PICARD GROUPS IN TRIANGULAR GEOMETRY AND APPLICATIONS TO MODULAR REPRESENTATION THEORY

PAUL BALMER

Abstract. For a tensor triangulated ℤ/p-category 𝒦, with spectrum Spc(𝒦), we construct an injective group homomorphism Ȟ^{1}(Spc(𝒦), 𝔾_{m}) ⊗ ℤ[1/p] ↪ Pic(𝒦)⊗ℤ[1/p], from the first Čech cohomology group into the Picard group of ⊗-invertible objects of 𝒦, after inverting multiplication by p on both groups. In modular representation theory, we prove that this homomorphism induces a rational isomorphism between the Picard group of the projective support variety and the group of endotrivial representations.

Introduction

The ultimate goal of this article is to prove Theorem 4.7 below, which says :

Theorem. Let G be a finite group and k be a field of characteristic p > 0. Let T (G) be the group of endotrivial kG-modules and V_{G} = Proj(H^{•}(G, k)) be the projective support variety of G over k. Then “gluing” induces an isomorphism

Pic(V_{G}) ⊗_{ℤ} ℚ ⟶^{∼} T (G) ⊗_{ℤ} ℚ

mapping O_{V_{G}}(d) ⊗ \frac{1}{d} to the kG-module Σ^{1}(k) = Ω^{−1}(k) for some suitable d > 0.

There is a rich literature on endotrivial modules, justified by their fundamental role in modular representation theory. For instance, Carlson and Thévenaz recently classified them over p-groups in [15, 16]. It is also worth mentioning that the rank of T (G), i.e. the dimension of the ℚ-vector space T (G) ⊗ ℚ, has been described by Alperin in [1]; see also Carlson [12] and Balmer-Benson-Carlson [6]. And of course, support varieties are extensively used in modular representation theory, including in the study of T (G). Despite all this, the above rational bijection between line bundles on V_{G} and endotrivial kG-modules seems to be new.

Perhaps even more interesting than the isomorphism itself, the technique for constructing it goes far beyond modular representation theory. It relies on the concept of gluing, which is very standard in algebraic geometry and which inspired the gluing construction of Balmer-Favi [7] in tensor triangular geometry.

Here, “tensor triangular geometry” refers to the study of tensor triangulated categories 𝒦 by geometric means. Initiated in [3], this theory got its proper foundation in [4] with the construction of Spc(𝒦), the spectrum of 𝒦, which is the universal topological space in which one can define reasonable closed supports supp(a) ⊂ Spc(𝒦) for all objects a ∈ 𝒦; see Def. 1.3. This space Spc(𝒦) can be equipped with a sheaf of commutative rings O_{𝒦} making it a locally ringed space

Spec(𝒦) = ( Spc(𝒦), O_{𝒦} ) .

Date: May 16, 2008.

1991 Mathematics Subject Classification. 18E30, 20C20.

Key words and phrases. Picard group, gluing, support variety, triangulated category.

Research supported by NSF grant 0654397.

In algebraic geometry, a quasi-compact and quasi-separated scheme X can be recovered as the spectrum of its ⊗-triangulated category of perfect complexes : Spec(D^{perf}(X)) ≅ X; see [4, Thm. 6.3 (a)] and [10, Thm. 9.5]. For instance, in the affine case, this reconstruction result reads Spec(K^{b}(R – proj)) ≅ Spec(R).

In modular representation theory, that is, in the study of representations of a finite group G over a field k when kG is not semisimple, i.e. when there are nonprojective kG-modules (i.e. when char(k) divides the order of G), there is a natural tensor triangulated category to consider, namely the stable module category

kG – stab = \frac{kG – mod}{kG – proj} (quotient as additive categories)

which exactly measures the non semi-simplicity of kG. Remarkably, in this case, the spectrum is isomorphic to the projective support variety : Spec(kG – stab) ≅ V_{G} =^{def} Proj(H^{•}(G, k)); see [4, Thm. 6.3 (b)]. This also holds for finite group schemes.

Other areas of mathematics are involved in tensor triangular geometry, i.e. produce important tensor triangulated categories, including stable homotopy theory, noncommutative geometry, motivic theory, and more. In general though, the spectrum Spec(𝒦) is simply a locally ringed space, not necessarily a scheme.

The first construction of this paper (Constr. 2.6) is a fraction of homomorphisms

\xymatrix{Ȟ^{1}(Spc(𝒦), 𝔾_{m}) & Pic_{loc.tr.}(𝒦) \ar[l]_{α} \ar@{^{(}->}[r] & Pic(𝒦)}

for any tensor triangulated category 𝒦, where Pic(𝒦) is the group of isomorphism classes of ⊗-invertible objects of 𝒦, where Pic_{loc.tr.}(𝒦) is the subgroup of locally trivial ⊗-invertible objects and finally, where Ȟ^{1}(Spc(𝒦), 𝔾_{m}) denotes the first Čech cohomology group of Spc(𝒦) with coefficients in the sheaf of units 𝔾_{m} = O_{𝒦}^{×} (see Constr. 1.11). Note that Ȟ^{1}(Spc(𝒦), 𝔾_{m}) is the Picard group of Spec(𝒦) when Spec(𝒦) is a scheme. The main result of the paper, Theorem 3.9, says that for 𝒦 in characteristic p, the map α is an isomorphism after inverting p. Its inverse is obtained via the gluing construction of [7]. Putting things together we get in particular :

Theorem. Suppose that 𝒦 is a tensor triangulated ℤ/p-category for some prime p. Assume also that 𝒦 is rigid (Def. 1.4) and idempotent complete (Def. 1.5) – which are mild assumptions. Then there exists a well-defined natural monomorphism

β : Ȟ^{1}(Spc(𝒦), 𝔾_{m}) ⊗_{ℤ} ℤ[1/p] ↪ Pic(𝒦) ⊗_{ℤ} ℤ[1/p]

obtained by gluing.

Section 1 is a two-page review of basic tensor triangular geometry. Section 2 recalls Picard groups, 𝔾_{m}-cocycles and the gluing technique. Section 3 is devoted to the proof of the above theorem and Section 4 focusses on modular representation theory, with the announced isomorphism Pic(V_{G}) ⊗ ℚ →^{∼} T (G) ⊗ ℚ. Let us stress that this is an application in the strict sense : The statement does not involve triangulated categories but the proof heavily does. Finally we show that this result cannot hold integrally, as explained in Example 4.9 for the quaternion group Q_{8}.

Acknowledgments : I wish to thank Dave Benson, Jon Carlson, Eric Friedlander, Nadia Mazza, Julia Pevtsova and Jacques Thévenaz for precious comments and numerous references.

1. Elements of tensor triangular geometry

Definition 1.1. A tensor triangulated category (𝒦, ⊗, 11) is a triangulated category 𝒦 (see Verdier [23, § II.1]) equipped with a symmetric monoidal structure (see Mac Lane [20, § XI.1]) ⊗ : 𝒦 × 𝒦 ⟶ 𝒦 with unit 11 ∈ 𝒦. The tensor ⊗ is also assumed exact in each variable. See more in Keller–Neeman [19].

