\underline{Contents}

On the notion of order in the stable module category^{∗}

Martin Langer

Abstract The notion of order in triangulated categories, as introduced by Schwede, is in-

vestigated in the case of the stable category of kG-modules, where k is a field of characteristic p and G is a finite group. For Tate cohomology classes ζ of even degree, we obtain bounds on the ζ-order which are amazingly similar to corresponding results on the p-order in the stable homotopy category.

On our way we introduce a power operation P_{1} on Tate cohomology which serves as an obstruction for the ζ-order to be larger than its minimal possible value. Furthermore, it enables us to compute certain higher Massey products explicitly.

Contents

1 Prerequisites  6
1.1 The stable module category . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 The graded center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 The notion of order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Tensor powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5 Symmetric and exterior powers . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.6 Symmetric and exterior powers in the stable category . . . . . . . . . . . . 15
2 The lower bound  16
3 The power operation  21
3.1 Negative Ext-groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2 A map of cochain complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3 Definitions and basic properties . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.4 The Cartan formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.5 Comparison with Steenrod operations . . . . . . . . . . . . . . . . . . . . . 38
3.6 The power operation for G = ℤ/pℤ × ℤ/pℤ . . . . . . . . . . . . . . . . . . 43
3.7 Further examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4 An obstruction for higher order  55
4.1 A commutative square . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2 Completion of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

∗This research was supported by the Deutsche Forschungsgemeinschaft within the graduate program ‘Homotopy and Cohomology’ (No. GRK 1150)

5 Toda brackets  62
5.1 Toda brackets and Massey products . . . . . . . . . . . . . . . . . . . . . . 63
5.2 Toda brackets and order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.3 Toda brackets and coherent modules . . . . . . . . . . . . . . . . . . . . . . 73
5.4 Example for equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6 A counterexample  81

Introduction

Introduction

Let G be a finite group and k be a field of prime characteristic p > 0, and let us denote by 𝔪𝔬𝔡-kG the category of finitely generated, right kG-modules. In this category, the classes of injective and projective modules coincide, and this allows us to form its stable category \underline{𝔪𝔬𝔡}-kG, whose objects are the same as in 𝔪𝔬𝔡-kG, and the morphisms are given by morphisms in 𝔪𝔬𝔡-kG modulo the subgroup of those morphisms factoring through a projective module. On \underline{𝔪𝔬𝔡}-kG we have the translation functor Σ which can be defined as follows: Choose an inclusion i : k ↪ P of the trivial kG-module k into a projective module P. For every object X define ΣX to be the cokernel of the (injective) map i ⊗ id_{X} : X → P ⊗ X. This functor serves as a translation functor of a triangulated structure on \underline{𝔪𝔬𝔡}-kG (where the exact triangles are ‘up to projectives’ those coming from short exact sequences in 𝔪𝔬𝔡-kG).

Let Ω denote the inverse of the shift functor Σ. Then we have

\underline{Hom}_{kG}(Ω^{n}X, Y) = \underline{𝔪𝔬𝔡}-kG(Ω^{n}X, Y) ≅ Êxt_{kG}^{n}(X, Y),

where Êxt denotes Tate Ext-groups. Suppose we are given a non-zero Tate cohomology class [ζ] ∈ Ĥ^{n}(G) = Êxt^{n}(k, k) represented by an unstable (surjective) map ζ : Ω^{n}k → k. Denote by L_{ζ} the kernel of the map ζ; then we get an exact triangle

⋯ → L_{ζ} → Ω^{n}k → k → Ω^{−1}L_{ζ} =: k/ζ → …

On k/ζ, we still have a multiplication by ζ, and the following theorem will be the starting point of our discussion.

Theorem (Carlson, [2]). If p is odd and n is even, then multiplication by ζ on k/ζ vanishes; i.e.,

Ω^{n}k ⊗ k/ζ ⟶^{ζ⊗id} k/ζ

is stably zero.

If p = 2 then this does not need to be true. For instance, one can take G = ℤ/2 × ℤ/2, then there is some non-zero [ζ] ∈ Ĥ^{−2}(G) such that multiplication by ζ does not vanish on k/ζ. But why is the prime 2 special here?

In topology, we have a similar phenomenon. Suppose we are given a triangulated category C, some object X ∈ C and a natural number m. On X, we have the ‘multiplication by m’, i.e., m · Id_{X} : X → X; denote by X/m some choice of cone of this map. On this cone, we also have a multiplication by m. If C is the stable homotopy category, and p is a prime, then the mod-p Moore spectrum S/p (where S denotes the sphere spectrum) has a multiplication by p, which is zero if p is odd, but non-zero if p = 2.

Motivated by his proof of the Rigidity Theorem [24], Schwede introduced the notion of m-order (see [25]), which measures ‘how strongly zero multiplication by m on some object is’. The m-order m-ord(X) of an object X ∈ C is an element of {0, 1, 2, …, ∞}, defined inductively by the following condition: m-ord(X) ≥ k if and only if for all objects K in C and all morphisms f : K → X there is an extension f̂: K/m → X such that for some (and hence any) cone Cf̂ of f̂, m-ord(Cf̂) ≥ k − 1. Here, extension means that the following diagram commutes:

\xymatrix{… \ar[r] & K \ar[r]^{m·} & K \ar[d]_{f} \ar[r] & K/m \ar@{-->}[dl]^{f̂} \ar[r] & … \\  &  & Y}

For instance, m-ord(X) ≥ 1 if and only if multiplication by m vanishes on X, which is what you would expect from a reasonable definition of order. It can be compared with, say, multiplicities of zeroes of real polynomials.

In topology, we have the following results:

Theorem (Schwede, [24] and [25]). Let p be a prime number and C be a topological triangulated category, i.e., the full subcategory of the homotopy category of a stable Quillen model category.

(t1) For any object X of C, the object X/p has p-order at least p − 2.

(t2) In the stable homotopy category SHC, the mod-p Moore spectrum S/p has p-order exactly p − 2.

(t3) We can also go one step further. If the morphism α_{1} ∧ X : Σ^{2p−3}X → X is divisible by p, then X/p has p-order at least p − 1. Here α_{1} : Σ^{2p−3}S → S is a generator of the p-torsion in the stable homotopy groups of spheres in the (2p − 3)-stem.

To a certain extent, statement (t2) explains the phenomenon described above: multiplication by p on S/p vanishes if and only if the p-order of S/p is at least 1, and this is the case exactly if p ≥ 3.

Let us turn back to the case C = \underline{𝔪𝔬𝔡}-kG and see what we get from the notion of order. It is certainly not very interesting to consider multiplication by m in our algebraic situation, so we extend the definition of order to elements in the graded center of C. In degree n, the graded center of the triangulated category C consists of all natural transformations ζ from the identity functor to the functor Σ^{n} which commute with the functor Σ up to the sign (−1)^{n}. The notion of order as defined above can be modified to work for arbitrary elements ζ in the graded center, so we obtain a number ζ-ord(X) for every object X in C. Now suppose we are given a cohomology class in Ĥ^{n}(G), represented by some unstable map ζ : Ω^{n}k → k, which in turn induces an element in the graded center, also denoted by ζ:

[figure: Ω^{n}X; ζ; ≃; X; ≃; Ω^{n}k ⊗ X; ζ⊗id; k ⊗ X]

This should be thought of as multiplication by the class [ζ]. With all this language at hand, Carlson’s theorem above reads ζ-ord(k/ζ) ≥ 1 for all primes p ≥ 3. This will be generalized by the following theorem whose proof is the main objective of this thesis.

Main Theorem. Suppose that k is a field of prime characteristic p and G is a finite group.

(a1) Let ζ ∈ Ĥ^{n}(G) be a Tate cohomology class of even degree n. For any object X in \underline{𝔪𝔬𝔡}-kG, the ζ-order of X/ζ is at least p − 2.

(a2) For every prime p and every field k of characteristic p, there exists a group G and a cohomology class ζ ∈ Ĥ^{∗}(G) of even degree such that the ζ-order of k/ζ is exactly p − 2.

(a3) Suppose that n > 0. Recall that the Steenrod reduced powers act on group cohomology H^{∗}(G, k). The first non-trivial Steenrod operation gives an obstruction for the ζorder of X/ζ to be at least p − 1. More precisely, if P_{1}ζ ⊗ X is divisible by ζ then ζ-ord(X/ζ) ≥ p − 1.

Here, P_{1} ζ is the first non-trivial Steenrod power operation, that is,

P_{1} ζ = \begin{cases}Sq_{1} ζ = Sq^{n−1} ζ & if p = 2, \\ βP^{\frac{n}{2} −1}ζ & if p is odd. \end{cases}

The plan of this thesis is as follows. As a first step, we recall several known facts about the objects we are going to work with. In the second section we prove the lower bound (a1) of our Main Theorem. In §3 we introduce a new power operation P_{1} on Tate cohomology which extends the Steenrod operation P_{1} above to negative degrees, at the price of introducing a certain indeterminacy. Basic properties of the new operation are shown, and the operation is computed for elementary abelian p-groups. In the fourth section we show the obstruction statement (a3) of the Main Theorem. In §5 we show that certain higher Massey products give upper bounds on the order, and we will use the new power operation to compute such Massey products and thereby find an example for (a2). The last section is devoted to the question what happens with the statement of (a1) if we allow ζ to be any element of even degree in the graded center of \underline{𝔪𝔬𝔡}-kG.

Acknowledgements In the first place, I would like to thank my advisor Stefan Schwede for suggesting the project, for his interest in my problems and my solutions occurring while finishing this thesis, and for the time he spent discussing the project’s details with me. I would also like to thank Dave Benson for his interest in the project and for the mathematical discussions we had during my stay in Aberdeen. I thank the BIGS and GRK1150 for all the financial support. Lots of thanks go to my fellow students in Bonn, above all to Matthias for listening to all these weird problems on modular representation theory.

Finally, I would like to thank my wife Britta for her patience and her love.

1. Prerequisites

1 Prerequisites

1.1 The stable module category

We begin with a brief introduction to the stable module category; a more detailed exposition can be found in [3]. Throughout this thesis, we will work with a fixed finite group G and a field k of characteristic p > 0. Consider the category 𝔪𝔬𝔡-kG of finitely generated right modules over the group algebra kG. On this category, we have a symmetric monoidal tensor product X ⊗_{k} Y defined for any two objects X, Y as follows. As a k-vector space, we take the usual tensor product X ⊗_{k} Y; the G-module structure is then given by the rule (x ⊗ y) · g = (x · g) ⊗ (y · g) for group elements g ∈ G. We will always drop the k from the notation and simply write ⊗ for the tensor product. Taking tensor products with a fixed module is an exact functor, since this is true for k-vector spaces and exactness does not depend on the G-module structure. The one-dimensional vector space k carries the structure of a trivial G-module by setting λ · g = λ for all λ ∈ k and g ∈ G. This trivial module serves as a unit object for the symmetric monoidal product ⊗ in the sense that there are natural isomorphisms X ⊗ k ≅ k ⊗ X ≅ X of kG-modules, given by x ⊗ 1 ↦ 1 ⊗ x ↦ x for all x ∈ X. Whenever X and Y are kG-modules, we can equip Hom_{k}(X, Y) with a G-module structure by setting (g · f)(x) = f(x · g^{−1}) · g for all g ∈ G, f ∈ Hom_{k}(X, Y). We will simply write Hom(X, Y) for this G-module and write X^{♯} = Hom(X, k) where k is the trivial G-module. The algebra kG is self-injective, and therefore in the category 𝔪𝔬𝔡-kG the classes of projective and injective modules coincide. This allows us to form the stable module category \underline{𝔪𝔬𝔡}-kG, defined as follows. The objects of \underline{𝔪𝔬𝔡}-kG are the same as in 𝔪𝔬𝔡-kG; for any two objects A, B, the group of morphisms is given by \underline{Hom}_{kG}(A, B) = Hom_{kG}(A, B)/PHom(A, B), where PHom(A, B) denotes the set of all morphisms A → B which factor through a projective module. It is easily verified that the set PHom(A, B) is actually a subgroup of Hom_{kG}(A, B) and that the construction of \underline{Hom} is compatible with composition, so one indeed obtains a category \underline{𝔪𝔬𝔡}-kG. We refer to 𝔪𝔬𝔡-kG and \underline{𝔪𝔬𝔡}-kG as the unstable and stable categories, respectively. There is a canonical functor from the unstable to the stable category which allows us to consider unstable maps as morphisms in the stable world. An unstable morphism will be called stable isomorphism if it maps to an isomorphism under the canonical functor. For instance, the projection X ⊕ P → X and the inclusion X → X ⊕ P are stable isomorphisms for projective modules P. We will sometimes denote stable isomorphisms by ≅_{st}.

The stable category carries the structure of a triangulated category. We will only sketch the construction here and leave the technical details to the textbooks. Let us begin with the shift functor Σ. For every module X, choose a short exact sequence X ↪ I(X) ↠ ΣX with an injective module I(X). Any map X → Y can be lifted to a map I(X) → I(Y) which in turn induces a map ΣX → ΣY, the stable class of which only depends on the class of the map we started with. This implies that Σ is a functor on the stable category. In general, Σ will be a self-equivalence of \underline{𝔪𝔬𝔡}-kG, but if we are careful enough when choosing the I(X) we can achieve that Σ is an automorphism of \underline{𝔪𝔬𝔡}-kG (see, e.g., [7], §2). We will denote by Ω the inverse functor of Σ.

If Σ′ is a self-equivalence arising from a construction as above, then Σ and Σ′ are

1.1 The stable module category

isomorphic as functors ([7], Remark on page 13). Let us apply this fact in a particular case. Whenever P is a projective module in 𝔪𝔬𝔡-kG and X is an arbitrary kG-module, then P ⊗ X is also projective (see [11], Theorem 3.2). This implies that the tensor product on 𝔪𝔬𝔡-kG descends to a symmetric monoidal tensor product on \underline{𝔪𝔬𝔡}-kG which we also denote by ⊗. Now the exact sequence k ↪ I(k) ↠ Σk yields an exact sequence X ↪ I(k) ⊗ X ↠ Σk ⊗ X. By the remarks above, the functors Σ and Σk ⊗ − are isomorphic.

Next we are going to define the exact triangles. Let f : X → Y be a morphism in 𝔪𝔬𝔡-kG. Then we obtain a diagram

[figure: X; i; f; I(X); ΣX; Y; g; C; h; ΣX]

where C is the pushout of f and i. The triangles X ⟶^{f} Y ⟶^{g} C ⟶^{h} ΣX in \underline{𝔪𝔬𝔡}-kG arising this way will be called standard triangles. A triangle in \underline{𝔪𝔬𝔡}-kG is called exact if and only if it is isomorphic to a standard triangle. Now one has to prove that this structure indeed satisfies the axioms of a triangulated category; the interested reader may find this in [6], Theorem 9.4. In fact, the closed symmetric monoidal structure on \underline{𝔪𝔬𝔡}-kG (given by ⊗ and Hom) is compatible with the triangulation in the sense of [10], appendix A.

We claim that short exact sequences A ↪ B ↠ C in 𝔪𝔬𝔡-kG induce exact triangles in \underline{𝔪𝔬𝔡}-kG. We can lift the identity map of A to a commutative diagram as follows:

[figure: 0; A; a; B; b; C; w; 0; 0; A; I(A); ΣA; 0]

Then, as is shown in Remark 2 in the proof of Theorem 9.4 in [6], we get an exact triangle A ⟶^{a} B ⟶^{b} C ⟶^{−w} ΣA in \underline{𝔪𝔬𝔡}-kG. By the rotation axiom, we also get an exact triangle

ΩC ⟶^{Ωw} A ⟶^{a} B ⟶^{b} C.

Now let n be a fixed integer; then all the k-vector spaces \underline{Hom}_{kG}(Ω^{n+m}X, Ω^{m}Y) for integers m are naturally isomorphic. This defines the Tate cohomology groups

Êxt_{kG}^{n}(X, Y) = \underline{Hom}_{kG}(Ω^{n}X, Y).

For any two morphisms f ∈ \underline{Hom}_{kG}(Ω^{n}X, Y) and g ∈ \underline{Hom}_{kG}(Ω^{m}Y, Z) we get a composition g ∘ Ω^{m}f ∈ \underline{Hom}_{kG}(Ω^{n+m}X, Z). This describes the composition product

Êxt_{kG}^{m}(Y, Z) ⊗ Êxt_{kG}^{n}(X, Y) → Êxt_{kG}^{m+n}(X, Z).

Let us write Ĥ^{∗}(G, M) = Êxt_{kG}^{∗}(k, M) for short; then Ĥ^{∗}(G, k) is a graded algebra, which we often denote by Ĥ^{∗}(G). As in the case of ordinary cohomology, Ĥ^{∗}(G) is graded commutative.

Note that for positive integers n, Tate cohomology Êxt_{kG}^{n}(X, Y) agrees with the usual Ext-groups Ext_{kG}^{n}(X, Y). For example, every short exact sequence A ↪ B ↠ C represents an element in Ext_{kG}^{1}(C, A); under the isomorphism Ext_{kG}^{1}(C, A) ≅ \underline{Hom}_{kG}(ΩC, A) this element corresponds to the stable map ΩC → A in the exact triangle ΩC → A → B → C. Remark 1.1. It is well-known that the group algebra kG is a cocommutative Hopf algebra, the comultiplication Δ : kG → kG ⊗ kG and the antipode ε : kG → kG being the k-linear maps given by

Δ(g) = g ⊗ g, ε(g) = g^{−1}

for all g ∈ G. Let us note that throughout this thesis we can replace the group algebra kG by an arbitrary (ungraded) finite-dimensional cocommutative Hopf algebra A over k. By a theorem of Larson and Sweedler [17], A is a Frobenius algebra and we are able to construct the stable module category \underline{𝔪𝔬𝔡}-A. Cocommutativity provides a symmetric monoidal tensor product ⊗ and will turn up later again when we discuss Steenrod operations. The tensor product of a projective module and an arbitrary module still is projective (see the last paragraph of §3 in [11]), and therefore the tensor product descends to a symmetric monoidal product on the stable module category with unit object k, the trivial module which gets its A-module structure via the counit of A.

We should also remark that we can generalize all results to the category 𝔐𝔬𝔡-A of arbitrary (not necessarily finite dimensional) A-modules and its corresponding stable category \underline{𝔐𝔬𝔡}-A.

1.2 The graded center

Suppose that C is a triangulated category with shift functor Σ. The graded center Z(C) of C consists of all natural transformations id_{C} → Σ^{n} which commute with Σ up to the sign (−1)^{n}; that is, the degree n-part of Z(C) is given by

Z^{n}(C) = {ζ : id_{C} → Σ^{n} | Σζ = (−1)^{n}ζΣ}.

