GLUING REPRESENTATIONS VIA IDEMPOTENT MODULES AND CONSTRUCTING ENDOTRIVIAL MODULES

PAUL BALMER, DAVID J. BENSON, AND JON F. CARLSON

Abstract. Let G be a finite group and k be a field of characteristic p. We show how to glue Rickard idempotent modules for a pair of open subsets of the cohomology variety along an automorphism for their intersection. The result is an endotrivial module. An interesting aspect of the construction is that we end up constructing finite dimensional endotrivial modules using infinite dimensional Rickard idempotent modules. We prove that a slight modification of this construction produces a subgroup of finite index in the group of endotrivial modules. More generally, we also show how to glue any pair of kG-modules.

Contents

Introduction 1 1. Stable categories and support varieties 2 2. Endotrivial modules 4 3. U-isomorphisms and localisation 6 4. Gluing finitely generated modules 9 5. Rickard’s idempotent modules 11 6. Gluing arbitrary modules 14 7. A gluing construction of endotrivial modules 18 8. First example: Rank two 21 9. Second example: The dihedral group D_{8} 23 10. Gluing along endomorphisms 24 11. The rank of the group of endotrivial modules 28 References 31

Introduction

Suppose that G is a finite group and k is a field of characteristic p > 0. The endotrivial kG-modules are the elements of the Picard group of invertible objects in the stable category of kG-modules. They form an important subgroup of the group of all self equivalences of the stable category. In addition, endotrivial modules play a significant role in the block theory and modular representation theory of G. There is now a complete classification of endotrivial modules in the case that the group G is a p-group. For the most part, the endotrivial modules over a p-group are all Heller shifts of the trivial module. In the cases where there are unusual endotrivial modules, various constructions for the exotic examples exist.

Date: January 30, 2007.

1991 Mathematics Subject Classification. 20C20.

In this paper we investigate another construction of endotrivial modules, which is not limited to p-groups. The method uses infinite dimensional techniques, based on a construction of Balmer and Favi [3], to produce finite dimensional endotrivial modules. We analyse the new method by comparing it to one of the two constructions of Carlson [12]. We show that the “cohomological pushout” method of [12] is a “gluing” in the sense of [3].

Geometrically, the “gluing” construction can be interpreted as taking the patching data for an invertible sheaf over the variety associated to the cohomology ring H^{∗}(G, k) and translating these data into patching data for infinite dimensional Rickard idempotent modules. The results are finite dimensional endotrivial modules. The significance of the idempotent modules is that each one we use is naturally isomorphic to the trivial module in a particular localisation associated to the patching data of the stable category. Note that there are problems with trying to do the patching with more than two open sets, but fortunately using two open sets already produces enough modules to generate a subgroup of finite index in the group of endotrivial modules, provided we are allowed to “glue along endomorphisms”.

In Sections 1 and 2, we present background material, definitions and notation on stable categories and on endotrivial modules. One notable result is that an invertible object in the stable category of all (not just finitely generated) kG-modules must be the direct sum of a finitely generated endotrivial module and a projective module. In Section 3, we give the fundamentals of localisations and verify that the inclusion functor of a localised category of finitely generated modules is fully faithful into the corresponding localised category of all kG-modules. The gluing process is introduced and defined in Section 4. There it is shown that the “cohomological pushouts” of [12] are in fact gluings. In the two sections that follow, we recall some facts about Rickard’s idempotent modules and show how they can be used to glue not necessarily finitely generated modules, via an appropriate weak pullback. In Section 7, we unfold this construction when the two glued modules are both the trivial module k, and we verify that the modules so constructed are endotrivial and coincide with the collection of modules which can be constructed by the pushout method of [12].

Sections 8 and 9 include some explicit calculations. The main question is what modules actually are constructed from a gluing or weak pullback along a chosen automorphism of an idempotent module. We answer the question first for groups of rank 2, and then illustrate the process with some specific automorphisms in the case of the Klein four group and the dihedral group of order eight. In Section 10, we propose some variation on the construction of Section 7, using endomorphisms instead of automorphisms. In the final section, we prove directly that the endotrivial modules constructed by this generalized gluing method give rise to a subgroup of finite index in the group of endotrivial modules.

1. Stable categories and support varieties

Throughout the paper we let G denote a finite group and k a field of characteristic p > 0. If M is a kG-module, let Ω(M) denote the kernel of a projective cover P ↠ M of M.

GLUING kG-MODULES 3

Likewise, let Ω^{−1}(M) be the cokernel of an injective hull M ↣ Q. Recall that a kG-module is projective if and only if it is injective. So for any n ∈ ℤ, we write Ω^{n}(M) for the appropriate iteration of Ω or Ω^{−1} applied to M.

Recall that the stable module category StMod (kG) is the additive quotient of the category Mod (kG) of all (left) kG-modules by the subcategory of projective modules. Explicitly, the category StMod (kG) has as objects the kG-modules, not necessarily finitely generated, and as morphisms the groups:

\underline{Hom}_{kG}(M, N) := \frac{Hom_{kG}(M, N)}{PHom_{kG}(M, N)}

where PHom_{kG}(M, N) is the subgroup of those kG-homomorphisms which factor through a projective module (which we can always choose to be the injective envelope of M or projective cover of N). The category StMod (kG) is triangulated with suspension functor Σ = Ω^{−1}, the inverse of the Heller loop functor. The category StMod (kG) carries a tensor structure given by M ⊗ N = M ⊗_{k} N with G acting diagonally.

We assume the reader has some minimal knowledge of triangulated categories in general, and at least of StMod (kG), as can be acquired in [6, 10, 25].

We denote by stmod (kG) the full subcategory of StMod (kG) on those M which are isomorphic in StMod (kG) to a finitely generated module. In fact, stmod (kG) is precisely the subcategory of “compact” objects of StMod (kG). This category stmod (kG) is equivalent to its more usual description as the additive quotient of the category mod(kG) of finitely generated kG-modules by the subcategory proj(kG) of projective objects.

The Krull-Schmidt Theorem holds for kG-modules. So, working in StMod (kG) simply consists in forgetting projective summands. The statement “M ≅ N in StMod (kG)” means M ⊕ (proj) ≅ N ⊕ (proj) in the usual notation. We systematically drop the “⊕ (proj)” in the sequel since this summand vanishes in our triangulated categories. For instance, for M ∈ StMod (kG), the statement “M is finitely generated” should be understood as M ∈ stmod (kG), i.e., M ≅ M_{0} ⊕ (proj) with M_{0} finitely generated.

The projective support variety of G over the field k is the projective variety (or rather the scheme)

V_{G} = Proj H^{∗}(G, k).

In general, if R is a graded commutative k-algebra, we define Proj R to be the scheme whose underlying space is the set of homogeneous prime ideals of R not containing the maximal ideal 𝔪 = R^{+} generated by elements of positive degree, with the Zariski topology. So if 𝔞 is a homogeneous ideal in R then V(𝔞) is the closed set {𝔭 ∈ Proj R | 𝔭 ⊇ 𝔞}.

If 𝔭 ∈ Proj R then we write R_{𝔭} for the homogeneous localisation whose elements of degree n are the quotients x/y with y ∉ 𝔭 and deg x − deg y = n. The degree zero part of this localisation gives the stalk O_{X,𝔭} of the structure sheaf O_{X} of X = Proj R (see Hartshorne [17] §II.2 for further details).

Every module M has a support variety V_{G}(M) ⊆ V_{G}. When M is finitely generated, V_{G}(M) is the closed set V_{G}(M) = V(𝔞) defined by the ideal 𝔞 = Ann_{H^{∗}(G,k)}(Ext_{kG}^{∗}(M, M)), where H^{∗}(G, k) = Ext_{kG}^{∗}(k, k) acts on Ext_{kG}^{∗}(M, M) in the natural way. (See for example [4], Chapter 5.) When M is not finitely generated, see [7].

The assignment M ↦ V_{G}(M) satisfies a few easy rules (see [10]). In the language of [2], the support variety V_{G} ≅ Spc(stmod (kG)) is the spectrum of the tensor triangulated category stmod (kG). This allows us to use the gluing technique of [3], which we briefly recall in Section 4.

We shall use Quillen’s Dimension Theorem [20, 21], which says that

V_{G} = ⋃_{E} res_{G,E}^{∗}(V_{E})

where the union is over a set of representatives of the maximal elementary abelian psubgroups of G. In particular, any image res_{G,E}^{∗}(V_{E}), for E a maximal elementary abelian p-subgroup, is a component of the variety Proj H^{∗}(G, k). Moreover, we have that if ζ ∈ H^{∗}(G, k) is an element whose restriction to every elementary abelian p-subgroup is zero, then ζ is nilpotent.

Example 1.1. We warn the reader that the generators of H^{∗}(G, k) are usually not all in the same degree, so that the usual intuitions from projective geometry might fail. Here is an example of this phenomenon.

Let p ≥ 5 be a prime, let G be the semidirect product (ℤ/p)^{3} ⋊ Σ_{3}, with the permutation action, and let k be a field of characteristic p. Then

H^{∗}(G, k) = k[x_{2}, x_{4}, x_{6}] ⊗ Λ(y_{1}, y_{3}, y_{5})

where the subscript indicates the degree of the generator and Λ(⋯) denotes an exterior algebra. So H^{∗}(G, k) modulo its radical is a polynomial ring k[x_{2}, x_{4}, x_{6}], and Proj H^{∗}(G, k) is equal to Proj k[x_{2}, x_{4}, x_{6}] as a variety (though not as a scheme because of the nilpotent part).

We claim that the variety Proj k[x_{2}, x_{4}, x_{6}] is singular. To see this, observe that the coordinate ring for the affine patch x_{4} ≠ 0 is generated by u = x_{2}^{2}/x_{4}, v = x_{6}^{2}/x_{4}^{3} and w = x_{2}x_{6}/x_{4}^{2}, which satisfy the relation uv = w^{2}. So there is a singularity at the point u = v = w = 0 of this patch, which is the projective point (0 : 1 : 0).

Example 1.2. In the case of an elementary abelian p-group G = (ℤ/p)^{r}, Proj H^{∗}(G, k) is projective space ℙ^{r−1}(k). In this case, Dade’s Theorem [15] states that every endotrivial module is isomorphic to Ω^{n}k for some n ∈ ℤ. This can be compared with the invertible sheaves on projective space: every invertible sheaf on ℙ^{r−1}(k) is isomorphic to O(n) for some n ∈ ℤ, see for instance Hartshorne [17, Corollary II.6.17].

2. Endotrivial modules

Endotrivial modules and endopermutation modules were first named in [15] by Dade who showed that the sources (in the context of the theory of vertices and sources) of irreducible module for p-nilpotent groups are endopermutation. In addition, the classes of endotrivial modules modulo projective modules make up the Picard group of invertible modules in stmod (kG) as well as playing an important role in block theory. In this section we give some discussion of endotrivial modules and end with a proof that any invertible object in StMod (kG) is actually an endotrivial module in stmod (kG).

As originally defined, a finitely generated module is endotrivial if its k-endomorphism ring is isomorphic in stmod (kG) to the trivial module. That is, M is endotrivial if

Hom_{k}(M, M) ≅ k ⊕ P

for some projective module P. Since for any finitely generated kG-modules M and N, Hom_{k}(M, N) ≅ M^{∗}⊗N, it is equivalent to say that M is endotrivial if M^{∗}⊗M is isomorphic to the trivial module in stmod (kG). A complete classification of the endotrivial modules for a p-group was finally completed in [12, 13]. The answer briefly is that the Picard group of endotrivial modules has no torsion unless G is cyclic or p = 2 and G is a quaternion or semi-dihedral group. Moreover, the torsion free part of the group is generated by Ω(k) unless G has at least two conjugacy classes of maximal elementary abelian subgroups and at least one of the classes has rank 2. This last fact which was first proved by Alperin [1] for p-groups, holds for any finite group.

Of relevance for this paper is a construction in [12]. We should emphasize that there are actually two methods for constructing endotrivial module given in that paper. The first method, which we might call the “sectional” method creates endotrivial module as sections U/V where U and V are very carefully chosen submodules

{0} ⊆ V ⊆ U ⊆ Ω^{n}(k).

The choice of U and V is dictated by a somewhat complicated formula determined by the structure of the cohomology ring H^{∗}(G, k). Every endotrivial module for a p-group can be constructed using the sectional method.

Of interest in this paper is the second method, which we call the “cohomological-pushout” method. It requires finding an element in H^{∗}(G, k) which has nontrivial restriction to the center of a Sylow p-subgroup of G, and then taking a pushout along a homomorphism whose existence is guaranteed by the support variety of the corresponding Carlson module. The method is described in Section 4 (see Diagram (4.1)) of this paper, where we prove that the construction is a gluing. It is important to note that not every endotrivial module can be obtained by this method. Specifically, we do not get any of the torsion modules for the quaternion or semi-dihedral groups and there are a few elements of infinite order that we don’t get (see Example 8.2 of loc. cit.). On the other hand, the method is guaranteed to produce generators for a subgroup of finite index of the group of endotrivial modules. And most importantly, it is applicable to all finite groups, not just p-groups.