Examples 1.2. Here are two important examples (details can be found in [5, § 4]) :

(a) For X a quasi-compact and quasi-separated scheme, e.g. X (topologically) noetherian, take 𝒦 = D^{perf}(X).

(b) For G a finite group (scheme) and k a field, take 𝒦 = D^{b}(kG – mod) or its quotient D^{b}(kG – mod)/ K^{b}(kG – proj) ≅ kG – stab. See more in § 4 below.

Definition 1.3. In [4], we define the spectrum, Spc(𝒦), of a tensor triangulated category 𝒦 as the set of proper thick triangulated ⊗-ideals 𝒫 ⊊ 𝒦 which are prime, i.e. such that a ⊗ b ∈ 𝒫 implies a ∈ 𝒫 or b ∈ 𝒫. We define for every object a ∈ 𝒦 an open subset U(a) := { 𝒫 ∈ Spc(𝒦) | a ∈ 𝒫 }. These form a basis of the topology on Spc(𝒦). This space is quasi-compact and quasi-separated, i.e. it has a basis of quasi-compact open subsets, namely precisely {U(a)}_{a∈𝒦} ; see [4, Prop. 2.14 (a)]. The support of an object a ∈ 𝒦 is the closed complement of U(a) :

supp(a) := { 𝒫 ∈ Spc(𝒦) | a ∉ 𝒫 } ⊂ Spc(𝒦) .

One has : supp(0) = ∅, supp(11) = Spc(𝒦), supp(a ⊕ b) = supp(a) ∪ supp(b), supp(Σa) = supp(a) for Σ : 𝒦 → 𝒦 the suspension, supp(c) ⊂ supp(a) ∪ supp(b) for every exact triangle a → b → c → Σa and supp(a ⊗ b) = supp(a) ∩ supp(b).

Definition 1.4. We say that the tensor triangulated category is rigid (in [5] and in [7], we said strongly closed) if there is a functor D : 𝒦^{op} → 𝒦 such that

Hom_{𝒦} (a ⊗ b, c) ≃ Hom_{𝒦} (a, D(b) ⊗ c) .

Definition 1.5. We say that an additive category 𝒦 is idempotent complete if for any idempotent e = e^{2} : a → a in 𝒦 there exists a decomposition of the object a = a_{1} ⊕ a_{2} such that e = ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ). It is easy to idempotent complete any additive category 𝒦 ↪^{ι} 𝒦̃ and 𝒦̃ is a (rigid) tensor triangulated category if 𝒦 was; see [8].

Assuming 𝒦 rigid and idempotent complete is a rather mild condition for applications : It usually holds for 𝒦 the category of compact objects in big enough ⊗-triangulated categories; see [5, Rem. 2.2]. These assumptions are used for the following general version of a theorem of Carlson in modular representation theory :

Theorem 1.6 ([5, Thm. 2.11]). Let 𝒦 be a rigid, idempotent complete, tensor triangulated category. Let a ∈ 𝒦 be an object such that supp(a) = Y_{1} ∪ Y_{2} with Y_{1} and Y_{2} disjoint closed subsets with quasi-compact open complements Spc(𝒦) ∖ Y_{i}. Then there exists a decomposition a ≅ a_{1} ⊕ a_{2} with supp(a_{i}) = Y_{i} for i = 1, 2.

Notation 1.7. The commutative ring End_{𝒦} (11) acts centrally on any morphism of 𝒦 as follows. Let f : 11 → 11 and let φ : a → b in 𝒦, one can define f · φ : a → b to be the composition from left to right in the commutative diagram :

(1.1)  \xymatrix{ & 11 ⊗ a \ar[r]^{f⊗φ} \ar[dd]_{≅} & 11 ⊗ b \ar[dr]^{≅} \ar[dd]_{≅} \\ a \ar[ur]^{≅} \ar[dr]_{≅} &  &  & b . \\  & a ⊗ 11 \ar[r]^{φ⊗f} & b ⊗ 11 \ar@{-->}[ur]_{≅}}

The left and right triangles commute by axioms of symmetric monoidal categories [20, XI.1.(4), p. 252], the middle square by naturality of the switch isomorphism. Plugging a = b = 11 in (1.1) shows that f · g and f ∘ g agree on End_{𝒦} (11) and also that End_{𝒦} (11) is commutative. It follows from the axioms of symmetric monoidal categories that composition and ⊗ are End_{𝒦} (11)-bilinear. So, for two morphisms φ_{i} : a_{i} → b_{i}, i = 1, 2, and for f ∈ End_{𝒦} (11), we have

(1.2)  (f · φ_{1}) ⊗ φ_{2} = f · (φ_{1} ⊗ φ_{2}) = φ_{1} ⊗ (f · φ_{2}) : a_{1} ⊗ a_{2} ⟶ b_{1} ⊗ b_{2} .

Construction 1.8. Recall the notion of Verdier localization of triangulated categories; see [23, § II.2]. Consider a thick triangulated subcategory 𝒥 ⊂ 𝒦. “Thick” means that for a, b ∈ 𝒦, if a ⊕ b ∈ 𝒥 then a, b ∈ 𝒥. Then 𝒦/𝒥 has the same objects as 𝒦 but new morphisms Hom_{𝒦/𝒥} (a, b) given by equivalence classes of fractions

\xymatrix{a & c \ar[l]_{s} \ar[r]^{f} & b ,}

where cone(s) ∈ 𝒥, and where two fractions are equivalent if they admit a common amplification. When 𝒦 is a (rigid) tensor triangulated category and 𝒥 is a ⊗-ideal, 𝒦/𝒥 inherits an obvious structure of (rigid) tensor triangulated category, such that the localization functor q : 𝒦 ↠ 𝒦/𝒥 is ⊗-triangulated; see [5, Prop. 2.15 (ii)]. For us, the important case of Verdier localization is when 𝒥 is equal to

𝒦_{Z} := { a ∈ 𝒦 | supp(a) ⊂ Z }

for a closed Z ⊂ Spc(𝒦), assuming that its open complement U := Spc(𝒦) ∖ Z is quasi-compact. We then define 𝒦(U) to be the idempotent completion of the corresponding Verdier localization. We have the commutative diagram, defining res_{U} :

(1.3)  [figure: 𝒦; q; ^{res^{U}}; { \atop 𝒦/𝒦_{Z}} =: 𝒦(U) .; 𝒦/𝒦_{Z}; ι]

The importance of the idempotent completion comes from Theorem 1.6 that we need to apply to 𝒦(U) as well. Indeed 𝒦(U) still is a tensor triangulated category with 11_{𝒦(U)} = res_{U}(11_{𝒦} ) and its spectrum is Spc(𝒦(U)) = U; see [7, Prop. 1.11].

Remark 1.9. This definition of 𝒦(U) originates in Thomason’s theorem saying that for a quasi-compact open subset U ⊂ X of a quasi-compact and quasi-separated scheme X, the above construction (1.3) applied to 𝒦 = D^{perf}(X) gives 𝒦(U) = D^{perf}(U); see Thomason-Trobaugh [22, § 5.2], as also explained in [3, Thm. 2.13].