There is an obvious pairing Z^{n}(C)×Z^{m}(C) → Z^{n+m}(C) mapping (φ, ψ) to the composition Σ^{m}φ ∘ ψ. We will ignore set theoretical issues and assume that Z^{∗}(C) is a set – this will be obvious in all the cases we are interested in. Then Z^{∗}(C) is a graded commutative ring (see, e.g., [18], §2).

Now let C be the stable module category \underline{𝔪𝔬𝔡}-kG. Suppose that we are given a cohomology class in Ĥ^{n}(G), represented by some map ζ : Ω^{n}k → k. Then multiplication by ζ induces an element of Z^{n}(C) also denoted by ζ:

[figure: Ω^{n}X; ζ_{Ω^{n}X}; ≃; X; ≃; Ω^{n}k ⊗ X; ζ⊗id; k ⊗ X]

Thus, we obtain a morphism of graded rings Ĥ^{∗}(G) → Z^{∗}(C). If we compose this with the evaluation at k we get a retraction

Ĥ^{∗}(G) → Z^{∗}(\underline{𝔪𝔬𝔡}-kG) ⟶^{ev_{k}} Ĥ^{∗}(G).

1.3 The notion of order

The first map is an inclusion of a direct summand, which will not be surjective in general. For example, take any element g in the center of G. Then the action of g on a given module is a natural morphism in 𝔪𝔬𝔡-kG and induces an element in Z^{0}(C). If g is not the unit element of G then g does not come from Ĥ^{∗}(G) since ev_{k}(g) = 1 ∈ Ĥ^{0}(G) but g ≠ 1 ∈ Z^{0}(C) if p divides the order of G.

More generally, let A be a symmetric algebra, i.e., a Frobenius algebra A whose corresponding nondegenerate bilinear form A×A → k is symmetric (as it is in the case A = kG). Define A^{op} to be the opposite algebra of A, and put B = A^{e} = A^{op} ⊗_{k} A, the enveloping algebra of A. Then B is also a symmetric algebra, and we can pass from 𝔪𝔬𝔡-B, whose objects are the same as A-A-bimodules, to the stable category \underline{𝔪𝔬𝔡}-B with shift-functor Σ_{B}. Then the Tate version of Hochschild cohomology is given by ĤH^{∗}(A) = \underline{Hom}_{B}(Ω_{B}^{∗}A, A). Given an element in ĤH^{n}(A) represented by some map f : Ω_{B}^{n}A → A in 𝔪𝔬𝔡-B, we obtain a natural transformation in \underline{𝔪𝔬𝔡}-A

Ω^{n}X ≅ Ω_{B}^{n}X ≅ Ω_{B}^{n}A ⊗_{A} X ⟶^{f⊗_{A}id} A ⊗_{A} X ≅ X,

where the first isomorphism comes from the fact that a projective resolution of B-modules is a projective resolution of A-modules. We therefore have constructed a map ĤH^{∗}(A) → Z^{∗}(\underline{𝔪𝔬𝔡}-A). In our case A = kG we have a factorization

Ĥ^{∗}(G) → ĤH^{∗}(kG) → Z(\underline{𝔪𝔬𝔡}-kG),

so the second map is more likely to be surjective than the whole composition. In fact, the central element given above not coming from Ĥ^{∗}(G) comes from ĤH^{∗}(kG). Nevertheless, the map ĤH^{∗}(kG) → Z^{∗}(\underline{𝔪𝔬𝔡}-kG) is not surjective in general. For example, in the case that G is a p-group of rank at least 2 and k is algebraically closed Linckelmann and Stancu [19] proved that Z^{∗}(\underline{𝔪𝔬𝔡}-kG) is infinite-dimensional in all degrees for p = 2 and in odd degrees for p > 2.

Remark 1.2. The map Ĥ^{∗}(G) → Z^{∗}(\underline{𝔪𝔬𝔡}-kG) is a special case of a more general setting. Suppose that C is a triangulated category with a symmetric monoidal product ∧ compatible with the triangulated structure. Let us denote by S the unit of the product. Then we have a canonical map from the graded ring [S, S]^{∗} of graded self-maps of S to the center Z^{∗}(C) by mapping φ : S → Σ^{n}S to the family of maps

X ≅ X ∧ S ⟶^{id_{X} ∧φ} X ∧ Σ^{n}S ≅ Σ^{n}X.

In the case of C = \underline{𝔪𝔬𝔡}-kG with product ⊗ and S = k being the trivial kG-module, we get our map Ĥ^{∗}(G) → Z^{∗}(\underline{𝔪𝔬𝔡}-kG). In the case of the stable homotopy category C = SHC with smash product ∧ and S being the sphere spectrum we get a map from the stable homotopy groups of spheres π_{−∗}^{s} → Z^{∗}(SHC).

1.3 The notion of order

Now we are going to define the notion of order. Let C be a triangulated category with shift functor Σ, and let ζ ∈ Z^{n}(C) be an element in the graded center of C. For every object K we denote by K/ζ some choice of cone of the map ζ_{Σ^{−n}K} : Σ^{−n}K → K, so we have exact triangles

⋯ → Σ^{−1}K/ζ → Σ^{−n}K ⟶^{ζ} K → K/ζ → Σ^{1−n}K → …

The ambiguity in choosing these cones will turn out to be irrelevant for our purposes. Now suppose that f : K → Y is some map in our category C; then we say that a map g : K/ζ → Y is an extension of f if the following diagram commutes:

\xymatrix{… \ar[r] & Σ^{−n}K \ar[r]^{ζ} & K \ar[r] \ar[d]_{f} & K/ζ \ar[r] \ar[dl]^{g} & … \\  &  & Y}

We are ready to define the main object of this thesis. The following is a straightforward generalization of a definition made by Schwede in [25].

Definition 1.3. For every object X in the category C we define the ζ-order of X, denoted ζ-ord(X), inductively by the following two conditions:

(i) For every X we have ζ-ord(X) ∈ {0, 1, 2, … } ∪ {∞}.

(ii) For every positive integer j and every object X we have ζ-ord(X) ≥ j if and only if for every object K and every morphism f : K → X there is some extension g of f such that for some (and hence any) choice of cone C_{g} of g we have ζ-ord(C_{g}) ≥ j−1.

We also define the ζ-order of the category C to be the ζ-order of some zero-object.

Remark 1.4. For every integer m and every object X in C we have multiplication by m on X, defined by the m-fold sum id + id + ⋯ + id : X → X. This is natural in X and commutes with the shift functor Σ, so we get an element m ∈ Z^{0}(C) and the definition applies. This way we recover the definition of m-order given in [25].

Lemma 1.5 (compare [25]). Let C and ζ be as above. The ζ-order enjoys the following properties:

(i) The ζ-order is invariant under isomorphisms and shifts.

(ii) An object X of C has ζ-order at least 1 if and only if ζ acts as zero on X.

(iii) The ζ-order of C equals 1 plus the minimum of the ζ-orders of all objects of the form X/ζ.

(iv) The ζ-order is invariant under equivalences of triangulated categories.

(v) If D ⊂ C is a full triangulated subcategory then we can restrict ζ to D and obtain an element in Z^{∗}(D). The notions of order in these two categories are related by the inequality ζ-ord^{D}(X) ≥ ζ-ord^{C}(X) for all objects X of D.

1.4 Tensor powers

Proof. Part (i) follows by a straightforward induction on j. At the same time the proof of the ‘isomorphism’ statement shows that the notion of order does not depend on the choices of cones in its definition. For (ii) note that if ζ is zero on X then there is some map h : X/ζ → X for which the composition X → X/ζ ⟶^{h} X is the identity of X. Then for every morphism f : K → X we have an extension K/ζ ⟶^{f/ζ} X/ζ ⟶^{h} X, which means that ζ-ord(X) ≥ 1. Conversely, if the order of X is at least 1, then we can apply the definition to the identity map f : X → X and obtain some extension g : X/ζ → X. Then ζ is given by the composite

Σ^{−n}X ⟶^{ζ} X → X/ζ ⟶^{g} X,

but the composition of the first two maps is zero. Part (iii) and (iv) follow directly from the definition. For (v) it is enough to show that ζ-ord^{C}(X) ≥ j implies ζ-ord^{D}(X) ≥ j which we show by induction on j, the case j = 0 being an empty statement. Suppose this is true for j − 1 an assume that ζ-ord^{C}(X) ≥ j. Let f : K → X be any map in D. By assumption, there is some extension g : K/ζ → X in C such that for some choice of cone C_{g} of g we have ζ-ord^{C}(C_{g}) ≥ j −1. Since C is a full triangulated subcategory, g and C_{g} lie in D, and by induction hypothesis ζ-ord^{D}(C_{g}) ≥ j − 1. This implies ζ-ord^{D}(X) ≥ j.

1.4 Tensor powers

Suppose that X and Y are kG-modules; then we have already defined a kG-module structure on the tensor product X ⊗ Y, coming from the comultiplication of the Hopf algebra structure on kG. We can therefore consider the n-fold tensor product X ⊗ X ⊗ ⋯ ⊗ X which we denote by X^{⊗n}. On this module we have an obvious Σ_{n}-action, where Σ_{n} denotes the symmetric group on n letters. The fact that the comultiplication Δ : kG → kG ⊗ kG is cocommutative shows that Σ_{n} acts by homomorphisms of kG-modules. Whenever we have a module X, we denote by T the map

X^{⊗p} → X^{⊗p}
x_{1} ⊗ x_{2} ⊗ ⋯ ⊗ x_{p} ↦ x_{p} ⊗ x_{1} ⊗ ⋯ ⊗ x_{p−1}.

Then T^{p} = id and (1−T)^{p} = 0. We will also write N = (1−T)^{p−1} = 1+T +T^{2}+⋯+T^{p−1}. All these notions generalize naturally to chain complexes of kG-modules. It should be noted that the action of Σ_{n} then involves a certain sign depending on the degrees of the elements, which can be deduced from the Koszul sign rule. Let us give the precise sign here. Whenever X is a chain complex of kG-modules and x ∈ X is a homogeneous element, then we denote by |x| the degree of x. The action of σ ∈ Σ_{n} on a tensor product x_{1} ⊗ ⋯ ⊗ x_{n} of homogeneous elements involves the sign (−1)^{m} with

m = ∑_{\begin{matrix}i<j, \\ σ(i)>σ(j) \end{matrix}} |x_{i}| · |x_{j}|.

For every chain complex X with differential ∂ we denote by X[m] the chain complex with modules X[m]_{n} = X_{n−m} and differential (−1)^{m}∂. When X is a kG-module, we can view X as a chain complex concentrated in degree 0; then X[1] is a chain complex concentrated in degree 1, and σ ∈ Σ_{n} acts on the tensor product X[1]^{⊗n} ≅ X^{⊗n}[n] via its signum (−1)^{σ}.

1.5 Symmetric and exterior powers

For a kG-module X, the module X^{⊗n} has two famous quotients; on the one hand, the symmetric power

S^{n}X = X^{⊗n}/ ⟨x_{1} ⊗ ⋯ ⊗ x_{n} − x_{σ(1)} ⊗ ⋯ ⊗ x_{σ(n)}⟩_{k} ,

and on the other hand, the exterior power

Λ^{n}X = X^{⊗n}/ ⟨x_{1} ⊗ ⋯ ⊗ x_{n} | x_{i} = x_{j} for some i ≠ j⟩_{k} .

Here we denoted by ⟨M⟩_{k} the k-vector space generated by the elements of the set M. As usual, we define X^{⊗0} = S^{0}X = Λ^{0}X = k, the trivial kG-module. The obvious multiplication maps

μ_{i,j} : S^{i}X ⊗ S^{j}X → S^{i+j}X, μ_{i,j} : Λ^{i}X ⊗ Λ^{j}X → Λ^{i+j}X

turn S^{∗}X and Λ^{∗}X into graded algebras. There are also comultiplications Δ : S^{∗}X → S^{∗}X ⊗ S^{∗}X and Δ : Λ^{∗}X → Λ^{∗}X ⊗ Λ^{∗}X which make S^{∗}X and Λ^{∗}X into graded Hopf algebras. For example, the former map is given by

Δ_{i,j} : S^{i+j}X ⟶ S^{i}X ⊗ S^{j}X
x_{1} ⋯ x_{i+j} ↦ ∑_{σ} x_{σ(1)} ⋯ x_{σ(i)} ⊗ x_{σ(i+1)} ⋯ x_{σ(i+j)},

where σ runs through all (i, j)-shuffles (see, e.g., [22] §1.3).

For any map f : X → Y let us denote by κ_{ij}^{f} the map

κ_{ij}^{f} : Λ^{i}X ⊗ S^{j}Y ⟶ Λ^{i−1}X ⊗ S^{j+1}Y
x_{1} ∧ ⋯ ∧ x_{i} ⊗ y_{1} ⋯ y_{j} ↦ ∑_{t=1}^{i}(−1)^{t−1}x_{1} ∧ ⋯ ∧ x̂_{t} ∧ ⋯ ∧ x_{i} ⊗ f(x_{t})y_{1} ⋯ y_{j}.

Notice that this map equals the composition

Λ^{i}X ⊗ S^{j}Y ⟶^{Δ⊗id} Λ^{i−1}X ⊗ X ⊗ S^{j}Y ⟶^{id ⊗f⊗id} Λ^{i−1}X ⊗ Y ⊗ S^{j}Y ⟶^{id ⊗μ} Λ^{i−1}X ⊗ S^{1+j}Y.

We will often drop the j from the notation and simply write κ_{i}^{f} for that map.

Lemma 1.6. Suppose that 0 → X ⟶^{ι} Y ⟶^{π} Z → 0 is an exact sequence of kG-modules, and let n be a positive integer. Then the sequences

0 → Λ^{n}X ⟶^{κ_{n}^{ι}} Λ^{n−1}X ⊗ S^{1}Y ⟶^{κ_{n−1}^{ι}} … ⟶^{κ_{2}^{ι}} Λ^{1}X ⊗ S^{n−1}Y ⟶^{κ_{1}^{ι}} S^{n}Y ⟶^{S^{n}π} S^{n}Z → 0

and

0 → Λ^{n}X ⟶^{Λ^{n}ι} Λ^{n}Y ⟶^{κ_{n}^{π}} Λ^{n−1}Y ⊗ S^{1}Z ⟶^{κ_{n−1}^{π}} … ⟶^{κ_{2}^{π}} Λ^{1}Y ⊗ S^{n−1}Z ⟶^{κ_{1}^{π}} S^{n}Z → 0

are exact.

1.5 Symmetric and exterior powers

Remark 1.7. For ι = id_{X} : X → X and Z = 0 this is a well-known fact (see, e.g., [22], Theorem 1.1); one obtains the so-called Koszul complex

0 → Λ^{n}X → Λ^{n−1}X ⊗ X → Λ^{n−2}X ⊗ S^{2}X → …

⋯ → X ⊗ S^{n−1}X → S^{n}X → 0. (1.8)

It is always exact, also for infinite dimensional modules X.

Proof. We only need to show a statement about exactness; we can therefore ignore the G-action and treat all modules as k-vector spaces. In particular, we can assume that Y = X ⊕ Z and ι and π are the obvious maps. Using the exponential property of the symmetric algebra functor S^{∗}(X ⊕ Z) ≅ S^{∗}X ⊗ S^{∗}Z (see 1.4.(2) in [22]) the first sequence decomposes into a direct sum of sequences

0 → Λ^{i}X ⊗ S^{n−i}Z → Λ^{i−1}X ⊗ S^{1}X ⊗ S^{n−i}Z → Λ^{i−2}X ⊗ S^{2}X ⊗ S^{n−i}Z → …

⋯ → Λ^{1}X ⊗ S^{i−1}X ⊗ S^{n−i}Z → S^{i}X ⊗ S^{n−i}Z → 0

which are exact by Remark 1.7, and one additional exact sequence 0 → S^{n}Z ⟶^{id} S^{n}Z → 0. Exactness of the second sequence is proved similarly.

Using the fact that there are natural isomorphisms Λ^{n}(X^{♯}) ≅ (Λ^{n}X)^{♯} for all n, and S^{n}(X^{♯}) ≅ (S^{n}X)^{♯} for n < p (see §1.5 in [22]), we obtain the following dual result:

Corollary 1.9. Suppose that 0 → X ⟶^{ι} Y ⟶^{π} Z → 0 is an exact sequence of kG-modules, and let n be an integer with n < p. Then the sequence

0 → S^{n}X ⟶^{S^{n}ι} S^{n}Y → Λ^{1}Z ⊗ S^{n−1}Y → ⋯ → Λ^{n−1}Z ⊗ S^{1}Y → Λ^{n}Z → 0

is exact, where the maps Λ^{i}Z ⊗ S^{n−i}Y → Λ^{i+1}Z ⊗ S^{n−i−1}Y are given by

z_{1} ∧ z_{2} ∧ ⋯ ∧ z_{i} ⊗ y_{1}y_{2} … y_{n−i} ↦ ∑_{j=1}^{n−i} z_{1} ∧ ⋯ ∧ z_{i} ∧ π(y_{j}) ⊗ y_{1} … ŷ_{j} … y_{n−i}.

Remark 1.10. Note that in the special case X = 0, Y = Z and π = id_{Y} we get a sequence

0 → S^{n}X → X ⊗ S^{n−1}X → Λ^{2}X ⊗ S^{n−2}X → …

⋯ → Λ^{n−1}X ⊗ X → Λ^{n}X → 0, (1.11)

which we call the de Rham complex. It is exact for n < p but not in general, compare with Lemma 1.2 of [22].

Another consequence is the following

Corollary 1.12. Suppose that 0 → X ⟶^{ι} Y ⟶^{π} k → 0 is an exact sequence of kG-modules. Then the map κ_{n}^{π} : Λ^{n}Y → Λ^{n−1}Y ⊗ S^{1}k ≅ Λ^{n−1}Y induces a map Λ^{n}Y → Λ^{n−1}X which we also denote by κ_{n}^{π}, and the sequence

0 → Λ^{n}X ⟶^{Λ^{n}ι} Λ^{n}Y ⟶^{κ_{n}^{π}} Λ^{n−1}X → 0  (1.13)

is exact.

Proof. By Lemma 1.6 we have a commutative diagram of exact sequences

[figure: 0; Λ^{n}X; Λ^{n}ι; Λ^{n}Y; Λ^{n−1}Y; …; Y; k; 0; 0; Λ^{n−1}X; Λ^{n−1}Y; …; Y; k; 0]

showing that the dotted arrow exists and that (1.13) is exact.

Remark 1.14. In the situation of the preceding corollary, consider the graded algebra Λ^{∗}X. Then Λ^{1+∗}X and Λ^{1+∗}Y are a graded right Λ^{∗}X-modules, and the corollary (and an easy to verify commutative diagram) show that we have an exact sequence

0 → Λ^{1+∗}X ⟶^{ι} Λ^{1+∗}Y ⟶^{κ} Λ^{∗}X → 0

of graded right Λ^{∗}X-modules.