We end the section with the promised result on the finite generation of endotrivial modules.

Theorem 2.1. Suppose that a module M ∈ 𝒯 = StMod (kG) is invertible in the sense that there exists a module N with M ⊗ N ≅ k in 𝒯. Then M and N belong to stmod (kG) and N ≅ M^{∗} in 𝒯.

Proof. The assumption M⊗N ≅ k implies that M⊗−: 𝒯 → 𝒯 is an equivalence of categories, with inverse N ⊗ −. So, for any object Y in 𝒯, we have an isomorphism, natural Y:

Hom_{𝒯}(k, N ⊗ Y) ≅ Hom_{𝒯}(M, M ⊗ N ⊗ Y) ≅ Hom_{𝒯}(M, Y)

using successively the fact that M ⊗ − is an equivalence and the assumption M ⊗ N ≅ k. Naturality implies in particular that if Y ↪ M is a submodule, the following diagram commutes (ignore 1_{M}, f, g and h which appear later in the proof):

\xymatrix{g ∈ Hom_{𝒯}(k, N ⊗ Y) \ar[r]^{≅} \ar[d] & Hom_{𝒯}(M, Y) ∋ h \ar[d] \\ f ∈ Hom_{𝒯}(k, N ⊗ M) \ar[r]^{≅} & Hom_{𝒯}(M, M) ∋ 1_{M} .}  (2.1)

Now take the identity 1_{M} ∈ Hom_{𝒯}(M, M). There is a morphism f: k → N ⊗ M which maps to 1_{M}. Since k is finitely generated, we can find a finitely generated submodule Y ⊆ M such that f factors as k ⟶^{g} N ⊗ Y ⟶ N ⊗ M. Pushing this element g ∈ Hom_{𝒯}(k, N ⊗ Y) into Hom_{𝒯}(M, Y), we find a morphism h: M → Y which maps to 1_{M}, i.e., h is a section of the inclusion Y ↪ M in 𝒯. This means that M is a direct summand of Y ∈ stmod (kG) and so M ∈ stmod (kG).

The isomorphism N ≅ M^{∗} is well known: the category stmod (kG) is closed symmetric monoidal and so any invertible object has its dual as inverse. See if necessary [18, Proposition A.2.8]. □

Remark 2.2. Alternatively to the above direct proof, one can use [18, Proposition A.2.8] at the cost of checking that StMod (kG) satisfies the hypotheses of loc. cit. and that stmod (kG) consists exactly of the compact objects of StMod (kG). These facts are of independent interest.

3. U-isomorphisms and localisation

In this section we establish a few basic facts and notations concerned with localisations. In particular, we show that the functor on the localised subcategories, induced by the inclusion stmod ↣ StMod is fully faithful (see 3.8).

Definition 3.1. Let W ⊆ V_{G} be a closed subset. Consider the following subcategories of StMod (kG):

𝒞_{W}  ⊆  𝒞_{W}^{⊕} ∩  ∩ stmod (kG) ⊆ StMod (kG),

where 𝒞_{W} is the full subcategory of stmod (kG) consisting of those objects whose support in contained in W. Note that 𝒞_{W} is a ⊗-ideal thick subcategory of stmod (kG). The category 𝒞_{W}^{⊕} is the subcategory of StMod (kG) generated by 𝒞_{W} in any of the following equivalent senses:

(1) 𝒞_{W}^{⊕} is the smallest triangulated subcategory of StMod (kG) containing 𝒞_{W} and closed under arbitrary coproducts (hence the notation, although 𝒞_{W}^{⨿} would be more precise).

(2) 𝒞_{W}^{⊕} is the subcategory of StMod (kG) of those objects which are filtered colimits in Mod (kG) of objects of 𝒞_{W}. (In the notation of [22, § 5, Theorem 5.17], 𝒞_{W}^{⊕} = 𝒞\overrightarrow{W}.)

(3) 𝒞_{W}^{⊕} is the subcategory of StMod (kG) consisting of those modules M which have the property that any morphism L → M with L finitely generated, factors through some object of 𝒞_{W}. (This is equivalent to the above by [6, Theorem 5.3].)

It follows from [22, Prop. 5.9] that 𝒞_{W}^{⊕} is a ⊗-ideal in StMod (kG).

Definition 3.2. Given a morphism s in StMod (kG) and an open subset U ⊆ V_{G}, we shall say that s is a U-isomorphism if the cone of s belongs to 𝒞_{W}^{⊕} where W = V_{G} ∖ U is the closed complement of U. For example, the morphism ε_{W} : k → F (W) of Theorem 5.1 below is a (prototypical) U-isomorphism. Also note that since 𝒞_{W}^{⊕} is a ⊗-ideal, we have that for every U-isomorphism s: M → N and for every object L ∈ StMod (kG), the morphism L ⊗ s: L ⊗ M → L ⊗ N is again a U-isomorphism, for its cone is L ⊗ cone(s) ∈ 𝒞_{W}^{⊕}.

The following is an easy application of the octahedral axiom (see [3, Lemma 1.13]):

Proposition 3.3. Consider, in a triangulated category, a distinguished triangle as follows:

X_{1} \underrightarrow{(\begin{matrix}f \\ g \end{matrix} )} X_{2} ⊕ X_{3} ⟶^{( h k)} X_{4} ⟶^{ℓ} ΣX_{1}.

Then cone(f) ≅ cone(k) and cone(g) ≅ cone(h). In particular, in the case of StMod (kG) and of U ⊆ V_{G} open, the morphism f is a U-isomorphism if and only if k is.

Definition 3.4. It is useful to say that a square

\xymatrix{X_{1} \ar[r]^{f} \ar[d]_{−g} & X_{2} \ar[d]^{h} \\ X_{3} \ar[r]_{k} & X_{4}}

is a weak pullback if there exists a distinguished triangle as in the above statement, for some morphism ℓ: X_{4} → Σ X_{1} , which will always remain of little relevance in the sequel. It is easy to check that X_{1} has the property of the pullback of X_{2} and X_{3} above X_{4}, except for the uniqueness of the corner morphism (hence the “weak”). Similarly, this square is automatically a weak push-out.

We want to invert the U-isomorphisms, both in stmod (kG) and in StMod (kG). Roughly speaking, one can understand the resulting categories as the parts of stmod (kG) and of StMod (kG) which “lie over U”, or equivalently, which survive after “killing” all objects which are supported on the closed complement of U. For the convenience of the reader, we recall some standard facts about localisations of triangulated categories.

Definition 3.5. We say that

[figure: 𝒥; j; 𝒦; q; ℒ]

is an exact sequence of triangulated categories if 𝒥 ⊆ 𝒦 is a thick subcategory of 𝒦 (i.e., for X, Y ∈ 𝒦, X ⊕ Y ∈ 𝒥 implies X ∈ 𝒥) and if

ℒ = 𝒦/𝒥

is the quotient of 𝒦 by 𝒥. The latter means that the functor q is the universal functor out of 𝒦 which sends 𝒥 to zero: q ∘ j = 0. Equivalently, ℒ can be constructed as a Verdier localisation, see [24], as follows. Consider S the class of those morphisms s: X → Y in 𝒦 whose cone belongs to 𝒥. Then ℒ = S^{−1}𝒦 is the localisation of 𝒦 with respect to the morphisms of S, i.e., the target of the universal functor out of 𝒦 which maps morphisms of S to isomorphisms.

Explicitly, we have a calculus of fractions. By this we mean that ℒ = S^{−1}𝒦 can be constructed as having the same objects as 𝒦 and morphisms between two objects X and Y being equivalence classes of left fractions X ⟵^{s} Z ⟶^{f} Y where s ∈ S. Two fractions are declared equivalent if they have a common amplification X ⟵^{s s′} Z′ ⟶^{f s′} Y for s′: Z′ → Z in S. Equivalently, we can work with classes of right fractions X ⟶^{g} W ⟵^{t} Y with t ∈ S. (The passage from left to right fractions is made by means of weak pushouts and weak pullbacks; see Proposition 3.3.)

The functor q: 𝒦 → S^{−1}𝒦 is the identity on objects and sends a morphism f: X → Y to the class of the fraction X ⟵^{1} X ⟶^{f} Y. A morphism f: X → Y becomes zero in S^{−1}𝒦 if and only if there exists an s: Z → X in S such that f s = 0 or equivalently if there exists t ∈ S such that t f = 0. A morphism f: X → Y becomes an isomorphism in S^{−1}𝒦 if and only if its cone belongs to 𝒥. The subcategory 𝒥 is exactly the kernel of q.

Notation 3.6. Let us give short names to the various subcategories and Verdier localisations of the stable category which will appear below. We abbreviate:

𝒯 := StMod (kG) and 𝒞 := stmod (kG)

and for any closed W ⊆ V_{G} with open complement U = V_{G} ∖ W:

𝒯_{W} := 𝒞_{W}^{⊕}  (see Definition 3.1)
𝒞(U) := 𝒞/𝒞_{W} = stmod (kG)/𝒞_{W}
𝒯(U) := 𝒯/𝒯_{W} = StMod (kG)/𝒞_{W}^{⊕} .

In other words, 𝒞(U) and 𝒯(U) are the Verdier localisations with respect to U-isomorphisms of 𝒞 and 𝒯 respectively. We have the following commutative diagram of functors:

[figure: 𝒞_{W}; 𝒞; 𝒞(U); 𝒯_{W}; 𝒯; 𝒯(U).]

The rows are exact sequences of triangulated categories (Definition 3.5) and the left-hand first two vertical functors are the (fully faithful) inclusions of subcategories. The right-hand functor is the induced functor, which is fully faithful, as we now check.

Lemma 3.7. Consider a full inclusion of triangulated categories 𝒦 ⊆ 𝒦̄ and classes of morphisms S ⊆ 𝒦 and S̄ ⊆ 𝒦̄ such that S ⊆ S̄. The induced functor S^{−1}𝒦 → S̄^{−1}𝒦̄ is fully faithful if the following condition holds: for any morphism s: Z̄ → X in S̄ with X ∈ 𝒦, there exists a morphism t: Z → Z̄ with Z ∈ 𝒦 and st ∈ S.

Proof. This is an easy exercise on calculus of fractions. The condition implies that any fraction X ⟵^{s} Z̄ ⟶^{f} Y with X, Y ∈ 𝒦 but with Z̄ ∈ 𝒦̄ can be amplified into a fraction X ⟵^{st} Z ⟶^{ft} Y with Z ∈ 𝒦. This proves S^{−1}𝒦 → S̄^{−1}𝒦̄ full. The condition also implies that if f ∘ s = 0 for some morphism f: X → Y in 𝒦 and for some morphism s ∈ S̄, one has f ∘ (st) = 0 with this time st ∈ S. This proves S^{−1}𝒦 → S̄^{−1}𝒦̄ faithful. □

Proposition 3.8. Let W ⊆ V_{G} be a closed subset with open complement U. The canonical functor 𝒞(U) ⟶ 𝒯(U) is fully faithful.

Proof. Let us check the condition of Lemma 3.7 for 𝒦 = 𝒞, 𝒦̄ = 𝒯 and S and S̄ the respective classes of U-isomorphisms. Let s: Z̄ → X be a U-isomorphism, i.e., a morphism whose cone belongs to 𝒯_{W}, and assume that X ∈ 𝒞, i.e., X is finitely generated. Consider a distinguished triangle:

Z̄ ⟶^{s} X ⟶^{s_{1}} Ȳ ⟶^{s_{2}} ΣZ̄.

The object Ȳ := cone(s) belongs to 𝒯_{W} by hypothesis. By Property (3) of 𝒯_{W} = 𝒞_{W}^{⊕} in Definition 3.1, the morphism s_{1} factors via some object Y ∈ 𝒞_{W}, say s_{1} = u v: X ⟶^{v} Y ⟶^{u} Ȳ. Let us write this in the middle square of the following diagram:

[figure: Z; v_{0}; ∃ t; X; v; Y; v_{2}; u; ΣZ; Σt; Z̄; s; X; s_{1}; Ȳ; s_{2}; ΣZ̄,]

where we also complete v into a distinguished triangle (first row). Let t: Z → Z̄ be a fill-in map as above. Of course, Z is finitely generated since Y and X are, and we have cone(st) = cone(v_{0}) = Y ∈ 𝒞_{W}, i.e., the map st is a U-isomorphism as desired. □

4. Gluing finitely generated modules

In this section we define the gluing process and show that the cohomological-pushout method of [12, § 4.5] is a gluing. For the entire section, assume that we have an open covering

V_{G} = U_{1} ∪ U_{2}

of the projective support variety. We contemplate the following commutative diagram of localisations of triangulated categories (see Notations 3.6):

\xymatrix{M ∈ 𝒞 = stmod (kG) \ar[r] \ar[d] & 𝒞(U_{1})  ∋ M_{1} \ar[d] \\ M_{2} ∈  𝒞(U_{2}) \ar[r] & 𝒞(U_{1} ∩ U_{2}) M_{1} ≅ M_{2}}

The objects M, M_{1}, M_{2} are the ones of the following definition.