Convention 1.10. For a quasi-compact open U ⊂ Spc(𝒦), we use the shorthand “on U” to mean “in 𝒦(U), after applying the appropriate functor res_{U}”.

Construction 1.11. Construction 1.8 can be used to define a presheaf of commutative rings pO_{𝒦} on Spc(𝒦), setting for every quasi-compact open U ⊂ Spc(𝒦)

(1.4)  pO_{𝒦} (U) := End_{𝒦(U)} (11) .

The sheafification of pO_{𝒦} defines the structure sheaf O_{𝒦} and gives us the ringed space Spec(𝒦) := (Spc(𝒦), O_{𝒦} ). There is a sub-presheaf p𝔾_{m} of pO_{𝒦} defined as the automorphisms of the unit :

(1.5)  p𝔾_{m}(U) := Aut_{𝒦(U)} (11) ⊂ pO_{𝒦} (U)

for U ⊂ Spc(𝒦) quasi-compact open. Sheafification gives the subsheaf 𝔾_{m} ⊂ O_{𝒦} .

2. invertible objects and cocycles

As above, 𝒦 is a tensor triangulated category, assumed rigid (Def. 1.4) and idempotent complete (Def. 1.5). Recall Convention 1.10.

Definition 2.1. An object x ∈ 𝒦 is ⊗-invertible if there exists y ∈ 𝒦 such that x ⊗ y ≃ 11. Since 𝒦 is rigid, this is equivalent to the obvious (evaluation) map x ⊗ D(x) → 11 being an isomorphism, i.e. we have y ≃ D(x). We denote by Pic(𝒦) the abelian group of isomorphism classes [x] of ⊗-invertible objects, with addition [x] + [y] = [x ⊗ y] and zero 0 = [11].

Remark 2.2. An object x ∈ 𝒦 which is locally ⊗-invertible, in the sense that Spc(𝒦) can be covered by open subsets U such that x is ⊗-invertible on U, is necessarily ⊗-invertible in 𝒦. Indeed, the natural map x ⊗ D(x) → 11 is a local isomorphism, hence an isomorphism; see [7, Lem. 6.2].

Definition 2.3. We say that a ⊗-invertible object x ∈ 𝒦 is locally trivial if Spc(𝒦) can be covered by quasi-compact open subsets U such that x ≃ 11 on U. (We call such a cover trivializing.) This defines a subgroup of Pic(𝒦), that we denote

Pic_{loc.tr.}(𝒦) := { [x] ∈ Pic(𝒦) | x is locally trivial } ⊂ Pic(𝒦) .

Example 2.4. For X a scheme, we have Pic_{loc.tr.}(D^{perf}(X)) = Pic(X). Indeed, Pic(D^{perf}(X)) = Pic(X) ⊕ ℤ^{m}, where m is the number of connected components of X and where ℤ^{m} comes from shifting complexes; see [7, Prop. 6.4].

Remark 2.5. We now use Čech cohomology of the topological space Spc(𝒦) with coefficients in the presheaf p𝔾_{m} or in the associated sheaf 𝔾_{m}; see 1.11. We only use the first group Ȟ^{1}(Spc(𝒦), p𝔾_{m}) ≅ Ȟ^{1}(Spc(𝒦), 𝔾_{m}), so let us specifically recall that one. Since the space Spc(𝒦) is quasi-separated (see Def. 1.3), we only consider finite covers by quasi-compact opens Spc(𝒦) = U_{1} ∪ … ∪ U_{n}. As always, we write

(2.1)  U_{i_{1}i_{2}...i_{r}} := U_{i_{1}} ∩ U_{i_{2}} ∩ ⋯ ∩ U_{i_{r}} for any 1 ≤ i_{1} , …, i_{r} ≤ n.

A “1-cocyle” for p𝔾_{m} over U_{∗} is the data of σ_{ji} ∈ p𝔾_{m}(U_{ij}) for all 1 ≤ i, j ≤ n satisfying the condition σ_{kj} · σ_{ji} = σ_{ki} in p𝔾_{m}(U_{ijk}). These cocycles form an abelian group under multiplication. For a collection τ_{i} ∈ p𝔾_{m}(U_{i}) we define the “1-boundary” dτ by dτ_{ji} = τ_{j}τ_{i}^{−1}. The quotient of the abelian group of 1-cocycles by the subgroup of 1-boundaries is Ȟ^{1}(U_{∗}, p𝔾_{m}). For any refinement V_{∗} of U_{∗}, there is an obvious map Ȟ^{1}(U_{∗}, p𝔾_{m}) → Ȟ^{1}(V_{∗}, p𝔾_{m}). One defines the abelian group Ȟ^{1}(Spc(𝒦), p𝔾_{m}) as the colimit of those groups. One can check directly that replacing p𝔾_{m} by its sheafification 𝔾_{m} does not change the result since the space Spc(𝒦) is quasi-separated.

Construction 2.6. There exists, as usual, a group homomorphism

α : Pic_{loc.tr.}(𝒦) → Ȟ^{1}(Spc(𝒦), 𝔾_{m}) ,

natural in 𝒦 and defined as follows. For a class ℓ ∈ Pic_{loc.tr.}(𝒦), choose some x ∈ ℓ. Choose a trivializing cover U_{∗} = (U_{i})_{i=1}^{n} of Spc(𝒦) by quasi-compact open subsets and choose isomorphisms ξ_{i} : x →^{∼} 11 on U_{i}. Define σ = σ(x, ξ) by σ_{ji} = ξ_{j} ∘ ξ_{i}^{−1} for all 1 ≤ i, j ≤ n. This defines an element σ_{ji} ∈ p𝔾_{m}(U_{i} ∩ U_{j}); see (1.5). The class of the cocyle σ = σ(x, ξ) in Ȟ^{1}(U_{∗}, p𝔾_{m}) is independent of the choice of the isomorphisms ξ_{i} : x →^{∼} 11, for other choices ξ′_{i} : x →^{∼} 11 give τ_{i} := ξ′_{i} ξ_{i}^{−1} ∈ p𝔾_{m}(U_{i}) such that σ(x, ξ′) · σ(x, ξ)^{−1} = dτ; see 1.7 if necessary. Hence [σ(x, ξ′)] − [σ(x, ξ)] = [dτ] = 0 in Ȟ^{1}(U_{∗}, p𝔾_{m}). Similarly, this class [σ(x)] ∈ Ȟ^{1}(U_{∗}, p𝔾_{m}) is independent of the chosen representative x ∈ ℓ. Indeed, for an isomorphism φ : y →^{∼} x, by the above, one can compute the class of σ(y) by using the trivialization ξ_{∗} ∘ φ, which gives the same cocycle ξ_{j} φ (ξ_{i} φ)^{−1} = ξ_{j} ξ_{i}^{−1}. Then α(ℓ) ∈ Ȟ^{1}(Spc(𝒦), 𝔾_{m}) is defined as the image of our class [σ] ∈ Ȟ^{1}(U_{∗}, p𝔾_{m}) in the colimit :

Ȟ^{1}(U_{∗}, p𝔾_{m}) ⟶ colim_{U_{∗}} Ȟ^{1}(U_{∗}, p𝔾_{m}) = Ȟ^{1}(Spc(𝒦), p𝔾_{m}) ≅ Ȟ^{1}(Spc(𝒦), 𝔾_{m})
[σ]  ⟼  α(ℓ)

replacing finally the coefficients in the presheaf p𝔾_{m} by the associate sheaf 𝔾_{m}. Sending [σ] in the colimit over the open covers U_{∗} (under refinement), makes the construction independent of the choice of the trivializing cover U_{∗}.

