A CHARACTERIZATION OF ENDOTRIVIAL MODULES OVER p-GROUPS

JON F. CARLSON^{∗}

Abstract. Let P be a finite p-group, and let k a field of characteristic p > 0. We prove that a finietly generated kG-module M is endotrivial if and only if

Dim(Hom_{kG}(M, M)/PHom_{kG}(M, M)) = 1,

where PHom(−, −) is the set of endomorphisms which factor through projective modules.

Suppose that G is a finite group and k is a field of characteristic p > 0. A kGmodule M is called an endotrivial module if M^{∗} ⊗ M ≅ Hom_{k}(M, M) ≅ k ⊕ P where P is a projective kP -module. Endotrivial modules were first introduced by Dade in [3]. Among other things, Dade showed that if G is an abelian p-group and M an endotrivial kG module then M ≅ Ω^{n}(k) for some n > 0. The group of endotrivial modules is a fundamental part of the system of endopermutation modules that play an important role in local representation theory (e.g. see [6]).

Endotrivial modules also determine equivalences of stable categories of modules modulo projectives. That is, if stmod(kG) is the stable category of kG-modules and M is an endotrivial module then the functor

M⊗_{k} : stmod(kG) ⟶ stmod(kG)

is an equivalence of the triangulated categories. The inverse functor is given by tensoring with the k-dual, M^{∗}.

The purpose of this paper is to give a direct characterization of endotrivial modules over p-groups in terms of their stable endomorphism ring. Specifically we show that a kG-module M is endotrivial if and only if Dim(Hom_{kG}(M, M)/PHom_{kG}(M, M)) = 1, where PHom(−, −) is the set of endomorphisms which factor through projective modules. This is the modular analog of Corollary 3.9 of [5] and the theorem can be used to prove a modular version of Theorem 3.1 of that paper. That is, we can follow the proof of [5] to show that any stable equivalence of Morita type between two p-groups P and Q is induced by tensoring with a module of the form U ⊗_{k} M where M is a kP−kQ-bimodule which induces a Morita equivalence and U is an endotrivial kP -module. An independent proof of this fact is given in [2]

∗ ∗ ∗ ∗ ∗

∗ partially supported by a grant from NSF.

2 JON F. CARLSON^{∗}

For the remainder of the paper, let G be a p-group and k a field of characteristic p > 0. Our main theorem is the following.

Theorem 1. Let M be a finitely generated kG-module and let

M ⊗_{k} M^{∗} ≅ Hom_{k}(M, M) ≅ L ⊕ P

where P is a projective kG-module and L has no projective submodules. Then M is an endotrivial module if and only if L has a simple socle, in which case L ≅ k.

Observe that Hom_{kG}(M, M) is the set of G-fixed points of Hom_{kG}(M, M) and that PHom_{kG}(M, M) is the set of G-fixed points of P in the above notation. Thus we have the characterization that was stated in the introduction.

Before beginning the proof we should notice a few things. First, then only if part of the theorem is obvious so we need only show that M is endotrivial if L has a simple socle. Also M ⊗ M^{∗} is self dual hence so is L. Thus L must also have a simple top.

The proof proceeds in several steps. We should emphasize that kG is a local ring and hence projective kG-modules are free.

Step 1. If p does not divide the dimension of M, then M is endotrivial. For in this case Hom_{k}(M, M) ≅ k ⊕ N where N is the submodule consisting of all matrices of trace zero and k is generated by the identity map. By the Krull-Schmidt Theorem L ≅ k.

So we may assume that p divides DimM. From here on, we assume by induction that the Theorem is true for all group with order less than that of G. If |G| = 1, then the result is obvious.

Step 2. We may assume that |G| > p. For if |G| = p then G is cyclic and all nonprojective indecomposable modules have dimension between 1 and p − 1.

Now let H be a maximal subgroup of G. The restriction of M ⊗ M^{∗} to H has the form

(Hom_{k}(M, M))_{H} = Hom_{k}(M_{H}, M_{H}) ≅ L_{1} ⊕ Q

where Q is a projective (free) kH-modoule. We have three cases to consider (A) L_{1} has a simple socle and L_{1} is not free. (B) L_{1} is free, i.e. L_{1} = (0). (C) L_{1} is not free and the dimension of the socle of L_{1} is greater than 1.

Step 3. Case (A) is impossible because then, by induction on |G|, L_{1} ≅ k_{H} and then p does not divide the dimension of M. So we are done by Step 1.

Step 4. Case (B) is impossible. Because L is a cyclic module and L is not projective, we have have that DimL < |G| = p^{n}. In particular, Dim(M^{∗} ⊗ M) is not divisible by p^{n}, since p^{n} divides DimP. On the other hand Hom_{k}(M_{H}, M_{H}) is a free kH-module. So M is a free kH-module since the identity map on M factors through a kH-projective module. Therefore DimM = t|H| = tp^{n−1} for some t. Now we have a contradiction because Dim(M^{∗} ⊗ M) = t^{2}p^{2n−2} is divisible by p^{n}.

So we are down to Case (C). Step 5. M is relatively kH-projective. Choose an element x ∈ G, x ∉ H. Let N = Hom_{kH}(L, L) be the kH-socle of L_{H}. We claim that as a k⟨x⟩-module N is a cyclic module. This follows from the fact that the k⟨x⟩-socle of N is contained in the kG-socle of L and hence it must have dimension 1.

Now suppose that α is an element in the kG-socle of L. Then α ∈ N and α : M ⟶ M is a kG-homomorphism. Moreover α must be invertible, because M is indecomposable and the nilpotent homomorphisms in Hom_{kG}(M, M) factor through projective modules. That is, the nilpotent homomorphism must be the elements of the socle of P.