We end this section with a construction that will be helpful later for checking commutativity of certain diagrams. Suppose that Y is any kG-module, M^{∗} is a graded right Λ^{∗}Y -module and N^{∗} is a graded left S^{∗}Y -comodule; so we have maps

μ : M^{∗} ⊗ Λ^{∗}Y → M^{∗}, ν : N^{∗} → S^{∗}Y ⊗ N^{∗}.

Then we have maps M^{i} ⊗ N^{j} ⟶^{d} M^{i+1} ⊗ N^{j−1} given by the composition

M^{i} ⊗ N^{j} ⟶^{id ⊗ν} M^{i} ⊗ Y ⊗ N^{j−1} ⟶^{μ⊗id} M^{i+1} ⊗ N^{j−1}.

We claim that d^{2} = 0 as we can read off from the following commutative diagram:

[figure: M^{i} ⊗ N^{j}; N is a S^{∗}Y -comodule; M^{i} ⊗ Y ⊗ N^{j−1}; μ⊗id; id ⊗ν; M^{i+1} ⊗ N^{j−1}; id ⊗ν; M^{i} ⊗ S^{2}Y ⊗ N^{j−2}; 0; M^{i} ⊗ Y ⊗ Y ⊗ N^{j−2}; μ⊗id; M is a Λ^{∗}Y -module; M^{i+1} ⊗ Y ⊗ N^{j−2}; M^{i} ⊗ Λ^{2}Y ⊗ N^{j−2}; M^{i+2} ⊗ N^{j−2}]

Definition 1.15. We call

… ⟶^{d} M^{i−1} ⊗ N^{j+1} ⟶^{d} M^{i} ⊗ N^{j} ⟶^{d} M^{i+1} ⊗ N^{j−1} ⟶^{d} …

the chain complex associated to (M, N).

This is a functorial construction in the sense that maps of graded right Λ^{∗}Y -modules and maps of graded left S^{∗}Y -comodules induce maps of chain complexes. The same construction works for N^{∗} ⊗M^{∗} when N is a graded right S^{∗}Y -comodule and M a graded left Λ^{∗}Y -module.

1.6 Symmetric and exterior powers in the stable category

Example 1.16. We have already seen instances of such chain complexes: suppose that π : Y → Z is a map of kG-modules, and let N = S^{∗}Y. Then M = Λ^{∗}Z becomes a Λ^{∗}Y -module via the map π. The chain complex associated to (Λ^{∗}Z, S^{∗}Y) is part of the exact sequence of Corollary 1.9.

We can also swap the roles of S^{∗} and Λ^{∗} and consider graded left S^{∗}Y -modules M and graded right Λ^{∗}Y -comodules; then we also get an associated sequence in a similar way.

Example 1.17. Suppose again that π : Y → Z is a map of kG-modules, and let M = Λ^{∗}Y. Similar to the example above, N = S^{∗}Z becomes an S^{∗}Y -module via the map π. Since S^{∗}π : S^{∗}Y → S^{∗}Z is a map of S^{∗}Y -modules, we obtain a morphism of the associated chain complexes:

\xymatrix{… & Λ^{j}Y ⊗ S^{i}Z \ar[l] & Λ^{j+1}Y ⊗ S^{i−1}Z \ar[l] & … \ar[l] \\ … & Λ^{j}Y ⊗ S^{i}Y \ar[l] \ar[u]^{id ⊗S^{i}π} & Λ^{j+1}Y ⊗ S^{i−1}Y \ar[l] \ar[u]_{id ⊗S^{i−1}π} & … \ar[l]}

1.6 Symmetric and exterior powers in the stable category

For every kG-module X and i < p, there are natural maps

S^{i}X → X^{⊗i} :  x_{1} ⋯ x_{i} ↦ \frac{1}{i!} ∑_{σ∈Σ_{i}} x_{σ_{1}} ⊗ ⋯ ⊗ x_{σ_{i}},  (1.18)

Λ^{i}X → X^{⊗i} :  x_{1} ∧ ⋯ ∧ x_{i} ↦ \frac{1}{i!} ∑_{σ∈Σ_{i}}(−1)^{σ}x_{σ_{1}} ⊗ ⋯ ⊗ x_{σ_{i}}.

Up to the scalar constants they are the i-fold iterated comultiplications of the graded Hopf algebra structures on S^{∗}X and Λ^{∗}X. We will refer to these maps as the canonical maps S^{i}X → X^{⊗i} and Λ^{i}X → X^{⊗i}. The compositions S^{i}X → X^{⊗i} → S^{i}X and Λ^{i}X → X^{⊗i} → Λ^{i}X with the projections are the identity maps. Therefore, S^{i}X and Λ^{i}X both are direct summands of X^{⊗i}. In particular, we have shown the following lemma which ensures that for i < p the functors S^{i} and Λ^{i} induce functors on the stable category.

Lemma 1.19. Suppose that i < p. If P is a projective kG-module, then so are S^{i}P and Λ^{i}P.

Remark 1.20. Note that the conclusion of the lemma fails in general if i ≥ p. For example, if G is the cyclic group of order p, then dim_{k} kG = p and hence dim_{k} Λ^{p}(kG) = 1. One can check that Λ^{p}(kG) ≅ k as kG-modules, which is not projective. Also, S^{p}(kG) ≅ k ⊕(free).

Lemma 1.21. Suppose that i < p and X is any kG-module. Then, in the stable module category, S^{i}ΩX ≅ Ω^{i}Λ^{i}X and Λ^{i}ΩX ≅ Ω^{i}S^{i}X.

Proof. Consider a short exact sequence 0 → ΩX → P → X → 0 in 𝔪𝔬𝔡-kG with a projective module P. By Corollary 1.9 there is an exact sequence

0 → S^{i}ΩX → S^{i}P → Λ^{1}X ⊗ S^{i−1}P → ⋯ → Λ^{i−1}X ⊗ S^{1}P → Λ^{i}X → 0.

2. The lower bound

But the S^{j}P ’s are projective by Lemma 1.19, so S^{i}ΩX is stably isomorphic to Ω^{i}Λ^{i}X. The other isomorphism is shown similarly using Lemma 1.6.

By induction, we conclude that

Λ^{i}Ω^{j}X ≅ \begin{cases}Ω^{ij}S^{i}X & if j is odd, \\ Ω^{ij}Λ^{i}X & if j is even \end{cases}

in the stable category. Since Ω^{m}X = 0 in \underline{𝔪𝔬𝔡}-kG implies that X is projective, we get the following corollary:

Corollary 1.22. If n is even, then Λ^{i}Ω^{n}k is projective for 1 < i < p.

2 The lower bound

From now on, let [ζ] ∈ Ĥ^{n}(G) be a non-zero Tate cohomology class of even degree n, represented by a map ζ : Ω^{n}k → k in 𝔪𝔬𝔡-kG. Since ζ is non-zero, it must be surjective and we get an exact sequence 0 → L_{ζ} ⟶^{ι} Ω^{n}k ⟶^{ζ} k → 0. This sequence yields an exact triangle

Ωk → L_{ζ} → Ω^{n}k → k

We denote by η an unstable representative of the stable map Ωk → L_{ζ}.

Definition 2.1. Let s be a non-negative integer. An s-coherent module X is a sequence X_{0}, X_{1}, X_{2}, …, X_{s} of modules together with maps

μ_{i,j} : Λ^{i}L_{ζ} ⊗ X_{j} → X_{i+j}

for all i ≥ 1 and j ≥ 0 with i + j ≤ s, satisfying the following conditions:

• (Unitality) For every 1 ≤ i < s, the composite

Ωk ⊗ X_{i} ⟶^{η⊗id} L_{ζ} ⊗ X_{i} ⟶^{μ_{1,i}} X_{1+i}

is a stable isomorphism.

• (Associativity) The square

\xymatrix{Λ^{i}L_{ζ} ⊗ Λ^{j}L_{ζ} ⊗ X_{m} \ar[r]^{id ⊗μ_{j,m}} \ar[d]^{μ_{i,j}⊗id} & Λ^{i}L_{ζ} ⊗ X_{j+m} \ar[d]^{μ_{i,j+m}} \\ Λ^{i+j}L_{ζ} ⊗ X_{m} \ar[r]^{μ_{i+j,m}} & X_{i+j+m}}

commutes for all 0 ≤ i, j, m with i + j + m ≤ s.

• (Exactness) The sequence

0 → S^{i}L_{ζ} ⊗ X_{0} → S^{i−1}L_{ζ} ⊗ X_{1} → ⋯ → S^{1}L_{ζ} ⊗ X_{i−1} → X_{i} → 0

is exact for all i ≤ s. Here the map S^{j}L_{ζ} ⊗ X_{m} → S^{j−1}L_{ζ} ⊗ X_{m+1} is given by the composition

S^{j}L_{ζ} ⊗ X_{m} ⟶^{Δ_{j−1,1}⊗} S^{j−1}L_{ζ} ⊗ L_{ζ} ⊗ X_{m} ⟶^{id ⊗μ_{1,m}} S^{j−1}L_{ζ} ⊗ X_{m+1}.

We call X_{1} the underlying object of X. A map of s-coherent modules f : X → Y is a family of maps f_{i} : X_{i} → Y_{i} compatible with the respective structure maps μ in the obvious sense. Such a map will be called injective (surjective) if each of these f_{i}’s is injective (surjective).

Remark 2.2. We have chosen a rather lengthy and very explicit way of defining s-coherent modules in order to stress the similarity to the definition of k-coherent M-modules of [24], Definition 2.1. Notice that we could equally well have defined s-coherent modules to be graded left Λ^{∗}L_{ζ}-modules X_{∗} with X_{i} = 0 for i outside {0, 1, …, s}, satisfying the unitality and exactness conditions. The sequence of the exactness condition is the chain complex associated to the right S^{∗}L_{ζ}-comodule S^{∗}L_{ζ} and the left Λ^{∗}L_{ζ}-module X_{∗} (in the sense of Definition 1.15).

Example 2.3. For every s < p, there is the tautological s-coherent module, defined as follows. Put X_{i} = Λ^{i}L_{ζ}, and define μ_{i,j} : Λ^{i}L_{ζ} ⊗ Λ^{j}L_{ζ} → Λ^{i+j}L_{ζ} to be the natural maps. Then we have associativity, and exactness holds due to Corollary 1.9. To prove unitality, note that we have a commutative diagram

[figure: 0; L_{ζ} ⊗ Λ^{i}L_{ζ}; Ω^{n}k ⊗ Λ^{i}L_{ζ}; Λ^{i}L_{ζ}; 0; 0; Λ^{1+i}L_{ζ}; Λ^{1+i}Ω^{n}k; Λ^{i}L_{ζ}; 0]

where the upper row is obtained by tensoring the exact sequence 0 → L_{ζ} → Ω^{n}k → k → 0 with Λ^{i}L_{ζ}; it is therefore exact. The bottom row is the exact sequence constructed in Corollary 1.12, and the middle vertical map is the composite Ω^{n}k ⊗ Λ^{i}L_{ζ} ⟶^{id ⊗Λ^{i}ι} Ω^{n}k ⊗ Λ^{i}Ω^{n}k ⟶^{·∧·} Λ^{i+1}Ω^{n}k. Passing to the stable category, we obtain a map of triangles:

[figure: Ωk ⊗ Λ^{i}L_{ζ}; ≅; (∗); L_{ζ} ⊗ Λ^{i}L_{ζ}; Ω^{n}k ⊗ Λ^{i}L_{ζ}; Λ^{i}L_{ζ}; ΩΛ^{i}L_{ζ}; Λ^{1+i}L_{ζ}; Λ^{1+i}Ω^{n}k; Λ^{i}L_{ζ}]

Since 0 < i < p − 1, we know (by Corollary 1.22) that Λ^{1+i}Ω^{n}k is projective and therefore isomorphic to 0 in the stable category. Hence, the first map of the bottom row is an isomorphism. The commutative diagram (∗) then shows the desired unitality condition.

More generally, if K is any module, we denote by F (K) the tautological s-coherent module over L_{ζ} ⊗ K, given by F (K)_{j} = Λ^{j}L_{ζ} ⊗ K. The maps and properties are obtained from the previous example by tensoring everything with K from the right. This is also a functorial construction in the sense that given any map f : K → L, we get an induced map F (K) ⟶^{F (f)} F (L), which is surjective if f is.

Lemma 2.4. Suppose that f : X → Y is a surjective map of s-coherent modules. Then the kernel of f is an s-coherent module in the natural way.

Proof. The kernel of f is defined levelwise by C_{i} = ker f_{i} : X_{i} → Y_{i}. The multiplication maps are defined by the following commutative diagram:

\xymatrix{Λ^{i}L_{ζ} ⊗ C_{j} \ar[r] \ar[d]^{μ_{i,j}} & Λ^{i}L_{ζ} ⊗ X_{j} \ar[r]^{id ⊗f_{j}} \ar[d]^{μ_{i,j}} & Λ^{i}L_{ζ} ⊗ Y_{j} \ar[d]^{μ_{i,j}} \\ C_{i+j} \ar[r] & X_{i+j} \ar[r]^{f_{i+j}} & Y_{i+j}}

Associativity follows immediately from associativity on X and Y. To prove unitality, consider the following commutative diagram:

\xymatrix{Ωk ⊗ C_{j} \ar[r] \ar[d] & Ωk ⊗ X_{j} \ar[r] \ar[d] & Ωk ⊗ Y_{j} \ar[d] \\ C_{1+j} \ar[r] & X_{1+j} \ar[r] & Y_{1+j}}

When we pass to the stable category, the exact rows turn into exact triangles, and the second and third vertical map turn into isomorphisms. By the five lemma in triangulated categories, the first map must be an isomorphism as well.

We are left with the exactness condition. Consider the following commutative diagram:

\xymatrix{0 \ar[r] & S^{i}L_{ζ} ⊗ C_{0} \ar[r] \ar[d] & S^{i−1}L_{ζ} ⊗ C_{1} \ar[r] \ar[d] & … \ar[r] & C_{i} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & S^{i}L_{ζ} ⊗ X_{0} \ar[r] \ar[d] & S^{i−1}L_{ζ} ⊗ X_{1} \ar[r] \ar[d] & … \ar[r] & X_{i} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & S^{i}L_{ζ} ⊗ Y_{0} \ar[r] & S^{i−1}L_{ζ} ⊗ Y_{1} \ar[r] & … \ar[r] & Y_{i} \ar[r] & 0}

Since the maps C_{i} → X_{i} are inclusions, we can deduce that the top row is a complex. Regarding the rows as chain complexes, the vertical maps form a short exact sequence of chain complexes, thus inducing a long exact sequence in homology. Together with the hypothesis that the second and the third row have zero homology, we obtain that the top row is an exact sequence.

Lemma 2.5. If X is an s-coherent module, then the multiplication maps μ_{i,j} : Λ^{i}L_{ζ} ⊗ X_{j} → X_{i+j} are surjective.

Proof. Let us prove this by induction on i. For i = 1, the map L_{ζ} ⊗ X_{j} → X_{1+j} is the same as the last non-trivial map in the exact sequence of the exactness condition; it is therefore surjective. For the inductive step, consider the following commutative diagram:

\xymatrix{L_{ζ} ⊗ Λ^{i}L_{ζ} ⊗ X_{j} \ar@{->>}[r] \ar[d] & L_{ζ} ⊗ X_{i+j} \ar@{->>}[d] \\ Λ^{1+i}L_{ζ} ⊗ X_{j} \ar[r] & X_{1+i+j}}

Since the upper-right composition is surjective, so is the lower-left one. This implies that the bottom map is onto.

If X is an s-coherent module, let X[1] be given by X[1]_{i} = X_{1+i} for i < s. Then X[1] is an (s − 1)-coherent module in the natural way. The multiplication maps of X induce a map of (s−1)-coherent modules F (X_{1}) → X[1] which is surjective by the previous lemma.

Proposition 2.6. If X is an s-coherent module with underlying object X_{1}, then we have that ζ-ord(X_{1}) ≥ s − 1.

Proof. We do this by induction on s, the case s = 1 being an empty statement. Now suppose that s ≥ 2. We start with a map K → X_{1} in the stable category and assume that it is given by a surjective map f : K → X_{1} in 𝔪𝔬𝔡-kG. Consider the composition of maps of (s − 1)-coherent modules

F (K) ⟶^{F (f)} F (X_{1}) ⟶ X[1].

By Lemma 2.5, the maps are surjective. Therefore, by Lemma 2.4, the kernel C of this map is an (s − 1)-coherent module. By induction hypothesis,

ζ-ord(C_{1}) ≥ s − 2.  (2.7)

Now consider the following commutative diagram in \underline{𝔪𝔬𝔡}-kG:

\xymatrix{Ωk ⊗ K \ar[r]^{η⊗id} \ar[d]_{id ⊗f} & L_{ζ} ⊗ K \ar[d]^{id ⊗f} \ar@{-->}[ddl] \\ Ωk ⊗ X_{1} \ar[r]^{η⊗id} \ar[d] & L_{ζ} ⊗ X_{1} \ar[dl]^{μ_{1,1}} \\ X_{2}}

By the unitality condition, the map Ωk ⊗ X_{1} → X_{2} is a stable isomorphism. Therefore the dashed arrow stably lifts to a map f̂ : L_{ζ} ⊗ K → Ωk ⊗ X_{1}. This is (up to a shift) an extension of f; we will show that some cone Cf̂ of f̂ satisfies ζ-ord(Cf̂) ≥ s − 2. It is enough to show that some cone of the dashed arrow has this property. Note that this map (in the unstable category) is surjective (because f and μ_{1,1} are). Therefore, up to a shift, the kernel of the composition

L_{ζ} ⊗ K ⟶^{id ⊗f} L_{ζ} ⊗ X_{1} ⟶^{μ_{1,1}} X_{2}

is a possible choice of cone of the dashed arrow. Since this kernel is the underlying object of C, we get the inductive step from (2.7).

Corollary 2.8. If X is any object in \underline{𝔪𝔬𝔡}-kG, then ζ-ord(X/ζ) ≥ p − 2. In particular, the ζ-order of \underline{𝔪𝔬𝔡}-kG is at least p − 1.

Proof. Since L_{ζ} ≅ Ω(k/ζ) in \underline{𝔪𝔬𝔡}-kG, we get ζ-ord(X/ζ) = ζ-ord(L_{ζ} ⊗ X). But L_{ζ} ⊗ X is the underlying object of the (p − 1)-coherent module F (X), so we get the result from Proposition 2.6.