As a partial converse to the above, we now recall the gluing technique of [7].

Definition 2.7. A gluing data is a triple (U_{∗}, x_{∗}, σ_{∗∗}), where U_{∗} = (U_{i})_{i=1}^{n} is a finite open cover of the spectrum Spc(𝒦) by quasi-compact open subsets, where x_{∗} = (x_{i})_{i=1}^{n} is a collection of objects x_{i} ∈ 𝒦(U_{i}) on each U_{i} and finally σ = (σ_{ij})_{1≤i,j≤n} is a collection of isomorphisms σ_{ji} : x_{i} →^{∼} x_{j} on U_{ij} (see (2.1) above) satisfying the cocyle condition : σ_{kj} ∘ σ_{ji} = σ_{ki} on U_{ijk} for every 1 ≤ i, j, k ≤ n.

We call gluing of the data (U_{∗}, x_{∗}, σ_{∗∗}) a pair (x, ξ_{∗}) consisting of an object x ∈ 𝒦 and a collection ξ_{∗} = (ξ_{i})_{i=1}^{n} of isomorphisms ξ_{i} : x →^{∼} x_{i} on U_{i} compatible with σ_{∗∗}, i.e. such that σ_{ji} ∘ ξ_{i} = ξ_{j} on U_{ij} for every 1 ≤ i, j ≤ n:

(2.2)  \xymatrix{ & x_{i} \ar[dd]^{σ_{ji}} \\ x \ar[ur]^{ξ_{i}} \ar[dr]_{ξ_{j}} \\  & x_{j} .}

We alternatively say that the gluing data (U_{∗}, x_{∗}, σ_{∗∗}) admits a solution (x, ξ_{∗}).

Suppose that (x, ξ_{∗}) and (y, η_{∗}) are two gluings for the same data (U_{∗}, x_{∗}, σ_{∗∗}). An isomorphism of gluings f : (x, ξ_{∗}) →^{∼} (y, η_{∗}) is an isomorphism f : x →^{∼} y in 𝒦, such that η_{i} ∘ f = ξ_{i} on U_{i} for all 1 ≤ i ≤ n.

Here, we shall only glue ⊗-invertible objects x_{i}, mostly x_{i} = 11 for all i although we need other x_{i}’s for induction arguments. In this situation, a gluing x ∈ 𝒦 as above must be ⊗-invertible in 𝒦 as well; see Remark 2.2. Using Theorem 1.6, one can prove :

Theorem 2.8 (Balmer-Favi [7, Cor. 5.10]). Let 𝒦 be a rigid (Def. 1.4) and idempotent complete (Def. 1.5) ⊗-triangulated category. Then the gluing problem for two open subsets admits a solution, unique up to possibly non-unique isomorphism.

Remark 2.9. The non-uniqueness of the isomorphism in Theorem 2.8 might sound secondary but it has consequences for the gluing of three pieces, i.e. when Spc(𝒦) = U_{1} ∪ U_{2} ∪ U_{3} and we want to glue three objects. What happens is that the gluing still exists but is not unique up to isomorphism anymore. Worst, with four pieces or more, the gluing simply does not exist in general. In the next section, we explain how to treat this problem in positive characteristic.

3. Gluing in characteristic p, up to p-th powers

As before, we assume for the whole section that 𝒦 is a rigid (Def. 1.4), idempotent complete (Def. 1.5), tensor triangulated category. We further assume that p = 0 in 𝒦 for some prime number p, that is, 𝒦 is a ℤ/p-category. This holds of course in modular representation theory, or for 𝒦 = D^{perf}(X) when X is a scheme over ℤ/p.

We consider an open cover of the spectrum by quasi-compact open subsets

(3.1)  Spc(𝒦) = U_{1} ∪ ⋯ ∪ U_{n}

and we denote by Z_{i} = Spc(𝒦) ∖ U_{i} their closed complements. Recall Convention 1.10. We start with a result which does not use p = 0 in 𝒦.

Proposition 3.1. If a morphism f : a → b in 𝒦 vanishes on each U_{i} then f^{⊗n} = 0 in 𝒦, as a morphism from the object a^{⊗n} := a ⊗ ⋯ ⊗ a (n factors) to b^{⊗n}.

Proof. This follows from [4, Prop. 2.21] but we sketch it for the reader’s convenience : Since ι : 𝒦/𝒦_{Z_{i}} ↪ 𝒦(U_{i}) is fully faithful, we have f = 0 in 𝒦/𝒦_{Z_{i}}. By a general fact of Verdier localization (see [4, Lem. 2.22] if necessary), f factors via an object c_{i} ∈ 𝒦_{Z_{i}}. Since ∩_{i} Z_{i} = ∅, the product c_{1} ⊗ ⋯ ⊗ c_{n} has empty support, hence vanishes ([5, Cor. 2.5]). So, f^{⊗n} factors via c_{1} ⊗ ⋯ ⊗ c_{n} = 0. □

Remark 3.2. We cannot deduce that if f = g on each U_{i} then f^{⊗r} = g^{⊗r} for r big enough. Set for instance f = id_{a⊕b} and g = ( \begin{matrix}1 & 0 \\ h & 1 \end{matrix} ), for h : a → b locally zero but not zero. Then g^{⊗r} ≠ id for all r ≥ 1. Hence the importance of the next result.

Lemma 3.3. Let x, y ∈ 𝒦 be ⊗-invertible objects (Def. 2.1) and let φ, ψ : x →^{∼} y be two isomorphisms such that φ = ψ on U_{i} for all 1 ≤ i ≤ n. Then φ^{⊗p^{r}} = ψ^{⊗p^{r}} : x^{⊗p^{r}} →^{∼} y^{⊗p^{r}} for r ≥ 0 large enough (p^{r} ≥ n will do).

Proof. For a ⊗-invertible object y, the functor − ⊗ y : 𝒦 → 𝒦 is an equivalence. We have in particular an isomorphism End_{𝒦} (11) →^{∼} End_{𝒦} (y), given by f ↦ f · id_{y}, using Notation 1.7. Therefore, the automorphism φ ∘ ψ^{−1} : y →^{∼} y is of the form f · id_{y} for some f : 11 →^{∼} 11, that we can write f = 1 + h, if we wish. The hypothesis φ = ψ on each U_{i} implies that h is zero on each U_{i}. So h^{⊗n} = 0 by Proposition 3.1. Now, on the ring End_{𝒦} (11), composition and ⊗ coincide (see 1.7), hence h^{n} = 0.

