r ≠ 0 and we have that V\overline{E}(\overline{M}) = V\overline{E}(k_{E}). Therefore, we have also that (infG ; \overline{G})^{∗} res_{G,E}^{∗}(V_{E}(k)) ⊆ V\overline{G}(\overline{M}). However, because (infG ; \overline{G})^{∗} res_{G,E}^{∗}(V_{E}(k)) is connected (in the sense that it can not be written as the union of two closed sets which intersect in {0}), we must have that (infG ; \overline{G})^{∗} res_{G,E}^{∗}(V_{E}(k)) ⊆ V_{i}, either for i = 1 or for i = 2. Hence, by what we have already proved, U_{E} is endotrivial. □

Our main applications of the construction use the formulation given in the next corollary.

Corollary 4.3. Assume the hypothesis and notation of Theorem 4.2. Suppose that E is an elementary abelian p-subgroup of G. Then one of the following holds.

(1) M_{E} ≅ k_{E} ⊕ (proj) in which case U_{E} ≅ k ⊕ (proj),

(2) (infG ; \overline{G})^{∗} res_{G,E}^{∗}(V_{E}(k)) ⊆ V_{1}, in which case U_{E} ≅ k_{E} ⊕ (proj), or

(3) (infG ; \overline{G})^{∗} res_{G,E}^{∗}(V_{E}(k)) ⊆ V_{2}, in which case M_{E} ≅ U_{E} ⊕ (proj).

Proof. It should be clear that the cases need only be checked on maximal elementary abelian subgroups of G. Hence we can assume that for any such E, ⟨z⟩ ⊆ E. Let \overline{E} = E/⟨z⟩. In the first case, notice that \overline{M}\overline{E} is projective and so (N_{1})\overline{E} and (N_{2})\overline{E} are projective modules. In general, we have that (infG ; \overline{G})^{∗} res_{G,E}^{∗}(V_{E}(k)) = res_{\overline{G},\overline{E}}^{∗}(infE ; \overline{E})^{∗}(V_{E}(k)). Therefore, in the event that M_{E} ≅ Ω^{n}(k_{E}) ⊕ (proj) for n ≠ 0, we must have that V\overline{E}(\overline{M}\overline{E}) = V\overline{E}(k). So then (infG ; \overline{G})^{∗} res_{G,E}^{∗}(V_{E}(k)) ⊆ V_{i} for some i. The reason is that otherwise res_{G,E}^{∗} V_{E}(k) would be the union of two closed sets that intersect only in \{0\}, which is impossible. Now note that if (infG ; \overline{G})^{∗} res_{G,E}^{∗}(V_{E}(k)) ⊆ V_{1}, then (infG ; \overline{G})^{∗} res_{G,E}^{∗}(V_{E}(k)) ∩ V_{2} = \{0\}, and vice versa. Now we apply Theorem 4.2. □

The construction given in the Theorem 4.2 is something of a disassembly method. There is also an assembly process by which we can create new endotrivial modules. The method is tensor product which is not new. However, the point is that sometimes we can make very specific statements about the tensor product of two endotrivial module. A complete proof of the following proposition can be found in [7] and we do not repeat it here.

Proposition 4.4. Suppose that G is a p-group with a central normal subgroup Z of order p. Suppose that M and N are indecomposable endotrivial modules such that M_{Z} ≅ k ⊕ (proj) and N_{Z} ≅ k ⊕ (proj). Assume further that V\overline{G}(\overline{M}) ∩ V\overline{G}(\overline{N}) = \{0\}. Then M ⊗ N ≅ L ⊕ (proj) where L is an indecomposable endotrivial module such that \overline{L} ≅ \overline{M} ⊕ \overline{N}.

We end this section with a variation on the construction given above. The variation looks rather different, and it is much easier to describe and explain. But unfortunately, it is not as general and is only a weaker form of the above construction. In particular, we will show in the last section that the there are group for which the variational construction will not yield a complete set of generators for the group of endotrivial modules. It will also not construct that torsion endotrivial modules for quaternion and semi-dihedral 2-groups in the next section. On the other hand, in the section concerned with the endotrivial modules for extraspecial groups we can get a complete set of generators using the variational method. The other advantage of the method is that it requires no hypothesis on the group. First we give the construction.

Let G be any finite group, and let ζ ∈ H^{m}(G, k) be an element of cohomology. Let V_{G}(ζ) ⊆ V_{G}(k) denote the set of all maximal ideals that contain ζ. Suppose that V_{G}(ζ) = V_{1} ∪ V_{2} where V_{1} and V_{2} are closed subvarieties such that V_{1} ∩ V_{2} = {0}. Then we have that L_{ζ} ≅ L_{1} ⊕ L_{2} where V_{G}(L_{i}) = V_{i} for i = 1, 2 (see Theorem 2.2(6)). Hence we have a diagram

[figure: 0; 0; L_{2}; L_{2}; 0; L_{ζ}; Ω^{m}(k); ζ̂; k; 0; 0; L_{1}; N; k; 0; 0; 0]

where N is the pushout.

Theorem 4.5. The module N in the above diagram is an endotrivial module. Moreover, its type can be determined by the following algorithm. Suppose that E is a maximal elementary abelian subgroup of G. Then V_{E}(L_{ζ}) = W_{1} ∪ W_{2} where res_{G,E}^{∗}(W_{i}) ⊆ V_{i}. (Thus, W_{1} ∩ W_{2} = {0}). So, res_{G,E}(ζ) = γ_{1}γ_{2} where V_{E}(γ_{i}) = W_{i}. Then we have that

N_{E} ≅ Ω^{d}(k) ⊕ (proj)

where d = deg(γ_{1}).

Proof. Because Ω^{m}(k)_{E} ≅ Ω^{m}(k_{E}) ⊕ (proj), we have an exact sequence

\xymatrix{0 \ar[r] & (L_{ζ})_{E} \ar[r] & Ω^{m}(k_{E}) ⊕ (proj) \ar[r]^{γ̂} & k \ar[r] & −1}

where γ̂ is a cocycle representing γ = res_{G,E}(ζ). Moreover, (L_{ζ})_{E} ≅ L_{γ} ⊕ (proj). Now, V_{E}(γ) = V_{E}(L_{γ}) = W_{1} ∪ W_{2}, for W_{1} and W_{2} as in the theorem. So it must be the case that γ = γ_{1}γ_{2} as asserted.

Actually we should give a little more explanation here. Notice first that if the rank r of E is three or more, then V_{E}(k) ≅ k^{r} and there is no way that V_{E}(γ) = W_{1} ∪ W_{2} with W_{1} ∩ W_{2} = {0} unless either W_{1} = {0} or W_{2} = {0}. If W_{1} = {0} then (L_{1})_{E} is projective and we can let γ_{1} = Id_{k} and γ_{2} = γ. If W_{2} = {0} then let γ_{1} = γ.

So it comes down to the case that the rank of E is 2. If p = 2 then H^{∗}(E, k) is a polynomial ring in two variables over an algebraically closed field and we must have that γ is a product of linear homogeneous polynomials with the factors having varieties as asserted. In the case that p > 2, the argument is similar but more complicated because of the nilpotent elements in H^{∗}(E, k). However it still works because we are dealing only with homogeneous polynomials. See Proposition 9.2 for complete details.

Now by the Krull-Schmidt Theorem, we have that (L_{1})_{E} ≅ L_{γ_{1}} ⊕ (proj) and (L_{2})_{E} ≅ L_{γ_{2}} ⊕ (proj). So consider the diagram

[figure: 0; 0; (L_{2})_{E}; (L_{2})_{E}; 0; (L_{ζ})_{E}; Ω^{m}(k_{E}) ⊕ (proj); γ̂; Ω^{d}(γ̂_{2}); k_{E}; 0; 0; (L_{1})_{E}; Ω^{d}(k_{E}) ⊕ (proj); γ̂_{1}; k_{E}; 0; 0; 0]

where d = deg(γ_{2}). Now we claim that N_{E} ≅ Ω^{d}(k_{E}) ⊕ (proj). This is basically the uniqueness of the pushouts. But for a proof with complete details, see Theorem 9.4 in the appendix.