Definition 4.1. Consider two objects M_{1}, M_{2} ∈ stmod (kG) for i = 1, 2 (though we think M_{i} ∈ 𝒞(U_{i}) as above). Suppose that we have an isomorphism σ: M_{1} →^{≅} M_{2} in 𝒞(U_{1} ∩ U_{2}).

A gluing of M_{1} and M_{2} along the isomorphism σ is an object of stmod (kG) which is locally isomorphic to M_{1} and M_{2} in a compatible way with σ. That is, a gluing along σ is a triple (M, σ_{1}, σ_{2}) where M is an object of 𝒞 and

σ_{1} : M →^{≅} M_{1} and σ_{2} : M →^{≅} M_{2}

are two isomorphisms in 𝒞(U_{1}) and 𝒞(U_{2}) respectively, such that the following diagram

\xymatrix{ & M_{1} \ar[dd]^{σ} \\ M \ar@{-->}[ur]^{σ_{1}} \ar[dr]_{σ_{2}} \\  & M_{2}}

commutes in 𝒞(U_{1} ∩ U_{2}). Such a gluing always exists in 𝒞 and is unique up to isomorphism by [3, Corollary 5.10].

Applying this to M_{1} = k and M_{2} = k, we obtain [3, Theorem 6.7], which is as follows.

Theorem 4.2 (Balmer-Favi). Consider Pic(𝒞) = T_{k}(G) the group of finitely generated endotrivial kG-modules, i.e., the group of invertible objects in 𝒞 = stmod (kG) with respect to ⊗. Denote by 𝔾_{m}(U_{1} ∩ U_{2}) := Aut_{𝒞(U_{1}∩U_{2})}(k) the group of automorphisms of k in 𝒞(U_{1} ∩ U_{2}). Gluing two copies of k along an automorphism α ∈ 𝔾_{m}(U_{1} ∩ U_{2}) defines a group homomorphism δ: 𝔾_{m}(U_{1} ∩ U_{2}) → Pic(𝒞).

Remark 4.3. Observe that we can glue all sorts of objects M_{1} and M_{2}, not necessarily copies of k, and not only endotrivial modules. Even for the construction of endotrivial modules, it can be interesting to glue Ω^{ℓ}k and Ω^{m}k along an isomorphism over U_{1} ∩ U_{2}. We illustrate this situation below.

We end the Section, with a proof that the construction [12, § 4.5] is indeed a gluing construction. For the setting, suppose that G has at least two conjugacy classes of maximal elementary abelian p-subgroups and that at least one of these is a class of maximal elementary abelian subgroups of rank 2 (order p^{2}). Then the center Z of a Sylow p-subgroup of G must be cyclic. Choose a homogeneous element ζ ∈ H^{m}(G, k) with the property that ζ restricts to a non-nilpotent element of H^{∗}(Z, k). Consider a morphism ζ: Ω^{m}k → k in stmod (kG), representing the cohomology element ζ.

With the assumptions on G and ζ (and only with these assumptions), we know that V (ζ) = W_{1} ∪ W_{2} decomposes into two disjoint non-empty closed subsets W_{1} ∩ W_{2} = ∅, which is the only property we need for this construction. By [9] there is an analogous decomposition of the module L_{ζ} ≅ L_{1} ⊕ L_{2} with the support of L_{i} being W_{i} for i = 1, 2. In other words, we have an exact sequence:

0 ⟶ L_{1} ⊕ L_{2} ⟶^{( υ_{1} υ_{2} )} Ω^{m}k ⟶^{ζ} k ⟶ 0.

Theorem 4.5 of [12] says that the module N obtained by the following push-out (marked ⌜):

[figure: 0; 0; L_{2}; _{(\begin{matrix}0 \\ 1 \end{matrix})}; L_{2}; υ_{2}; 0; L_{ζ}; _{(_{υ_{1} υ_{2}} )}; _{(_{1 0} )}; ⌜; Ω^{m}k; ζ; ρ; k; 0; 0; L_{1}; ρ υ_{1}; N; σ_{1}; k; 0; 0; 0]  (4.1)

is endotrivial. In triangular terms, N ≅ cone(υ_{2}). Of particular interest are the morphisms σ_{1} and ρ which appear in the above diagram and which satisfy σ_{1} ρ = ζ.

Theorem 4.4. Consider the open complements U_{i} = V_{G}∖W_{i} of the above closed subsets W_{i}, for i = 1, 2. We have by assumption an open covering V_{G} = U_{1} ∪ U_{2}. With the above notations, the module N is the gluing (Definition 4.1) of k and Ω^{m}k along the isomorphism

ζ^{−1} : k →^{≅} Ω^{m}k in 𝒞(U_{1} ∩ U_{2}).

Proof. The exact sequences of diagram (4.1) yield corresponding triangles in stmod (kG). Since V_{G}(L_{ζ}) = W_{1} ∪ W_{2}, the morphism ζ: Ω^{m}(k) → k is a U_{1} ∩ U_{2}-isomorphism (Definition 3.2), i.e., ζ is an isomorphism in 𝒞(U_{1} ∩ U_{2}). So, the statement makes sense. Since V_{G}(L_{i}) = W_{i} for i = 1, 2, we see that σ_{1} : N → k is a U_{1}-isomorphism and that ρ: Ω^{m}k → N is a U_{2}-isomorphism. Let us define σ_{2} := ρ^{−1} : N →^{≅} Ω^{m}(k) in 𝒞(U_{2}). These are the desired isomorphisms σ_{1} : N →^{≅} k in 𝒞(U_{1}) and σ_{2} : N →^{≅} Ω^{m}k in 𝒞(U_{2}). By (4.1), we have σ_{1} ρ = ζ. In 𝒞(U_{1} ∩ U_{2}), these three morphisms are isomorphisms by the above comments and hence the relation σ_{1} ρ = ζ yields ζ^{−1}σ_{1} = ρ^{−1} = σ_{2}:

\xymatrix{ & k \ar[dd]^{ζ^{−1}} \\ N \ar[ur]^{σ_{1}} \ar[dr]_{σ_{2}} \\  & Ω^{m}(k).}

This shows that N is the gluing of k and Ω^{m}(k) along ζ^{−1}, as in Definition 4.1. □

5. Rickard’s idempotent modules

In this section, we recall some basic facts about idempotent modules. The definition, which is really an existence theorem is a part of the following.

Theorem 5.1 (Rickard [22]). Suppose that W is a closed subset of V_{G}.

(1) There exist two kG-modules E(W) and F (W) and a distinguished triangle

E(W) ⟶^{η_{W}} k ⟶^{ε_{W}} F (W) ⟶^{θ_{W}} ΣE(W)

in StMod (kG) such that E(W) ∈ 𝒞_{W}^{⊕} and such that F (W) is 𝒞_{W}^{⊕} -local, i.e.,

Hom(X, F (W)) = 0

for any X ∈ 𝒞_{W}^{⊕}.

(2) For any object M ∈ StMod (kG), the morphism M ⊗ η_{W} : M ⊗ E(W) → M ⊗ k ≅ M is the universal morphism from an object of 𝒞_{W}^{⊕} to M, and dually M ⊗ ε_{W} is the universal morphism from M to a 𝒞_{W}^{⊕} -local object. (In particular, M ⊗ F (W) is 𝒞_{W}^{⊕} -local.)

(3) If W_{1}, W_{2} ⊆ V_{G} are closed subsets, then there are unique isomorphisms

E(W_{1} ∩ W_{2}) ≅ E(W_{1}) ⊗ E(W_{2}) and F (W_{1} ∪ W_{2}) ≅ F (W_{1}) ⊗ F (W_{2})

such that η_{W_{1}∩W_{2}} = η_{W_{1}} ⊗ η_{W_{2}} and such that ε_{W_{1}∪W_{2}} = ε_{W_{1}} ⊗ ε_{W_{2}}. In particular E(W) ⊗ E(W) ≅ E(W) and F (W) ⊗ F (W) ≅ F (W) (hence the name “idempotent” modules). Moreover, there exists two Mayer–Vietoris triangles

E(W_{1} ∩ W_{2}) ⟶ E(W_{1}) ⊕ E(W_{2}) ⟶ E(W_{1} ∪ W_{2}) ⟶ ΣE(W_{1} ∩ W_{2})  (5.1)
F (W_{1} ∩ W_{2}) ⟶ F (W_{1}) ⊕ F (W_{2}) ⟶ F (W_{1} ∪ W_{2}) ⟶ ΣF (W_{1} ∩ W_{2})  (5.2)

The proof is Rickard’s beautiful insight. The picky reader might observe that the key Lemma 4.2 in [22], is not correctly proved, and might even be incorrect as stated. With the same notation, the conclusion of that Lemma should be corrected to read: Then there is a distinguished triangle

X′ → hocolim Y_{i} → hocolim Z_{i} → Σ(X′)

for some object X′ which fits into a distinguished triangle ⨁_{i} X_{i} → ⨁_{i} X_{i} → X′ → ⨁_{i} ΣX_{i}. The problem is that X′ might not be hocolim X_{i}. The proof of this statement is exactly the one given by Rickard in loc. cit.(Note that Verdier’s result does not control the third morphism.) This Lemma is the starting point of the construction of idempotent modules and the reader can check that the above formulation suffices to prove the other statements of [22]. In particular, in the proof of Proposition 5.4, loc. cit., ℰ_{𝒞}(X) might not be hocolim(A_{i}) but still belongs to 𝒞^{⊕}. The proof of Proposition 5.5, loc. cit., remains the same.

Rickard’s idempotent modules are extremely useful in that they allow the description of morphisms in the localisation 𝒯(U) in terms of usual morphisms in the stable category 𝒯.

Proposition 5.2. Let W ⊆ V_{G} be a closed subset with open complement U. Let M, N ∈ 𝒯 = StMod (kG). We have an isomorphism

Hom_{𝒯}(M, N ⊗ F (W)) →^{≅} Hom_{𝒯(U)}(M, N ⊗ F (W))

given by the localisation functor 𝒯 → 𝒯(U).

Proof. This is a general fact coming from the situation guaranteed by Rickard’s Theorem 5.1. In fact, the functor − ⊗ F (W) maps 𝒯 to the subcategory of 𝒯_{W} -local objects and is a left adjoint to the inclusion of 𝒯_{W} -local objects in 𝒯. Then, the functor from 𝒯_{W} -local objects to 𝒯/𝒯_{W} is an equivalence, or equivalently, one can realise the localisation functor as the functor − ⊗ F (W). Let us translate this in down-to-earth terms.

The key property of 𝒯_{W} -local objects like F (W) or N ⊗ F (W) is the following: Let F ∈ 𝒯 be a 𝒯_{W} -local object and let t: F → X be a U-isomorphism from our F to some object X. Then t is a split monomorphism. This is immediate from a distinguished triangle over t:

\xymatrix{F \ar[r]^{t} & X \ar[r]^{t_{1}} \ar@{-->}[l]^{∃r} & cone(t) \ar[r]^{t_{2}=0} & ΣF.}

By definition of F being 𝒯_{W} -local, the morphism t_{2} must be zero because cone(t) ∈ 𝒯_{W}. So, there exists r: X → F such that rt = 1.

With this in hand, the isomorphism of the statement is easy to prove. Let F = N ⊗F (W). Any right fraction M ⟶^{g} X ⟵^{t} F can be amplified by a retraction r: X → F as above, giving the equivalent fraction M ⟶^{rg} F ⟵^{1} F, that is a morphism coming from 𝒯. Injectivity is proved similarly: if tf = 0 then f = r t f = 0. □

Corollary 5.3. Let W ⊆ V_{G} be closed with open complement U and let M, N ∈ 𝒯. Then

Hom_{𝒯} (M ⊗ F (W), N ⊗ F (W)) ≅ Hom_{𝒯(U)}(M, N) .

This isomorphism maps α: M ⊗F (W) → N ⊗F (W) to (N ⊗ε_{W})^{−1} ∘α∘(M ⊗ε_{W}): M → N in 𝒯(U). In particular, this isomorphism respects the composition operation and therefore, if we denote by Isom ⊆ Hom the subsets of isomorphisms, we have an induced bijection

Isom_{𝒯} (M ⊗ F (W), N ⊗ F (W)) ≅ Isom_{𝒯(U)}(M, N) .

Proof. ¿From Proposition 5.2, we know that localisation yields an isomorphism between Hom_{𝒯}(M ⊗ F (W), N ⊗ F (W)) and Hom_{𝒯(U)}(M ⊗ F (W), N ⊗ F (W)). In the latter group, we can replace M ⊗ F (W) by M and N ⊗ F (W) by N, since they are isomorphic in 𝒯(U) via M ⊗ ε_{W} and N ⊗ ε_{W} respectively. The isomorphism is exactly as announced in the statement. Hence it preserves composition. Therefore invertible elements, i.e., isomorphisms, are also preserved. □

Corollary 5.4. Let W ⊆ V_{G} be a closed subset with open complement U. Then

End_{𝒯}(F (W)) ≅ End_{𝒞(U)}(k) and Aut_{𝒯} (F (W)) ≅ Aut_{𝒞(U)}(k)

and these bijections send α: F (W) → F (W) onto ε_{W}^{−1} α ε_{W} : k → k in 𝒞(U).