In short, φ = (1 + h) · ψ for h ∈ End_{𝒦} (11) such that h^{p^{r}} = 0 for r >> 0. Then

φ^{⊗p^{r}} = ((1 + h) · ψ)^{⊗p^{r}} =^{(1.2)} (1 + h)^{p^{r}} · ψ^{⊗p^{r}} = ψ^{⊗p^{r}}.

since (1 + h)^{p^{r}} = 1 + h^{p^{r}} as p = 0 in the ring End_{𝒦} (11), by assumption. □

Lemma 3.4. Let x, y ∈ 𝒦 be ⊗-invertible objects and let φ_{i} : x →^{∼} y be isomorphisms on U_{i} such that φ_{i} = φ_{j} on U_{ij}. Then there exist r ≥ 0 large enough and an isomorphism φ : x^{⊗p^{r}} →^{∼} y^{⊗p^{r}} such that φ = (φ_{i})^{⊗p^{r}} on each U_{i}.

Proof. It is a result of [7] that for a cover by two open subsets Spc(𝒦) = U ∪ V there exists a Mayer-Vietoris long exact sequence, which says in particular that for every objects a, b ∈ 𝒦 the following sequence is exact in the middle :

(3.2) [figure: Hom_{𝒦} (a, b); ^{( \begin{matrix}res \\ res \end{matrix} )}; Hom_{𝒦(U)} (a, b) ⊕ Hom_{𝒦(V)} (a, b); ^{(^{− res res} )}; Hom_{𝒦(U∩V)} (a, b)]

In general, the left-hand map is not injective but we circumvent this via Lemma 3.3. The proof proceeds by induction on the number n of open in (3.1). The case n = 1 is trivial, so let us assume n ≥ 2. Let V = U_{2} ∪ ⋯ ∪ U_{n}. By induction hypothesis, there exist r ≥ 0 and an isomorphism ψ : x^{⊗p^{r}} →^{∼} y^{⊗p^{r}} on V such that ψ = φ_{i}^{⊗p^{r}} on each U_{i}, for 2 ≤ i ≤ n. We have U_{1} ∩ V = U_{12} ∪ ⋯ ∪ U_{1n} and the morphisms φ_{1}^{⊗p^{r}} and ψ agree on each little piece U_{1i} for 2 ≤ i ≤ n by construction of ψ and since φ_{1} and φ_{i} agree on U_{1i}. Applying Lemma 3.3 to the category 𝒦(U_{1} ∩ V), we see that there exists s ≥ 1 such that φ_{1}^{⊗p^{r+s}} = ψ^{⊗p^{s}} on U_{1} ∩ V. By Mayer-Vietoris (3.2), applied to U = U_{1} and to our V, there exists a morphism φ : x^{⊗p^{r+s}} → y^{⊗p^{r+s}} in 𝒦 such that φ = φ_{1}^{⊗p^{r+s}} on U_{1} and such that φ = ψ^{⊗p^{s}} on V, which implies φ = ψ^{⊗p^{s}} = φ_{i}^{⊗p^{r+s}} on each U_{i}, for 2 ≤ i ≤ n. This morphism φ is locally an isomorphism so it is an isomorphism. □

Remark 3.5. Let (U_{∗}, x_{∗}, σ_{∗∗}) and (U_{∗}, x′_{∗}, σ′_{∗∗}) be gluing data as in Definition 2.7 for the same open cover U_{∗}. Then we can define a new gluing data (U_{∗}, x_{∗}⊗x′_{∗}, σ_{∗∗}⊗σ′_{∗∗})

with objects x_{i} ⊗ x′_{i} on U_{i} and isomorphisms σ_{ji} ⊗ σ′_{ji} : x_{i} ⊗ x′_{i} →^{∼} x_{j} ⊗ x′_{j} on U_{ij}.

Similarly, if (x, ξ_{∗}) and (x′, ξ′_{∗}) are gluings of the original data, respectively, then (x ⊗ x′, ξ_{∗} ⊗ ξ′_{∗}) is a gluing of the new one. In particular, for r ≥ 1, if (x, ξ_{∗}) is a

gluing of (U_{∗}, x_{∗}, σ_{∗∗}) then (x^{⊗r}, ξ⊗r_{∗}) is a gluing of (U_{∗}, x⊗r_{∗}, σ⊗r_{∗∗}).

Proposition 3.6 (Uniqueness of gluing up to p^{th} powers). Suppose that (x, ξ_{∗}) and (y, η_{∗}) are two gluings for the same data (U_{∗}, x_{∗}, σ_{∗∗}). Suppose that the objects x_{i} are ⊗-invertible in 𝒦(U_{i}). Then there exist r ≥ 0 and an isomorphism of gluings

(x^{⊗p^{r}}, ξ⊗p^{r}_{∗}) →^{∼} (y^{⊗p^{r}}, η⊗p^{r}_{∗}) for the data (U_{∗}, x⊗p^{r}_{∗}, σ⊗p^{r}_{∗∗}).

Proof. The isomorphisms φ_{i} := η_{i}^{−1} ∘ ξ_{i} : x ⟶^{∼} y on U_{i} agree on U_{ij} since

\xymatrix{ & x_{i} \ar[dd]^{σ_{ji}} \\ x \ar[ur]^{ξ_{i}} \ar[dr]_{ξ_{j}} &  & y \ar[ul]_{η_{i}} \ar[dl]^{η_{j}} \\  & x_{j}}

commutes on U_{ij} by definition, see (2.2). We conclude by Lemma 3.4. □

Proposition 3.7 (Existence of gluing up to p^{th} powers). Let (U_{∗}, x_{∗}, σ_{∗∗}) be a gluing data (Def. 2.7). Suppose that the objects x_{i} are ⊗-invertible in 𝒦(U_{i}). Then there exists r ≥ 0 such that the data (U_{∗}, x⊗p^{r}_{∗}, σ⊗p^{r}_{∗∗}) admits a gluing.

Proof. The proof goes by induction on the number n of open in (3.1) as above. The case n = 1 is trivial, so let us assume n ≥ 2. Consider as before V = U_{2} ∪ ⋯ ∪ U_{n} and let U ′_{∗} = (U_{i})_{i=2}^{n}. By induction hypothesis, there exists r ≥ 0 such that the data (U ′_{∗}, x⊗p^{r}_{∗}, σ⊗p^{r}_{∗∗}) admits a gluing, say, (y, η_{∗}) on V. In order to apply Theorem 2.8 to the cover Spc(𝒦) = U_{1}∪V and to x_{1}^{⊗p^{r}} ∈ 𝒦(U_{1}) and y ∈ 𝒦(V), we would need them to be isomorphic on U_{1} ∩ V, which fails a priori. Nevertheless, both (x_{1}^{⊗p^{r}}, σ_{∗1}^{⊗p^{r}}) and (y, η_{∗}) are gluings on U_{1} ∩ V for the same gluing data (U ′′_{∗}, x⊗p^{r}_{∗}, σ⊗p^{r}_{∗∗}) where U ′′_{∗} = (U_{1i})_{i=2}^{n}. By Proposition 3.6 applied to the category 𝒦(U_{1} ∩ V), there exist s ≥ 0 and an isomorphism ψ : x_{1}^{⊗p^{r+s}} →^{∼} y^{⊗p^{s}} on U_{1} ∩ V compatible with σ_{i1}^{⊗p^{r+s}} and η_{i}^{⊗p^{s}} on U_{1i} for 2 ≤ i ≤ n. We conclude by Theorem 2.8 that there exist x ∈ 𝒦 and isomorphisms ξ_{1} : x →^{∼} x_{1}^{⊗p^{r+s}} on U_{1} and ξ′ : x →^{∼} y^{⊗p^{s}} on V compatible with ψ on U_{1} ∩ V. We deduce an isomorphism ξ_{i} := η_{i}^{⊗p^{s}} ∘ res_{U_{i}}(ξ′) : x →^{∼} x_{i}^{⊗p^{r+s}} on U_{i} for all 2 ≤ i ≤ n. We have the wanted gluing (x, ξ_{∗}) of the data (U_{∗}, x⊗p^{r+s}_{∗}, σ⊗p^{r+s}_{∗∗}).