The fact that N_{E} is endotrivial for any elementary abelian p-subgroup E implies that N is an endotrivial module. □

In order to state our primary application of Theorem 4.5 we adopt the notation of Proposition 3.5 and the corollaries that follow it. In particular we want to assume that G has maximal elementary abelian p-subgroups of rank 2 and more than one conjugacy class of maximal elementary abelian p-subgroups. For notation, let v_{i} ∈ ℤ^{n} be the standard vector of dimension n with 1 in the i^{th} position and 0 elsewhere.

Corollary 4.6. Suppose that Z is a subgroup of order p that is in the center of a Sylow p-subgroup of G. Suppose that for m > 0, H^{2m}(G, k) has an element ζ such that res_{G,Z}(ζ) ≠ 0. Then G has an endotrivial module of type 2mv_{i} for every i.

Proof. By Quillen’s Theorem 2.1, we have that V_{G}(k) = ∪_{i=1}^{n}W_{i} where W_{i} = res_{G,E_{i}}^{∗} V_{E_{i}}(k) in all cases except when i = n and the p-rank of G is at least 3. In the last case W_{n} = ∪ res_{G,E}^{∗}(V_{E}(k)) where the union is over all elementary abelian p-subgroups E having rank at least 3. In any event, for i ≠ j, the intersection W_{i} ∩ W_{j} = res_{G,Z}^{∗}(V_{Z}(k)) is the line corresponding to the center of the group.

Now V_{G}(L_{ζ}) = V_{G}(ζ) is a closed subvariety of V_{G}(k) that is transverse to the line res_{G,Z}^{∗}(V_{Z}(k)), since the restriction of ζ to the cohomology ring of Z is not nilpotent. It follows that V_{G}(L_{ζ}) = ∪_{i=1}^{n}V_{i} where V_{i} = W_{i} ∩ V_{G}(ζ). Moreover, V_{i} ∩ V_{j} = {0} if i ≠ j. Hence, for any i we get a decomposition V_{G}(L_{ζ}) = V_{i} ∪ V′_{i} where V′_{i} is the union of all V_{j} for j ≠ i. Also, V′_{i} intersects V_{i} trivially. We can construct the module N corresponding to this decomposition as in Theorem 4.5. Then because V_{i} = res_{G,E_{i}}(V_{E_{i}}(L_{ζ})), we have that N_{E_{i}} ≅ Ω^{2m}(k) ⊕ (proj). On the other hand, for j ≠ i, we have that V_{i} ∩ res_{G,E_{j}}^{∗}(V_{E_{j}}(k)) = {0}. So N_{E_{j}} ≅ k ⊕ (proj), and the type of N is as asserted. □

5. Endotrivial module for quaternion and semi-dihedral groups

In this section we show that the construction of the Theorem 4.2 is sufficient to produce generators of T (G) in the cases that G is a quaternion or semi-dihedral 2-group. The issue here is the torsion elements in T (G), and for that reason, we separate these two cases from the more general discussion of the torsion free generators in the next two sections. In fact, the construction of the generators in the case of the quaternion group of order 8 has been known for several years. It was an important insight leading to the proof of the main theorem in [8]. One thing of interest is that this special case does not require Theorem 2.3. The construction for the semi-dihedral groups and for the larger quaternion groups were realized only recently and both use 2.3.

Consider the quaternion group G = ⟨g, h| g^{2^{n−1}} = 1, h^{2} = g^{2^{n−2}}, hgh^{−1} = g^{−1}⟩ for n ≥ 3.. Let the field k have characteristic 2. We wish to look at the module Ω^{2}(k). More specifically, we are interested in the support variety V\overline{G}(\overline{Ω^{2}(k)}), where the notation is the same as that at the beginning of the last section. Of course, z = g^{2^{n−2}} is the generator for the center of G. We know T (G) ≅ ℤ/4ℤ ⊕ ℤ/2ℤ where the first factor is generated by Ω(k).

The quaternion group of order 8. First we consider the case that n = 3 and the order of G is 8. Then \overline{G} is a fours group, an elementary abelian group of order 4. So we know that for any subgroup of order 4 such as C = ⟨g⟩ in G, Ω^{2}(k)_{↓C} ≅ k ⊕(kC)^{2}, since Ω^{2}(k_{C}) ≅ k. In particular, (\overline{Ω^{2}(k)})\overline{C} is a free k\overline{C}-module. The implication is that V\overline{G}(\overline{Ω^{2}(k)}) is not all of V\overline{G}(k) = k^{2}. Hence it must have dimension one and must be a union of lines.

The cohomology ring of H^{∗}(G, k) is well known and it is easy enough to compute that the dimension of Ω^{2}(k) is 9. This means that the dimension of \overline{Ω^{2}(k)} is 4. Now if the support variety V\overline{G}(\overline{Ω^{2}(k)}) is a union of r lines, then by Theorem 2.2, \overline{Ω^{2}(k)} is a direct sum of r modules each having dimension at least 2. Consequently, it must be that the support variety is a union of at most two lines.

Next we notice that the three 𝔽_{2}-rational lines in V\overline{G}(k) correspond to the actual subgroups of order 2 of \overline{G} which act freely on \overline{Ω^{2}(k)}. So they are not in the support variety. On the other hand Ω^{2}(k) is certainly defined over 𝔽_{2}, and so its variety is 𝔽_{2}-rational. That is, it is the zero set in k^{2} of a polynomial with coefficients in 𝔽_{2}. Hence, the only choice is that V\overline{G}(\overline{Ω^{2}(k)}) consists of the two lines through the points (1, α) and (1, α^{2}) where α is a cube root of unity in 𝔽_{4}. Then using the Construction Theorem 4.2, we obtain two endotrival KG-modules of dimension 5. It can be shown by methods in [8] (see Theorem 4.4) that the tensor product of the two modules is again Ω^{2}(k). So T (G) is generated by Ω(k) and one of these modules (and either one will do).

We should note that the modules constructed above, were discovered by Dade [14] using direct computations some thirty years ago. These are the only endotrivial modules for any p-group that can not be defined over the prime field 𝔽_{p}.

Generalized quaternion groups. Assume that n > 3 and G is a quaternion group of order 2^{n}. This time the quotient group \overline{G} ≅ G/Z is a dihedral, where Z = ⟨z⟩. So in the spectral sequence of the group extension,

E_{2}^{r,s} = H^{r}(\overline{G}, H^{s}(Z, k)) ≅ H^{r}(\overline{G}, k) ⊗ H^{s}(Z, k) ⇒ H^{r+s}(G, k),

we have that E_{2}^{∗,0} ≅ k[z, y, x]/(zy) exactly as in Example 2.4. Similarly we have a polynomial ring E_{2}^{0,∗} = H^{∗}(Z, k) ≅ k[u]. As in the example the elements z, y and u are in degree 1, while x is in degree 2. The cohomology ring of the quaternion group is very well known [12] and has the form H^{∗}(G, k) ≅ k[z, y, x]/(z^{2}+zy+y^{2}, z^{2}y+zy^{2}). By an analysis that is very similar to that of Example 2.4, we conclude that d_{2}(u) = x + z^{2} + y^{2}. Also, we can conclude that d_{3}(w) = z′x′ + y′x′ where w, z′, y′ and x′ are the classes of u^{2}, z, y and x on the E_{3} page of the spectral sequence. Note that the class of u is zero on the E_{3} page and hence w is not a square. Likewise, the element of x in H^{4}(G, k) is represented by u^{4} on the E_{4} page of the spectral sequence. But this is not our concern. The point is that V\overline{G}(\overline{Ω^{2}(k)}) is the spectrum of the ring H^{∗}(\overline{G}, k)/(x + z^{2} + y^{2}). Consequently the spectrum is disconnected and consists of two lines V_{1} and V_{2} corresponding to the two prime ideals (z, x + z^{2} + y^{2}) and (y, x + z^{2} + y^{2}).