Proof. Apply Corollary 5.3 to M_{1} = M_{2} = k ∈ 𝒞 and replace Hom_{𝒯(U)} by Hom_{𝒞(U)} using the fact that 𝒞(U) ↪ 𝒯(U) is fully faithful by Proposition 3.8. □

Remark 5.5. It is known (see for example Benson and Gnacadja [8, §5.2]) that the endomorphism rings of Rickard idempotent modules are graded commutative.

Here is a useful example of the endomorphism ring of an idempotent module corresponding to a principal closed subset of V_{G}.

Proposition 5.6. Let ζ ∈ H^{d}(G, k) be a homogeneous element and consider the closed subset W = V_{G}(ζ) ⊆ V_{G} with open complement U. Then End_{𝒯}(F (W)) is isomorphic to (H^{∗}(G, k)[ζ^{−1}])^{0}, the degree zero part of the cohomology ring localised at ζ. Via Corollary 5.4,

a fraction \frac{η}{ζ^{i}} for η ∈ H^{di}(G, k) corresponds to the fraction k ⟵^{ζ^{i}} Ω^{di}k ⟶^{η} k in 𝒞(U).

Proof. See Rickard [22, § 6] or Friedlander and Pevtsova [16, Prop. 7.4]. □

Remark 5.7. Let U′ ⊆ U ⊆ V_{G} be open subsets with closed complements W′ ⊃ W respectively. Note that 𝒞_{W}^{⊕} ⊆ 𝒞_{W′}^{⊕} and that therefore U-isomorphisms are U′-isomorphisms. Consider the induced localisation functor 𝒯(U) → 𝒯(U′). For any pair of objects M, N ∈ 𝒯, the induced homomorphism Hom_{𝒯(U)}(M, N) → Hom_{𝒯(U′)}(M, N) gives a homomorphism

Hom_{𝒯}(M ⊗ F (W), N ⊗ F (W)) ⟶ Hom_{𝒯}(M ⊗ F (W′), N ⊗ F (W′))

by Corollary 5.3. This homomorphism can simply be described as follows

(M ⊗ F (W) ⟶^{α} N ⊗ F (W)) ⟼ (M ⊗ F (W′) ⟶^{α⊗F (W′)} N ⊗ F (W′))

using the identification F (W) ⊗ F (W′) ≅ F (W′) of Theorem 5.1. This verification is left to the reader. See [22].

6. Gluing arbitrary modules

We now explain how to glue any pair of not necessarily finitely generated kG-modules. At this stage, always assuming V_{G} = U_{1} ∪ U_{2}, we abandon stmod (kG) (in the right-hand diagram below) and consider instead the left-hand commutative diagram of localisations of larger triangulated categories (see Notation 3.6):

\xymatrix{𝒯 = StMod (kG) \ar[r] \ar[d] & 𝒯(U_{1}) \ar[d] & 𝒞 = stmod (kG) \ar[r] \ar[d] & 𝒞(U_{1}) \ar[d] \\ 𝒯(U_{2}) \ar[r] & 𝒯(U_{1} ∩ U_{2}) & 𝒞(U_{2}) \ar[r] & 𝒞(U_{1} ∩ U_{2}) .}

The definition of a gluing M ∈ 𝒯 of two objects M_{1} ∈ 𝒯(U_{1}) and M_{2} ∈ 𝒯(U_{2}) along an isomorphism σ: M_{1} →^{≅} M_{2} in 𝒯(U_{1} ∩ U_{2}) is exactly the same as in Definition 4.1, except of course that we allow M to live in the big category 𝒯. Before proving existence and uniqueness of the gluing, let us unfold what happens to Rickard’s idempotent modules in this situation.

Let us denote by W_{1} = V_{G} ∖ U_{1} and W_{2} = V_{G} ∖ U_{2} the closed complements of the two open subsets covering V_{G}. By assumption, we have that W_{1} ∩ W_{2} = ∅. Hence, 𝒞_{W_{1}∩W_{2}} = 0 = 𝒞_{W_{1}∩W_{2}}^{⊕}. So we get from Theorem 5.1 that E(W_{1} ∩ W_{2}) = 0 and that F (W_{1} ∩ W_{2}) = k, as well as a Mayer–Vietoris distinguished triangle

k \underrightarrow{(\begin{matrix}ε_{1} \\ −ε_{2} \end{matrix} )} F (W_{1}) ⊕ F (W_{2}) ⟶^{( ε_{12} ε_{21} )} F (W_{1} ∪ W_{2}) ⟶^{γ} Σk,  (6.1)

where the first two morphisms ε_{i} are the ε_{W_{i}} of Theorem 5.1 and where ε_{12} and ε_{21} are characterised by the commutativity of the following diagram:

[figure: k; ε_{1}; ε_{2}; ε_{W_{1}∪W_{2}}; F (W_{1}); ε_{12}; F (W_{2}); ε_{21}; F (W_{1} ∪ W_{2}) .]  (6.2)

In the notation of [22, Def., p. 164], ε_{12} = ε_{W_{1},W_{1}∪W_{2}} and ε_{21} = ε_{W_{2},W_{1}∪W_{2}}, or, using idempotence, ε_{12} = ε_{2} ⊗F (W_{1}) and ε_{21} = ε_{1} ⊗F (W_{2}). It will sometimes be convenient to abbreviate ε := ε_{W_{1}∪W_{2}}.

So, returning to our gluing problem, let M_{1} and M_{2} be objects of 𝒯 (thought of as objects of 𝒯(U_{1}) and 𝒯(U_{2}) respectively) and let σ: M_{1} →^{≅} M_{2} be an isomorphism in 𝒯(U_{1} ∩ U_{2}). By Proposition 5.3, there exists an isomorphism α: M_{1} ⊗ F (W_{1} ∪ W_{2}) →^{≅} M_{2} ⊗ F (W_{1} ∪ W_{2}) in 𝒯 such that the following diagram of isomorphisms commutes in 𝒯(U_{1} ∩ U_{2}):

\xymatrix{ & M_{1} \ar[r]^{σ} \ar[dl]_{M_{1}⊗ε} & M_{2} \ar[dr]^{M_{2}⊗ε} \\ M_{1} ⊗ F (W_{1} ∪ W_{2}) \ar[rrr]_{α} &  &  & M_{2} ⊗ F (W_{1} ∪ W_{2}) .}  (6.3)

Using this isomorphism α, we can now give our main construction.

Definition 6.1. Let M_{1}, M_{2} ∈ 𝒯 and let α: M_{1} ⊗ F (W_{1} ∪ W_{2}) →^{≅} M_{2} ⊗ F (W_{1} ∪ W_{2}) be an isomorphism in 𝒯. Consider the following morphism:

(M_{1} ⊗ F (W_{1})) ⊕ (M_{2} ⊗ F (W_{2})) \underrightarrow{(^{α∘(M_{1}⊗ε_{12})} ^{M_{2}⊗ε_{21}} )} M_{2} ⊗ F (W_{1} ∪ W_{2})

and complete it into a distinguished triangle:

X_{α} \underrightarrow{(\begin{matrix}ε_{1}^{α} \\ −ε_{2}^{α} \end{matrix} )} (M_{1} ⊗ F (W_{1})) ⊕ (M_{2} ⊗ F (W_{2})) ⟶^{( α∘(M_{1}⊗ε_{12}) M_{2}⊗ε_{21} )} M_{2} ⊗ F (W_{1} ∪ W_{2}) ⟶^{γ^{α}} ΣX_{α}

for some object X_{α} ∈ 𝒯 and morphisms ε_{1}^{α}, ε_{2}^{α} and γ^{α} as above. (In this notation, we only indicate dependence on α but not on M_{1} and M_{2}, nor on W_{1} and W_{2}, for obvious reasons.)

Note that the module X_{α} is only well-defined up to (non-unique) isomorphism.

Remark 6.2. We have the following weak pullback (Definition 3.4):

\xymatrix{X_{α} \ar[r]^{ε_{1}^{α}} \ar[d]_{ε_{2}^{α}} & M_{1} ⊗ F (W_{1}) \ar[d]^{α∘(M_{1}⊗ε_{12})} \\ M_{2} ⊗ F (W_{2}) \ar[r]_{M_{2}⊗ε_{21}} & M_{2} ⊗ F (W_{1} ∪ W_{2}) .}  (6.4)

Indeed, this characterises X_{α}, as we shall see in Lemma 6.5. The reader might prefer the following more symmetric “square”:

\xymatrix{ & X_{α} \ar@{-->}[dl]_{ε_{1}^{α}} \ar@{-->}[dr]^{ε_{2}^{α}} \\ M_{1} ⊗ F (W_{1}) \ar@{-->}[dr]^{M_{1}⊗ε_{12}} &  & M_{2} ⊗ F (W_{2}) \ar@{-->}[dl]_{M_{2}⊗ε_{21}} \\  & M_{1} ⊗ F (W_{1} ∪ W_{2}) ⟶_{∼}^{α} M_{2} ⊗ F (W_{1} ∪ W_{2})}

Lemma 6.3. Recall the notion of U-isomorphism from Definition 3.2.

(1) The morphism ε_{12} is a U_{2}-isomorphism and ε_{21} is a U_{1}-isomorphism.

(2) For every objects M_{1}, M_{2} ∈ 𝒯 and every isomorphism α: M_{1} ⊗ F (W_{1} ∪ W_{2}) →^{≅} M_{2} ⊗ F (W_{1} ∪ W_{2}), the morphism ε_{i}^{α} is a U_{i}-isomorphism for i = 1, 2.

Proof. We know that ε_{i} is a U_{i}-isomorphism by Theorem 5.1 and the statement for ε_{12} and ε_{21} follows by Proposition 3.3 and the Mayer–Vietoris triangle (6.1). The second part of the statement is a consequence of the same Proposition and the distinguished triangle of Definition 6.1. Recall from Definition 3.1 that M_{1} ⊗ ε_{12} is still a U_{1}-isomorphism. □

Proposition 6.4. We have 𝒞_{W_{1}}^{⊕} ∩ 𝒞_{W_{2}}^{⊕} = 0. In particular, if a morphism f: L → M in StMod (kG) is both a U_{1}-isomorphism and a U_{2}-isomorphism, then f is an isomorphism.

Proof. Let N ∈ 𝒞_{W_{i}}^{⊕} for i = 1, 2. Then N ⊗ F (W_{i}) = 0. (This follows from Theorem 5.1 or can be found explicitly as [22, Proposition 5.15].) A fortiori, N ⊗ F (W_{1} ∪ W_{2}) = N ⊗ F (W_{1}) ⊗ F (W_{2}) = 0. But then, tensoring N with the Mayer–Vietoris triangle (6.1), we see that N = N ⊗ k = 0, as claimed. This proves the first statement. The second follows from the first since cone(f) ∈ 𝒞_{W_{1}}^{⊕} ∩ 𝒞_{W_{2}}^{⊕}. □

Lemma 6.5. Let M_{1}, M_{2} ∈ 𝒯 and let α: M_{1} ⊗ F (W_{1} ∪ W_{2}) →^{≅} M_{2} ⊗ F (W_{1} ∪ W_{2}) be an

isomorphism. Consider a commutative square

\xymatrix{X \ar[r]^{f_{1}} \ar[d]_{f_{2}} & M_{1} ⊗ F (W_{1}) \ar[d]^{α∘(M_{1}⊗ε_{12})} \\ M_{2} ⊗ F (W_{2}) \ar[r]_{M_{2}⊗ε_{21}} & M_{2} ⊗ F (W_{1} ∪ W_{2}).}

Then the following conditions are equivalent:

(1) The above square is a weak pullback (see Definition 3.4).

(2) f_{i} is a U_{i}-isomorphism for i = 1, 2. When these conditions hold, the module X is isomorphic to the X_{α} of Definition 6.1.

Proof. If the square is a weak pullback, Proposition 3.3 insures that f_{1} is a U_{1}-isomorphism since ε_{21} is a U_{1}-isomorphism by Lemma 6.3. Similarly, f_{2} is a U_{2}-isomorphism.

Conversely, suppose that f_{i} is a U_{i}-isomorphism for i = 1, 2, and construct the weak pullback X_{α} (see Remark 6.2):

\xymatrix{X \ar[drr]^{f_{1}} \ar[ddr]_{f_{2}} \ar@{-->}[dr]^{f} \\  & X_{α} \ar[r]^{ε_{1}^{α}} \ar[d]_{ε_{2}^{α}} & M_{1} ⊗ F (W_{1}) \ar[d]^{α∘(M_{1}⊗ε_{12})} \\  & M_{2} ⊗ F (W_{2}) \ar[r]_{M_{2}⊗ε_{21}} & M_{2} ⊗ F (W_{1} ∪ W_{2}) .}

Since the outer diagram commutes, there exists a corner morphism f: X → X_{α} making the whole diagram commute. This is the weak pullback property. By Lemma 6.3, ε_{i}^{α} is a U_{i}-isomorphism for i = 1, 2. By two-out-of-three, we see that f is a U_{i}-isomorphism as well, for i = 1, 2. Hence, by Proposition 6.4, f is an isomorphism. □

Theorem 6.6. Let M_{1}, M_{2} ∈ 𝒯 and let α: M_{1} ⊗ F (W_{1} ∪ W_{2}) →^{≅} M_{2} ⊗ F (W_{1} ∪ W_{2}) be

an isomorphism, which corresponds, via Corollary 5.3, to an isomorphism σ: M_{1} →^{≅} M_{2} in 𝒯(U_{1} ∩U_{2}). Then the object X_{α} constructed in Definition 6.1 is a gluing of M_{1} and M_{2} along the isomorphism σ. Moreover, this gluing is unique up to isomorphism in 𝒯.