Remark 2.9. It should be noted that, for the proof of Proposition 2.6, the exactness condition can be dropped from the definition of s-coherent modules. The reason why we introduced it was that we needed the surjectivity of the composite L_{ζ}⊗K → L_{ζ}⊗X_{1} → X_{2}, because then the kernel of this map serves as a fiber in the stable category. We could equally well have used a functorial mapping cone construction on 𝔪𝔬𝔡-kG as follows: for an unstable map f : X → Y define the mapping cone C(f) to be the cokernel of the injective map

X ≅ X ⊗ k ⟶^{(id ⊗i,f)} X ⊗ kG ⊕ Y.

Here, i : k → kG denotes the usual inclusion. The construction is very similar to the mapping cone for maps f : X → Y of simplicial sets, given as C(f) = Δ[1] ∧ X ∪_{1×X} Y where Δ[1] is the standard 1-simplex, pointed by the 0-vertex.

The mapping cone is sufficiently well-behaved for our purposes: we have natural isomorphisms C(id_{Z} ⊗f) ≅ Z ⊗ C(f), and for every commutative diagram

\xymatrix{X \ar[r]^{f} \ar[d] & Y \ar[d] \\ X′ \ar[r]_{f′} & Y′}

we get an induced map C(f) → C(f′) making the obvious diagrams commute. Using these facts, it is straightforward to formulate an alternative proof of Proposition 2.6 not using the exactness condition of s-coherent modules.

Nevertheless, the way we have done it turns out to be useful in the proof of the more general statement in §4.

Remark 2.10. Let us note that Corollary 2.8 is not true for arbitrary non-zero even-degree elements ζ of the graded center. We will work out an example in the case p = 3 with ζ-ord(k/ζ) = 0 in §6.

3. The power operation

3 The power operation

In this section we define a power operation P_{1} on the Tate cohomology Ĥ^{∗}(G). The operation will serve as an obstruction for the ζ-order of k/ζ to be at least p − 1. At the same time, it enables us to compute certain Massey products explicitly, and we will make use of this fact later on in order to prove ζ-ord(k/ζ) = p − 2 for certain groups G and certain classes ζ. Let us gather several interesting properties of P_{1} in one big theorem.

Theorem 3.1. There is a power operation P_{1} on Tate cohomology Ĥ^{∗}(G) satisfying the following properties:

(i) For every Tate cohomology class ζ ∈ Ĥ^{n}(G) (with n even if p is odd) the operation P_{1}(ζ) is a coset of ζ^{p} · Ĥ^{−(2p−3)}(G) in Ĥ^{pn−(2p−3)}(G).

(ii) The operation is linear in the sense that P_{1}(ζ + φ) ⊆ P_{1}(ζ) + P_{1}(φ) and P_{1}(c · ζ) = c^{p} · P_{1}(ζ) for all ζ, φ ∈ Ĥ^{n}(G) and c ∈ k.

(iii) If p = 2 then for every ordinary cohomology class ζ ∈ H^{n}(G) we have that the Steenrod square Sq^{n−1}(ζ) is an element of P_{1}(ζ). If p is odd then for every ordinary cohomology class ζ ∈ H^{n}(G) of even degree we get that βP^{\frac{n}{2} −1}(ζ) ∈ P_{1}(ζ).

(iv) The operation is natural with respect to injective group homomorphisms i : G ↪ H: for every ζ ∈ Ĥ^{∗}(H) (of even degree if p is odd) we get i^{∗}P_{1}(ζ) ⊆ P_{1}(i^{∗}ζ).

(v) For any ζ, φ ∈ Ĥ^{∗}(G) the Cartan formula P_{1}(ζφ) ⊆ P_{1}(ζ)φ^{p} + ζ^{p}P_{1}(φ) holds.

Example 3.2. Property (iii) says that our power operation extends the Steenrod operations Sq_{1} and P_{1} that we have on ordinary cohomology to Tate cohomology, at the price of getting a certain indeterminacy. Let us demonstrate the effect of the indeterminacy for two examples of groups. Let p be an odd prime. We begin with G = ℤ/pℤ, the cyclic group of order p. The structure of Ĥ^{∗}(G) is known (see [4], §XII.7) to be Ĥ^{∗}(G) ≅ k[u, v^{±1}] where u is an exterior class of degree 1 and v is a Laurent polynomial class of degree 2. For every non-zero ζ of even degree we get that ζ^{p} · Ĥ^{−(2p−3)}(G) = Ĥ^{pn−(2p−3)}(G), the indeterminacy is the whole cohomology group, so that P_{1} does not store any information at all. As a second example let us study G = ℤ/pℤ × ℤ/pℤ. The structure of Ĥ^{∗}(G) is known and we will recall it in §3.6. In particular we have that ζφ = 0 for all classes ζ, φ whose degrees satisfy |ζ| < 0, |φ| < 0 or |ζ| < 0, |ζφ| ≥ 0. This implies that P_{1}(ζ) has zero indeterminacy for all ζ (except in the case |ζ| = 0).

Example 3.3. Let us do an example of a non-commutative group which can be worked out completely using Theorem 3.1 only. Let p = 2 and G = Q_{8}, the quaternion group with 8 elements. The Tate cohomology ring is known (see, e.g., [4], XII.11, and [1], Lemma IV.2.10) to be

Ĥ^{∗}(G) ≅ k[x, y, s^{±1}]/(x^{2} + xy + y^{2}, x^{3})

with degrees |x| = |y| = 1 and |s| = 4. In order to describe P_{1} we first have a look at the indeterminacy. The vector space Ĥ^{−1}(G) is generated by xy^{2}s^{−1}, and for every homogeneous z ∈ Ĥ^{∗}(G) we have xy^{2}s^{−1}z^{2} = 0 unless |z| is divisible by 4. Therefore, P_{1}(s^{n}) = Ĥ^{8n−1}(G) for all integers n, and in all other degrees P_{1} does not have any indeterminacy. We know by part (iii) of the theorem that P_{1}(x) = Sq_{1}(x) = Sq^{0}(x) = x and P_{1}(y) = y. The Cartan formula implies P_{1}(x^{2}) = 0, P_{1}(y^{2}) = 0, and P_{1}(x^{2}y) = 0, and finally

\begin{aligned}P_{1}(s^{n}x) & = \underbrace{P_{1}(s^{n})x^{2}}_{0} +s^{2n}P_{1}(x) = s^{2n}x, \\ P_{1}(s^{n}y) & = s^{2n}y, \end{aligned}  \begin{aligned}P_{1}(s^{n}x^{2}) & = 0, \\ P_{1}(s^{n}y^{2}) & = 0, \\ P_{1}(s^{n}x^{2}y) & = 0. \end{aligned}

The plan of this section is as follows. The first two subsections have a preparative character. In the first one we introduce a new description of negative Ext-groups via complexes of projectives, whereas in the second one we prove the existence of a cochain map between certain cochain complexes. The definition of our power operation and proofs of the most obvious properties are given in the third subsection. In §3.4 we prove the Cartan formula, and in §3.5 we show that the new operation contains the first non-trivial Steenrod operation in positive degrees. In the last two subsections we compute P_{1} in the case of elementary abelian p-groups.

3.1 Negative Ext-groups

Let n > 0. It is well-known that Ext_{kG}^{n}(A, B) = \underline{Hom}_{kG}(Ω^{n}A, B) admits a description via extensions of A by B. We will now give a similar description of Êxt_{kG}^{−n}(A, B) ≅ \underline{Hom}_{kG}(A, Ω^{n}B). Let us define a category K_{n}(A, B), whose objects are all the chain complexes

C : A ⟶ P_{n} ⟶ P_{n−1} ⟶ … ⟶ P_{1} ⟶ B

with projective modules P_{1}, P_{2}, …, P_{n}, and a morphism of two such complexes is a commutative diagram as follows:

[figure: C; A; P_{n}; P_{n−1}; …; P_{1}; B; C′; A; P′_{n}; P′_{n−1}; …; P′_{1}; B]

For objects C and C′, let us write C ≈ C′ if there is a morphism C → C′ in K_{n}(A, B). Define the relation ∼ on K_{n}(A, B) to be the equivalence relation generated by ≈, and put K_{n}(A, B) = K_{n}(A, B)/ ∼, the connected components of K_{n}(A, B). We will sometimes write K_{n}^{kG}(A, B) if we want to emphasize that we are working over the algebra kG.

Let us fix a projective resolution of B:

P :  0 ⟶ Ω^{n}B ⟶^{i} P_{n} ⟶ P_{n−1} ⟶ … ⟶ P_{1} ⟶ B ⟶ 0  (3.4)

Theorem 3.5. The map Φ : Hom_{kG}(A, Ω^{n}B) → K_{n}(A, B) which associates to each map f : A → Ω^{n}B the complex A ⟶^{i∘f} P_{n} → P_{n−1} → ⋯ → P_{1} → B induces a bijection

\underline{Hom}_{kG}(A, Ω^{n}B) ⟷^{1:1} K_{n}(A, B).

3.1 Negative Ext-groups

To prove this, we need the following lemma.

Lemma 3.6. Suppose we are given two finite chain complexes A = (0 → A_{n+1} → ⋯ → A_{0} → 0) and B = (0 → B_{n+1} → ⋯ → B_{0} → 0), where A_{i} is projective for i = 1, 2, …, n, and B is exact. Let f, g : A → B be chain maps satisfying f_{0} = g_{0} : A_{0} → B_{0}. Then the classes of f_{n+1} and g_{n+1} in \underline{Hom}_{kG}(A_{n+1}, B_{n+1}) are the same.

Proof. This is a standard fact from homological algebra. We can assume that g = 0. Then f_{0} = 0, hence d_{B} f_{1} = 0. Since A_{1} is projective and B is exact, there exists some h_{1} : A_{1} → B_{2} such that d_{B} h_{1} = f_{1}. Inductively, one can find h_{j} : A_{j} → B_{j+1} satisfying d_{B} h_{j} + h_{j−1} d_{A} = f_{j} for all j = 2, 3, …, n. But then

d_{B} h_{n} d_{A} = (f_{n} − h_{n−1} d_{A}) d_{A} = f_{n} d_{A} = d_{B} f_{n+1},

and since d_{B} : B_{n+1} → B_{n} is injective, we get h_{n} d_{A} = f_{n+1}. Therefore, f_{n+1} factors through a projective module (namely A_{n}).

Proof of Theorem 3.5: As a first step, we show that Φ induces a map \underline{Hom}_{kG}(A, Ω^{n}B) → K_{n}(A, B). Suppose we are given f′ ∈ Hom_{kG}(A, Ω^{n}B) such that f′ − f factors through some projective module R:

\xymatrix{f′ − f : & A \ar[r]^{u} & R \ar[r]^{w} & Ω^{n}B}

Then the complexes Φ(f) and Φ(f′) differ in their first map only; let us denote these by α, α′ : A → P_{n}, respectively. From the commutative diagram

[figure: A; α; P_{n}; P_{n−1}; P_{n−2}; …; B; A; ( α u); P_{n} ⊕ R; _{( \begin{matrix}id \\ 0 \end{matrix} )}; _{( \begin{matrix}d \\ 0 \end{matrix} )}; _{( \begin{matrix}id \\ d∘w \end{matrix} )}; P_{n−1}; P_{n−2}; …; B; A; α′; P_{n}; P_{n−1}; P_{n−2}; …; B]

we get that Φ(f) ∼ Φ(f′). Therefore, we obtain a map \underline{Hom}_{kG}(A, Ω^{n}B) → K_{n}(A, B) which we also denote by Φ.

To construct an inverse for Φ, start with some object C = (A → Q_{∗} → B) ∈ K_{n}(A, B). Since the Q_{i}’s are projective and (3.4) is exact, we can lift the identity on B to a map of chain complexes f : C → P:

[figure: A; Q_{n}; Q_{n−1}; …; Q_{1}; B; Ω^{n}B; P_{n}; P_{n−1}; …; P_{1}; B]

By Lemma 3.6, the stable class of the resulting map f_{n+1} : A → Ω^{n}B is independent of the choice of the lift; let us write Ψ(C) = f_{n+1} ∈ \underline{Hom}_{kG}(A, B). Suppose we are given a morphism g : C′ → C in K_{n}(A, B). Then f ∘ g is a lift of the identity on B to a map of chain complexes C′ → P. Since g_{n+1} = id_{A}, we have Ψ(C′) = (f ∘ g)_{n+1} = f_{n+1} = Ψ(C). Therefore, we have constructed a map Ψ : K_{n}(A, B) → \underline{Hom}_{kG}(A, Ω^{n}B). The proofs of Ψ ∘ Φ = id and Φ ∘ Ψ = id are immediate.

Let us investigate the additive structure more closely.

Lemma 3.7. Suppose that A ⟶^{i} P_{n} → ⋯ → P_{1} ⟶^{f} B and A ⟶^{j} Q_{n} → ⋯ → Q_{1} ⟶^{g} B are

two complexes representing classes κ, λ ∈ Êxt_{kG}^{−n}(A, B), respectively. Then the complex

A \underrightarrow{( \begin{matrix}i \\ j \end{matrix} )} P_{n} ⊕ Q_{n} → P_{n−1} ⊕ Q_{n−1} → ⋯ → P_{1} ⊕ Q_{1} ⟶^{( f g)} B

represents the class κ + λ.

Corollary 3.8. Suppose that A ⟶^{i} P_{∗} ⟶^{f} B and A ⟶^{i} P_{∗} ⟶^{g} B are complexes representing classes κ, λ ∈ Êxt_{kG}^{−n}(A, B). Then A ⟶^{i} P_{∗} ⟶^{f+g} B is a complex representing κ + λ.

We omit the straightforward proof of the lemma; the corollary is deduced by using the commutative diagram

[figure: A; i; P_{∗}; Δ; f+g; B; A; _{( \begin{matrix}i \\ i \end{matrix} )}; P_{∗} ⊕ P_{∗}; _{(_{f g})}; B]

where Δ is the diagonal map.

Proposition 3.9. Suppose we have a commutative diagram

\xymatrix{ & A \ar[r] \ar[d]_{g} & P_{n} \ar[r] \ar[d] & … \ar[r] & P_{1} \ar[r] \ar[d] & B \ar[d]^{f} \\ 0 \ar[r] & D \ar[r] & E_{n} \ar[r] & … \ar[r] & E_{1} \ar[r] & C \ar[r] & 0}

in 𝔪𝔬𝔡-kG. Assume further that the P_{i}’s are projective, so that the upper row represents some element α ∈ \underline{Hom}_{kG}(A, Ω^{n}B), and assume that the lower row is exact, therefore representing some element β ∈ \underline{Hom}_{kG}(Ω^{n}C, D). Then the diagram

\xymatrix{A \ar[r]^{α} \ar[d]_{g} & Ω^{n}B \ar[d]^{Ω^{n}f} \\ D & Ω^{n}C \ar[l]^{β}}

commutes stably.

Proof. Choose projective resolutions Ω^{n}B → Q_{∗} → B and Ω^{n}C → R_{∗} → C. By the usual ‘projective to acyclic’-argument, we get a diagram

[figure: A; ᾱ; P_{∗}; B; Ω^{n}B; Ω^{n}f; Q_{∗}; B; f; Ω^{n}C; β̄; R_{∗}; C; D; E_{∗}; C]

where ᾱ and β̄ are unstable representatives of α and β, respectively. The result follows from Lemma 3.6.

Remark 3.10. Suppose we have an exact sequence A ↪ P_{n} → ⋯ → P_{1} ↠ B with projective modules P_{1}, …, P_{n}. Then we can view this as an extension representing some stable isomorphism Ω^{n}B → A; but we can also consider this as an element of K_{n}(A, B), representing some stable isomorphism A → Ω^{n}B; by the previous proposition, the two maps are stable inverses of each other.

We have a composition product K_{n}(B, C) × K_{m}(A, B) → K_{n+m}(A, C) similar to the Yoneda splice: given E : A → P_{∗} → B and E′ : B → Q_{∗} → C we define E′ ∘ E to be the complex

\xymatrix{E′ ∘ E : & A \ar[r] & P_{∗} \ar[rr] \ar@{-->}[dr] &  & Q_{∗} \ar[r] & C. \\  &  &  & B \ar@{-->}[ur]}

This product is compatible with the equivalence relation ∼ and therefore induces a product

K_{n}(B, C) × K_{m}(A, B) → K_{n+m}(A, C).

Lemma 3.11. The composition products on K_{∗} and Êxt_{kG}^{−∗} coincide under the bijection of Theorem 3.5.

Proof. Let us start with complexes A → P_{∗} → B and B → Q_{∗} → C representing stable maps α : A → Ω^{m}B and β : B → Ω^{n}C, respectively. Choose projective resolutions Ω^{n}C → R_{∗} → C and Ω^{n+m}C → T_{∗} → Ω^{n}C. Then we can lift the identity map on C to commutative diagrams as follows:

[figure: A; γ̄; P_{∗}; B; β̄; B; β̄; Q_{∗}; C; Ω^{n+m}C; T_{∗}; Ω^{n}C; Ω^{n}C; R_{∗}; C]

Here, β̄ and γ̄ are unstable representatives of β and some γ. Note that the extension Ω^{n+m}C → T_{∗} → Ω^{n}C represents the identity map id ∈ \underline{Hom}_{kG}(Ω^{m}Ω^{n}C, Ω^{n+m}C). By Proposition 3.9, the left diagram shows that γ = βα. After splicing the two diagrams the result follows from Lemma 3.6.

There is also a way of composing an element x ∈ Êxt_{kG}^{−n}(A, B) given as a complex A → P_{∗} → B with an element of y ∈ Êxt_{kG}^{m}(B, C) (with m > 0) given as an extension C ↪ M_{∗} ↠ B:

Lemma 3.12. Suppose m < n. The identity map of B can be lifted to a diagram

[figure: A; P_{n}; …; P_{m+1}; P_{m}; …; P_{1}; B; C; M_{m}; …; M_{1}; B]

and for any such lifting, the complex \xymatrix{A \ar[r] & P_{n} \ar[r] & … \ar[r] & P_{m+1} \ar[r] & C} represents the composition y · x ∈ Êxt_{kG}^{m−n}(A, C).

Proof. Existence of the lifting is common homological algebra. For the second statement choose a projective resolution Ω^{n−m}C → R_{∗} → C; then we have the following commutative diagram:

[figure: A; γ̄; P_{n}; …; P_{m+1}; P_{m}; …; P_{1}; B; C; Ω^{n−m}C; R_{n−m}; …; R_{1}; M_{m}; …; M_{1}; B]

The complex in question represents the stable class of the map γ̄. The bottom row represents y ∈ \underline{Hom}_{kG}(Ω^{n}B, Ω^{n−m}C), the upper row represents x ∈ \underline{Hom}_{kG}(A, Ω^{n}B). The result follows from Proposition 3.9.