Now notice that the two ideals correspond to two different elementary abelian subgroups of \overline{G}, in the sense that each contains the kernel of the restriction to exactly one of these subgroups. If we denote the subgroups \overline{E}_{1} and \overline{E}_{2}, then we have that V_{1} ∈ res_{\overline{G},\overline{E}_{1}}^{∗}(V_{\overline{E}_{1}}(k)) and V_{1} ∩ res_{\overline{G},\overline{E}_{2}}^{∗}(V_{\overline{E}_{2}}(k)) = \{0\} while the situation is reversed for V_{2}. Now, E_{1} and E_{2} (the inverse images of \overline{E}_{1} and \overline{E}_{2} under the map G ⟶ \overline{G}) are quaternion groups. By Theorem 4.2, we have an endotrivial modules M_{1} with the properties that (M_{1})_{E_{1}} ≅ k ⊕ (proj), and (M_{1})_{E_{2}} ≅ Ω^{2}(k) ⊕ (proj). So the classes of Ω(k) and M_{1} generate T (G).

Semi-dihedral groups. The construction and proof here are very similar to that for the generalized quaternion group. We use the notation of Example 2.4. Suppose that \overline{E}_{1} and \overline{E}_{2} are the two noncyclic maximal subgroups of the dihedral group \overline{G}. From the example we know that V\overline{G}(\overline{Ω^{2}(k)}) = V_{1} ∪V_{2} where V_{1} ∩V_{2} = \{0\}. Moreover we can assume, by choosing the numbering, that

V_{1} ∩ res_{\overline{G},\overline{E}_{2}}^{∗}(V_{\overline{E}_{2}}(k)) = \{0\} and V_{2} ∩ res_{\overline{G},\overline{E}_{1}}^{∗}(V_{\overline{E}_{1}}(k)) = \{0\}.

So if E_{1} and E_{2} are the inverse images of \overline{E}_{1} and \overline{E}_{2} under the quotient map G ⟶ \overline{G}, then by the Theorem 4.2, there exists an endotrivial module U with the property that U_{E_{1}} ≅ k ⊕ (proj) while U_{E_{2}} ≅ Ω^{2}(k) ⊕ (proj). However, we may assume that E_{1} is a dihedral group and that E_{2} is a quaternion group. Because, E_{1} contains all elements of order 2 in G, we know that U is a torsion element of T (G). Also we have that

(U ⊗ U)_{E_{2}} ≅ Ω^{4}(k) ⊕ (proj) ≅ k ⊕ (proj) .

Hence U ≅ k ⊕ (proj) (see [7]). It follows that the classes of Ω(k) and U generated T (G).

6. The endotrivial modules for extraspecial groups

In this section we describe the group T (G) for G an extraspecial or almost extraspecial p-group that is not a quaternion group. For a quick definition, a p-group G is extraspecial or almost extraspecial if it has a unique normal subgroup Z of order p and if the quotient by that subgroup G/Z is elementary abelian. Such groups are described in [7] and in many other references. In only a few cases do we need to know many details. Most of our cases are covered in the following.

Proposition 6.1. Suppose that G is an extraspecial or almost extraspecial group of order p^{n}. Assume that n ≥ 5 if p is odd and that n ≥ 6 if p = 2. Then T (G) ≅ ℤ and is generated by the class of the module Ω(k).

Proof. The main points are that the torsion part of T (G) is trivial by Theorem 3.3, and all of the maximal elementary abelian subgroups of G have rank at least 3. Hence by Proposition 3.4 the rank of the torsion free part of T (G) is one. □

Note that groups of order p^{2} are abelian and we know T (G) by Dade’s Theorem 3.2. So we have only to treat the cases that G has order p^{3} and p^{4} for p odd and order 8, 16, or 32 for p = 2. The cases are considered individually. For the purposes of the argument, we adopt the notation of Proposition 3.5. In every one of the cases, the group G has rank two and so the subgroups E_{1}, E_{2}, …, E_{n} form a complete set of representatives of the conjugacy classes of maximal elementary abelian subgroups of G.

Most of the proofs in this section follow very similar patterns. For that reason, we prove the first case, below, in some detail and present only sketches for the remaining cases.

Extraspecial groups of order p^{3} for p odd. The extraspecial groups of order p^{3} come in two isomorphism types. The first type has the form

G ≅ ⟨g, h|g^{p^{2}} = 1 = h^{p}, hgh^{−1} = g^{p+1}⟩

and has exponent p^{2}. It has a unique maximal elementary abelian subgroup (⟨g^{p}, h⟩). Hence T (G) ≅ ℤ by Corollary 3.6.

Proposition 6.2. Let G be an extraspecial group of order p^{3} and exponent p^{2} for p odd. Then T (G) ≅ ℤ

The other extraspecial group is isomorphic to the Sylow p-subgroup of SL(3, p). It has exponent p and is presented as

G ≅ ⟨g, h, z|g^{p} = h^{p} = z^{p} = 1, z = ghg^{−1}h^{−1}, gz = zg, zh = hz⟩.

It has exactly p+1 maximal elementary abelian subgroups, E_{i} = ⟨z, g^{i}h⟩, i = 1, …, p and E_{p+1} = ⟨z, g⟩, all of which are normal. Consequently, T (G) has rank at most p + 1.

On the other hand, the cohomology ring H^{∗}(G, k) was computed several years ago by Ian Leary [18] (see [4] for a report). The main result that we want to know is there exist an elements ζ ∈ H^{2p}(G, k) such that res_{G,Z}(ζ) ≠ 0 where Z = ⟨z⟩ is the center of G. It is easy to construct ζ as the norm of an element of degree 2 in any of the E_{i}’s, or alternatively, it is easy to see that ζ exists from the spectral sequence of the extension of G/Z by Z. Hence, by Corollary 4.6 there is an endotrivial module N_{i} of type 2pv_{i} for any i = 1, …, p + 1.

Proposition 6.3. Let G be an extraspecial group E_{p^{3}} of order p^{3} and exponent p for p odd. Then T (G) is a free group of rank of rank p+1 and is generated by the classes of Ω(k) and N_{1}, …, N_{p}. The image of the injective map Type : T (G) ⟶ ℤ^{p+1} is the set of all tuples (a_{1}, …, a_{p+1}) such that a_{i} − a_{j} ∈ 2pℤ for all i and j.

Proof. Clearly, the elements Ω(k) and N_{1}, …, N_{p} generate a subgroup S of T (G) of the correct rank. It remains only to show that there are no other elements outside of this group. For the purposes of the proof we consider S to be a subgroup of ℤ^{p+1} by identifying each element with its type.

If α : G ⟶ G is an automorphism of G, then α permutes the elementary abelian subgroups of G. Hence, α(E_{i}) = E_{σ(i)} where σ is some element in the symmetric group S_{p+1}. Now α has an action on T (G). That is, if M is an endotrivial module of type ∑_{i=1}^{p+1} a_{i}v_{i} then M^{α} is an endotrivial module of type ∑_{i=1}^{p+1} a_{i}v_{σ(i)}. In particular, the class of N_{i}^{α} is the class of N_{σ(i)}. It is easy to see that the subgroup W = 2p(ℤ^{p+1}) is a subset of S and is invariant under any automorphism α.