Indeed, the remaining compatibility between ξ_{∗} and σ⊗p^{r+s}_{∗∗} is visible in the following diagram of isomorphisms :

\xymatrix{ & x_{1}^{⊗p^{r+s}} \ar[d]^{ψ}_{(U_{1}∩V)} \ar[ddr]^{σ_{i1}^{⊗p^{r+s}}}^{(U_{1i})} \\ x \ar[ur]^{ξ_{1}}|{(U_{1})} \ar[r]^{ξ′}_{(V)} \ar[dr]^{ξ_{j}}_{(U_{j})} \ar[drr]_{ξ_{i}}_{(U_{i})} & y^{⊗p^{s}} \ar[d]^{η_{j}^{⊗p^{s}}}_{(U_{j})} \ar[dr]^{η_{i}^{⊗p^{s}}}|{(U_{i})} \\ & x_{j}^{⊗p^{r+s}} \ar[r]^{σ_{ij}^{⊗p^{r+s}}}_{(U_{ij})} & x_{i}^{⊗p^{r+s}}} for 2 ≤ i, j ≤ n.

Under each isomorphism, we indicate between parentheses on which open it is defined. Each triangle commutes where all its edges are defined. □

Proposition 3.8 (Vanishing on boundaries up to p^{th} powers). Suppose that (x, ξ_{∗}) is a gluing of a data (U_{∗}, 11_{∗}, σ_{∗∗}), i.e. every x_{i} = 11. Suppose that [σ_{∗∗}] = 0 in Ȟ^{1}(U_{∗}, p𝔾_{m}). Then there exist r ≥ 0 and an isomorphism x^{⊗p^{r}} →^{∼} 11 in 𝒦.

Proof. There exists τ_{∗} ∈ Aut_{𝒦(U_{∗})} (11) such that σ_{ji} = (dτ_{∗})_{ji} = τ_{j} τ_{i}^{−1} on U_{ij} for all 1 ≤ i, j ≤ n. Define φ_{i} := τ_{i}^{−1} ξ_{i} : x →^{∼} 11 on each U_{i}. The above relation and the compatibility of ξ_{∗} with σ_{∗∗} in (2.2) imply that φ_{i} = φ_{j} on U_{ij}. By Lemma 3.4, some p-th power of these φ_{i} glue into an isomorphism φ : x^{⊗p^{r}} →^{∼} 11^{⊗p^{r}} = 11. □

Putting things together, we have our central result :

Theorem 3.9. Let 𝒦 be a rigid, idempotent complete, tensor triangulated ℤ/pcategory. Let U_{∗} = (U_{i})_{i=1}^{n} be an open cover of Spc(𝒦) as in (3.1). Let σ_{ij} ∈ Aut_{𝒦(U_{ij})} (11) be a cocycle : σ_{kj} · σ_{ji} = σ_{ki} on U_{ijk} for all 1 ≤ i, j, k ≤ n. Then there exists r ≥ 0 such that the data (U_{∗}, 11_{∗}, σ⊗p^{r}_{∗∗}) admits a gluing x(r) ∈ 𝒦. Two such gluings x_{1}(r_{1}) and x_{2}(r_{2}) have isomorphic p-th powers; more precisely, there exists r_{3} ≥ 0 with x_{1}(r_{1})^{⊗p^{r_{2}+r_{3}}} ≃ x_{2}(r_{2})^{⊗p^{r_{1}+r_{3}}} . In particular, the following class in Pic_{loc.tr.}(𝒦) ⊗_{ℤ} ℤ[1/p] ⊂ Pic(𝒦) ⊗_{ℤ} ℤ[1/p] is independent of the choice of r:

β(U_{∗}; σ_{∗∗}) := [x(r)] ⊗ \frac{1}{p^{r}} .

This defines an isomorphism β : Ȟ^{1}(Spc(𝒦), 𝔾_{m})⊗_{ℤ}ℤ[1/p] →^{∼} Pic_{loc.tr.}(𝒦)⊗_{ℤ}ℤ[1/p] natural in 𝒦 and inverse to the homomorphism α ⊗ id_{ℤ[1/p]} from Construction 2.6.

Proof. Existence of the gluing x(r) is Proposition 3.7. Uniqueness is Proposition 3.6 applied to x_{1}(r_{1})^{⊗p^{r_{2}}} and x_{2}(r_{2})^{⊗p^{r_{1}}} which are two gluings of the same data (U_{∗}, 11_{∗}, σ⊗p^{r_{1}+r_{2}}_{∗∗}). Hence β(U_{∗}, σ_{∗∗}) ∈ Pic_{loc.tr.}(𝒦) ⊗ ℤ[1/p] is well-defined. It is multiplicative in the cocycle by Remark 3.5 and vanishes on boundaries by Proposition 3.8. Hence we have a well-defined homomorphism Ȟ^{1}(U_{∗}, p𝔾_{m}) ⊗ ℤ[1/p] ⟶ Pic_{loc.tr.}(𝒦) ⊗ ℤ[1/p] for all open cover U_{∗} of Spc(𝒦). This is obviously compatible with the refinement of open covers (a solution of the gluing for U_{∗} gives a solution for any refinement). So, we get the announced homomorphism β. It is straightforward to check that β is, by its very construction, a two-sided inverse of α ⊗ id : Pic_{loc.tr.}(𝒦) ⊗ ℤ[1/p] ⟶ Ȟ^{1}(Spc(𝒦), 𝔾_{m}) ⊗ ℤ[1/p]. □

4. Applications to modular representation theory

Let G be a finite group (scheme) and V_{G} = V_{G}(k) = Proj(H^{•}(G, k)) its projective support variety over a field k, whose characteristic p divides the order of G. Let 𝒦 = kG – stab be the stable category, which has finitely generated kG-modules as objects and, as morphisms, kG-homomorphisms modulo those which factor via a projective. The tensor product of two kG-modules is M ⊗_{k} N with diagonal action of G. This 𝒦 is a rigid (Def. 1.4) with usual dual D(M) = M^{∗}, idempotent complete (Def. 1.5), ⊗-triangulated ℤ/p-category. See more in Carlson [11] or Benson [9].