Proof. Recall from Corollary 5.3 that in 𝒯(U_{1} ∩ U_{2}), we have σ = (M_{2} ⊗ ε)^{−1} ∘ α ∘ (M_{1} ⊗ ε), where ε = ε_{W_{1}∪W_{2}} , as presented in Diagram (6.3).

Let us first check that X_{α} is indeed a gluing. For i = 1, 2, define the isomorphisms

σ_{i}: X_{α} →^{≅} M_{i} in 𝒯(U_{i}) by σ_{i} := (M_{i} ⊗ ε_{i})^{−1} ∘ ε_{i}^{α}

\xymatrix{X_{α} \ar[rr]^{σ_{i}} \ar[r]_{ε_{i}^{α}} & M_{i} ⊗ F (W_{i}) & M_{i} . \ar[l]^{M_{i}⊗ε_{i}}}

Recall from Lemma 6.3 that ε_{i} and ε_{i}^{α} are U_{i}-isomorphisms, hence M_{i} ⊗ ε_{i} as well. Recall that we have two commutative Diagrams (6.2) and (6.4) in 𝒯 which are respectively:

(6.2) : \xymatrix{k \ar[r]^{ε_{1}} \ar[d]_{ε_{2}} \ar[dr]^{ε} & F (W_{1}) \ar[d]^{ε_{12}} \\ F (W_{2}) \ar[r]_{ε_{21}} & F (W_{1} ∪ W_{2})} and (6.4) :\xymatrix{X_{α} \ar[r]^{ε_{1}^{α}} \ar[d]_{ε_{2}^{α}} & M_{1} ⊗ F (W_{1}) \ar[d]^{α∘(M_{1}⊗ε_{12})} \\ M_{2} ⊗ F (W_{2}) \ar[r]_{M_{2}⊗ε_{21}} & M_{2} ⊗ F (W_{1} ∪ W_{2})}

and all morphisms in sight are (U_{1} ∩ U_{2})-isomorphisms by Lemma 6.3. Now we compute in 𝒯(U_{1} ∩ U_{2}) using the commutativity of the above squares:

σ σ_{1} =^{def} (M_{2} ⊗ ε)^{−1} α (M_{1} ⊗ ε) (M_{1} ⊗ ε_{1})^{−1} ε_{1}^{α} =^{(6.2)} (M_{2} ⊗ ε)^{−1} α (M_{1} ⊗ ε_{12}) ε_{1}^{α} =

=^{(6.4)} (M_{2} ⊗ ε)^{−1} (M_{2} ⊗ ε_{21}) ε_{2}^{α} =^{(6.2)} (M_{2} ⊗ ε_{2})^{−1} ε_{2}^{α} =^{def} σ_{2}.

This proves that (X_{α}, σ_{1}, σ_{2}) is a gluing of M_{1} and M_{2} along σ.

Let us now turn to uniqueness. Let (X, τ_{1}, τ_{2}) be another gluing of M_{1} and M_{2} along σ, that is, τ_{i}: X →^{≅} M_{i} in 𝒯(U_{i}) for i = 1, 2 and σ ∘ τ_{1} = τ_{2} in 𝒯(U_{1} ∩ U_{2}). Consider the morphisms (M_{i} ⊗ ε_{i}) ∘ τ_{i}: X → M_{i} ⊗ F (W_{i}) in 𝒯(U_{i}) for i = 1, 2. By Proposition 5.2, there exit two morphisms f_{i}: X → M_{i} ⊗ F (W_{i}), i = 1, 2, which give the above morphisms (M_{i} ⊗ ε_{i}) ∘ τ_{i} under localisation, i.e., such that the following diagram commutes in 𝒯(U_{i}):

\xymatrix{X \ar[rr]^{τ_{i}} \ar[r]_{f_{i}} & M_{i} ⊗ F (W_{i}) & M_{i} . \ar[l]^{M_{i}⊗ε_{i}}}

In particular, it is immediate that f_{i} is a U_{i}-isomorphism. We now want to apply Lemma 6.5 to the triple (X, f_{1}, f_{2}):

\xymatrix{X \ar[r]^{f_{1}} \ar[d]_{f_{2}} & M_{1} ⊗ F (W_{1}) \ar[d]^{α∘(M_{1}⊗ε_{12})} \\ M_{2} ⊗ F (W_{2}) \ar[r]_{M_{2}⊗ε_{21}} & M_{2} ⊗ F (W_{1} ∪ W_{2}).}

We have already checked condition (2) of that Lemma and it only remains to check that the above square really commutes. To see this, note that the lower right object, the target of both compositions, is of the form (⋯) ⊗ F (W_{1} ∪ W_{2}). So, using Proposition 5.3, it is enough to check the commutativity of that square in the localisation 𝒯(U_{1} ∩ U_{2}). There, it becomes easy, for it exactly amounts to the condition στ_{1} = τ_{2} , as can be readily verified using Diagram (6.2) again. □

Corollary 6.7. Let M ∈ 𝒯 = StMod (kG) and suppose that M is finitely generated on U_{1} and U_{2}, that is, M is isomorphic in 𝒯(U_{i}) to an object of 𝒞(U_{i}) ⊆ 𝒯(U_{i}) for i = 1, 2. Then M is finitely generated, that is, M is isomorphic to an object of 𝒞 = stmod (kG).

Proof. Consider M_{i} ∈ 𝒞(U_{i}) and σ_{i}: M →^{≅} M_{i} in 𝒞(U_{i}) for i = 1, 2. Define the isomorphism σ = σ_{2} σ_{1}^{−1} : M_{1} →^{≅} M_{2} in 𝒞(U_{1} ∩ U_{2}) – here we use Proposition 3.8. Then, obviously, M is the gluing of M_{1} and M_{2} along σ in 𝒯. As already mentioned, we know from [3, Cor. 5.10] that the gluing is possible in 𝒞, that is, there exists a gluing M′ ∈ 𝒞 of M_{1} and M_{2} along σ. Since the gluing is unique in the big category 𝒯, we must have M ≅ M′. □

7. A gluing construction of endotrivial modules

We now unfold the general gluing construction of Section 6 in the special case of

M_{1} = M_{2} = k .

The outcome, in that case, is an endotrivial (finitely generated) module. As before, we assume that we have an open covering V_{G} = U_{1} ∪ U_{2} of the projective support variety and we denote by W_{i} = V_{G} ∖ U_{i} the closed complements i = 1, 2. Here, Definition 6.1 becomes:

Definition 7.1. Let α ∈ Aut_{𝒯}(F (W_{1} ∪ W_{2})) be an automorphism of F (W_{1} ∪ W_{2}) in 𝒯 = StMod (kG), i.e., a unit in \underline{End}_{kG}(F (W_{1} ∪ W_{2})). Consider the morphism

F (W_{1}) ⊕ F (W_{2}) ⟶^{( α∘ε_{12} ε_{21} )} F (W_{1} ∪ W_{2})

which differs from the middle map of the Mayer-Vietoris triangle (6.1), only in that we twist the first component by the automorphism α. Completing this morphism to a distinguished triangle defines a module X_{α} and morphisms ε_{1}^{α}, ε_{2} and γ^{α} as follows:

X_{α} \underrightarrow{(\begin{matrix}ε_{1}^{α} \\ −ε_{2}^{α} \end{matrix} )} F (W_{1}) ⊕ F (W_{2}) ⟶^{( α ε_{12} ε_{21} )} F (W_{1} ∪ W_{2}) ⟶^{γ^{α}} ΣX_{α}.  (7.1)

As before, the module X_{α} is only well-defined up to (non-unique) isomorphism. We shall only be interested in the isomorphism class of X_{α} in StMod (kG). If α: k → k is the identity then the Mayer–Vietoris triangle (6.1) shows that X_{Id} ≅ k. It should also be pointed out that the definition of X_{α} depends on the ordering of the two disjoint closed subsets W_{1} and W_{2} of the support variety. In 7.6 we see what happens if we interchange W_{1} and W_{2}.

Remark 7.2. As in Remark 6.2, we have a weak pullback:

\xymatrix{X_{α} \ar[r]^{ε_{1}^{α}} \ar[d]_{ε_{2}^{α}} & F (W_{1}) \ar[d]^{α ε_{12}} \\ F (W_{2}) \ar[r]_{ε_{21}} & F (W_{1} ∪ W_{2}) ,}  (7.2)

which characterises X_{α}, by Lemma 6.5. Taking two automorphisms α and β, we can tensor the above square with the similar square for β. Using idempotence (Theorem 5.1), we get

\xymatrix{X_{α} ⊗ X_{β} \ar[r]^{ε_{1}^{α}⊗ε_{1}^{β}} \ar[d]_{ε_{2}^{α}⊗ε_{2}^{β}} & F (W_{1}) \ar[d]^{αβ ε_{12}} \\ F (W_{2}) \ar[r]_{ε_{21}} & F (W_{1} ∪ W_{2}) .}

By Lemma 6.5 (2), the latter square is a weak pullback and therefore X_{α} ⊗ X_{β} ≅ X_{αβ}. In particular X_{α} ⊗ X_{α^{−1}} ≅ k and Theorem 2.1 forces X_{α} to belong to stmod (kG), i.e., X_{α} is a finitely generated endotrivial module. We give another proof of these facts below.

Theorem 7.3. Let α ∈ \underline{End}_{kG}(F (W_{1} ∪ W_{2})) be an automorphism in StMod (kG). Then the module X_{α} of Definition 7.1 is isomorphic in StMod (kG) to a finite dimensional endotrivial module, that is, X_{α} ∈ 𝒞 = stmod (kG).

Moreover, if σ ∈ Aut_{𝒞(U_{1}∩U_{2})}(k) is the automorphism of k over U_{1} ∩ U_{2} corresponding to α (see Corollary 5.4), then the module X_{α} ∈ stmod (kG) is a gluing (Definition 4.1) of two copies of k along the isomorphism σ: k →^{≅} k in 𝒞(U_{1} ∩ U_{2}).

Proof. We already know by Theorem 6.6, applied to M_{1} = M_{2} = k, that X_{α} is a gluing of two copies of k along σ in the big category 𝒯. Corollary 6.7 tells us that X_{α} ∈ stmod (kG). It is endotrivial because it is locally endotrivial. That is, the evaluation map (X_{α})^{∗} ⊗ X_{α} → k is an isomorphism in 𝒞(U_{i}) for i = 1, 2, hence is an isomorphism in 𝒞 (its cone has empty support). Another proof of the latter fact was given in Remark 7.2 above. (Alternatively, see [3, Lem. 6.2].) □

We now have the following dictionary with the terminology of [3]:

Corollary 7.4. Consider T_{k}(G) = Pic(𝒞), the group of finitely generated endotrivial kGmodules, i.e., the group of invertible objects in 𝒞 = stmod (kG) with respect to ⊗. Consider the map ξ: Aut_{𝒯}(F (W_{1}∪W_{2})) ⟶ T_{k}(G) given by the above construction, α ↦ X_{α}. Consider the homomorphism δ: 𝔾_{m}(U_{1} ∩ U_{2}) = Aut_{𝒞(U_{1}∩U_{2})}(k) → Pic(𝒞) of Theorem 4.2. Then the two maps ξ and δ are equal. More precisely, the following diagram commutes:

[figure: Aut_{𝒯}(F (W_{1} ∪ W_{2})); ξ; (Cor. 5.4); ∼; 𝔾_{m}(U_{1} ∩ U_{2}); δ; T_{k}(G); Pic(𝒞).]

In particular, if α and β are two invertible elements in \underline{End}_{kG}(F (W_{1} ∪ W_{2})) then

X_{α} ⊗ X_{β} ≅ X_{α∘β} .

Proof. This is simply a condensed form of the previous results. Note that ξ is a homomorphism because δ is already known to be one, hence X_{α} ⊗ X_{β} ≅ X_{α∘β}. (We sketched a direct proof of the latter in Remark 7.2.) □

Corollary 7.5. Let α ∈ \underline{End}_{kG}(F (W_{1} ∪W_{2})) be an automorphism. Then X_{α} ≅ k is trivial if and only if there exists automorphisms α_{i} ∈ \underline{End}_{kG}(F (W_{i})) for i = 1, 2 such that α = α_{1} ⊗α_{2} under the identification F (W_{1} ∪ W_{2}) ≅ F (W_{1}) ⊗ F (W_{2}).