Remark 3.13. We can also compose an element x ∈ Êxt_{kG}^{0}(A, B) = \underline{Hom}_{kG}(A, B) given by an unstable map f : A → B with an element y ∈ Êxt_{kG}^{−n}(B, C) given by a complex B ⟶^{i} P_{∗} ⟶^{π} C; the composition y · x ∈ Êxt_{kG}^{−n}(A, C) is then given by the complex A ⟶^{i∘f} P_{∗} ⟶^{π} C. Similarly, for any z ∈ Êxt_{kG}^{0}(C, D) represented by some unstable map g : C → D, the complex B ⟶^{i} P_{∗} ⟶^{g∘π} D represents the product z · y ∈ Êxt_{kG}^{−n}(B, D). In fact, these statements are special cases of the following Proposition.

Proposition 3.14. Suppose that we have a commutative diagram

\xymatrix{A \ar[r] \ar[d]_{f} & P_{n} \ar[r] \ar[d] & … \ar[r] & P_{1} \ar[r] \ar[d] & B \ar[d]^{g} \\ A′ \ar[r] & Q_{n} \ar[r] & … \ar[r] & Q_{1} \ar[r] & B′}

with projective modules P_{i}, Q_{i} for i = 1, 2, …, n. Then the rows represent maps x : A → Ω^{n}B and y : A′ → Ω^{n}B′, respectively, and y ∘ f = Ω^{n}(g) ∘ x in \underline{𝔪𝔬𝔡}-kG.

Proof. Choose a projective resolution Ω^{n}B′ → R_{∗} → B′. By usual homological algebra, we get a diagram

[figure: A; f′; P_{∗}; B; g; A′; y; Q_{∗}; B′; Ω^{n}B′; R_{∗}; B′]

and then the result follows from Proposition 3.9.

Remark 3.15. There is a similar statement for extensions. Suppose that we have a diagram as in Proposition 3.14, but this time with exact rows and the P_{i}’s and Q_{i}’s are not necessarily projective. Then the rows represent maps x : Ω^{n}B → A and y : Ω^{n}B′ → A′, and f ∘ x = y ∘ Ω^{n}(g) in \underline{𝔪𝔬𝔡}-kG.

We end this part by mentioning that the bijection of Theorem 3.5 is natural with respect to injective group homomorphisms. Suppose that H is another group and i : H ↪ G is an injective group homomorphism. Then the functor i^{∗} : 𝔪𝔬𝔡-kG → 𝔪𝔬𝔡-kH is exact and maps projective kG-modules to projective kH-modules. This implies that i^{∗}(Ω_{G}^{n}X) ≅ Ω_{H}^{n}(i^{∗}X) in \underline{𝔪𝔬𝔡}-kH for every kG-module X. Also, we get induced maps K_{n}^{kG}(A, B) → K_{n}^{kH}(i^{∗}A, i^{∗}B) and \underline{Hom}_{kG}(A, Ω^{n}B) → \underline{Hom}_{kH}(i^{∗}A, Ω^{n}(i^{∗}B)). Naturality of the map Φ then shows the following lemma.

Lemma 3.16. The diagram

\xymatrix{K_{n}^{kG}(A, B) \ar[r]^{1:1} \ar[d] & \underline{Hom}_{kG}(A, Ω^{n}B) \ar[l] \ar[d] \\ K_{n}^{kH}(i^{∗}A, i^{∗}B) \ar[r]_{1:1} & \underline{Hom}_{kH}(i^{∗}A, Ω^{n}(i^{∗}B)) \ar[l]}

commutes.

Example 3.17. Suppose that p divides the order of the group G. Then it is known that Ĥ^{−1}(G) ≅ \underline{Hom}_{kG}(k, Ωk) is isomorphic to k. Under the bijection of Theorem 3.5, a canonical generator of that vector space is given by the complex

\xymatrix{k \ar[r]^{∑_{g∈G} g} & kG \ar[r]^{ϵ} & k}

where ϵ is the augmentation of kG. The previous lemma shows that this complex is invariant under every group automorphism of G.

Remark 3.18. The previous lemma generalizes to those morphisms C → C′ of Hopf algebras for which C′ becomes a projective C-module. In the case when C = kH and C′ = kG are group algebras and the morphism comes from a morphism f : H → G of groups, it is a nice exercise to show that kG is projective as a kH-module if and only if p does not divide the order of the kernel of f.

3.2 A map of cochain complexes

Let X be any kG-module. Consider the complex

S :  X^{⊗p} ⟶^{1−T} X^{⊗p} ⟶^{N} X^{⊗p} ⟶^{1−T} X^{⊗p} ⟶^{N} … ⟶^{N} X^{⊗p} ⟶^{1−T} X^{⊗p}  (3.19)

consisting of 2p − 2 objects and 2p − 3 morphisms, which we refer to as the Steenrod complex. On the other hand, we can splice the complexes (1.11) and (1.8) to obtain the de Rham-Koszul complex

[figure: RK :; X ⊗ S^{p−1}X; …; Λ^{p−1}X ⊗ X; Λ^{p−1}X ⊗ X; …; X ⊗ S^{p−1}X; Λ^{p}X]

also consisting of 2p − 2 objects and 2p − 3 morphisms. We consider both complexes S and RK as cochain complexes.

Proposition 3.20. There is a natural map of cochain complexes β : S → RK such that

(i) the last map β_{2p−2} : X^{⊗p} → X ⊗ S^{p−1}X is given by x_{1} ⊗ x_{2} ⊗ ⋯ ⊗ x_{p} ↦ x_{1} ⊗ x_{2}x_{3} ⋯ x_{p}, and

(ii) the first map β_{1} : X^{⊗p} → X ⊗ S^{p−1}X is given by β_{1} = −β_{2p−2}.

Proof. The statement is obvious for p = 2, so let us assume p ≥ 3 from now on. Because of naturality, we can forget the G-module structures and consider this as a statement on k-vector spaces X. Let us recall some standard notions. Suppose that X, Y are k-vector spaces, and let H be a group acting on X from the right and on Y from the left. Then we have the usual definitions

X ⊗_{H} Y = X ⊗ Y/ ⟨(x · h) ⊗ y − x ⊗ (h · y) | x ∈ X, y ∈ Y, h ∈ H⟩_{k} ,
X/H = X/ ⟨x · h − x | x ∈ X, h ∈ H⟩_{k} ≅ X ⊗_{H} k,

where k denotes the trivial H-module. Whenever X is a right H-module and H_{1} ⊂ H is a subgroup, then X is a right H_{1}-module and there is a natural map X/H_{1} → X/H. For every vector space X, the symmetric group Σ_{n} acts on X^{⊗n} from the right by permuting the factors. The cyclic subgroup C_{n} = ℤ/nℤ ⊂ Σ_{n} acts by cyclic permutation of the factors, and we define the cyclic product to be the quotient C^{n}(X) = X^{⊗n}/C_{n}. Then we have a natural map C^{n}(X) = X^{⊗n}/C_{n} → X^{⊗n}/Σ_{n} = S^{n}(X).

All these notions generalize naturally to (co)chain complexes C, where one has to note that the action of Σ_{n} on C^{⊗n} involves signs depending on the degrees of the permuted elements. For instance, if C = X[1] is the chain complex with X concentrated in degree 1, then S^{n}(C) = (Λ^{n}X)[n] (here we use that p is odd), and C^{n}(C) = (C^{n}(X))[n], so there is also a natural map C^{n}(X) → Λ^{n}X.

We will use the following result of Swan. Let m be any integer, and define M to be the cochain complex over k generated by two elements x, y of degrees m+1 and m, respectively, subject to the condition ∂y = x. That is, M is isomorphic to ⋯ → 0 → k ⟶^{id} k → 0 → . . ..

3.2 A map of cochain complexes

Proposition 3.21 (Swan [27], Lemma 22.2 and Remark 22.1). Let X be the kC_{p}-cochain complex given by the sequence

0 → k ⟶^{N} kC_{p} ⟶^{1−T} kC_{p} ⟶^{N} … ⟶^{1−T} kC_{p} ⟶^{ε} k → 0,

where T is a generator of C_{p} and N = 1 + T + ⋯ + T^{p−1}, and the k’s are in degrees mp and mp + p. Then there is a C_{p}-equivariant homotopy equivalence ψ : X → M^{⊗p} such that ψ_{mp+p}(1) = x^{⊗p} and ψ_{mp}(1) = c · y^{⊗p} with the constant c = (−1)^{m· \frac{p−1}{2}} (\frac{p−1}{2})!.

It follows from the proposition’s proof that we can impose one of the following two extra conditions on ψ: either ψ_{mp+p−1}(1) = y ⊗ x^{⊗(p−1)} or ψ_{mp+1}(1) = c · x ⊗ y^{⊗(p−1)}.

Now let X be any k-vector space. Since ψ : X → M^{⊗p} is C_{p}-equivariant, we can form the composite

X ⊗_{C_{p}} X^{⊗p} ⟶^{ψ⊗_{Cp}id} M^{⊗p} ⊗_{C_{p}} X^{⊗p} → M^{⊗p} ⊗_{Σ_{p}} X^{⊗p} ≅ (M ⊗ X)^{⊗p}/Σ_{p} ≅ S^{p}(M ⊗ X).

If we now put m = 0, then this composite yields a map of cochain complexes

\xymatrix{C^{p}X \ar[r]^{N} \ar[d] & X^{⊗p} \ar[r]^{1−T} \ar[d] & X^{⊗p} \ar[r]^{N} \ar[d] & … \ar[r]^{1−T} & X^{⊗p} \ar[r] \ar[d] & C^{p}X \ar[d] \\ Λ^{p}X \ar[r] & Λ^{p−1}X ⊗ X \ar[r] & Λ^{p−2}X ⊗ S^{2}X \ar[r] & … \ar[r] & X ⊗ S^{p−1}X \ar[r] & S^{p}X}  (3.22)

where we can assume that X^{⊗p} → X ⊗ S^{p−1}X is the map β_{2p−2}. On the other hand we can put m = 1 to get

\xymatrix{C^{p}X \ar[r]^{N} \ar[d] & X^{⊗p} \ar[r]^{1−T} \ar[d] & X^{⊗p} \ar[r]^{N} \ar[d] & … \ar[r]^{1−T} & X^{⊗p} \ar[r] \ar[d] & C^{p}X \ar[d] \\ S^{p}X \ar[r] & X ⊗ S^{p−1}X \ar[r] & Λ^{2}X ⊗ S^{p−2}X \ar[r] & … \ar[r] & Λ^{p−1}X ⊗ X \ar[r] & Λ^{p}X}  (3.23)

with the map X^{⊗p} → X ⊗ S^{p−1}X being equal to β_{1}. By splicing the two complexes (3.22) and (3.23) after multiplying with suitable non-zero constants we obtain a proof of Proposition 3.20.

Remark 3.24. To determine the constant c in Proposition 3.21 it seems to be necessary to write down a suitable cochain map ψ explicitly. This is not done in Swan [27], and we will also not do it here. In an earlier proof of Proposition 3.20, the author of this thesis constructed the cochain map β explicitly, thereby showing that the multiplicative constants are indeed as given. However, the proof is rather lengthy and not very enlightening, which is why we omit it here. For the main results of this thesis it is actually enough to know that β_{1} is a non-zero multiple of β_{2p−2}, which follows from Swan’s proof. Then, certain diagrams will only commute up to non-zero scalars, but this is enough for our purposes.

3.3 Definitions and basic properties

Now we give the definition of the power operation we are interested in. Let n be an integer. We assume that n is even if p is odd. Let [ζ] ∈ Ĥ^{n}(G) be a Tate cohomology class; as a first step, we will define a subset D_{i}([ζ]) of Ĥ^{pn−i}(G) for all positive integers i = 1, 2, …

To do so, choose an unstable map ζ : Ω^{n}k → k representing the class [ζ], and consider all commutative diagrams of the form

[figure: (Ω^{n}k)^{⊗p}; P_{1}; P_{2}; …; P_{i}; k; (Ω^{n}k)^{⊗p}; 1−T; (Ω^{n}k)^{⊗p}; N; (Ω^{n}k)^{⊗p}; 1−T; …; (∗); (Ω^{n}k)^{⊗p}; ζ^{⊗p}; k]  (3.25)

in which the upper row is a complex with projective modules P_{j}. It therefore defines an element of Êxt_{kG}^{−i}((Ω^{n}k)^{⊗p}, k) ≅ Ĥ^{pn−i}(G). The set of all elements obtained this way is denoted by D_{i}(ζ). Note that the map (∗) in the diagram is either N or 1 − T, depending on the parity of i.

Lemma 3.26. The set D_{i}(ζ) defined above does not depend on the chosen representative ζ for the cohomology class [ζ]. We can therefore write D_{i}([ζ]) = D_{i}(ζ).

Proof. If we assume that Ω^{n}k arises from a minimal resolution ⋯ → P_{n} → …, then we know that PHom_{kG}(Ω^{n}k, k) = 0. To see this, note that every morphism Ω^{n}k → k which factors through a projective module also factors as Ω^{n}k ↪ P_{n−1} → k. Since we have chosen a minimal resolution, the differential of the complex Hom_{kG}(P_{∗}, k) vanishes and therefore the composition P_{n} → P_{n−1} → k is zero. From the surjectivity of P_{n} → Ω^{n}k we then get that our original map must be zero. Therefore, PHom_{kG}(Ω^{n}k, k) = 0 and there is no ambiguity in choosing ζ.

Now suppose we have another version of Ω^{n}k and call it Ω̃^{n}k, and let ζ̃: Ω̃^{n}k → k be a map representing the same cohomology class as ζ. Then we have that Ω̃^{n}k ≅ Ω^{n}k ⊕ R for some projective module R. We have retraction maps Ω^{n}k ⟶^{ι} Ω̃^{n}k ⟶^{r} Ω^{n}k and we know that ζ̃ι = ζ. We want to show that the sets D_{i}(ζ) and D_{i}(ζ̃) agree under the isomorphisms

\xymatrix{Êxt_{kG}^{−i}((Ω^{n}k)^{⊗p}, k) \ar[r]^{(r^{⊗p})^{∗}} & Êxt_{kG}^{−i}((Ω̃^{n}k)^{⊗p}, k) \ar[l]^{(ι^{⊗p})^{∗}}} .

Let us suppress these isomorphisms from the notation and simply prove D_{i}(ζ̃) = D_{i}(ζ).

Note that (Ω̃^{n}k)^{⊗p} ≅ (Ω^{n}k)^{⊗p} ⊕ S for some projective module S, and under this isomorphism, the map T decomposes as the direct sum of the map T on (Ω^{n}k)^{⊗p} and some map on S. This implies that there is an isomorphism of complexes

[figure: (Ω̃^{n}k)^{⊗p}; (Ω̃^{n}k)^{⊗p}; …; (Ω̃^{n}k)^{⊗p}; ζ̃^{⊗p}; k; (Ω^{n}k)^{⊗p} ⊕ S; (Ω^{n}k)^{⊗p} ⊕ S; …; (Ω^{n}k)^{⊗p} ⊕ S; (ζ^{⊗p},α); k]

3.3 Definitions and basic properties

where the lower row (without the last map) is a direct sum of two complexes. Now suppose we have a diagram

[figure: (Ω̃^{n}k)^{⊗p}; d_{0}; P_{1}; d_{1}; (f_{1},g_{1}); …; d_{i−1}; P_{i}; d_{i}; (f_{i},g_{i}); k; (Ω^{n}k)^{⊗p} ⊕ S; (Ω^{n}k)^{⊗p} ⊕ S; …; (Ω^{n}k)^{⊗p} ⊕ S; k]

defining some element in D_{i}(ζ̃). Then we obtain a diagram

[figure: (Ω^{n}k)^{⊗p}; d_{0}∘ι^{⊗p}; P_{1}; d_{1}; f_{1}; …; d_{i−1}; P_{i}; d_{i}−αg_{i}; f_{i}; k; (Ω^{n}k)^{⊗p}; (Ω^{n}k)^{⊗p}; …; (Ω^{n}k)^{⊗p}; ζ; k]

defining some element in D_{i}(ζ). By Corollary 3.8 the difference of that element and the element we started with is the cohomology class represented by the upper row of the diagram

[figure: (Ω^{n}k)^{⊗p}; 0; P_{1}; g_{1}; …; P_{i}; αg_{i}; g_{i}; k; S; S; …; S; α; k]

and since the left-most vertical map is zero, we get that this difference vanishes by Proposition 3.14.

Conversely, let us start with some diagram

[figure: (Ω^{n}k)^{⊗p}; P_{1}; f_{1}; …; P_{i}; g; f_{i}; k; (Ω^{n}k)^{⊗p}; (Ω^{n}k)^{⊗p}; …; (Ω^{n}k)^{⊗p}; ζ; k]

defining some element of D_{i}(ζ); then we obtain a diagram

[figure: (Ω̃^{n}k)^{⊗p} ≅ (Ω^{n}k)^{⊗p} ⊕ S; P_{1} ⊕ S; f_{1}⊕id; …; P_{i} ⊕ S; (g,α); f_{i}⊕id; k; (Ω̃^{n}k)^{⊗p} ≅ (Ω^{n}k)^{⊗p} ⊕ S; (Ω^{n}k)^{⊗p} ⊕ S; …; (Ω^{n}k)^{⊗p} ⊕ S; k]

defining an element in D_{i}(ζ̃) which is the same as we started with because of the following commutative diagram:

[figure: (Ω^{n}k)^{⊗p}; ι^{⊗p}; P_{1}; …; P_{i}; g; k; (Ω̃^{n}k)^{⊗p} ≅ (Ω^{n}k)^{⊗p} ⊕ S; P_{1} ⊕ S; …; P_{i} ⊕ S; (g,α); k]

We could have equally well started with a slightly modified definition:

Lemma 3.27. Every class in D_{i}(ζ) comes from a diagram of the form (3.25) in which the sequence 0 → (Ω^{n}k)^{⊗p} → P_{1} → ⋯ → P_{i} is exact.

Proof. Choose an injective resolution (Ω^{n}k)^{⊗p} ↪ Q_{1} → ⋯ → Q_{i}. Given any diagram of the form (3.25), we can lift the identity map on (Ω^{n}k)^{⊗p} to a map of complexes Q_{∗} → P_{∗} (because the P_{j}’s are injective modules and the resolution is an exact sequence). Finally define Q_{i} → k to be the composite Q_{i} → P_{i} → k.

An immediate consequence is the following lower bound on the indeterminacy of D_{i}(ζ).

Corollary 3.28. If a ∈ D_{i}(ζ), then a + ζ^{p} · Ĥ^{−i}(G) ⊆ D_{i}(ζ).