A kG-module M is ⊗-invertible in kG – stab (see Def. 2.1) if and only if M⊗M^{∗} ≃ k in kG – stab, which is equivalent to End_{k}(M) ≃ k⊕(proj) in kG – mod. The latter has been dubbed “endotrivial” in the literature. So, the group of stable isomorphism classes of endotrivial modules T (G) is nothing but Pic(kG – stab).

By [4, Thm. 6.3 (b)], we have an isomorphism Spec(kG – stab) ≃ V_{G}(k), natural in G (see Rem. 4.3), and which is compatible with the support varieties of modules, namely for every finitely generated kG-module M, we have supp(M) = V_{G}(M).

Remark 4.1. At this stage, we can make sense of what should be “the category kG – stab on some open U ⊂ V_{G}”, by means of triangular geometry; see Section 1. (Note that V_{G} is noetherian, so all open subsets are quasi-compact.) The answer is the ⊗-triangulated category 𝒦(U) for 𝒦 = kG – stab, as in Construction 1.8. We also have the restriction functor res_{U} : kG – stab → 𝒦(U). By Remark 1.9, in algebraic geometry, the new category 𝒦(U) has the “same nature” as the original 𝒦. This is far from being the case in modular representation theory :

Proposition 4.2. Let G be a finite group and k be an algebraically closed field. Set 𝒦 := kG – stab. Let U ⊊ V_{G} be a proper principal open of Krull dimension at least one. Consider 𝒦(U) as in Construction 1.8. Then 𝒦(U) cannot be equivalent to kG′ – stab, as tensor triangulated category, for any finite group G′.

Proof. Let ζ ∈ H^{d}(G, k) with d > 0. The corresponding principal open is

(4.1)  U = D(ζ) := { 𝔭 ∈ Proj(H^{•}(G, k)) | ζ ∉ 𝔭 } ⊂ V_{G} .

In 𝒦 = kG – stab, we can consider ζ as a morphism ζ : Ω^{d}(k) → k, where Ω = Σ^{−1} is the Heller loop functor. The cone of ζ is the suspension of Carlson’s module L_{ζ} which has support V (ζ), the closed complement of D(ζ); see [9, Prop. II.5.9.1, p. 186]. Hence res_{U}(ζ) is an isomorphism in 𝒦(U), by the very construction of 𝒦(U). This means that 𝒦(U) is periodic : Σ^{d}(11) ≃ 11 in 𝒦(U) for some d ≠ 0.

Let now G′ be a finite group. If, ab absurdo, the stable category kG′ – stab is equivalent to 𝒦(U), as a ⊗-triangulated category, then kG′ – stab must be periodic as well, i.e. one has Σ^{d}(11) ≃ 11 in kG′ – stab, for some d ≠ 0. But 11 = k in kG′ – stab, and this forces Spc(kG′ – stab) ≃ V_{G′}(k) to have Krull dimension zero, see [9, Cor. II.5.10.3, p. 191]. This contradicts Spc(kG′ – stab) ≃ Spc(𝒦(U)) ≅ U which has dimension at least one by assumption. □

Remark 4.3. Given a subgroup H ⊂ G, restriction defines a ⊗-triangulated functor res_{G,H} : kG – stab → kH – stab. This induces a group homomorphism res_{G,H} : T (G) → T (H) and a morphism of schemes res_{G,H}^{∗} : V_{H} → V_{G}. Note finally that Pic(−) is contravariant on schemes, so res_{G,H} also induces res_{G,H} : Pic(V_{G}) → Pic(V_{H}). This is what we mean here by “naturality” in the group G, not more.

Construction 2.6 and Theorem 3.9 have the following incarnation here :

Corollary 4.4. There exists a fraction of homomorphisms, natural in G (Rem. 4.3)

(4.2)  \xymatrix{Pic(V_{G}) & Pic_{loc.tr.}(kG – stab) \ar[l]_{α} \ar@{^{(}->}[r] & T (G)}

and α becomes an isomorphism after inverting p on these groups. The resulting natural injective homomorphism β : Pic(V_{G}) ⊗_{ℤ} ℤ[1/p] ↪ T (G) ⊗_{ℤ} ℤ[1/p] can be described by the gluing of 𝔾_{m}-cocycles as in Section 3. □

Example 4.5. By the Evens-Venkov Theorem, the k-algebra H^{•}(G, k) is finitely generated. So, there exist d ≥ 1 and ζ_{1}, …, ζ_{n} ∈ H^{d}(G, k) such that V_{G} = U_{1} ∪ ⋯ ∪ U_{n} where U_{i} := D(ζ_{i}) are the principal open subsets of V_{G} defined by ζ_{i} as in (4.1) above. We refer to such ζ_{1}, …, ζ_{n} as a system of parameters in H^{d}(G, k). As in the proof of Proposition 4.2, each ζ_{i} : Ω^{d}(k) → k is an isomorphism “on U_{i}” in the sense of triangular geometry, see Construction 1.8. This proves that Ω^{d}(k) is locally trivial, i.e. defines a class [Ω^{d}(k)] ∈ Pic_{loc.tr.}(𝒦); see Def. 2.3. By Construction 2.6, the cocycle σ_{∗∗} associated to [Ω^{d}(k)] under the homomorphism α : Pic_{loc.tr.}(𝒦) → Ȟ^{1}(U_{∗}, p𝔾_{m}) is simply σ_{ji} = ζ_{j} ∘ ζ_{i}^{−1} ∈ Aut_{𝒦(U_{ij})} (11). On the other hand, it is easy to check that this class [σ] ∈ Ȟ^{1}(V_{G}, p𝔾_{m}) ≅ Pic(V_{G}) corresponds to the line bundle O_{V_{G}}(−d) over V_{G}. Indeed, each ζ_{i} ∈ H^{d}(G, k) defines a global homomorphism O_{V_{G}}(−d) → O_{V_{G}} (see Hartshorne [18, § II.5, p. 116]) which is an isomorphism on U_{i}. We have the commutative diagram

(4.3)  \xymatrix{ & O_{V_{G}} \ar[dd]^{σ_{ji}=ζ_{j} ζ_{i}^{−1}} \\ O_{V_{G}}(−d) \ar@{-->}[ur]^{ζ_{i}}_{≃} \ar@{-->}[dr]_{ζ_{j}}^{≃} \\  & O_{V_{G}}}

of sheaves on U_{ij} = D(ζ_{i} ζ_{j}), analogous to (2.2) in 𝒦(U_{ij}). This shows that the sheaf O_{V_{G}}(−d) is the gluing corresponding to our cocycle σ_{∗∗} as announced; see [18, Ex. II.1.22]. This proves that α([Ω^{d}(k)]) = [O_{V_{G}}(−d)] for that specific integer d ≥ 1. Now, computing in Pic(V_{G}) ⊗ ℚ, we have

(α ⊗ id_{ℚ})([Ω^{1}(k)] ⊗ 1) = (α ⊗ id_{ℚ})([Ω^{d}(k)] ⊗ \frac{1}{d}) = [O_{V_{G}}(−d)] ⊗ \frac{1}{d}

and (β⊗id_{ℚ})(O_{V_{G}}(−d)⊗ \frac{1}{d}) = (α⊗id_{ℚ})^{−1}(O_{V_{G}}(−d)⊗ \frac{1}{d}) = [Ω^{1}(k)]⊗1 in T (G)⊗ℚ.