Proof. Transcribe in modular representation theoretic terms the exactness of the sequence

⋯ ⟶ 𝔾_{m}(U_{1}) ⊕ 𝔾_{m}(U_{2}) ⟶ 𝔾_{m}(U_{1} ∩ U_{2}) ⟶^{δ} Pic(stmod (kG))

established in [3, Thm. 6.7] as part of the Mayer-Vietoris long exact sequence. □

Remark 7.6. The definition of X_{α} (Definition 7.1) is not symmetric in the two closed subsets W_{1} and W_{2} and this might lead to some confusion. Strictly speaking, we should write X_{α} = X(α, W_{1}, W_{2}). Switching the order of W_{1} and W_{2} inverts the module X_{α} (i.e., gives the dual (X_{α})^{∗} instead). This can be easily checked, for instance from the gluing property of Theorem 7.3, which says that the following left-hand diagram commutes in 𝒞(U_{1} ∩ U_{2}):

\xymatrix{ & k \ar[dd]^{σ} \\ X_{α} \ar@{-->}[ur]^{σ_{1}} \ar[dr]_{σ_{2}} \\ & k} ⟹ \xymatrix{ & k \ar[dd]^{σ^{−1}} \\ X_{α} \ar@{-->}[ur]^{σ_{2}} \ar[dr]_{σ_{1}} \\ & k .}

Hence the right-hand diagram commutes as well and by Theorem 7.3 again but applied to (W_{2} , W_{1}), we obtain X(α^{−1}, W_{2}, W_{1}) = X_{α} which implies X(α, W_{2}, W_{1}) = X_{α^{−1}} = (X_{α})^{∗}.

Remark 7.7. Of course, the definition of X_{α} given in Definition 7.1 also makes sense if α ∈ \underline{End}_{kG}(F (W_{1} ∪ W_{2}) is a non-invertible endomorphism. The problem is that the module X_{α} will not be endotrivial in general. Take for instance α = 0. Then we have (α ε_{12} ε_{21}) = (0 ε_{21}): F (W_{1}) ⊕ F (W_{2}) → F (W_{1} ∪ W_{2}), and the defining triangle (7.1) becomes:

F (W_{1}) ⊕ (E(W_{1}) ⊗ F (W_{2})) \underrightarrow{(\begin{matrix}1 & 0 \\ 0 & η_{W_{1}}⊗1 \end{matrix})} F (W_{1}) ⊕ F (W_{2}) ⟶^{( 0 ε_{21} )} F (W_{1} ∪ W_{2}) \underrightarrow{(\begin{matrix}0 \\ θ_{W_{1}}⊗1 \end{matrix})} Σ(...).

To see that this triangle is distinguished, apply −⊗F (W_{2}) to the original triangle for E(W_{1}) and F (W_{1}), from Theorem 5.1 (1), and then add the trivial triangle

F (W_{1}) ⟶^{1} F (W_{1}) ⟶ 0 ⟶ ΣF (W_{1}).

So, the module X_{0} is F (W_{1}) ⊕ (E(W_{1}) ⊗ F (W_{2})) which is not even in stmod (kG) in general.

Remark 7.8. There is an extreme situation where our construction produces an endotrivial module for any endomorphism α, even the most trivial α = 0. Namely, this happens if W_{1} = ∅. Indeed, in that case, F (W_{1}) = k and the weak pullback (7.2) becomes:

[figure: X_{α}; ε_{1}^{α}; ε_{2}^{α}; k; ∀α; F (W_{2}); F (W_{2}) ,]

which forces ε_{1}^{α}: X_{α} →^{≅} k to be an isomorphism. This rather trivial remark will be useful at the end of Section 10.

8. First example: Rank two

Suppose that ζ_{1} and ζ_{2} are elements of H^{n}(G, k) such that V_{G}(ζ_{1}) ∩ V_{G}(ζ_{2}) = ∅. Note that this implies that H^{n}(G, k) is a finitely generated module over k[ζ_{1}, ζ_{2}], so that it has Krull dimension two, which by Quillen’s Dimension Theorem forces the group G to have p-rank two. If p > 2, then it also requires that ζ_{1} and ζ_{2} have even degree, as otherwise ζ_{1} and ζ_{2} are nilpotent. In particular, ζ_{1} and ζ_{2} must commute with each other. Letting W_{1} = V_{G}(ζ_{1}) and W_{2} = V_{G}(ζ_{2}), we are in precisely the situation of Sections 7. Note that W_{1} ∪ W_{2} = V_{G}(ζ_{1} ζ_{2}).

The endomorphism ring \underline{End}_{kG}(F (W_{1})) consists of the degree zero elements of the localisation H^{∗}(G, k)[ζ_{1}^{−1}] by Proposition 5.6, and similarly for F (W_{2}) and H^{∗}(G, k)[ζ_{2}^{−1}] and for F (W_{1} ∪ W_{2}) and H^{∗}(G, k)[ζ_{1}^{−1}ζ_{2}^{−1}].

The element ζ_{2}/ζ_{1} ∈ (H^{∗}(G, k)[ζ_{1}^{−1}])^{0} ≅ \underline{End}_{kG}(F (W_{1})) is not invertible, but it becomes invertible in H^{∗}(G, k)[ζ_{1}^{−1}ζ_{2}^{−1}] with inverse ζ_{1}/ζ_{2}. So it is an automorphism of F (W_{1}∪W_{2}) and is a candidate for α in the construction of our module X_{α} (Definition 7.1). Note however that we can consider other automorphisms of F (W_{1} ∪ W_{2}). This particular α = ζ_{1}/ζ_{2} is in some sense the “trivial” choice since ζ_{1} and ζ_{2} are precisely the defining equations of W_{1} = V_{G}(ζ_{1}) and W_{2} = V_{G}(ζ_{2}). In Section 9, we shall consider an example with another α.

Proposition 8.1. With the above notation, the module X_{ζ_{1}/ζ_{2}} is isomorphic to Σ^{n}k = Ω^{−n}k.

Proof. It is clear that each ζ_{i}: k → Σ^{n}k is a U_{i}-isomorphism (Definition 3.2) since its cone is a delooping of the Carlson module L_{ζ_{i}} which has support exactly W_{i} (see [4]). Let σ_{i} = (ζ_{i})^{−1} : Σ^{n}k →^{≅} k be the inverse isomorphism in the localisation 𝒞(U_{i}). In the final localisation 𝒞(U_{1} ∩ U_{2}), the following diagram obviously commutes:

\xymatrix{ & k \ar[dd]^{ζ_{1}/ζ_{2}} \\ Σ^{n}k \ar@{-->}[ur]^{σ_{1}} \ar@{-->}[dr]_{σ_{2}} \\  & k.}

In the language of [3], this means that Σ^{n}k is the gluing of two copies of k along the automorphism ζ_{1}/ζ_{2} ∈ Aut_{𝒞(U_{1}∩U_{2})}(k), i.e., we have X_{α} = Σ^{n}k by Theorem 7.3. □

It is also possible to prove the previous proposition by describing the idempotent modules, F (W_{1}), F (W_{2}) and F (W_{1} ∪ W_{2}), as colimits, giving the various morphisms ε_{1} , ε_{2} , ε_{12} , ε_{21} as maps on the colimits, and then making the distinguished triangle (7.1) explicit. However, the proof using the gluing technique that we have given here is considerably shorter.

Example 8.2. We unfold our construction of the endotrivial modules with a very explicit example in the case of the fours group G = ⟨g, h⟩ ≅ (ℤ/2)^{2} over a field k of characteristic two. We have H^{∗}(G, k) = k[x, y] with deg(x) = deg(y) = 1. Here, the basis x, y of H^{1}(G, k) is dual to the basis g − 1, h − 1 of H_{1}(G, k) ≅ J(kG)/J^{2}(kG). The support variety is V_{G} = Proj k[x, y] = ℙ^{1}(k). Let W_{1} = V_{G}(x), W_{2} = V_{G}(y), the zero loci of x and y respectively. We use the open covering by the two affine sets U_{i} = V_{G} ∖ W_{i}, i = 1, 2 defined by x and y, with intersection U_{1} ∩ U_{2} = V_{G} ∖ V_{G}(xy) and union U_{1} ∪ U_{2} = V_{G}. The module F_{x} = F (W_{1}) is represented by the following diagram:

[figure: ·; g−1; ·; h−1; ·; ·] F_{x} = [figure: ·; ·; ·; ·; ·; ·; 1; y/x; y^{2}/x^{2}] …

Here, we have labeled basis elements of the socle of F_{x} with the corresponding elements of \underline{Hom}_{kG}(k, F_{x}), namely degree zero elements of H^{∗}(G, k)[x^{−1}]. Similarly, we have

F_{y} = … [figure: ·; ·; ·; ·; ·; ·; x^{2}/y^{2}; x/y; 1] F_{xy} = … [figure: ·; ·; ·; ·; ·; ·; ·; x/y; 1; y/x] …

The maps ε_{12} : F_{x} ⟶ F_{xy} and ε_{21} : F_{y} ⟶ F_{xy} correspond to the obvious inclusions of diagrams.

The endomorphism ring of F_{xy} consists of the degree zero elements of H^{∗}(G, k)[x^{−1}y^{−1}] by Proposition 5.6. So x/y is an automorphism with inverse y/x. The action of x/y on F_{xy} is a shift one place to the left, while y/x is a shift one place to the right. Let α = x/y. Then we have a diagram

\xymatrix{ & F_{x} \ar[d]^{(x/y) ε_{12}} \\ F_{y} \ar[r]_{ε_{21}} & F_{xy}}

Since the sum of the two maps is surjective, the weak pullback of this diagram is the same as the ordinary pullback, namely the submodule corresponding to the intersection of the subdiagrams. This gives the diagram

[figure: ·; ·; ·; x/y; 1]

for X_{α} ≅ Ω^{−1}(k), in accordance with Proposition 8.1. On the other hand, if we use the endomorphism α^{−1} = y/x, then the subdiagrams do not intersect, and the weak pullback is not the same as the pullback. We must add a projective to make the sum of the maps surjective, and then take the pullback to obtain X_{α^{−1}} ≅ Ω(k).

9. Second example: The dihedral group D_{8}

In this section we discuss the example of the dihedral group G = D_{8} over a field k of characteristic two. The arguments that we use here are a model for what appears in the Section 11.

The cohomology ring of G has the form

H^{∗}(G, k) = k[x, y, z]/(xy)

where deg(x) = deg(y) = 1 and deg(z) = 2. Consider ζ_{1} = x^{2} +y^{2} +z and ζ_{2} = z in H^{2}(G, k) and define W_{1} = V_{G}(ζ_{1}) and W_{2} = V_{G}(ζ_{2}) as in Section 8. We claim that W_{1} ∩ W_{2} = ∅. Indeed, a homogenous prime ideal 𝔭 of H^{∗}(G, k) = k[x, y, z]/(xy) containing x^{2} + y^{2} and z necessarily contains x, y and z (it contains xy = 0 hence x or y hence both since it contains x^{2} + y^{2}). But then 𝔭 contains the maximal ideal (x, y, z) which is exluded in Proj (H^{∗}(G, k)). So, we have the open covering necessary for our construction of endotrivial modules of the form X_{α} as in Section 7

V_{G} = U_{1} ∪ U_{2}

where U_{i} = V_{G}∖W_{i} for i = 1, 2. Now we want to produce an automorphism α of F (W_{1}∪W_{2}) in a more subtle way than in Section 8, that is, different from ζ_{1}/ζ_{2} = (x^{2} + y^{2} + z)/z.

Because (x^{2} + z)(y^{2} + z) = (x^{2} + y^{2} + z) z in H^{∗}(G, k), we have that

\frac{x^{2} + z}{x^{2} + y^{2} + z} . \frac{y^{2} + z}{z} = 1

in H^{∗}(G, k)[(x^{2} + y^{2} + z)^{−1}z^{−1}] = H^{∗}(G, k)[(ζ_{1}ζ_{2})^{−1}]. So (y^{2} + z)/z is invertible in this localisation with inverse (x^{2} + z)/(x^{2} + y^{2} + z).

As before we use the notation F_{u} = F (V_{G}(u)), where u ∈ H^{∗}(G, k). We set α = (y^{2}+z)/z, so that X_{α} is the weak pullback

\xymatrix{X_{α} \ar[r] \ar[d] & F_{x^{2}+y^{2}+z} \ar[d]^{\frac{y^{2}+z}{z} ε_{12}} \\ F_{z} \ar[r]^{ε_{21}} & F_{(x^{2}+y^{2}+z)z}}

We can characterize the endotrivial module X_{α} by restricting it to the two subgroups isomorphic to ℤ/2 × ℤ/2 and by applying Proposition 8.1. On one of these subgroups, H_{1}, x restricts to zero and y does not, and on the other, H_{2}, y restricts to zero and x does not. The element z restricts to the product of the remaining two nonzero elements of H^{1}(H_{1}, 𝔽_{2}) ≅ 𝔽_{2}^{2} and of H^{1}(H_{2}, 𝔽_{2}) ≅ 𝔽_{2}^{2}. So the restriction of α to H_{2} is the identity element, while on H_{1} it is a ratio of two degree-two elements with no common factor, and is the same as the restriction of ζ_{1}/ζ_{2} = (x^{2} + y^{2} + z)/z. So X_{α}↓_{H_{2}} ≅ k while X_{α}↓_{H_{1}} is isomorphic to Ω^{−2}k by Proposition 8.1. This module X_{α} is one of two well known five dimensional endotrivial modules, and has the following diagram:

[figure: ·; ·; ·; ·; ·]

See [10] for more details on the diagrams.