Proof. We assume that a is given by a diagram

[figure: (Ω^{n}k)^{⊗p}; P_{1}; …; d_{i−1}; P_{i}; f_{i}; g; k; (Ω^{n}k)^{⊗p}; (Ω^{n}k)^{⊗p}; …; (Ω^{n}k)^{⊗p}; ζ^{⊗p}; k]

in which the upper row is as in Lemma 3.27. Then the cokernel of d_{i−1} is a choice of Σ^{i}(Ω^{n}k)^{⊗p}. If we are given some class b ∈ Ĥ^{−i}(G) represented by some unstable map h : coker d_{i−1} → (Ω^{n}k)^{⊗p} we can form the composition w : P_{i} → coker d_{i−1} ⟶^{h} (Ω^{n}k)^{⊗p}. Replace f_{i} by f_{i} + w and g by g + ζ^{⊗p} ∘ w, then we obtain a new diagram showing that a + ζ^{p}b ∈ D_{i}(ζ).

Let us fix some non-zero map ζ : Ω^{n}k → k. Let A_{1} → A_{2} → ⋯ → A_{2p−2} be the de Rham-Koszul complex introduced in §3.2, so that A_{j} = A_{2p−1−j} = Λ^{j}(Ω^{n}k) ⊗ S^{p−j}(Ω^{n}k) for all j = 1, 2, …, p − 1. Furthermore, let K be the kernel of the surjective map ζ^{⊗p} : (Ω^{n}k)^{⊗p} → k. Then we get an exact triangle

Ωk → K → (Ω^{n}k)^{⊗p} → k.

Let us denote the stable map Ωk → K by υ. We also get a map λ : A_{2p−3} → K from the commutative diagram in 𝔪𝔬𝔡-kG

\xymatrix{A_{2p−3} \ar[r] \ar@{-->}[d]_{λ} & A_{2p−2} = Ω^{n}k ⊗ S^{p−1}(Ω^{n}k) \ar[r] \ar[d] & S^{p}(Ω^{n}k) \ar[d]^{S^{p}ζ} \\ K \ar[r] & (Ω^{n}k)^{⊗p} \ar[r]^{ζ^{⊗p}} & k}

in which the middle vertical map is the tensor product of the identity with the canoncial inclusion (1.18).

Lemma 3.29. If i < 2p − 3 then the composition (Ω^{n}k)^{⊗p} ⟶^{a} Ω^{i}k ⟶^{Ω^{i−1}υ} Ω^{i−1}K vanishes for every a ∈ D_{i}(ζ). For i = 2p − 3, the composition equals the composition of the canonical map (Ω^{n}k)^{⊗p} → Ω^{n}k ⊗ S^{p−1}(Ω^{n}k) = A_{1}, multiplied by a minus sign, with the class represented by the complex A_{1} → A_{2} → ⋯ → A_{2p−3} ⟶^{λ} K.

Corollary 3.30. Suppose that i ≤ 2p − 3, and let us assume that D_{i}(ζ) ≠ ∅. Then D_{i}(ζ) is a coset of ζ^{p} · Ĥ^{−i}(G) in Ĥ^{pn−i}(G), and 0 ∈ D_{i}(ζ) unless i = 2p − 3.

Proof of the Corollary. From the lemma we know that the composition

(Ω^{n}k)^{⊗p} ⟶^{a} Ω^{i}k ⟶^{Ω^{i−1}υ} Ω^{i}K

does not depend on the choice of a ∈ D_{i}(ζ). Therefore, if we are given a, a′ ∈ D_{i}(ζ), then a − a′ vanishes when postcomposed with Ω^{i−1}υ. Thus, a − a′ factors as (Ω^{n}k)^{⊗p} →

Ω^{i}(Ω^{n}k)^{⊗p} ⟶^{ζ^{⊗p}} Ω^{i}k, which proves a − a′ ∈ ζ^{p} · Ĥ^{−i}(G). Together with Corollary 3.28 we

get the first part. If i < 2p − 3 then the composite (Ω^{n}k)^{⊗p} ⟶^{a} Ω^{i}k ⟶^{Ω^{i−1}υ} Ω^{i}K vanishes for a ∈ D_{i}(ζ); thus a is divisible by ζ^{p} which implies 0 ∈ D_{i}(ζ).

We will show in Proposition 3.47 that D_{i}(ζ) is non-empty for all positive integers i.

Proof of Lemma 3.29. Let a ∈ D_{i}(ζ) be defined by the first two rows of the following diagram:

[figure: (Ω^{n}k)^{⊗p}; P_{1}; …; P_{i−1}; P_{i}; k; (Ω^{n}k)^{⊗p}; γ; (Ω^{n}k)^{⊗p}; …; (Ω^{n}k)^{⊗p}; (Ω^{n}k)^{⊗p}; β; k; A_{2p−2−i}; A_{2p−1−i}; …; A_{2p−3}; λ; A_{2p−2}; (∗); K; (Ω^{n}k)^{⊗p}; k]  (3.31)

The connection between the second and the third row is given by the cochain map constructed in Proposition 3.20. In particular, β is the canonical map, and if i = 2p − 3 then also −γ is the canonical map. The diagram (∗) commutes due to the computation

u ⊗ u_{1} ⊗ ⋯ ⊗ u_{p−1} ↦ u ⊗ u_{1}u_{2} … u_{p−1} ↦ \frac{1}{(p − 1)!} ∑_{σ∈Σ_{p−1}} u ⊗ u_{σ_{1}} ⊗ ⋯ ⊗ u_{σ_{p−1}}
↦ ζ(u)ζ(u_{1}) … ζ(u_{p−1}).

Therefore, the diagram (3.31) commutes. Since the bottom row represents υ, Lemma 3.12

says that the composition (Ω^{n}k)^{⊗p} ⟶^{a} Ω^{i}k ⟶^{Ω^{i−1}υ} Ω^{i−1}K is represented by the complex

(Ω^{n}k)^{⊗p} → P_{1} → ⋯ → P_{i−1} → K,

which in turn by Proposition 3.14 is the same as the composition of γ with the class in Êxt_{kG}^{−i+1}(A_{2p−2−i}, K) represented by the complex

A_{2p−2−i} → A_{2p−1−i} → ⋯ → A_{2p−3} → K.

If i < 2p − 3 the latter class vanishes because A_{2p−2−i} is projective.

We conclude this section with the proof of two nice properties of the D_{i}’s.

Lemma 3.32. For all ζ and all c ∈ k we have that D_{i}(c · ζ) = c^{p} · D_{i}(ζ). Let us assume that D_{i+1}(ξ) ≠ ∅ for some ξ. Then the operation D_{i} is additive in the sense that D_{i}(ζ + φ) ⊆ D_{i}(ζ) + D_{i}(φ) for all ζ, φ of the same degree.

Proof. Note that the statements given fit with the indeterminacy of the D_{j}’s. The first statement is an immediate consequence of the definition. We will prove the second statement for odd numbers i only; the proof for even numbers i is similar. Take any diagram of the form

[figure: (Ω^{n}k)^{⊗p}; P_{1}; …; P_{i}; P_{i+1}; (Ω^{n}k)^{⊗p}; (Ω^{n}k)^{⊗p}; …; (Ω^{n}k)^{⊗p}; (Ω^{n}k)^{⊗p}]

which exists due to our assumption. It is enough to show that the upper row of the diagram

[figure: (Ω^{n}k)^{⊗p}; P_{1}; …; P_{i}; k; (Ω^{n}k)^{⊗p}; (Ω^{n}k)^{⊗p}; …; 1−T; (Ω^{n}k)^{⊗p}; λ; k]

represents the zero map, where λ = (ζ + φ)^{⊗p} − ζ^{⊗p} − φ^{⊗p}. We claim that λ factors as (Ω^{n}k)^{⊗p} ⟶^{N} (Ω^{n}k)^{⊗p} → k. To see this, note that λ is the sum over all p-fold tensor products of ζ’s and φ’s except ζ^{⊗p} and φ^{⊗p}, and the ℤ/pℤ-action (by cyclic permutation) on these tensor products is free. Therefore, we get a diagram as follows:

[figure: (Ω^{n}k)^{⊗p}; P_{1}; …; P_{i}; k; P_{i+1}; (Ω^{n}k)^{⊗p}; (Ω^{n}k)^{⊗p}; …; 1−T; (Ω^{n}k)^{⊗p}; λ; N; k; (Ω^{n}k)^{⊗p}]

But now the upper row represents the composition of a map Σ^{i}(Ω^{n}k)^{⊗p} → P_{i+1} with a map P_{i+1} → k, which is stably trivial since P_{i+1} is projective.

Lemma 3.33. Suppose that i ≤ 2p − 3. The D_{i}’s are natural in the following sense: whenever we are given an injective group homomorphism f : H → G and a cohomology class [ζ] ∈ Ĥ^{n}(G), we have f^{∗}D_{i}([ζ]) ⊆ D_{i}(f^{∗}[ζ]) as subsets of Ĥ^{pn−i}(H).

Proof. The functor f^{∗} : 𝔪𝔬𝔡-kG → 𝔪𝔬𝔡-kH is exact and maps projective modules to projective modules (here we use the injectivity of f). Also, it maps the trivial G-module k to the trivial H-module k. This implies that f^{∗}(Ω_{G}^{n}k) ≅ Ω_{H}^{n}k in \underline{𝔪𝔬𝔡}-kH. Now we start with a diagram of the form (3.25) defining some element a in D_{i}(ζ) and apply the functor f^{∗}. Then we obtain a diagram showing that f^{∗}a ∈ D_{i}(f^{∗}ζ).

3.4 The Cartan formula

Remark 3.34. For the general case of a Hopf algebra, see Remark 3.18 which applies verbatim to the previous lemma.

Let us finally define our power operation.

Definition 3.35. For all Tate cohomology classes ζ (of even degree if p is odd) we put P_{1}(ζ) = −D_{2p−3}(ζ).

For us the minus sign occurring in the definition is mainly motivated by the minus sign in the statement of Lemma 3.29. It is the same sign that shows up in any construction of the Steenrod operations, compare with (5.2) in [21].

3.4 The Cartan formula

Put D_{0}(ζ) = ζ^{p} (as a set with exactly one element). We are now going to prove the Cartan formula in the following form.

Proposition 3.36. Let s be a positive integer with s ≤ 2p − 3, and let ζ, φ be cohomology classes (of even degree if p is odd). Then

D_{s}(ζ · φ) ⊆ ∑_{\begin{matrix}i+j=s \\ i,j≥0 \end{matrix}} D_{i}(ζ) · D_{j}(φ).

We start with some auxiliary constructions and lemmas. As before let C_{p} be the cyclic group of order p, and let A be the algebra k(C_{p} × C_{p}) with augmentation ϵ : A → k and augmentation ideal I = ker ϵ.

Let g be a generator of C_{p}, then we have a well-known exact sequence of free kC_{p}- modules

kC_{p} ⟶^{1−g} kC_{p} ⟶^{(1−g)^{p−1}} kC_{p} ⟶^{1−g} kC_{p} ⟶^{(1−g)^{p−1}} …

which we consider as a cochain complex with the first kC_{p} sitting in degree 0. When we tensor two such complexes we obtain an exact sequence A of free kC_{p} ⊗ kC_{p} = A-modules

A : A → A^{⊕2} → A^{⊕3} → A^{⊕4} → …

where the i-th summand of A^{⊕s} comes from the tensor product of the two kC_{p}’s sitting in degree s − i and i − 1, respectively. Let us denote the generators of the two factors of C_{p} × C_{p} by g and h; then we also have a cochain complex

B : A ⟶^{1−g⊗h} A ⟶^{(1−g⊗h)^{p−1}} A ⟶^{1−g⊗h} A ⟶^{(1−g⊗h)^{p−1}} …,

which is actually exact, but we do not need this.

Lemma 3.37. There is a map of cochain complexes of A-modules γ : A → B lifting the identity of A in degree 0 with the following property: when considered as a matrix, in degree s the map γ_{s} : A^{⊕(s+1)} → A is of the form

• (1 1 + I 1 … 1 1 + I) if s is odd, • (1 1 + I 1 … 1 + I 1) if s is even and p = 2,

• (1 I 1 … I 1) if s is even and p is odd.

Proof. We use the identification A = k(C_{p} × C_{p}) = k[x, y]/(x^{p}, y^{p}) given by x = 1 − g and y = 1 − h. The augmentation ideal I is the ideal generated by x and y, and 1 − gh corresponds to w = x + y − xy. Let us define z = (1 − x)y; then w = x + z. The upper row in the diagram

[figure: A; ^{(\begin{matrix}x \\ y \end{matrix})}; A^{2}; ^{( \begin{matrix}x^{p−1} \\ −y & x \\ & y^{p−1} \end{matrix} )}; _{(\begin{matrix}1 \\ & 1−x \end{matrix})}; A^{3}; ^{( \begin{matrix}x \\ y & x^{p−1} \\ & −y^{p−1} & x \\ & & y \end{matrix} )}; _{( \begin{matrix}1 \\ & 1−x \\ & & 1 \end{matrix} )}; A^{4}; _{( \begin{matrix}1 \\ & 1−x \\ & & 1 \\ & & & 1−x \end{matrix} )}; …; A; _{(\begin{matrix}x \\ z \end{matrix})}; A^{2}; _{( \begin{matrix}x^{p−1} \\ −z & x \\ & z^{p−1} \end{matrix} )}; A^{3}; _{( \begin{matrix}x \\ z & x^{p−1} \\ & −z^{p−1} & x \\ & & z \end{matrix} )}; A^{4}; …]

equals A, and the diagram commutes because of the identities x^{p−1}(1 − x) = x^{p−1} and (1 − x)z^{p−1} = (1 − x)^{p}y^{p−1} = y^{p−1}. Let us put u = x^{p−2} − x^{p−3}z + − ⋯ − z^{p−2} ∈ I for p ≥ 3 and u = 1 if p = 2. Then the diagram

[figure: A; ^{(\begin{matrix}x \\ z \end{matrix})}; A^{2}; ^{( \begin{matrix}x^{p−1} \\ −z & x \\ & z^{p−1} \end{matrix} )}; _{(_{1 1} )}; A^{3}; ^{( \begin{matrix}x \\ z & x^{p−1} \\ & −z^{p−1} & x \\ & & z \end{matrix} )}; _{(_{1 u 1} )}; A^{4}; _{(_{1 1 1 1} )}; …; A; w; A; w^{p−1}; A; w; A; …]

commutes because of the equalities

w^{p−1} = x^{p−1} − zu = xu + z^{p−1},
uw = x^{p−1} − z^{p−1}.

If we put the two diagrams together we get the desired cochain map, which in degree s is given by the matrix (^{1} ^{1−x} ^{...} ^{1} ^{1−x}) if s is odd and by the matrix (^{1 (1−x)u 1 ... (1−x)u 1} ) if s is even.

Suppose we are given cochain complexes X → P_{1} → P_{2} → ⋯ → P_{i+j} and Y → Q_{1} → Q_{2} → ⋯ → Q_{i+j} with projective modules P_{s} and Q_{s}, and X and Y are sitting in degree 0. The tensor product of these two complexes is a complex X ⊗ Y → R_{1} → R_{2} → … with projective modules R_{s}. Let α : P_{i} → X′ and β : Q_{j} → Y′ be maps such that the sequences X → P_{1} → ⋯ → P_{i} → X′ and Y → Q_{1} → ⋯ → Q_{j} → Y′ are complexes, thus representing some classes a ∈ Êxt_{kG}^{−i}(X, X′) and b ∈ Êxt_{kG}^{−j}(Y, Y′), respectively.

Lemma 3.38. In the situation above, the sequence X ⊗Y → R_{1} → ⋯ → R_{i+j} → X′⊗Y′, in which the last map is the composite R_{i+j} ↠ P_{i} ⊗ Q_{j} ⟶^{α⊗β} X′ ⊗ Y′, is a complex and represents a ⊗ b ∈ Êxt_{kG}^{−(i+j)}(X ⊗ Y, X′ ⊗ Y′).

Proof. We know that a ⊗ b = (a ⊗ id_{Y′})(id_{X} ⊗b) is represented by the complex

\xymatrix{X ⊗ Y \ar[r] & X ⊗ Q_{1} \ar[r] & … \ar[r] & X ⊗ Q_{j} \ar[rr] \ar@{-->}[dr] &  & P_{1} ⊗ Y′ \ar[r] & … \ar[r] & P_{i} ⊗ Y′ \ar[r] & X′ ⊗ Y′. \\  &  &  &  & X ⊗ Y′ \ar@{-->}[ur]}

On the other hand we have a commutative diagram

[figure: X ⊗ Y; R_{1}; …; R_{j}; R_{j+1}; …; R_{i+j}; X′ ⊗ Y′; X ⊗ Y; X ⊗ Q_{1}; …; X ⊗ Q_{j}; P_{1} ⊗ Y′; …; P_{i} ⊗ Y′; X′ ⊗ Y′]

given by the projection maps R_{s} → X ⊗ Q_{s} for all s = 1, 2, …, j and by the compositions R_{s+j} → P_{s} ⊗ Q_{j} → P_{s} ⊗ Y′ for s = 1, 2, …, i. The diagram also shows that its upper row is indeed a complex.