Remark 4.6. Note that the O_{V_{G}}-module O_{V_{G}}(m) is not necessarily a line bundle for all m ∈ ℤ. This comes from H^{•}(G, k) not being generated by H^{1}(G, k). Therefore, we cannot replace the above O_{V_{G}}(−d) ⊗ \frac{1}{d} by O_{V_{G}}(−1) ⊗ 1, a priori.

Theorem 4.7. Let G be a finite group and k be a field of characteristic p. Then the gluing map β of Corollary 4.4 induces an isomorphism, natural in G (see Rem. 4.3),

β : Pic(V_{G}) ⊗ ℚ ⟶^{∼} T (G) ⊗ ℚ .

Moreover, this isomorphism maps [O_{V_{G}}(−d)] ⊗ \frac{1}{d} to [Ω(k)] ⊗ 1 for every d ≥ 1 such that H^{d}(G, k) contains a system of parameters (see Ex. 4.5).

Proof. By Corollary 4.4, it suffices to prove that the subgroup Pic_{loc.tr.}(kG – stab) has finite index in T (G). In [6, § 10], we proved that T (G) admits a subgroup of finite index generated by endotrivial modules X which are the result of gluing two copies of k over a cover of V_{G} by two open pieces. These modules X belong to the subgroup Pic_{loc.tr.}(kG – stab) by definition (Def. 2.3), hence the result. □

Remark 4.8. For experts, we outline the construction, given in [6], of enough locally trivial modules to prove that Pic_{loc.tr.}(kG – stab) is of finite index in T (G). When G has p-rank one, T (G) is finite and the result is obvious. So let us assume that G has p-rank at least two. Then there exist elementary abelian p-subgroups E_{1}, …, E_{n} ⊂ G of p-rank two, such that :

(a) when G has p-rank two, E_{1}, …, E_{n} form a complete set of representatives of the conjugacy classes of maximal elementary abelian p-subgroups;

(b) when G has p-rank three or more, E_{1}, …, E_{n−1} form a complete set of representatives of the conjugacy classes of maximal elementary abelian psubgroups of p-rank two and every maximal elementary abelian p-subgroup of p-rank three or more contains a conjugate of E_{n}.

The homomorphism ρ : T (G) → ∏_{i=1}^{n} T (E_{i}) ≃ ℤ^{n} has finite kernel. (This n will be the dimension of T (G)⊗ℚ.) It now suffices to prove that for each i = 1, ..., n we can produce an endotrivial module M_{i} ∈ Pic_{loc.tr.}(kG – stab) whose image under ρ is (k, …, k, Ω^{r_{i}}(k), k …, k) in ∏_{i=1}^{n} T (E_{i}) for some r_{i} ≠ 0. Moreover, by Example 4.5, we have Ω^{d}(k) ∈ Pic_{loc.tr.}(kG – stab) for some d > 0, whose image under ρ is (Ω^{d}(k), …, Ω^{d}(k)), so it suffices to produce the above M_{i} only for i = 1, …, n − 1 and we can assume i = 1. Then, using Quillen’s Theorem, one shows the existence of two elements y, z ∈ H^{r_{1}}(G, k) for some r_{1} > 0, such that y restricts to zero on every maximal elementary abelian p-subgroup not conjugate to E_{1} and such that y, z restrict to a system of parameters in H^{•}(E_{1}, k) (see Ex. 4.5). One then considers the disjoint closed subsets W_{1} = V (z) ∩ res_{G,E_{1}}^{∗}(V_{E_{1}}) and W_{2} = V (y). Their open complements (U_{i} := V_{G}∖W_{i})_{i=1,2} form an open cover of V_{G} = Spc(kG – stab). The support of the cone of y : Ω^{r_{1}}(k) → k is exactly W_{2}, hence y is an isomorphism on U_{2} and a fortiori on U_{1} ∩ U_{2}. Moreover, by construction of y, V (y) = W_{2} contains all irreducible components of V_{G} except res_{G,E_{1}}^{∗}(V_{E_{1}}). So, V (z) ⊂ W_{1} ∪ W_{2}, meaning that z : Ω^{r_{1}}(k) → k is an isomorphism on U_{1}∩U_{2}. In short, we have an isomorphism

\xymatrix{k & Ω^{r_{1}}(k) \ar[l]_{z} \ar[r]^{y} & k}

on U_{1} ∩ U_{2} that we can use to glue two copies of k, that is, x_{i} = k on U_{i} for i = 1, 2, along the isomorphism σ_{21} = y ∘ z^{−1} on U_{1} ∩ U_{2}; see Thm. 2.8. We proved in [6, Thm. 10.2] that the resulting gluing x ∈ Pic_{loc.tr.}(kG – stab) is the wanted endotrivial module M_{1} , that is, its image under ρ is (Ω^{r_{1}}(k), k, …, k).

Example 4.9. Theorem 4.7 does not hold integrally. Take for instance G = Q_{8} the quaternion group of order 8 and p = 2. Then G has 2-rank one, hence V_{G} is a point and Pic(V_{G}) = 0, but T (G) = ℤ/4 when k contains no cubic root of unity and T (G) = ℤ/4 ⊕ ℤ/2 otherwise. See Dade [17] or Carlson–Thévenaz [14, Thm. 6.3].

We end the paper with an obvious application of Theorem 4.7 :

Corollary 4.10. Let H ⊂ G be a subgroup inducing an isomorphism V_{H} →^{∼} V_{G} on support varieties (e.g. H controls p-fusion, see Mislin [21] or Alperin [2]). Then res_{G,H} ⊗ id_{ℚ} : T (G) ⊗ ℚ →^{∼} T (H) ⊗ ℚ is an isomorphism. □

Example 4.11. Corollary 4.10 does not hold integrally either. Let p > 2 be an odd prime and let n be an integer 2p ≤ n < p^{2}. Let G be the symmetric group on n letters. As explained in Carlson–Mazza–Nakano [13, Thm. A (c)], we have T (G) ≃ ℤ ⊕ (ℤ/2)^{2} when 2p ≤ n < 3p and T (G) ≃ ℤ ⊕ ℤ/2 when 3p ≤ n < p^{2}. On the other hand, the p-Sylow subgroup P ⊂ G is abelian (even elementary abelian of p-rank [\frac{n}{p}]). So, by a celebrated theorem of Burnside, its normalizer H := N_{G}(P) controls p-fusion. Yet, the torsion part of T (H) is ℤ/(p − 1) ⊕ ℤ/2 by [13, § 6].

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Paul Balmer, Mathematics Department, UCLA, Los Angeles, CA 90095-1555, USA

E-mail address: balmer@math.ucla.edu
URL: http://www.math.ucla.edu/∼balmer