Notice that if M is an endotrivial module of type (a_{1}, …, a_{p+1}) then a_{i} − a_{j} is always an even integer. The reason is that M_{Z} ≅ Ω^{e}(k)⊕(proj) where a_{i} ≡ e mod 2 for all i. Hence we need only consider T̂ = T (G) ∩ U where U = (2ℤ)^{p+1}. Now U/W ≅ (ℤ/pℤ)^{p+1} has an induced action of the automorphism group of G which is isomorphic to SL(2, p). As such it is the direct sum of two submodules. One is the set of all multiples of (2, 2, …, 2) + W and the other is the set of all vectors (2a_{1}, … 2a_{p+1}) + W such that ∑ a_{i} is a multiple of p. It is easy to check that both of these are irreducible modules for the automorphism group. Specifically, the first is the trivial module and the second is the Steinberg module. Consequently, if T̂/W has an element that is not in the form (2a, 2a, …, 2a) + W then there exits an endotrivial module of type (2, 0, 0, …, 0). The rest of the proof is aimed at showing that this can not happen, and hence, that S = T (G).

So suppose that there exists an endotrivial module M = M_{1} of type (2, 0, …, 0) = 2v_{1}. Then applying the automorphism group we can get an endotrivial module M_{i} of type 2v_{i} for every i. Because M_{E_{i}} ≅ k ⊕ (proj), we have that V_{\overline{E}_{i}}(\overline{M}) = \{0\} for all i ≠ 1. Therefore, V\overline{G}(\overline{M}) ⊂ V\overline{G}(k) ≅ k^{2} is a closed subvariety of dimension one and hence is a union of a finite number of lines that intersect in the origin of k^{2}. One of those lines is res_{\overline{G},\overline{E}_{1}}^{∗}(V_{\overline{E}_{1}}(k)). If there were another line, then we could employ the methods of Theorem 4.2 to obtain an endotrivial module N with exactly the same restrictions to the E_{i}’s and hence the exactly the same type. As this is not possible, we must have that V\overline{G}(\overline{M}) = res_{\overline{G},\overline{E}_{1}}^{∗}(V_{\overline{E}_{1}}(k)). Likewise we have that V\overline{G}(\overline{M}_{i}) = res_{\overline{G},\overline{E}_{i}}^{∗}(V_{\overline{E}_{i}}(k)). Therefore the tensor product N = M_{1} ⊗ ⋯ ⊗ M_{p+1} is an endotrivial module of type (2, 2, …, 2), hence N ≅ Ω^{2}(k) ⊕ (proj) . From Theorem 4.4 we have that

V\overline{G}(\overline{N}) = V\overline{G}(\overline{Ω^{2}(k)}) = ∪_{i=1}^{p+1} res_{\overline{G},\overline{E}_{i}}^{∗}(V_{\overline{E}_{i}}(k)).

Finally we consider the spectral sequence of the group extension,

E_{2}^{r,s} = H^{r}(G/Z, H^{s}(Z, k)) ⇒ H^{r+s}(G, k).

From Theorem 2.3 we have that the variety of \overline{Ω^{2}(k)} is the variety of the partial inflation onto the E_{3} page of the spectral sequence. That is, it is variety of the ring E_{3}^{∗,0}. The spectral sequence is well understood (see [4]). Because \overline{G} is elementary abelian of order p^{2}, its cohomology ring has the form H^{∗}(\overline{G}, k) ≅ k[ζ_{1}, ζ_{2}] ⊗ Λ(η_{1}, η_{2}) where Λ is an exterior algebra, η_{1}, η_{2} are in degree 1, and ζ_{1}, ζ_{2} are in degree 2. The kernel of the partial inflation map H^{∗}(\overline{G}, k) ⟶ H^{∗}(G, k) is generated only by the extension class η_{1}η_{2}, which is nilpotent. Hence V\overline{G}(\overline{Ω^{2}(k)}) = k^{2}. This contradicts our assumption of the existence of an endotrivial module of type 2v_{1}. So the proof is complete. □

The almost extra special group AE_{p^{4}} of order p^{4}, for p odd. There is only one isomorphism class of (almost) extraspecial groups of order p^{4}, and it is almost extraspecial. It can be presented as

G ≅ ⟨g, h, w|g^{p} = h^{p} = w^{p^{2}} = 1, w^{p} = ghg^{−1}h^{−1}, gw = wg, wh = hw⟩.

It is the central product of a group of type E_{p^{3}} and a cyclic group of order p^{2}. The center of G is generated by w while the commutator subgroup is Z = ⟨z⟩, for z = w^{p}. The important thing to notice in this case is that the subgroup H = ⟨g, h⟩ is an extraspecial group of order p^{3} and exponent p. Moreover, H contains all of the elements of order p in G. So again, there are p + 1 maximal elementary abelian subgroup all having order p^{2}.

We also know that there is an element ζ ∈ H^{2p}(G, k) such that res_{G,Z}(ζ) is not nilpotent. So there exist endotrivial modules N_{1}, …, N_{p+1} such that N_{i} has type 2pv_{i}. Moreover the restriction of N_{i} to H must be the module N_{i} of Proposition 6.3 plus a projective module. So we have the following.

Proposition 6.4. Let G be an almost extraspecial group AE_{p^{4}} of order p^{4} for p odd, and let H be the maximal subgroup generated by the elements of order p. The restriction map T (G) ⟶ T (H) is an isomorphism and the classes of the endotrivial modules Ω(k) and N_{1}, …, N_{p} generate T (G).

Dihedral group D_{8} of order 8. This case is well known and is documented in [7] as well as other places. We add only that there is an element ζ ∈ H^{2}(G, k) whose restriction to the center is not zero. Hence, we have the following.

Proposition 6.5. Let G be a dihedral group D_{8} of order 8. Then T (G) ≅ ℤ^{2} is generated by Ω(k) and by an endotrivial module of type (2, 0).

Almost extraspecial group AE_{16} of order 16. Let G be the group given by the presentation

G = ⟨g, h, w|g^{2} = h^{2} = (gh)^{2} = w^{4} = 1, gw = wg, hw = wh⟩.

Let z = w^{2}. Then G has three maximal elementary abelian subgroups ⟨g, z⟩, ⟨h, z⟩ and ⟨ghw, z⟩, and any pair of the elementary abelian subgroups generates a dihedral subgroup.

The cohomology ring of G is well known [22] (see also [4]). It has the form

H^{∗}(G, k) = k[z, y, x, w]/(zy + x^{2}, z^{2}y + zy^{2})

where the elements z, y, x, w have degrees 1, 1, 1, 4, respectively. Moreover the restriction of w to the subgroups Z is not zero. Hence there exist endotrivial modules of type (4, 0, 0), (0, 4, 0) and (0, 0, 4). Note that if (a_{1}, a_{2}, a_{3}) is the type of an endotrivial module then a_{i} − a_{j} must be even. Also, if there is an endotrivial module of type (2, 2, 0) then there is one of type (0, 0, 2) = (2, 2, 2) − (2, 2, 0). However the partial inflation of H^{∗}(\overline{G}, k) onto E_{3}^{∗,0} is k[z, y, x]/(zy + x^{2}). Hence the variety of \overline{Ω^{2}(k)} is irreducible and by the arguments of Proposition 6.3, there is no endotrivial module of type (2, 0, 0). Therefore we have the following.

Proposition 6.6. Let G be an almost extraspecial group AE_{16} of order 16. Then T (G) is generated by the classes of Ω(k) and endotrivial modules of type (4, 0, 0) and (0, 4, 0).

Extraspecial groups of order 32. There are (up to isomorphism) two extraspecial groups of order 32. One of them is a central product of two dihedral groups. For this group, G ≅ D_{8} ∗ D_{8}, we have that T (G) ≅ ℤ because all of the maximal elementary abelian subgroups of G have rank 3. Hence this case matches the conclusion of Proposition 6.1.

Proposition 6.7. Suppose that G ≅ D_{8} ∗ D_{8}. Then T (G) ≅ ℤ is generated by the class of Ω(k).

Hence we need only investigate the group of endotrivial modules for the central produce G ≅ D_{8} ∗ Q_{8} of a dihedral group of order 8 and a quaternion group of order 8. The group is presented as

G = ⟨e, f, g, h|e^{2} = f^{2} = g^{4} = 1, h^{2} = (gh)^{2} = g^{2} = (ef)^{2},
eg = ge, eh = he, fg = gf, fh = hf⟩.