Proof of Proposition 3.36. The statement is trivial unless s = 2p − 3, so we restrict attention to that case. Let ζ : Ω^{n}k → k and φ : Ω^{m}k → k be given; then we will be interested in the complex

(Ω^{n}k ⊗ Ω^{m}k)^{⊗p} ⟶^{1−T} (Ω^{n}k ⊗ Ω^{m}k)^{⊗p} ⟶^{(1−T)^{p−1}} (Ω^{n}k ⊗ Ω^{m}k)^{⊗p} ⟶^{1−T} …,

which is isomorphic to the complex

(Ω^{n}k)^{⊗p} ⊗ (Ω^{m}k)^{⊗p} ⟶^{1−T ⊗T} (Ω^{n}k)^{⊗p} ⊗ (Ω^{m}k)^{⊗p} ⟶^{(1−T ⊗T)^{p−1}} …

via the obvious permutation maps. Let us put X = (Ω^{n}k)^{⊗p} and Y = (Ω^{m}k)^{⊗p}. Then C_{p} acts on X and Y via the endomorphism T given by cyclic permutation. Let commutative diagrams

[figure: X; P_{1}; f_{1}; …; P_{s}; f_{s}; Y; Q_{1}; g_{1}; …; Q_{s}; g_{s}; X; 1−T; X; N; …; 1−T; X; Y; 1−T; Y; N; …; 1−T; Y]

be given. Let us denote the cochain complexes in the bottom row by X and Y, respectively. We write X ⊗ Y → R_{1} → ⋯ → R_{s} → … for the tensor product (X → P_{∗}) ⊗ (Y → Q_{∗}). Then we obtain a cochain map (X ⊗ Y → R_{∗}) ⟶^{f⊗g} X ⊗ Y. Furthermore we have that

X ⊗ Y ≅ X ⊗_{C_{p}} A ⊗_{C_{p}} Y.  (3.39)

This is easy to see, but the right hand side needs some explanation. The objects of A are free k(C_{p} × C_{p})-modules. On these we have a left C_{p} action given by the inclusion C_{p} → C_{p} × C_{p} of the first factor, and we have a right action given by the inclusion of the second factor, and the actions commute. On X and Y we have the C_{p}-actions given by the endomorphism T. When we endow the modules of A with the trivial G-action, (3.39) is an isomorphism of cochain complexes of kG-modules. Now Lemma 3.37 tells us that there is a certain map of cochain complexes

X ⊗_{C_{p}} γ ⊗_{C_{p}} Y : X ⊗_{C_{p}} A ⊗_{C_{p}} Y → X ⊗_{C_{p}} B ⊗_{C_{p}} Y,

where the latter complex is isomorphic to

X ⊗ Y ⟶^{1−T ⊗T} X ⊗ Y ⟶^{(1−T ⊗T)^{p−1}} X ⊗ Y ⟶^{1−T ⊗T} …

Putting things together we get the following diagram:

[figure: X ⊗ Y; R_{1}; …; R_{s}; k; X ⊗ Y; X ⊗ Y; …; X ⊗ Y; ζ^{⊗p}⊗φ^{⊗p}; k]

In order to say which map is represented by the complex in the upper row, we need to investigate the map R_{s} → k. Take a direct summand P_{i} ⊗ Q_{j} of R_{s} (with i + j = s) and consider the composite P_{i} ⊗ Q_{j} ↪ R_{s} → X ⊗ Y. The description of γ in Lemma 3.37 says that this map is given as a composite

P_{i} ⊗ Q_{j} ⟶^{f_{i}⊗g_{j}} X ⊗ Y ⟶^{id +X⊗_{Cp}θ⊗_{Cp}Y} X ⊗ Y

where θ : A → A is some map having its image in the augmentation ideal I of A. In particular, the composition

X ⊗ Y ⟶^{X⊗_{Cp}θ⊗_{Cp}Y} X ⊗ Y ⟶^{ζ^{⊗p}⊗φ^{⊗p}} k

vanishes and the map P_{i} ⊗ Q_{j} → k equals (ζ^{⊗p} ∘ f_{i}) ⊗ (φ^{⊗p} ∘ g_{j}). Therefore, if we replace the last map of

X ⊗ Y → R_{1} → ⋯ → R_{s} → k

by its restriction to the direct summand P_{i} ⊗ Q_{j}, Lemma 3.38 tells us that we obtain a complex representing some element of D_{i}(ζ) ⊗ D_{j}(φ). Furthermore, if we replace the last map R_{s} → k by its restriction to the direct summand X ⊗ Q_{s}, the resulting complex represents some element of ζ^{⊗p} ⊗D_{s}(φ), and similarly for P_{s} ⊗Y. Now the Cartan formula follows from Corollary 3.8.

3.5 Comparison with Steenrod operations

3.5 Comparison with Steenrod operations

We are now going to show that our power operation extends certain Steenrod operations in the following sense: for every ζ ∈ H^{n}(G) of positive degree n we have that Sq^{n−1} ζ ∈ P_{1}(ζ) if p = 2 and βP^{\frac{n}{2} −1}ζ ∈ P_{1}(ζ) if p is odd and n is even. To do so, let us recall the definition of Steenrod operations on H^{∗}(G). There are several ways of constructing these operations which lead to the same result. One can use the isomorphism H^{∗}(G) ≅ H^{∗}(BG; k) where BG denotes the classifying space of G, and then work in the topological setting where we have Steenrod operations on the cohomology of a space. On the other hand, we can use the fact that kG is a cocommutative Hopf algebra, and there is a general construction of Steenrod operations on the cohomology of cocommutative Hopf algebras (see [21], §11). In either case, one usually constructs operations D_{i} : H^{n}(G) → H^{pn−i}(G) for all nonnegative integers i, then proves that some of these operations vanish, and finally defines the Steenrod operations P^{∗} and βP^{∗} to be the non-vanishing ones. Let us go through this process more precisely; we will take the topological path, even though in the case of a cocommutative Hopf algebra the purely algebraic way is more appropriate.

Consider the simplicial set EG which can be obtained by forming the nerve of the category whose objects are the elements of G, and there is exactly one morphism for every ordered pair of objects. Then EG is a contractible simplicial set with a free G-action. We denote by P_{∗} the corresponding chain complex over k, i.e., P_{n} is the k-vector space over the set (EG)_{n} of n-simplices, and we have the differential ∂ = ∑_{i} ∂_{i} : P_{n} → P_{n−1}. The right G-action on EG gives a right G-action on P_{n} turning this into a kG-module. Naturality of the construction shows that ∂ is a kG-module map. The P_{n}’s are in fact free kG-modules, and k ⟵ P_{∗} is a projective resolution of the trivial kG-module k. In what follows, we choose Ω^{n}k to be the cokernel of the map ∂ : P_{n+1} → P_{n}, and write p_{n} : P_{n} → Ω^{n}k for the obvious map. Then p_{n} extends to a map π : P_{∗} → (Ω^{n}k)[n] of chain complexes.

We also have a complex Q_{∗} of kG-modules defined by Q_{n} = P_{n}^{⊗p}, the differential given by ∑_{i}(∂_{i})^{⊗p}. The diagonal of EG induces a map D : P_{n} → Q_{n} of kG-chain complexes. Furthermore, we have the Alexander-Whitney map ξ : Q_{∗} → (P^{⊗p})_{∗}. Putting things together, we get a composition like this:

\xymatrix{P_{∗} \ar[r]^{D} & (P_{∗})^{⊗p} \ar[r]^{ξ} & (P^{⊗p})_{∗} \ar[r]^{π^{⊗p}} & (Ω^{n}k)[n]^{⊗p} = (Ω^{n}k)^{⊗p}[pn]}

Restricting these chain maps to degree pn, we obtain the following diagram:

\xymatrix{P_{pn} \ar[dr]^{π^{⊗p} ξ D} \ar[d]_{p_{pn}} \\ Ω^{pn}k \ar[r] & (Ω^{n}k)^{⊗p}}  (3.40)

The lower row is a stable isomorphism; in what follows, we will always mean this map whenever we write Ω^{pn}k ≅ (Ω^{n}k)^{⊗p}.

Recall that T : (P^{⊗p})_{∗} → (P^{⊗p})_{∗} is the chain map given by

T (a_{1} ⊗ a_{2} ⊗ ⋯ ⊗ a_{p}) = ± a_{p} ⊗ a_{1} ⊗ ⋯ ⊗ a_{p−1},  (3.41)

with the usual sign convention; in particular, the sign is + if the degrees of all the a_{i}’s are the same. Let us write Δ_{0} = ξ D. From the general theory we know that (1 − T)Δ_{0} is null-homotopic (in fact, naturally) via some homotopy Δ_{1}, i.e.,

(1 − T)Δ_{0} = Δ_{1}∂ + ∂Δ_{1}.

Proceeding inductively, we find (natural) maps Δ_{i} of degree i satisfying

(1 − T)Δ_{i} = Δ_{i+1}∂ + ∂Δ_{i+1}  if i is even,  (3.42)
(1 + T + ⋯ + T^{p−1})Δ_{i} = Δ_{i+1}∂ − ∂Δ_{i+1}  if i is odd.  (3.43)

Naturality of the Δ_{i}’s implies that they are kG-linear. We assume that these Δ_{i} are chosen as in the construction of Steenrod operations in [21], §7 (or, for the case of a general cocommutative Hopf algebra, §11 in the same paper). From now on, we assume that p · n is even.

Lemma 3.44. Define α_{i} = π^{⊗p}Δ_{i} : P_{pn−i} → (Ω^{n}k)^{⊗p}. Then α is a cochain map:

\xymatrix{P_{pn} \ar[r] \ar[d]^{α_{0}} & P_{pn−1} \ar[r] \ar[d]^{α_{1}} & P_{pn−2} \ar[r] \ar[d]^{α_{2}} & P_{pn−3} \ar[r] \ar[d]^{α_{3}} & … \\ (Ω^{n}k)^{⊗p} \ar[r]^{1−T} & (Ω^{n}k)^{⊗p} \ar[r]^{N} & (Ω^{n}k)^{⊗p} \ar[r]^{1−T} & (Ω^{n}k)^{⊗p} \ar[r]^{N} & …}

Proof. Note that the following diagram commutes:

\xymatrix{(P^{⊗p})_{pn} \ar[r]^{T} \ar[d]^{π^{⊗p}} & (P^{⊗p})_{pn} \ar[d]^{π^{⊗p}} \\ (Ω^{n}k)^{⊗p} \ar[r]^{T} & (Ω^{n}k)^{⊗p}}

To see this, start with some element a = a_{1} ⊗ ⋯ ⊗ a_{p} in (P^{⊗p})_{pn}. If it does not belong to (P_{n})^{⊗p} ⊂ (P^{⊗p})_{pn}, both a and T a will be mapped to zero under π^{⊗p}. If a lies in (P_{n})^{⊗p}, the sign occurring in (3.41) is +, because the degrees of all the a_{j}’s are equal.

Using (3.42), we obtain

(1 − T)α_{i} = (1 − T)π^{⊗p}Δ_{i} = π^{⊗p}(1 − T)Δ_{i}

= π^{⊗p}Δ_{i+1}∂ + \underbrace{π^{⊗p}∂}_{0} Δ_{i+1} = α_{i+1}∂

for i even. A similar argument using (3.43) for odd i completes the proof of the lemma.

Let [ζ] ∈ H^{n}(G) be given by a map ζ : Ω^{n}k → k, and let i be any positive integer. Define λ to be the composition P_{pn−i} ⟶^{α_{i}} (Ω^{n}k)^{⊗p} ⟶^{ζ^{⊗p}} k. Then

λ ∂ = ζ^{⊗p} α_{i}∂ = \begin{cases}\underbrace{ζ^{⊗p} (1 − T)}_{0} α_{i−1} = 0 & if i is odd, \\ \underbrace{ζ^{⊗p} N}_{0} α_{i−1} = 0 & if i is even. \end{cases}

Therefore, λ induces a map Ω^{pn−i}k → k which represents D_{i}([ζ]), where D_{i} is defined as in [21] (proof of Theorem 11.8, together with Definitions 2.2).

Lemma 3.45. Viewed as an element in \underline{Hom}_{kG}(Ω^{pn}k, Ω^{i}k), the upper row of the commutative diagram

[figure: (Ω^{n}k)^{⊗p}; Ω^{pn}k; ^{≅^{st}}; P_{pn−1}; P_{pn−2}; …; P_{pn−i}; λ; k; (Ω^{n}k)^{⊗p}; (Ω^{n}k)^{⊗p}; (Ω^{n}k)^{⊗p}; …; (Ω^{n}k)^{⊗p}; ζ^{⊗p}; k]

represents Ω^{i}D_{i}([ζ]). In particular, D_{i}([ζ]) ∈ D_{i}(ζ).

Proof. Commutativity of the diagram follows from the diagram (3.40), Lemma 3.44, and the definition of λ. The upper row is the composition of the complex

\xymatrix{Ω^{pn}k \ar@{^{(}->}[r] & P_{pn−1} \ar[r] & P_{pn−2} \ar[r] & … \ar[r] & P_{pn−i} \ar@{->>}[r] & Ω^{pn−i}k}

representing the identity in \underline{Hom}_{kG}(Ω^{pn}k, Ω^{i}Ω^{pn−i}k) with D_{i}([ζ]) : Ω^{pn−i}k → k. The result follows from Remark 3.13.

Corollary 3.46. We have that P_{1}(ζ) ∈ P_{1}(ζ).

Proof. Checking the signs in the formulas (1) and (2) of §5 in [21], we get that indeed P_{1}(ζ) = −D_{2p−3}(ζ).

Proposition 3.47. For all non-negative integers i and all Tate cohomology classes ζ (of even degree if p is odd) the set D_{i}(ζ) is non-empty.

Proof. We have already done this for ordinary cohomology classes ζ, and now we need to extend this result to negative degrees. To do so, we have to extend our projective resolution P_{∗} and the maps Δ_{i} to the negative range. Recall that Δ_{i} was defined as a map (not a chain map in general) P_{∗} → (P^{⊗p})[−i]_{∗}.

The first step is easy: we simply extend P_{∗} to a complete projective resolution of k. To define Δ_{i}, it turns out that we need to modify the notion of tensor product for our purposes. The usual tensor product of two chain complexes X_{∗} and Y_{∗} is defined by (X ⊗ Y)_{n} = ⨁_{i+j=n} X_{i} ⊗ Y_{j} with a certain differential, but this is not what we want. For every complex X, let us write X^{+} and X^{−} for the truncated complexes

X_{n}^{+} = \begin{cases}X_{n} & if n ≥ 0, \\ 0 & otherwise, \end{cases}

X_{n}^{−} = \begin{cases}X_{n} & if n < 0, \\ 0 & otherwise, \end{cases}

with the obvious differentials. Now we introduce a new product ⊠ for two given chain complexes X and X′ as follows. Define K to be the kernel of the differential X_{−1} → X_{−2}; then we obtain a complex X^{−} ← K with K sitting in degree 0, and we also get an augmented complex K ← X^{+}. Similarly, we get the complexes X^{′−} ← K′ and K′ ← X^{′+}. We can now form the tensor product of complexes (X^{−} ← K) ⊗ (X^{′−} ← K′) and the tensor product of augmented complexes K ⊗ K′ ← X^{+} ⊗ X^{′+}. Connecting these two at their common object K ⊗ K′ we obtain a new complex which we denote by X ⊠ X′:

⋯ ← X_{−1} ⊗ K′ ⊕ K ⊗ X_{−1} ← X_{0} ⊗ X′_{0} ← X_{1} ⊗ X′_{0} ⊕ X_{0} ⊗ X′_{1} ← …

One should bear in mind that the elements of K and K′ are considered to be of degree 0; then the degree of an element of the form a ⊗ b equals |a| + |b|. The operation ⊠ is a symmetric monoidal product on the category of (unbounded) chain complexes. If X and Y are complete projective resolutions of the modules K and L, respectively, then by the Künneth theorem X ⊠ Y is a complete projective resolution of K ⊗ L. Furthermore, it is an extension of the usual tensor product of non-negatively graded chain complexes in the sense that (X ⊠ Y)^{+} = X^{+} ⊗ Y^{+}. Also notice that (X_{n})^{⊗p} is a direct summand of (X^{⊠p})_{pn} for every integer n.

We can extend T to a chain map (P^{⊠p})_{∗} → (P^{⊠p})_{∗} by putting T (a_{1} ⊗ a_{2} ⊗ ⋯ ⊗ a_{p}) = ±a_{p} ⊗ a_{1} ⊗ ⋯ ⊗ a_{p−1} with the same sign rule as in §1.4. For every non-negative integer i, let us extend the maps Δ_{i} : P_{∗}^{+} → ((P^{+})^{⊗p})_{∗+i} to maps Δ_{i} : P_{∗} → (P^{⊠p})_{∗+i} satisfying the following properties:

0 = Δ_{0}∂ − ∂Δ_{0},
(1 − T)Δ_{i} = Δ_{i+1}∂ + ∂Δ_{i+1}  if i is even,
(1 + T + ⋯ + T^{p−1})Δ_{i} = Δ_{i+1}∂ − ∂Δ_{i+1}  if i is odd.

For i = 0, we need to extend a chain transformation Δ_{0} which is defined on a large (in fact, infinite) range to the whole projective resolutions. It is a standard fact from homological algebra that this is possible. To do the inductive step, note that the term on the lefthand side is already defined everywhere and is a chain transformation P → (P^{⊠p})[−i]. Furthermore, Δ_{i+1} is a partially defined null-homotopy for that chain map, and again usual homological algebra tells us that we can find a suitable extension of Δ_{i+1} to the whole projective resolution. Let p_{n} : P_{n} → Ω^{n}k be the obvious map for every integer n. For any fixed integer n, this map extends to a chain map π : P_{∗} → (Ω^{n}k)[n]. We obtain a commutative diagram

\xymatrix{P_{pn} \ar[dr]^{π^{⊠p}Δ_{0}} \ar[d]_{p_{pn}} \\ Ω^{pn}k \ar[r] & (Ω^{n}k)^{⊗p}}

in which the vertical arrow is a stable isomorphism which we refer to as ‘the’ isomorphism Ω^{pn}k ≅ (Ω^{n}k)^{⊗p}. If we put α_{i} = π^{⊠p}Δ_{i} : P_{pn−i} → (Ω^{n}k)^{⊗p} we get a cochain map α:

\xymatrix{P_{pn} \ar[r] \ar[d]^{α_{0}} & P_{pn−1} \ar[r] \ar[d]^{α_{1}} & P_{pn−2} \ar[r] \ar[d]^{α_{2}} & P_{pn−3} \ar[r] \ar[d]^{α_{3}} & … \\ (Ω^{n}k)^{⊗p} \ar[r]^{1−T} & (Ω^{n}k)^{⊗p} \ar[r]^{N} & (Ω^{n}k)^{⊗p} \ar[r]^{1−T} & (Ω^{n}k)^{⊗p} \ar[r]^{N} & …}

3.6 The power operation for G = ℤ/pℤ × ℤ/pℤ

The proof of this fact is exactly the same as the one of Lemma 3.44. For every ζ : Ω^{n}k → k we get a commutative diagram

[figure: (Ω^{n}k)^{⊗p}; Ω^{pn}k; ^{≅^{st}}; P_{pn−1}; P_{pn−2}; …; P_{pn−i}; k; (Ω^{n}k)^{⊗p}; (Ω^{n}k)^{⊗p}; (Ω^{n}k)^{⊗p}; …; (Ω^{n}k)^{⊗p}; ζ^{⊗p}; k]

proving that D_{i}(ζ) ≠ ∅.

Proof of Theorem 3.1. This is essentially a table of contents for the previous results. Part (i) follows from Proposition 3.47 and Corollary 3.30. Part (ii) is Lemma 3.32, Part (iii) follows from Corollary 3.46, Part (iv) is Lemma 3.33. The Cartan formula (v) is shown in Proposition 3.36.