10. Gluing along endomorphisms

Let us recall the general Mayer-Vietoris situation of Sections 4, 6 and 7, that is, we assume that we have an open covering of the projective support variety

V_{G} = U_{1} ∪ U_{2}

and we denote by W_{1} = V_{G} ∖ U_{1} and W_{2} = V_{G} ∖ U_{2} the closed complements.

Recall our quest, initiated in Remark 7.7, for good conditions under which the gluing automorphism α: k →^{≅} k on U_{1} ∩ U_{2} could be replaced by a general endomorphism, in our original construction of X_{α} in Definition 7.1. We already gave a rather trivial answer to this question in Remark 7.8 when W_{1} = ∅.

In the present section, we are going to say something under the assumption that the gluing automorphism extends to one of the open pieces, say U_{1}. This extension need not be an automorphism anymore. In this case, the gain is that we can replace one of the idempotent modules of Definition 7.1, namely F (W_{2}), simply by k. More precisely, this goes as follows.

Definition 10.1. Let β ∈ End_{𝒯}(F (W_{1})) be an endomorphism of F (W_{1}) in 𝒯 = StMod (kG). Define an object X̃_{β} in 𝒯 and two morphisms ε̃_{1}^{β} and ε̃_{2}^{β} by the following weak pullback (Definition 3.4):

\xymatrix{X̃_{β} \ar[r]^{ε̃_{1}^{β}} \ar[d]_{ε̃_{2}^{β}} & F (W_{1}) \ar[d]^{β} \\ k \ar[r]_{ε_{1}} & F (W_{1}) ,}  (10.1)

which should be compared to (7.2). Equivalently, we have a distinguished triangle:

X̃_{β} \underrightarrow{(\begin{matrix} & ε̃_{1}^{β} \\ −ε̃_{2}^{β} \end{matrix} )} F (W_{1}) ⊕ k ⟶^{( β ε_{1} )} F (W_{1}) ⟶ ΣX̃_{β} .  (10.2)

Recall from Corollary 5.3, that idempotent modules are primarily designed for the property that End_{𝒯}(F (W_{1})) ≅ End_{𝒞(U_{1})}(k). In particular, we can restrict our given endomorphism β ∈ End_{𝒯}(F (W_{1})) to U_{1} ∩ U_{2} and see what happens if it becomes an automorphism :

[figure: β; ∈; End_{𝒯}(F (W_{1})); End_{𝒞(U_{1})}(k); localisation; Aut_{𝒯}(F (W_{1} ∪ W_{2})); ⊆; End_{𝒯}(F (W_{1} ∪ W_{2})); End_{𝒞(U_{1}∩U_{2})}(k) .]

Proposition 10.2. Let β be an endomorphism of F (W_{1}) in 𝒯 and assume that it becomes an isomorphism on U_{1} ∩ U_{2}. Let α ∈ Aut_{𝒯}(F (W_{1} ∪ W_{2})) be the restriction of β. Then the object X̃_{β} of Definition 10.1 is isomorphic to the endotrivial module X_{α} of Definition 7.1.

Proof. By Remark 5.7, the hypothesis that the localisation of β is equal to α boils down to the commutativity of the following diagram in 𝒯:

\xymatrix{F (W_{1}) \ar[r]^{β} \ar[d]_{ε_{12}} & F (W_{1}) \ar[d]^{ε_{12}} \\ F (W_{1} ∪ W_{2}) \ar[r]_{α}^{≅} & F (W_{1} ∪ W_{2}) .}  (10.3)

By Lemma 6.3, the morphism ε_{12} is a U_{2}-isomorphism and so is α of course. From this, we deduce by two-out-of-three that β is a U_{2}-isomorphism. Therefore, by Proposition 3.3 applied to the distinguished triangle (10.2), the morphism ε̃_{2}^{β} is also a U_{2}-isomorphism. By the same proposition for the same triangle, since ε_{1} is a U_{1}-isomorphism, so is ε̃_{1}^{β}. We now have two isomorphisms in 𝒯(U_{1}) and 𝒯(U_{2}) respectively, σ_{1} := ε_{1}^{−1} ∘ ε̃_{1}^{β} and σ_{2} := ε̃_{2}^{β}.

σ_{1} : X̃_{β} ⟶^{ε̃_{1}^{β}} F (W_{1}) ⟵^{ε_{1}} k and σ_{2} : X̃_{β} ⟶^{ε̃_{2}^{β}} k .

Computing in 𝒯(U_{1} ∩ U_{2}), where all morphisms in sight become isomorphisms, we have

σ_{2} ∘ σ_{1}^{−1} =^{def} ε̃_{2}^{β} ∘ (ε̃_{1}^{β})^{−1} ∘ ε_{1} =^{(10.1)} ε_{1}^{−1} ∘ β ∘ ε_{1} =^{(10.3)} (ε_{12}ε_{1})^{−1} ∘ α ∘ (ε_{12}ε_{1}) =: σ .

Since ε_{12} ε_{1} = ε_{W_{1}∪W_{2}} : k → F (W_{1} ∪ W_{2}) by (6.2), the above morphism σ: k → k is the automorphism of k in 𝒞(U_{1} ∩ U_{2}) which corresponds to α ∈ Aut_{𝒯}(F (W_{1} ∪ W_{2})), see Corollary 5.4. So, we have proved that the object X̃_{β} is isomorphic to k on U_{1} and on U_{2}, via the isomorphisms σ_{1} and σ_{2} respectively, and we have σ_{2} ∘ σ_{1}^{−1} = σ. This means that X̃_{β} is the the gluing of two copies of k along σ. But we already know from Theorem 7.3 that this gluing is X_{α}. Therefore, X̃_{β} ≅ X_{α} by uniqueness of the gluing. □

Examples 10.3. In both examples given in Sections 8 and 9, the automorphism α can be extended to one of the pieces. This gives the following.

(1) Let G be a group of p-rank two and keep notations as in Section 8. The endomorphism β := ζ_{2}/ζ_{1} ∈ \underline{End}_{kG}(F (W_{1})) obviously extends α = ζ_{2}/ζ_{1} to U_{1}. By Propositions 10.2 and 8.1, there is a weak pullback diagram as follows:

\xymatrix{Ω^{d}k \ar[d]_{ζ_{2}} \ar[r]^{ε_{W_{1}}ζ_{1}} & F (W_{1}) \ar[d]^{ζ_{2}/ζ_{1}} \\ k \ar[r]_{ε_{W_{1}}} & F (W_{1}).}

The determination of the precise values of ε̃_{1}^{β} = ε_{W_{1}}ζ_{1} and ε̃_{2}^{β} = ζ_{2} are left to the reader (and will not be used below).

(2) Let G = D_{8} and keep notations as in Section 9. The automorphism α = (y^{2} + z)/z does not extend to an endomorphism on U_{1} = V_{G}(x^{2} + y^{2} + z) but α^{−1} does. Indeed β := \frac{x^{2}+z}{x^{2}+y^{2}+z} ∈ \underline{End}_{kG}(F_{x^{2}+y^{2}+z}) restricts to α^{−1} on U_{1} ∩ U_{2}. By Proposition 10.2 and the results of Section 9, we have a weak pullback as follows:

\xymatrix{X_{α}^{∗} \ar[d] \ar[r] & F (W_{1}) \ar[d]^{β} \\ k \ar[r]_{ε_{W_{1}}} & F (W_{1}) ,}

where X_{α}^{∗} = X_{α^{−1}} is the dual of the five-dimensional endotrivial module X_{α} of Section 9.

We now combine the above modified construction with the trivial Remark 7.8, to get the following statement. This final construction happens to be very useful, as we shall see in Section 11, for it easily applies even when the p-rank of G is greater than two. For the sake of clarity, we repeat all hypotheses.

Corollary 10.4. Let G be a finite group and let W_{1}, W_{2} ⊆ V_{G} be disjoint closed subsets of its support variety. Let γ: F (W_{1}∪W_{2}) → F (W_{1}∪W_{2}) be an endomorphism in 𝒯 = StMod (kG).

Define an object X̂_{γ} ∈ StMod (kG) and two morphisms ε̂_{1}^{γ} and ε̂_{2}^{γ} by the following weak pullback (Def. 3.4):

\xymatrix{X̂_{γ} \ar[r]^{ε̂_{1}^{γ}} \ar[d]_{ε̂_{2}^{γ}} & F (W_{1}) \ar[d]^{γ ε_{12}} \\ F (W_{2}) \ar[r]_{ε_{21}} & F (W_{1} ∪ W_{2}) .}  (10.4)

Let W_{3} ⊂ V_{G} be another closed subset, disjoint from W_{1}. Suppose that for every maximal elementary abelian p-subgroup E ⊆ G, at least one of the following two conditions holds true:

(1) W_{1} ∩ res_{G,E}^{∗}(V_{E}) = ∅.

(2) W_{2} ∩ res_{G,E}^{∗}(V_{E}) = ∅ and, if we denote by W′_{i} the preimage of W_{i} in V_{E} via the map res_{G,E}^{∗}: V_{E} → V_{G} for i = 1, 3, the morphism res_{G,E}(γ): F (W′_{1}) → F (W′_{1}) is a U-isomorphism (Def. 3.2) where U is the open V_{E} ∖ (W′_{1} ∪ W′_{3}) in V_{E}.

Then X̂_{γ} is an endotrivial kG-module.

Its restriction to a subgroup E ⊆ G as above is trivial if E satisfies Condition (1). If E satisfies Condition (2), the restriction res_{G,E}(X̂_{γ}) is isomorphic to the kE-module X_{α} obtained from Definition 7.1 for the group E, for the disjoint closed subsets W′_{1} and W′_{3} of V_{E} and for the automorphism α := res_{G,E}(γ) ⊗ F (W′_{3}) ∈ Aut_{StMod (kE)} (F (W′_{1} ∪ W′_{3})).

Proof. Let us restrict the weak pulback of the statement to a maximal elementary abelian p-subgroup E ⊆ G. Assume first that E satisfies Condition (1) then it is clear that res_{G,E}(X̂_{γ}) ≅ k (see Remark 7.8). On the other hand, suppose that E satisfies Condition (2). Note that we then have res_{G,E}(F (W_{2})) = k. The restriction to E of the weak pullback (10.4) is isomorphic to

\xymatrix{res_{G,E}(X̂_{γ}) \ar[r] \ar[d] & F (W′_{1}) \ar[d]^{res_{G,E}(γ)} \\ k \ar[r]_{ε_{1}} & F (W′_{1}) .}

So, res_{G,E}(X̂_{γ}) is a module of the form X̃_{res_{G,E}(γ)} as in Definition 10.1, applied to the group E, to the endomorphism β = res_{G,E}(γ), and to the open covering of V_{E} given by the complements of W′_{1} and W′_{3}, which are obvious disjoint since W_{1} ∩ W_{3} = ∅. Proposition 10.2 shows that this kE-module is endotrivial and coincides with the announced module X_{α}.

We have proved that res_{G,E}(X̂_{γ}) is endotrivial for all E ⊆ G as above. This is indeed enough by the following folklore result. □

Proposition 10.5. Let M ∈ StMod (kG) such that the restriction of M to every (maximal) elementary abelian p-subgroup of G is finitely generated (resp. endotrivial), then so is M.

Proof. Use Chouinard’s Theorem [14] and Frobenius reciprocity to show that the modules induced from elementary abelian subgroups generate the stable module category, see more in [11]. Then use Frobenius reciprocity again to see that an object in the stable category is compact if and only if its restriction to every elementary abelian subgroup is compact. □

Remark 10.6. In Corollary 10.4, it is not necessary to have W_{3} defined “globally”. It is enough to assume that a W′_{3} = W′_{3}(E) satisfying Condition (2) exists for every E ⊆ G such that W_{1} ∩ res_{G,E}^{∗}(V_{E}) ≠ ∅.

11. The rank of the group of endotrivial modules

In this section, we demonstrate that the construction of Section 7, as enhanced in Corollary 10.4, yields a sufficiency of modules to generate a subgroup of finite index in the group T (G) of endotrivial kG-modules. Hence, we have another proof of the rank of the torsion free subgroup of T (G). The rank was first obtained in [1], though Alperin’s original proof was only meant to apply to the case that G is a p-group. The determination of the rank of T (G) in [12] is valid for all finite groups. The proof given here follow roughly the lines of that one, and like that proof, it relies heavily on the context for the problem laid out and proved in [1]. Specifically, we have the following.

Suppose that the Sylow p-subgroup of G is not a dihedral group. There exists a collection E_{1}, …, E_{n} of elementary abelian subgroups with the property that

(1) Every E_{i} has p-rank 2,

(2) The subgroup E_{n} is normal in a Sylow p-subgroup of G,

(3) If G has p-rank 2, then the subgroups E_{1}, …, E_{n} are a complete set of representatives of the conjugacy classes of maximal elementary abelian p-subgroups of G, and

(4) If G has p-rank greater than 2, then E_{1}, …, E_{n−1} is a complete set of representatives of the conjugacy classes of maximal elementary abelian p-subgroups of G of rank 2, and E_{n} is conjugate to a subgroup of any maximal elementary abelian p-subgroup of G that has p-rank greater than 2.