Hence ⟨e, f⟩ is a dihedral group, ⟨g, h⟩ is a quaternion group, and the centers of the two subgroups coincide. Let z = g^{2} be the generator of the center. The group G has five maximal elementary abelian subgroups: ⟨e, z⟩, ⟨f, z⟩, ⟨efg, z⟩, ⟨efh, z⟩, and ⟨efgh, z⟩, every one of which is normal. The centralizer of each of these groups is an almost extraspecial group of order 16, and all of the maximal elementary abelian subgroups are in the same orbit of the action of the automorphism group of G.

The cohomology ring of G has the form

H^{∗}(G, k) = k[z, y, x, w, v]/I,

where

I = (zy + x^{2} + xw + w^{2}, z^{2}y + zy^{2} + x^{2}w + xw^{2}, z^{4}y + zy^{4} + x^{4}w + xw^{4})

and where the degrees of z, y, x, w, v are 1, 1, 1, 1, 8, respectively. The element v in degree 8 has nonzero restriction to the center, and hence there must exist endotrivial modules of type (8, 0, 0, 0, 0), (0, 8, 0, 0, 0), . . .. Because the centralizers are almost extraspecial, if (a_{1}, a_{2}, a_{3}, a_{4}, a_{5}) is the type of an endotrivial module then a_{i} − a_{j} is a multiple of 4. If there exists an endotrivial of type (4, 4, 4, 4, 0) then there is one of type (4, 0, 0, 0, 0) and if there is one of type (4, 4, 4, 0, 0) then there is one of type (0, 0, 0, 4, 4) and one of type (0, 4, 4, 4, 4). So the issue of the generators of T (G) comes down to whether the variety V\overline{G}(\overline{Ω^{4}(k)}) can be written as the union of closed subvarieties which intersect only in {0}. This variety is the maximal ideal spectrum of the ring

k[z, y, x, w]/(zy + x^{2} + xw + w^{2}, z^{2}y + zy^{2} + x^{2}w + xw^{2})

which is the partial inflation onto the E_{5} page of the spectral sequence of the extension of \overline{G} by Z (see [22] or the survey [4]). Hence the variety is irreducible. Therefore we have the following.

Proposition 6.8. Let G be the central product D_{8} ∗ Q_{8} of a dihedral group of order 8 and a quaternion group of order 8. There exist endotrivial modules N_{1}, …, N_{5} of type 8v_{i} for all i = 1, …, 5. Then T (G) is a free abelian group of rank 5 and is generated by the classes of Ω(k) and N_{1}, …, N_{4}. The image of the injective map Type : T (G) ⟶ ℤ^{5} is the set of all tuples (a_{1}, …, a_{5}) such that a_{i} − a_{j} ∈ 8ℤ for all i and j.

We end the section with a general observation that will be useful later. It is really an extension of the above proofs.

Proposition 6.9. Let G be one of the groups E_{p^{3}}, AE_{p^{4}}, D_{8}, AE_{16} or D_{8} ∗ Q_{8}. If G is E_{p^{3}} or AE_{p^{4}} for p odd, then let a = 2p. If G is one of the groups D_{8}, AE_{16} or D_{8} ∗ Q_{8} then let a = |G|/4. Then

V\overline{G}(\overline{Ω^{a}(k)}) = ∪_{F} res_{\overline{G},\overline{F}} (V\overline{F} (k))

where the intersection is over the maximal elementary abelian subgroups F of G.

Proof. Suppose for example that G ≅ E_{p^{3}} an extraspecial group of order p^{3} and exponent p. Then, the variety of \overline{Ω^{a}(k)} is the variety of the partial inflation onto the E_{2p+1} page of the spectral sequence of the group extension. But in this case, as in the cases of all of the other groups under consideration, this page is also the E_{∞} page. The variety of the kernel of the inflation is the zero set of the polynomial ζ_{1}^{p}ζ_{2}−ζ_{1}ζ_{2}^{p} in the polynomial ring k[ζ_{1}, ζ_{2}] (see [4]). It consists of the p+1 𝔽_{p}-rational lines in k^{2} which correspond precisely to the p + 1 maximal subgroups.

For the group G ≅ AE_{p^{4}} the variety is the zero set of the polynomials ζ_{3} and ζ_{1}^{p}ζ_{2} − ζ_{1}ζ_{2}^{p} in k[ζ_{1}, ζ_{2}, ζ_{3}] and we get the same result. For the dihedral group D_{8}, the variety is determined by the polynomial ζ_{1}ζ_{2} in k[ζ_{1}, ζ_{2}], while for AE_{16} it is given by ζ_{1}ζ_{2} + ζ_{3}^{2} and ζ_{1}^{2}ζ_{2} + ζ_{1}ζ_{2}^{2} in k[ζ_{1}, ζ_{2}, ζ_{3}]. Finally, in D_{8} ∗ Q_{8} the variety is the zero set of the polynomials ζ_{1}ζ_{2} + ζ_{3}^{2} + ζ_{3}ζ_{4} + ζ_{4}^{2}, ζ_{1}^{2}ζ_{2} + ζ_{1}ζ_{2}^{2} + ζ_{3}^{2}ζ_{4} + ζ_{3}ζ_{4}^{2} and ζ_{1}^{4}ζ_{2} + ζ_{1}ζ_{2}^{4} + ζ_{3}^{4}ζ_{4} + ζ_{3}ζ_{4}^{4}. In every case the result is as stated in the proposition. □

Remark 6.10. We should note that the module \overline{Ω^{a}(k)} is the same as that of \overline{L} = L/(z − 1)L where L = L_{ζ} for a suitably chosen ζ ∈ H^{a}(G, k) with res_{G,Z}(ζ) ≠ 0 as in Theorem 4.5. Hence, the proposition given above could have been proved by a method similar to the one used in the proof of Lemma 9.4 of [8]. The argument can be sketched as follows. Suppose that M is any k\overline{G}-module with the property that V\overline{G}(M) ∩ res_{\overline{G},\overline{F}} (V\overline{F} (k)) = \{0\}. Then as a kG-module it can be seen that V_{G}(M) = res_{G,Z}(V_{Z}(k)) is the line corresponding to Z. The proof of this last fact uses Quillen’s Theorem 2.1 and an analysis of V_{F} (M). But L is free on restriction to Z and hence its variety V_{G}(L) avoids the line res_{G,Z}(V_{Z}(k)). The consequence is that L ⊗ M must be a projective G-module. But the only way this can happen is if \overline{L} ⊗ M = \overline{L ⊗ M} is a projective k\overline{G}-module. So V\overline{G}(\overline{L}) ∩ V\overline{G}(M) = \{0\}. As this must happen for any such M, we get a proof of Proposition 6.9

7. Endotrivial modules for p-groups

In this section, we prove the general theorem on the generators of the group of endotrivial modules. The proof relies heavily on the results of the last section. That is, basically, the problem of finding the generators of T (G) reduces to the problem of finding the generators of T (E) for all extraspecial and almost extraspecial subgroups E of G. Indeed, the proof is reasonably easy, given what has been demonstrated in the earlier sections.

For the record, we begin by listing the results that we know about T (G) for G a p-group.

Theorem 7.1. Let G be a p-group.

(1) If G is cyclic of order 2, then T (G) has order 1.

(2) If G is cyclic with |G| > 2, then T (G) ≅ ℤ/2ℤ.

(3) If G is a quaternion 2-group, then T (G) ≅ ℤ/4ℤ ⊕ ℤ/2ℤ.

(4) If G is a semi-dihedral 2-group, then T (G) ≅ ℤ ⊕ ℤ/2ℤ.

(5) Suppose that G is not one of the above. If G has only one conjugacy class of maximal elementary abelian subgroups or if all maximal elementary abelian subgroups have rank at least three, then T (G) ≅ ℤ.