By hypothesis, DimM ≥ 2 and there must be an element β ∈ N such that (x − 1)β = α. Here, of course, ((x − 1)β)(m) = xβ(x^{−1}m) − β(m) for m ∈ M. Moreover β is a kH-homomorphism. Let

ψ : (Hom_{kH}(M, M))^{⊗^{p−1}} ⟶ Hom_{kH}(M, M)

be the composition ψ(γ_{1} ⊗ ⋯ ⊗ γ_{p−1}) = γ_{1} ∘ γ_{2} ∘ ⋯ ∘ γ_{p−1}. It is an easy check that ψ is a k⟨x⟩-homormophism.

Let ζ = β ⊗ β ⊗ ⋯ ⊗ β ∈ (Hom_{kH}(M, M))^{⊗^{p−1}}. There are two things to notice about ζ. First (x − 1)^{p−1}ζ = (x − 1)β ⊗ ⋯ ⊗ (x − 1)β = α ⊗ ⋯ ⊗ α. This is a consequence of the usual identity on tensor products that (x − 1)(m ⊗ n) = (x − 1)m ⊗ n + m ⊗ (x − 1)n + (x − 1)m ⊗ (x − 1)n. Then it follows that

ψ((x − 1)^{p−1}ζ) = (x − 1)^{p−1}ψ(ζ) = α^{p−1}.

Hence ψ(ζ) is a kH-homomorphism with the property that

Tr_{H}^{G}(ψ(ζ)) = ∑_{i=0}^{p−1} x^{i}ψ(ζ) = (x − 1)^{p−1}ψ(ζ) = α^{p−1}

is not invertible. Here Tr_{H}^{G} is the relative trace [4]. Because the nilpotent homomorphism in Hom_{kG}(M, M) are all in the image of Tr_{H}^{G} we must have that Hom_{kG}(M, M) = Tr_{H}^{G}(Hom_{kH}(M, M)). In particular the identity map on M is a relative trace and M is relatively kH-projective.

Step 6. Case (C) is not possible. We now know that M ≅ U^{↑G} for some indecomposable kH-module U. That is, we may extend the field if necessary, so that U is absolutely indecomposable and then U^{↑G} is absolutely indecomposable by Green’s Indecomposability Theorem [4].

So M_{H} ≅ U_{0} ⊕ ⋯ ⊕ U_{p−1} where U_{i} = x^{i} ⊗ U. By the Nakayama relations

Hom_{k}(M, M) ≅ M^{∗} ⊗ M ≅ (U^{∗})^{↑G} ⊗ M

≅ ⊕ ∑_{i=0}^{p−1}(U^{∗} ⊗ U_{i})^{↑G}

Clearly U^{∗} ⊗ U_{0} ≅ Hom_{k}(U, U) is not a free kH-module. That is, Hom_{k}(U, U) ≅ V ⊕ Q where Q is a free kH-module and V ≠ (0) has no projective submodules. Then V^{↑G} has no projective summands and it must have a simple socle because it is a direct summand of Hom_{k}(M, M). Therefore U satisfies the conditions of the Theorem and must be an endotrivial module by induction.

The conclusion that we draw from all of this is that the dimension of U is relatively prime to p. The same must also hold for U_{1}, …, U_{p−1} and for U^{∗}. So it can not be the case that U^{∗} ⊗ U_{i} is a projective module for any i. Consequently the dimension of socle of the nonprojective part of Hom_{k}(M, M) is at least p. The contradiction eliminates the possibility of case (C) and the proof of the Theorem is complete.

We should point out that the theorem is not true if G is not a p-group. One example is the 2 dimensional standard module for SL(2, 2^{n}), with n > 1 and p = 2. Here M^{∗} ⊗ M has simple socle and top but is not projective and contains no direct summand isomorphic to k.

Let Êxt_{kG}^{∗}(M, N) denote the Tate cohomology groups. It is known [1] that the almost split sequence for M represents a class in the End(M) socle of the cohomology group Êxt_{kG}^{1}(M, Ω^{2}(M)) ≅ Êxt_{kG}^{−1}(M, M). By Tate duality

(Êxt_{kG}^{−1}(M, M))^{∗} ≅ Êxt_{kG}^{0}(M, M) = Hom_{kG}(M, M)/PHom_{kG}(M, M).

Hence if this group has dimension 1 as a k-space then M must be endotrivial by the Theorem. The converse also holds. Therefore we have the following.

Corollary 1. Suppose that G is a p-group and M is a kG-module. Then M is endotrivial if and only if every nonsplit extension of the form

0 ⟶ Ω^{2}(M) ⟶ B ⟶ M ⟶ 0,

for some B, is almost split.

References

[1] M. Auslander, I. Reiten and S. O. Smalo, “Representation Theory of Artin Algebras”, Cambridge Univ. Press, 1995.

[2] J. F. Carlson and R. Rouquier, Self-equivalences of stable module categories, (submitted)

[3] E. C. Dade, Endo-permutation modules over p-groups, I, II, Ann. Math. 107 (1978), 317–345.

[4] C. Curtis and I. Reiner “Methods of Representation Theory with Applications to Finite Groups and Orders, I”, Wiley Interscience, 1981.

[5] M. Linckelmann, Stable equivalences of Morita type for self-injective algebras and p-groups, Math. Z. 223 (1996), 87–100.

[6] L. Puig, Une correspondance de modules pour les blocs à groupes de défaut abéliens, Geom. Dedicata 37 (1991), 9–43.

Department of Mathemtics, University of Georgia, Athens, GA 30602

  E-mail address: jfc@sloth.math.uga.edu; http://www.math.uga.edu/∼jfc