Notice here that if G has no maximal elementary abelian p-subgroup of p-rank 2, then n = 1.

Proposition 11.1. (See [19] or [13].) The kernel of the product of the restriction maps

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

is finite.

Our object is to give a new proof that the rank of T (G) is the number n of subgroups in the list E_{1}, …, E_{n}. To this end, it is only necessary to show that the image of the product of the restriction maps has finite index in ∏_{i=1}^{n} T (E_{i}) ≅ ℤ^{n} (for E elementary abelian p-group of rank at least 2, we know from Dade’s Theorem [15] that ℤ →^{≅} T (E) via m ↦ Ω^{m}k for instance). Since obviously ∏ res_{G,E_{i}}(Ω^{1}k) = (1, …, 1) ∈ ℤ^{n}, it is enough for us to prove the following.

Theorem 11.2. For each i = 1, …, n − 1, there exists a number d and an endotrivial kG-module M = M(E_{i}) with the property that res_{G,E_{i}}(M) ≅ Ω^{−d}k in StMod (kE_{i}) while res_{G,E_{j}}(M) ≅ k in StMod (kE_{j}) for 1 ≤ j ≤ n and j ≠ i.

Therefore the classes of the modules Ωk, M(E_{1}), …, M(E_{n−1}) generate a subgroup of finite index in the group of endotrivial modules.

We first need the folowing result.

Lemma 11.3. Suppose that G is a finite group that has at least two classes of maximal elementary abelian p-subgroups and has a maximal elementary abelian p-subgroup of rank 2. Let r be the p-rank of G. Let E_{1}, …, E_{n} subgroups of G as above. Then there exists a number d and elements z, y_{1}, y_{2}, x_{3}, …, x_{r} in H^{d}(G, k) such that the following hold.

(1) res_{G,Z}(z) ≠ 0 where Z is the center of a Sylow p-subgroup of G.

(2) y_{1}y_{2} = 0 and moreover res_{G,E_{1}}(y_{2}) = 0 and res_{G,E}(y_{1}) = 0 for E any maximal elementary abelian p-subgroup which is not conjugate to E_{1}.

(3) The set {res_{G,E_{1}}(z), res_{G,E_{1}}(y_{1})} is a system of parameters for the ring H^{∗}(E_{1}, k).

(4) For j = 2, …, n, the set {res_{G,E_{j}}(z), res_{G,E_{j}}(y_{2})} is a system of parameters for the ring H^{∗}(E_{j}, k).

(5) For any maximal elementary abelian p-subgroup E of rank s > 2, the set

{res_{G,E}(z), res_{G,E}(y_{2}), res_{G,E}(x_{3}), … res_{G,E}(x_{s})}

is a system of parameters for the ring H^{∗}(E, k).

Proof. Notice first that the hypotheses on G require that the center Z of a Sylow p-subgroup of G be cyclic. It is a straightforward exercise in the application of Quillen’s Dimension Theorem [20, 21] (recalled in Section 1) to find elements which satisfy all of the restriction conditions on systems of parameters and on the structure of varieties. The process can be described as follows.

For any elementary abelian p-subgroup E let J_{E} denote the ideal \sqrt{Kernel res_{G,E}}. Then J_{E} is a prime ideal because the ring H^{∗}(E, k)/(Rad H^{∗}(E, k)) is an integral domain. By Quillen’s Dimension Theorem, the minimal prime ideals are the ideals J_{E} where E is a maximal elementary abelian subgroup. The first element, z is chosen so the that z is not in J_{Z}. Now the second element y_{1} is chosen to be in the intersection of all J_{E} for E not conjugate to a subgroup of E_{1}, but y_{1} not in J_{E_{1}}. In addition we want the two elements res_{G,E_{1}}(z) and res_{G,E_{1}}(y_{1}) to be a system of parameters for H^{∗}(E_{1}, k) which means that res_{G,E_{1}}(y_{1}) can not be contained in any of the finite number of maximal ideals that contain res_{G,E_{1}}(z). We can find such an element y_{1} by the following process. Let E be a maximal elementary abelian p-subgroup which is not conjugate to E_{1}. Choose a homogeneous element y_{E} in J_{E} such that res_{G,E_{1}}(y_{E}) is not contained in any of the maximal ideals that contain res_{G,E_{1}}(z). Quillen’s Theorem guarantees that we can do this. Now let y_{1} be the product of the y_{E} where E runs through set of representative of the conjugacy classes of maximal elemetary abelian subgroups that are not conjugate to E_{1}. Then, y_{1} satisfies the properties (2) and (3) of the lemma.

Now for the element y_{2} we can proceed as follows. For each elementary abelian subgroup E of rank 2, which contains Z and is not conjugate to E_{1}, choose a homogeneous element u_{E} with the properties that u_{E} ∈ J_{F} for every elementary abelian subgroup F which has rank 2, contains Z and which is not conjugate to E. Also we want that res_{G,E}(u_{E}) is not in any of the finite number of maximal ideals of H^{∗}(E, k) that contain res_{G,E}(z). This is accomplished exactly as in the previous paragraph. Next, it should be noticed that u_{E}^{m} has the same properties. Hence, there is a number d such that for every elementary abelian subgroup E of rank 2, containing Z and not conjugate to E_{1}, there is such an element u_{E} in H^{d}(G, k) having the properties specificied above. So finally, let y_{2} be the sum of the elements u_{E} where E runs through a set of representatives of the conjugacy classes of elementary abelian subgroup of having rank 2, containing Z and not conjugate to E_{1}.

By the construction, the elements y_{1} and y_{2} satisfy conditions (2), (3) and (4) except possibly for the requirement that y_{1}y_{2} = 0. However, we do know that res_{G,E}(y_{1}y_{2}) = 0 for every maximal elementary abelian subgroup E of G. Hence, by Quillen’s Theorem, y_{1}y_{2} is in the Jacobson radical of H^{∗}(G, k) and hence is nilpotent. So there is some m such that (y_{1}y_{2})^{m} = 0. We can now check that the elements z, y_{1}^{m} and y_{2}^{m} satisfy all of the first four conditions of the lemma.

The elements x_{3}, …, x_{r} can be chosen to satisfy the required condition (5), by similar arguments. We leave this part of the proof to the reader. □

Proof of Theorem 11.2. Without loss of generality, we can assume that i, in the theorem, is equal to one. We can assume that G satisfies the hypotheses, and hence also the conclusion, of Lemma 11.3. So, let z, y_{1}, y_{2}, x_{3}, …, x_{r} be as in Lemma 11.3.

We are going to construct an endotrivial module X̂_{γ} as in Corollary 10.4, for which we need to define three closed subsets W_{1}, W_{2}, W_{3} of V_{G} and an endomorphism γ: F (W_{1} ∪ W_{2}) → F (W_{1} ∪ W_{2}).

Let us denote by E the family of maximal elementary abelian p-subgroups E which are not conjugate to E_{1}. We can now set

W_{1} = V_{G}(z) ∩ res_{G,E_{1}}^{∗}(V_{E_{1}})
W_{2} = V_{G}(z) ∩ ⋃_{E∈E} res_{G,E}^{∗}(V_{E})
W_{3} = V_{G}(y_{1}) .

Note that W_{1} ∩ W_{2} = ∅ because res_{G,Z}(z) ≠ 0 by Lemma 11.3 (1). Also note that W_{1} ∩ W_{3} = ∅ because res_{G,E_{1}}(z), res_{G,E_{1}}(y_{1}) is a system of parameters in H^{∗}(E_{1}, k) by Lemma 11.3 (3).

Observe also that W_{1} ∪ W_{2} = V_{G}(z) and recall from Proposition 5.6 that End_{𝒯}(F (W_{1} ∪ W_{2})) = End_{𝒯}(F (V_{G}(z))) = (H^{∗}(G, k)[z^{−1}])^{0}. So, we can define the endomorphism

γ := y_{1}/z: F (W_{1} ∪ W_{2}) → F (W_{1} ∪ W_{2}) .

Let us see that γ satisfies the hypotheses of Corollary 10.4. Let E ⊆ G be an elementary abelian p-subgroup of G.

On the one hand, suppose then that E ∈ E. Then W_{1} ∩ res_{G,E}^{∗}(V_{E}) ⊆ W_{1} ∩ W_{2} = ∅. So, Condition (1) of Corollary 10.4 is satisfied. On the other hand, E_{1} itself satisfies Condition (2) of Corollary 10.4 by construction. Indeed, let ζ_{1} = res_{G,E_{1}}(z) and ζ_{2} = res_{G,E_{1}}(y_{1}) in H^{d}(E_{1}, k). We have W′_{1} = V_{E_{1}}(ζ_{1}) and W′_{3} = V_{E_{1}}(ζ_{2}). Obviously, res_{G,E_{1}}(γ) = ζ_{2}/ζ_{1} is a U-isomorphism where U = V_{E_{1}} ∖ (W′_{1} ∪ W′_{3}).

Then, by Corollary 10.4, M = X̂_{γ} is an endotrivial kG-module, whose restriction to each E_{i}, 2 ≤ i ≤ n is k and whose restriction to E_{1} is X_{ζ_{2}/ζ_{1}} = (X_{ζ_{1}/ζ_{2}})^{∗} = Ω^{d}k by Proposition 8.1. See also Example 10.3 (1), if necessary. □

References

[1] J. L. Alperin, A construction of endo-permutation modules, J. Group Theory 4 (2001), 3–10.

[2] P. Balmer, The spectrum of prime ideals in tensor triangulated categories, J. Reine & Angew. Math. 588 (2005), 149–168.

[3] P. Balmer and G. Favi, Gluing techniques in triangular geometry, Q. J. Math., to appear.

[4] D. J. Benson, Representations and Cohomology II: Cohomology of groups and modules, Cambridge Studies in Advanced Mathematics, vol. 31, Cambridge University Press, 1991, reprinted in paperback, 1998.

[5] D. J. Benson and J. F. Carlson, Diagrammatic methods for modular representations and cohomology, Commun. in Algebra 15 (1987), 53–121.

[6] D. J. Benson, J. F. Carlson, and J. Rickard, Complexity and varieties for infinitely generated modules, I, Math. Proc. Camb. Phil. Soc. 118 (1995), 223–243.

[7] , Complexity and varieties for infinitely generated modules, II, Math. Proc. Camb. Phil. Soc. 120 (1996), 597–615.

[8] D. J. Benson and G. Ph. Gnacadja, Phantom maps and purity in modular representation theory, I, Fundamenta Mathematicae 161 (1999), 37–91.

[9] J. F. Carlson, The variety of an indecomposable module is connected, Invent. Math., 77 (1984), 291–299.

[10] , Modules and Group Algebras, Lectures in Mathematics, ETH Zürich, Birkhäuser, 1996.

[11] , Cohomology and induction from elementary abelian subgroups, Q. Math. J. 51 (2000), 169–181.

[12] J. F. Carlson, Constructing endotrivial modules, J. Pure and Appl. Algebra, 206(2006), 83–110.

[13] J. F. Carlson and J. Thévenaz, The classification of endo-trivial modules, Invent. Math. 158 (2004), 389–411.

[14] L. Chouinard, Projectivity and relative projectivity over group rings, J. Pure & Applied Algebra 7 (1976), 278–302.

[15] E. C. Dade, Endo-permutation modules over p-groups, II, Ann. of Math. 108 (1978), 317–346.

[16] E. Friedlander and J. Pevtsova, Π-supports for modules for finite group schemes, Duke Math. J., to appear.

[17] R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, vol. 52, Springer-Verlag, Berlin/New York, 1977.

[18] M. Hovey, J. H. Palmieri, and N. P. Strickland, Axiomatic stable homotopy theory, Mem. Amer. Math. Soc., vol. 128, American Math. Society, 1997.

[19] L. Puig, Affirmative answer to a question of Feit, J. Algebra 131 (1990), 513–526.

[20] D. G. Quillen, The spectrum of an equivariant cohomology ring, I, Ann. of Math. 94 (1971), 549–572.

[21] , The spectrum of an equivariant cohomology ring, II, Ann. of Math. 94 (1971), 573–602.

[22] J. Rickard, Idempotent modules in the stable category, J. London Math. Soc. 178 (1997), 149–170.

[23] C. M. Ringel, The indecomposable representations of the dihedral 2-groups, Math. Ann. 214 (1975), 19–34.

[24] J.-L. Verdier, Des catégories dérivées des catégories abéliennes, Astérisque, vol. 239, Société Math. de France, 1996.

[25] C. A. Weibel, Introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, 1994.

Department of Mathematics, ETH Zürich, 8092 Zürich, Switzerland

  E-mail address: \p/a\u/l\ (remove slashes) dot balmer at math dot ethz dot ch

Department of Mathematics, University of Aberdeen, Aberdeen AB24 3UE, Scotland

  E-mail address: \b/e\n/s\o/n\d/j\ (without the slashes) at maths dot abdn dot ac dot uk

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

  E-mail address: /j\f/c\ (no diagonal lines) at math dot uga dot edu