In (2) and (5), T (G) is generated by the class of Ω(k). In cases (3) and (4), T (G) is generated by Ω(k) and a module constructed as in Section 5.

Proof. The first four cases are treated in detail in [7]. The last case is a consequence of the theorems in [1] and [8] as noted in Proposition 3.4. □

The next theorem describes the generators for T (G) for all other p-groups. We use the notation of Proposition 3.5. Note that dihedral 2-groups are not included in the hypothesis of Proposition 3.5. If G is a dihedral 2-group, then let E_{1} and E_{2} be representatives of two conjugacy classes of maximal elementary abelian subgroups. Both E_{1} and E_{2} have rank 2. If the order of G is 16 or more, then neither E_{1} nor E_{2} is normal (as contrasted to the situation in 3.5). However, the special case will not affect the conclusion of the theorem.

Theorem 7.2. Let G be a p-group which has at least one maximal elementary abelian subgroup of rank 2 and which has more than one class of maximal elementary abelian subgroups. Suppose that Z is the unique normal subgroup of order p in G. For each i = 1, …, n − 1, there exists an endotrivial module N_{i} of type a_{i}v_{i} where a_{i} is determined by the following algorithm.

(1) If p is odd, then a_{i} = 2p.

(2) If p = 2, then let \overline{A}_{i} ⊆ \overline{G} be a maximal elementary abelian subgroup of largest

order that contains \overline{E}_{i} = E_{i}/Z. Then a_{i} = |\overline{A}_{i}|/2. The abelian group T (G) is freely generated by Ω(k) and N_{1}, …, N_{n−1}.

Proof. Generators for T (G) are described in detail in [7] in the case that the group G is a dihedral group. Generators can also be constructed using the methods of Proposition 6.5 or by the methods of Theorem 4.2. In the dihedral case, T (G) is generated by the class of Ω(k) and the class of an endotrivial module of type (2, 0). Notice here that for either i = 1 or i = 2, the order of \overline{A}_{i} is 4, since \overline{G} is either a fours group (if |G| = 8) or a dihedral group. Hence the conclusion of the theorem holds in the dihedral case.

For the remainder of the proof we assume that G is not dihedral. So the conclusions of Proposition 3.5 are valid. We claim the following.

Claim: For every i = 1, …, n − 1, there is a decomposition of the variety

V\overline{G}(\overline{Ω^{a_{i}}(k)}) = V_{i} ∪ V′_{i}

such that V_{i} ∩ V′_{i} = {0} and

V_{i} = res_{\overline{G},\overline{E}_{i}}(V_{\overline{E}_{i}}(k)) and res_{\overline{G},\overline{E}_{j}}(V_{\overline{E}_{j}}(k)) ⊆ V′_{i}

for all j ≠ i.

First we should notice that the claim proves the existence of the endotrivial module N_{i} that is asserted in the theorem. That is, it follows from Corollary 4.3 that there is an endotrivial module N_{i} such that (N_{i})_{E_{i}} ≅ Ω^{a_{i}}(k_{E_{i}}) ⊕ (proj) and for j ≠ i, (N_{i})_{E_{j}} ≅ k ⊕ (proj). This is exactly what is claimed for N_{i} in the theorem.

Next we prove the claim. Let \overline{A}_{i} be an elementary abelian subgroup of \overline{G} such that \overline{A}_{i} has maximal order among all of the elementary abelian subgroups that contain \overline{E}_{i}. Let A_{i} ⊆ G be subgroup such that Z ⊆ A_{i} and A_{i}/Z = \overline{A}_{i}. Because E_{n} is normal in G, \overline{E}_{n} = E_{n}/Z is in the center of \overline{G}. Hence, E_{n} ⊆ A_{i}. It follows that A_{i} must be an extraspecial or almost extraspecial group which contains at least two classes of maximal elementary abelian subgroups and at least one class of maximal elementary abelian subgroups of rank 2. Therefore, A_{i} is isomorphic to one of the groups E_{p^{3}}, AE_{p^{4}}, D_{8}, AE_{16}, or D_{8} ∗ Q_{8}, one of the groups considered in the last section.

We have that

V_{\overline{A}_{i}}((\overline{Ω^{a_{i}}(k)})_{A_{i}}) = V_{\overline{A}_{i}}(\overline{Ω^{a_{i}}(k_{A_{i}})})

since (Ω^{a_{i}}(k))_{A_{i}} ≅ Ω^{a_{i}}(k_{A_{i}}) ⊕ (proj). In every one of the cases of the groups E_{p^{3}}, AE_{p^{4}}, D_{8}, AE_{16}, we have that

V_{\overline{A}_{i}}(\overline{Ω^{a_{i}}(k)}) = ∪_{F} res_{\overline{G},\overline{F}} (V\overline{F} (k))

where F runs through the maximal elementary abelian subgroups of A_{i}, by Proposition 6.9. Now, we know that that

V\overline{G}(\overline{Ω^{a_{i}}(k)}) = ∪\overline{A} res_{\overline{G},\overline{A}}(V\overline{A}(\overline{Ω^{a_{i}}(k)})),

where the union is over the maximal elementary abelian subgroups \overline{A} of \overline{G}. Hence, V_{i} = res_{\overline{G},\overline{E}_{i}}(V_{\overline{E}_{i}}(k)) is a component of V\overline{G}(\overline{Ω^{a_{i}}(k)}). This is because if \overline{E}_{i} ∩ \overline{E} = \{1\} for an elementary abelian subgroup \overline{E}, then V_{i} ∩ res_{\overline{G},\overline{E}}^{∗}(V\overline{E}(k)) = \{0\}. This proves the claim.

We have demonstrated the existence of the modules N_{1}, …, N_{n−1} as asserted in the theorem. It remains only to show that there are no endotrivial modules that are not in the subgroup generated by the classes of Ω(k) and N_{1}, …, N_{n−1}. For this purpose let M be any endotrivial module, suppose that its type is (b_{1}, …, b_{n}). By replacing M by Ω^{−b_{n}}(k) ⊗ M we may assume that b_{n} = 0. Now for each i, consider the restriction of M to A_{i}. Remember that A_{i} contains both E_{i} and E_{n}, and A_{i} is isomorphic to one of the groups E_{p^{3}}, AE_{p^{4}}, D_{8}, AE_{16}, or D_{8} ∗Q_{8}. Then by Propositions 6.3, 6.4, 6.5, 6.6 and 6.8, b_{i} = u_{i}a_{i} for some integer u_{i}. Now replacing M by M ⊗ (N_{i}^{∗})^{⊗u_{i}} we may assume that b_{i} = 0. Hence we can assume that M has type (0, …, 0), and the restriction of M to any elementary abelian subgroup is a trivial module plus a projective. By Theorem 3.3 we are done. □

8. Examples and questions

There are a number of questions that are raised by the constructions of this paper. One of them certainly is the question of how we compare the different constructions. As an example consider the following. Let G be an extraspecial group of order p^{3} and exponent p. Then an indecomposable endotrivial module M of type (2p, 0, …, 0) can be constructed as in 6.3. In Alperin’s construction [1], we begin by taking a cyclic noncentral subgroup C = ⟨x⟩ in the first elementary abelian subgroup E_{1}. Then we take the second relative syzygy of k relative to the subgroup C. This is the module N = Ω_{G/C}^{2}(k). We know that this module has type (2 − 2p, 2, , 2, …, 2) (see [9]). Hence Ω^{−2}(N) is indecomposable and has type (−2p, 0, …, 0). It follows that M ≅ N^{∗}. However, we only know that these two modules are isomorphic because they are both indecomposable endotrivial modules of the same type. A direct connection between the constructions is not known.

Question 8.1. Is it possible to find a direct connection between the construction of this paper and the relative syzygies of Alperin?

In this paper we have only constructed generators for the group of endotrivial modules. The question might be asked as to whether all endotrivial module can be constructed by carving up copies of Ω^{n}(k) for some n, in one way or another. The answer is yes, but the procedure is often not easy to envision. As an example, suppose that G ≅ E_{p^{3}} is an extraspecial group of order p^{3} and exponent p. Then in H^{2p}(G, k) there is an element ζ whose restriction to the center in not zero. Consequently, as in the proof of Proposition 6.3, the generators N_{1}, …, N_{p+1} can be constructed as pushouts of the exact sequence

\xymatrix{0 \ar[r] & L_{ζ} \ar[r] & Ω^{2p}(k) \ar[r]^{ζ′} & k \ar[r] & 0}

where ζ′ is a cocycle representing ζ. Then any of the modules Ω^{n}(N_{1}), …, Ω^{n}(N_{p+1}) can be constructed as pushouts of the translated sequence

\xymatrix{0 \ar[r] & Ω^{n}(L_{ζ}) \ar[r] & Ω^{2p+n}(k) ⊕ (proj) \ar[r]^{Ω^{n}(ζ′)} & Ω^{n}(k) \ar[r] & 0.}

Similarly, products of such elements can be constructed in this way. It is also possible to translate the construction in Theorem 4.2. The problem is to find the correct filtration to replace the one given by the powers of (z − 1).

The final item of the section is an example that shows that the method of Theorem 4.5 is not sufficient to produce a full set of generators for the group T (G) of endotrivial modules. The example that we use is the group of order 64 with Hall-Senior number 257 and small-group-library number 152. The group would be found under the small group number in MAGMA or GAP. It’s complete cohomology ring can be found in the appendix of [10] and on the author’s web site (http://www.math.uga.edu/˜ lvalero/cohointro.html).

Example 8.2. Let G be the group whose power commutator presentation is given as

G = ⟨g_{1}, g_{2}, g_{3}, g_{4}, g_{5}, g_{6}| g_{1}^{2} = g_{4}g_{6}, g_{3}^{2} = g_{4}g_{6}, g_{4}^{2} = g_{6},
g_{1}^{−1}g_{2}g_{1} = g_{2}g_{4}, g_{2}^{−1}g_{3}g_{2} = g_{3}g_{5},
g_{2}^{−1}g_{4}g_{2} = g_{4}g_{6}, g_{3}^{−1}g_{5}g_{3} = g_{5}g_{6}⟩

Notice that in this presentation only the nontrivial relations are given. For example, there are no relations for g_{2}^{2} or for g_{1}^{−1}g_{3}g_{1}. Thus it is to be assumed that g_{2}^{2} = 1 and g_{1}^{−1}g_{3}g_{1} = g_{3}. Notice that G has two conjugacy classes of elementary abelian subgroups represented by E_{1} = ⟨g_{1}g_{3}g_{4}, g_{6}⟩ and ⟨g_{2}g_{4}, g_{5}, g_{6}⟩. The first of these has rank 2 and the second has rank 3. If we let E_{2} = ⟨g_{5}, g_{6}⟩, then E_{1} and E_{2} satisfy the conditions of Theorem 3.5.

Now let \overline{G} = G/Z where Z = ⟨g_{6}⟩ denotes the center of G. The elementary abelian subgroup of maximal order that contains \overline{E}_{1} = E_{1}/Z is F = ⟨\overline{g}_{1}\overline{g_{3}}, \overline{g_{4}}, \overline{g_{5}}⟩. This is a maximal elementary abelian subgroup in \overline{G} because it is a maximal elementary abelian subgroup in its centralizer which is ⟨\overline{g}_{1}, F ⟩. Hence Theorem 7.2 says that T (G) is a free group of rank 2 with generators Ω(k) and an endotrivial module of type (4, 0).

The group has cohomology ring H^{∗}(G, k) ≅ k[z, y, x, w, v, u, t, s, r, q]/I, where the degree of the variables are 1, 1, 1, 2, 4, 5, 5, 5, 8, 8 and the ideal I is generated by 19 elements. The generators for I can be found on page 740 of [10]. The actual details are not as important to us as the inflation map from \overline{G} the quotient by the center of the group. The one thing that it is important to know about H^{∗}(G, k) is that only the generator q in degree 8 has a nontrivial restriction to the center of G. Consequently, Corollary 4.6 only guarantees an endotrivial module of type (8, 0) and is not sufficient to construct one of type (4, 0).

To understand what is happening here, it is necessary to consider the cohomology ring of the quotient group \overline{G} and its inflation into the cohomology of G. The group \overline{G} has Hall-Senior number 36 and small-group-library number 28, and its cohomology is found on page 380 of [10]. The cohomology ring of \overline{G} is relatively simple and is given as H^{∗}(\overline{G}, k) ≅ k[z, y, x, w, v]/J where J is the ideal generated by the elements

zy + y^{2}, zx.

The kernel of the inflation of G/Z is generated by the three elements

y^{2} + w + v, yw + yv + xv, xv^{2}

which are in degrees 2, 3 and 5. The E_{2} term of the spectral sequence has the form E_{2}^{r,s} = H^{r}(\overline{G}, k) ⊗ H^{s}(Z, k), where H^{s}(Z, k) = k[u] is a polynomial ring. The fact that there is no element of H^{∗}(G, k) that restricts to Z nontrivially before degree 8 implies that d_{2}(u), d_{3}(u^{2}), and d_{5}(u^{4}) are not zero. The data on the inflation map confirms this precisely. That is, we must have that

d_{2}(u) = y^{2} + w + v, d_{3}(u^{2}) = yw + yv + xv, and d_{5}(u^{4}) = xv^{2}.

Here is the observation of note. The partial inflation of H^{∗}(\overline{G}, k) onto the E_{4} page of the spectral sequence has image k[z, y, x, w, v]/J_{4} where J_{4} is the ideal generated by the elements zy +y^{2}, zx, y^{2} +w+v, and yw+yv +xv. It is contained in the prime ideals P_{1} = ⟨z, y, w, v⟩ and P_{2} = ⟨x, y, w + v⟩. The variety of k[z, y, x, w, v]/P_{1} has dimension 1 while that of k[z, y, x, w, v]/P_{2} is 2. These are the two components of V\overline{G}(\overline{Ω^{4}(k)}) and they explain why \overline{Ω^{4}(k)} decomposes as a direct sum of two submodules. It is all consistent with Theorem 4.2.

The thing to remember is that d_{5}(u^{4}) = xv^{2}. Now xv^{2} is obviously not in the ideal J_{4}. On the other hand,

(xv^{2})^{2} = xv^{3}(xv) ≡ xv^{3}(y(w + v)) ≡ xv^{3}(y(y^{2})) ≡ xv^{3}y(zy) ≡ 0 (modulo J_{4}).

Hence the addition of the element xv^{2} to the kernel of the inflation from \overline{G} to G affects the module theory, but has only a minor impact on the geometry of the cohomology ring.

9. Appendix: Two messy technicalities

In this section we prove two facts that are needed in the proof of Theorem 4.5. They have been separated from the rest of the proof, not only because they are somewhat disjoint from the spirit of the rest of the proof, but also because they may have some independent interest. They are rather technical in nature.

For the first, let R = k[x, y] be a polynomial ring in two (commuting) variables over an algebraically closed field. Assume that x and y both have degree 1.

Lemma 9.1. Suppose that γ_{1} and γ_{2} are homogeneous elements in R of degrees r and s respectively. Assume also that γ_{1} and γ_{2} are relatively prime in the sense of having no nonunit common factors. If γ_{3} is any homogeneous element of R of degree r + s − 1 then there exist (unique) homogeneous elements u and v in R such that γ_{3} = uγ_{1} + vγ_{2}

Proof. Let V_{n} denote the k-space of homogeneous polynomials of degree n. The dimension of V_{n} is the number of distinct monomials of degree n, which is Dim V_{n} = n + 1. In V_{r+s−1}, the subspace W_{1} of all multiples of γ_{1} is γ_{1}V_{s−1} and has dimension s. Likewise the subspace W_{2} of all multiples of γ_{2} in V_{r+s−1} has dimension r. Hence, Dim W_{1} + Dim W_{2} = r + s = Dim V_{r+s−1}. Consequently, we are finished if we can show that W_{1} ∩ W_{2} = {0}. However, this part is clear, since R is a unique factorization domain and γ_{1} and γ_{2} are relatively prime. □

Proposition 9.2. Let E be an elementary abelian p-group of rank 2, and let k be an algebraically closed field of characteristic p. Suppose that we have an element ϑ ∈ H^{n}(G, k) for some n, which we assume to be even in the case that p > 2. Assume that V_{G}(ϑ) = V_{1} ∪ V_{2} where V_{1}, V_{2} are closed subvarieties such that V_{1} ∩ V_{2} = {0}. Then ϑ = ϑ_{1}ϑ_{2} where ϑ_{1} and ϑ_{2} are homogeneous element such that V_{G}(ϑ_{1}) = V_{1} and V_{G}(ϑ_{2}) = V_{2}.

Proof. If p = 2, then H^{∗}(E, k) ≅ k[ζ_{1}, ζ_{2}] is a polynomial ring. In this case the proof is obvious. That is, ϑ factors as a product of linear polynomials each having the form aζ_{1} + bζ_{2} and each defining a line in V_{G}(ϑ) ⊆ V_{G}(k) = k^{2}. Hence, ϑ_{1} is the product of all of the factors whose corresponding line is in V_{1} and ϑ_{2} is the product of the rest of the factors.

Suppose that p > 2. Then H^{∗}(E, k) = k[ζ_{1}, ζ_{2}] ⊗ Λ(η_{1}, η_{2}) where Λ is an exterior algebra, deg(ζ_{i}) = 2 and deg(η_{i}) = 1 for i = 1, 2. In this case it must be that ϑ = ϑ_{1}ϑ_{2} + η_{1}η_{2}ϑ_{3} where ϑ_{1}.ϑ_{2}, ϑ_{3} ∈ k[ζ_{1}, ζ_{2}] and V_{G}(ϑ_{1}) = V_{1}, V_{G}(ϑ_{2}) = V_{2} as in the characteristic 2 case. Now by the Lemma 9.1, there exist homogeneous elements μ, ν ∈ k[ζ_{1}, ζ_{2}] such that ϑ_{3} = μϑ_{1} + νϑ_{2}. Hence,

ϑ = ϑ_{1}ϑ_{2} + η_{1}η_{2}ϑ_{3} = (ϑ_{1} + η_{1}η_{2}ν)(ϑ_{2} + η_{1}η_{2}μ)

as desired. □

The other result that we need in this appendix concerns the uniqueness of splittings and pushouts for certain modules. First a lemma. Assume for the rest of the section that G is a finite group and k is a field of characteristic p. A homomorphism ϕ : M ⟶ N is left split if there is a homomorphism θ : N ⟶ M such that θϕ is the identity on M. Such a homomorphism must be injective.

Lemma 9.3. Suppose that L = L_{1}⊕L_{2} is a kG module such that V_{G}(L_{1})∩V_{G}(L_{2}) = {0}. Let σ, τ : L_{1} ⟶ L be left split homomorphisms. Then there is an automorphism ϕ : L_{1} ⟶ L_{1} such that σϕ−τ factors through a projective module. Moreover, L/σ(L_{1}) ≅ L_{2} ≅ L/τ(L_{1}).

Proof. Write L_{1} as L′_{1} ⊕ P where P is a projective module and L′_{1} has no projective submodules. The first observation is that we can assume that P = {0}. This is because any two maps from P into L differ by a map that factors through a projective module, and since P is also injective, the Krull-Schmidt Theorem implies that L/P is defined up to isomorphism regardless of the embedding. So assume that L_{1} = L′_{1} has no projective submodules. We can also assume that τ is the natural injection of L_{1} into the direct sum L = L_{1} ⊕ L_{2}.

For notation, we write elements of L in the form (u, v) for u ∈ L_{1}, v ∈ L_{2}. Then for any m ∈ L_{1}, σ(m) = (μ(m), ν(m)) where μ : L_{1} ⟶ L_{1} and ν : L_{1} ⟶ L_{2} are homomorphisms. Our first claim is that ν factors through a projective module. The reason is that Hom_{kG}(L_{1}, L_{2}) ≅ Hom_{kG}(k, L_{1}^{∗} ⊗ L_{2}). But L_{1}^{∗} ⊗ L_{2} is a projective module because the varieties of L_{1} and L_{2} intersect trivially. Consequently, any homomorphism from L_{1} to L_{2} factors through a projective module. The map μ must be an isomorphism by the Krull-Schmidt Theorem. That is, no indecomposable direct summand of L_{1} is isomorphic to any indecomposable direct summand of L_{2}. Thus the usual proof of the Krull-Schmidt Theorem implies that the composition of σ with the projection of L onto L_{1} is an isomorphism. Another way of seeing this is to notice that if β is the splitting of σ then Id_{L_{1}} = βμ + βν, but βν factors through a projective module. Hence βμ must be an isomorphism.

From all of this it follows that σμ^{−1} − τ factors through a projective module. The last statement of the lemma is a direct consequence of the Krull-Schmidt Theorem.

□

Theorem 9.4. Let G be a finite group and let k be an algebraically closed field of characteristic p. Suppose that we are given an exact sequence

0 ⟶ L_{1} ⊕ L_{2} ⟶^{ι} M ⟶^{γ} N ⟶ 0

of kG-modules and maps such that V_{G}(L_{1})∩V_{G}(L_{2}) = {0}. Let σ, τ : L_{1} ⟶ L_{1} ⊕L_{2} be two embeddings. Then in the two induced sequences.

0 ⟶ L_{2} ⟶ M/σ(L_{1}) ⟶^{γ_{1}} N ⟶ 0,

0 ⟶ L_{2} ⟶ M/τ(L_{1}) ⟶^{γ_{2}} N ⟶ 0

we have that M/σ(L_{1}) ≅ M/τ(L_{1}) and the two maps γ_{1} and γ_{2} can be assumed to differ by a map that factors through a projective.

Proof. By the last lemma, there exists an automorphism ϕ : L_{1} ⟶ L_{1} such that σϕ − τ = μν where ν : L_{1} ⟶ P and μ : P ⟶ L_{1} ⊕ L_{2} for some projective module P. Then because P is injective, there exist maps θ and ψ such that the diagram

\xymatrix{L_{1} \ar[d]_{ν} \ar[r]^{τ} & L_{1} ⊕ L_{2} \ar[r]^{ι} \ar[dl]_{θ} & M \ar[dll]^{ψ} \\ P}

commutes. Hence we get a commutative diagram

\xymatrix{L_{1} \ar[r]^{τ} \ar[d]^{ϕ} & L_{1} ⊕ L_{2} \ar[r]^{ι} \ar[d]^{1+μθ} & M \ar[d]^{1+ιμψ} \\ L_{1} \ar[r]^{σ} & L_{1} ⊕ L_{2} \ar[r]^{ι} & M.}

Now we factor out the images of L_{1} to get a commutative diagram of induced sequences

[figure: 0; L_{2}; ι_{1}; ω_{1}; M/τ(L_{1}); γ_{1}; ω_{2}; N; 0; 0; L_{2}; ι_{2}; M/σ(L_{1}); γ_{2}; N; 0]

where ω_{1} and ω_{2} are induced by 1 + μθ and 1 + ιμψ respectively. But notice now that γ(1 + ιμψ) = γ. Hence we can assume that γ_{1} = γ_{2}. □

A lot of the proof of the above theorem is simply a part of the verification that the stable module category stmod(kG), of finitely generated kG-modules modulo projectives, is a triangulated category.

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Department of Mathematics, University of Georgia, Athens, Georgia 30602, USA

  E-mail address: jfc@math.uga.edu