3.6 The power operation for G = ℤ/pℤ × ℤ/pℤ

In this section we will describe the power operation P_{1} on the Tate cohomology of the group G = ℤ/pℤ × ℤ/pℤ. Let us start with the case p = 2. It is well-known that the ordinary cohomology ring H^{∗}(G) is the graded algebra generated by two elements u_{1}, u_{2} of degree 1. Recall from Example 3.17 that Ĥ^{−1}(G) ≅ k has a certain canonical generator which is invariant under automorphisms of G. Now we use Tate duality (see, e.g., §4 of [28]) saying that the natural bilinear form

Ĥ^{−1−n}(G) ⊗ Ĥ^{n}(G) → Ĥ^{−1}(G) ≅ k

given by multiplication in the Tate cohomology is non-degenerate and therefore induces an isomorphism (Ĥ^{−1−n}(G))^{∗} ≅ Ĥ^{n}(G). For a fixed non-negative degree n, the monomials u_{1}^{i}u_{2}^{j} of degree n form a k-linear basis of Ĥ^{n}(G), and we denote the dual basis (which is a basis of Ĥ^{−1−n}(G)) by φ_{i,j}. From [16], Proposition 4.21, we know that Ĥ^{∗}(G) is the graded commutative algebra generated by the elements u_{1}, u_{2}, φ_{i,j} (with i, j ≥ 0) subject to the relations

φ_{i,j}u_{1} = \begin{cases}φ_{i−1,j} & if i > 0, \\ 0 & otherwise, \end{cases}

φ_{i,j}u_{2} = \begin{cases}φ_{i,j−1} & if j > 0, \\ 0 & otherwise, \end{cases}
φ_{i,j}φ_{i′,j′} = 0.

Now let p be an odd prime; then the ordinary cohomology ring H^{∗}(G) is the graded commutative algebra generated by two exterior classes u_{1}, u_{2} of degree 1 and two polynomial classes v_{1}, v_{2} of degree 2. For a fixed non-negative degree n, we can consider all monomials of the form v_{1}^{a}v_{2}^{b}u_{1}^{c}u_{2}^{d} of degree n with a, b, c, d ≥ 0 and c, d ≤ 1; these form a basis of Ĥ^{n}(G), and we denote the dual basis elements by φ_{2a+c,2b+d}. Again from [16], Proposition 4.21, we get that Ĥ^{∗}(G) is the graded algebra generated by elements u_{1}, u_{2}, v_{1}, v_{2} and φ_{i,j} for all i, j ≥ 0, of degrees |u_{1}| = |u_{2}| = 1, |v_{1}| = |v_{2}| = 2 and |φ_{i,j}| = −1 − (i + j), subject to the relations

φ_{i,j}u_{1} = \begin{cases}φ_{i−1,j} & if i is odd, \\ 0 & otherwise, \end{cases}

φ_{i,j}u_{2} = \begin{cases}(−1)^{i}φ_{i,j−1} & if j is odd, \\ 0 & otherwise, \end{cases}

φ_{i,j}v_{1} = \begin{cases}φ_{i−2,j} & if i ≥ 2, \\ 0 & otherwise, \end{cases}

φ_{i,j}v_{2} = \begin{cases}φ_{i,j−2} & if j ≥ 2, \\ 0 & otherwise, \end{cases}
φ_{i,j}φ_{i′,j′} = 0.

(in fact, [16] obtains different signs due to a slightly different chosen basis in negative degrees). As an immediate consequence of the multiplicative structure of Ĥ^{∗}(G) we get the following:

Lemma 3.48. Outside degree 0, the operation P_{1} on Ĥ^{∗}(G) has zero indeterminacy.

We therefore obtain a map P_{1} : Ĥ^{n}(G) → Ĥ^{pn−(2p−3)}(G) for every non-zero integer n (with n even unless p = 2), and our main objective will be to describe this map for all negative values of n. The following proposition does the first step.

Proposition 3.49. Let p be an odd prime. Then there is some constant t ∈ k with P_{1}(φ_{1,0}) = t · φ_{2p−1,2p−3} and P_{1}(φ_{0,1}) = −t · φ_{2p−3,2p−1}.

Proof. The result will follow from the naturality of P_{1} with respect to automorphisms of G, as established in Lemma 3.33. Suppose we are given an automorphism ψ of G which we think of as a matrix ( \begin{matrix}a & b \\ c & d \end{matrix} ) in the general linear group GL_{2}(𝔽_{p}). We have a natural isomorphism H^{1}(G) ≅ Hom_{groups}(G, k) ≅ k^{2} which implies that ψ^{∗} is given on H^{1}(G) ≅ k{u_{1}, u_{2}} by the transposed matrix, i.e.,

ψ^{∗}(u_{1}) = au_{1} + cu_{2}, ψ^{∗}(u_{2}) = bu_{1} + du_{2}.

Furthermore, we know that the Bockstein homomorphism β is natural and maps u_{i} to v_{i}; hence ψ^{∗}(v_{1}) = av_{1} + cv_{2} and ψ^{∗}(v_{2}) = bv_{1} + dv_{2}. This determines the morphism ψ^{∗} of graded algebras uniquely on ordinary cohomology H^{∗}(G). Together with naturality of Tate duality we get that ψ^{∗} is determined on Tate cohomology Ĥ^{∗}(G). Now we will exploit this fact for several morphisms ψ.

Let us begin with a diagonal matrix ( \begin{matrix}a & 0 \\ 0 & b \end{matrix} ) with a, b ∈ 𝔽_{p}^{∗}. The action on Ĥ^{1}(G) ≅ k{u_{1}, u_{2}} is given by the same matrix, and therefore the action on Ĥ^{−2}(G) ≅ k{φ_{1,0}, φ_{0,1}} is given by its transpose, which is again the same matrix, so that ψ^{∗}(φ_{1,0}) = aφ_{1,0} and similarly for φ_{0,1}. We are interested in the action of ψ^{∗} on P_{1}(φ_{0,1}) which lives in degree −4p + 3, so let us determine ψ^{∗} on the dual space H^{4p−4}(G). The latter space has the basis elements v_{1}^{i}v_{2}^{j} with i + j = 2p − 2 and v_{1}^{i}v_{2}^{j}u_{1}u_{2} with i + j = 2p − 3. The action of ψ^{∗} is given by the formula

v_{1}^{i}v_{2}^{j} ↦ a^{i}b^{j}v_{1}^{i}v_{2}^{j},
v_{1}^{i}v_{2}^{j}u_{1}u_{2} ↦ a^{i+1}b^{j+1}v_{1}^{i}v_{2}^{j}u_{1}u_{2},

which is a diagonal matrix. Therefore, the action of ψ^{∗} on Ĥ^{−4p+3}(G) is given by

φ_{2i,2j} ↦ a^{i}b^{j}φ_{2i,2j},
φ_{2i+1,2j+1} ↦ a^{i+1}b^{j+1}φ_{2i+1,2j+1}.

On the other hand we know that P_{1} is a natural construction, i.e.,

ψ^{∗}P_{1}(φ_{1,0}) = P_{1}(ψ^{∗}(φ_{1,0})) = P_{1}(aφ_{1,0}) = a^{p}P_{1}(φ_{1,0}) = aP_{1}(φ_{1,0}).

Hence P_{1}(φ_{1,0}) is a linear combination of φ_{2i,2j}’s with a^{i}b^{j} = a for all a, b ∈ 𝔽_{p}^{∗} and φ_{2i+1,2j+1}’s with a^{i+1}b^{j+1} = a for all a, b ∈ 𝔽_{p}^{∗}. Since 𝔽_{p}^{∗} is the cyclic group of order p − 1, we get from a^{i}b^{j} = a that j is divisible by p − 1; from i + j = 2p − 2 we get that i is also divisible by p − 1, but then a^{i}b^{j} = 1 ≠ a in general. From a^{i+1}b^{j+1} = a we can deduce that j + 1 is divisible by p − 1, so that j ∈ {p − 2, 2p − 3}. We have therefore shown that

P_{1}(φ_{1,0}) = s · φ_{1,4p−5} + t · φ_{2p−1,2p−3}  (3.50)

for some constants s, t ∈ k.

The next matrix we consider is ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ); again, denote by ψ the corresponding automorphism of G. Using the same arguments as before we get:

u_{1} ↦ u_{1} + u_{2}  v_{1} ↦ v_{1} + v_{2}  φ_{1,0} ↦ φ_{1,0}
u_{2} ↦ u_{2}  v_{2} ↦ v_{2}  φ_{0,1} ↦ φ_{1,0} + φ_{0,1}

Therefore, ψ acts on H^{4p−4}(G) via

v_{1}^{i}v_{2}^{j} ↦ ∑_{m=0}^{i} \binom{i}{m}v_{1}^{m}v_{2}^{i+j−m},  v_{1}^{i}v_{2}^{j}u_{1}u_{2} ↦ ∑_{m=0}^{i} \binom{i}{m}v_{1}^{m}v_{2}^{i+j−m}u_{1}u_{2}.

Passing to the dual space Ĥ^{−4p+3}(G) we get

φ_{2i+1,2j+1} ↦ ∑_{m=i}^{2p−3} \binom{m}{i}φ_{2m+1,2(i+j−m)+1}.

We know that P_{1}(φ_{1,0}) must be ψ^{∗}-invariant. The element φ_{2p−1,2p−3} is ψ^{∗}-invariant because

ψ^{∗}(φ_{2p−1,2p−3}) = ∑_{m=p−1}^{2p−3} \binom{m}{p − 1}φ_{2m+1,4p−5−2m}

and \binom{m}{p−1} is non-zero (mod p) if and only if m ≡ −1 (mod p), which is only true for m = p − 1. On the other hand, φ_{1,4p−5} is not ψ^{∗}-invariant because ψ^{∗}(φ_{1,4p−5}) = ∑_{m=0}^{2p−3} φ_{2m+1,4p−5−2m}. Hence s = 0 in (3.50).

As a last step we use the automorphism ψ given by the matrix ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) to find P_{1}(φ_{0,1}). We know that ψ acts as follows:

u_{1} ↔ u_{2}  v_{1} ↔ v_{2}  φ_{1,0} ↔ φ_{0,1}

The action on H^{4p−4}(G) is then given by v_{1}^{i}v_{2}^{j} ↦ v_{1}^{j}v_{2}^{i} and v_{1}^{i}v_{2}^{j}u_{1}u_{2} ↦ v_{2}^{i}v_{1}^{j}u_{2}u_{1} = −v_{1}^{j}v_{2}^{i}u_{1}u_{2}. From this we deduce that ψ acts on Ĥ^{−4p+3}(G) via φ_{2i,2j} ↦ φ_{2j,2i} and φ_{2i+1,2j+1} ↦ −φ_{2j+1,2i+1}. Together with P_{1}(φ_{1,0}) = t · φ_{2p−1,2p−3} we finally obtain P_{1}(φ_{0,1}) = −t · φ_{2p−3,2p−1}.

In the case p = 2 we can also determine P_{1}(φ_{0,0}) using similar methods.

Proposition 3.51. If p = 2 then P_{1}(φ_{0,0}) = t · φ_{1,1} for some constant t ∈ k.

Proof. As before, start with some matrix ( \begin{matrix}a & b \\ c & d \end{matrix}) in GL_{2}(𝔽_{2}). Written as a matrix with respect to the monomial basis u_{1}^{2}, u_{1}u_{2}, u_{2}^{2}, the action on H^{2}(G) is given by the first of the following two matrices:

(\begin{matrix}a^{2} & ac & c^{2} \\ 0 & ad + bc & 0 \\ b^{2} & bd & d^{2} \end{matrix})  (\begin{matrix}a^{2} & 0 & b^{2} \\ ac & 1 & bd \\ c^{2} & 0 & d^{2} \end{matrix})  (3.52)

Note that ad+bc = ad−bc = 1. Therefore, with respect to the basis φ_{2,0}, φ_{1,1}, φ_{0,2} we get that ψ^{∗} acts as the right hand matrix of (3.52) on Ĥ^{−3}(G). Note that φ_{0,0} is ψ^{∗}-invariant for every ψ. Therefore, P_{1}(φ_{0,0}) must be ψ^{∗}-invariant for all ψ. Using these facts for the matrix ( \begin{matrix}0 & 1 \\ 1 & 1 \end{matrix} ) one gets that P_{1}(φ_{0,0}) = t · φ_{1,1} for some constant t.

Together with the Cartan formula we are able to determine the power operation completely, up to the scalar constant. For this we need a technical lemma.

Lemma 3.53. Let x, y ∈ Ĥ^{n}(G) be elements of negative degree n. If p = 2 then assume that n ≤ −4, xu_{1}^{2} = yu_{1}^{2} and xu_{2}^{2} = yu_{2}^{2}, and if p is odd then assume that n ≤ −4p, xv_{1}^{p} = yv_{1}^{p} and xv_{2}^{p} = yv_{2}^{p}. Then it follows that x = y.

Proof. We may assume that y = 0. Let us start with p = 2. Then xu_{1}^{2} = 0 implies x ∈ ⟨φ_{0,−n−1}, φ_{1,−n−2}⟩_{k} and xu_{2}^{2} = 0 implies x ∈ ⟨φ_{−n−2,1}, φ_{−n−1,0}⟩_{k}. Since n ≤ −4 we get x = 0.

Now let p be an odd prime. Then

xv_{1}^{p} = 0  implies  x ∈ ⟨φ_{0,−n−1}, φ_{1,−n−2}, …, φ_{2p−1,−n−2p}⟩_{k}  and
xv_{2}^{p} = 0  implies  x ∈ ⟨φ_{−n−2p,2p−1}, φ_{−n−2p+1,2p−2}, …, φ_{−n−1,0}⟩_{k} .

From n ≤ −4p we can deduce that x = 0.

Proposition 3.54. If p = 2 then there is some constant t ∈ k such that P_{1}(φ_{i,j}) = t · φ_{2i+1,2j+1} for all i, j ≥ 0. If p is odd, then there is some constant t ∈ k such that

P_{1}(φ_{2i+1,2j}) = t · φ_{2pi+2p−1,2pj+2p−3},
P_{1}(φ_{2i,2j+1}) = −t · φ_{2pi+2p−3,2pj+2p−1}.

Proof. Let us start with p = 2 and proceed by induction on i + j, the case i = j = 0 being covered by Proposition 3.49. The Cartan formula Theorem 3.1.(v) yields

P_{1}(φ_{i,j})u_{1}^{2} + \underbrace{φ_{i,j}^{2}}_{0} P_{1}(u_{1}) = P_{1}(φ_{i,j}u_{1}) = \begin{cases}t · φ_{2i−1,2j+1} & if i ≥ 1, \\ 0 & otherwise, \end{cases}

P_{1}(φ_{i,j})u_{2}^{2} + \underbrace{φ_{i,j}^{2}}_{0} P_{1}(u_{2}) = P_{1}(φ_{i,j}u_{2}) = \begin{cases}t · φ_{2i+1,2j−1} & if j ≥ 1, \\ 0 & otherwise. \end{cases}

Therefore, t · φ_{2i+1,2j+1} and P_{1}(φ_{i,j}) satisfy the conditions of Lemma 3.53 and hence are equal.

The case of an odd prime p runs similarly. Let us prove the first equation only. Again we use induction on i + j with i = j = 0 being already done. Using the Cartan formula we get

P_{1}(φ_{2i+1,2j})v_{1}^{p} + \underbrace{φ_{2i+1,2j}^{p}}_{0}P_{1}(v_{1}) = P_{1}(φ_{2i+1,2j}v_{1})

= \begin{cases}t · φ_{2p(i−1)+2p−1,2pj+2p−3} & if i ≥ 1, \\ 0 & otherwise, \end{cases}

P_{1}(φ_{2i+1,2j})v_{2}^{p} + \underbrace{φ_{2i+1,2j}^{p}}_{0}P_{1}(v_{2}) = P_{1}(φ_{2i+1,2j}v_{2})

= \begin{cases}t · φ_{2pi+2p−1,2p(j−1)+2p−3} & if j ≥ 1, \\ 0 & otherwise. \end{cases}

As before, t · φ_{2pi+2p−1,2pj+2p−3} and P_{1}(φ_{2i+1,2j}) satisfy the conditions of Lemma 3.53, and we are done.

Remark 3.55. The previous proposition shows that the formulae for P_{1} given for odd primes p are actually also true for p = 2.

It remains to find the constants t in Proposition 3.54.

Proposition 3.56. The constants in Proposition 3.54 equal t = 1.

Proof. Let us denote by g and h generators of the two cyclic factors of the group G. Recall that, as an augmented algebra, kG = k[x, y]/(x^{p}, y^{p}) = L, where x = 1 − g and y = 1 − h. The action of x on a tensor product of kG-modules is given by the formula

(a ⊗ b) · x = (ax) ⊗ b + a ⊗ (bx) − (ax) ⊗ (bx),  (3.57)

and similarly for y. Let us define Ω^{i}k for small values of |i| via the following (partially given) complete resolution:

\xymatrix{… &  & L^{2} \ar[ll] \ar@{-->}[dl] &  & L \ar[ll]_{^{(\begin{matrix}x \\ y \end{matrix})}} \ar[dl] &  & L \ar[ll]_{x^{p−1}y^{p−1}} \ar[dl] &  & L^{2} \ar[ll]_{( x y)} \ar[dl] &  & … \ar[ll] \\  & Ω^{−2}k \ar[ul] &  & Ω^{−1}k \ar@{-->}[ul] &  & k \ar[ul] &  & Ωk \ar@{-->}[ul] &  & Ω^{2}k \ar@{-->}[ul]}

We consider Ωk as the submodule of L generated by x and y, Ω^{−1}k as the quotient L/x^{p−1}y^{p−1}L, and the leftmost L^{2} has the basis e_{1}, e_{2}. We also view Ω^{−2}k as the quotient L^{2}/(xe_{1} +ye_{2})L. With these notions, the map Ω^{−1}k → L^{2} sends the class [1] to xe_{1} +ye_{2}. The cohomology class u_{1} is represented by the L-linear map Ωk → k that sends x to 1 and y to 0. Similarly u_{2} can be described by mapping x to 0 and y to 1. This implies that φ_{1,0} is represented by the map Ω^{−1}k → Ωk sending [1] ∈ L/x^{p−1}y^{p−1}L = Ω^{−1}k to x. This map lifts to a diagram

\xymatrix{Ω^{−1}k \ar[r]^{[1]↦xe_{1}+ye_{2}} \ar[d]_{[1]↦x} & L^{2} \ar[d]_{( 1 0 )} \ar[r] & Ω^{−2}k \ar[d]^{[e_{1}]↦1,[e_{2}]↦0} \\ Ωk \ar[r] & L \ar[r] & k}

in which the rightmost vertical map represents φ_{1,0} and will be denoted by φ. Now we are interested in P_{1}(φ). Define the module M = Ω^{−2}k/([e_{2}] · L), that is, we divide out the submodule generated by [e_{2}]. Then φ factors uniquely as Ω^{−2}k → M → k, and we denote the latter map by φ̃. Starting with some element of −P_{1}(φ) = D_{2p−3}(φ) we obtain the following commutative diagram:

[figure: (Ω^{−2}k)^{⊗p}; P_{1}; …; P_{2p−3}; k; (Ω^{−2}k)^{⊗p}; (Ω^{−2}k)^{⊗p}; …; (Ω^{−2}k)^{⊗p}; φ^{⊗p}; k; M^{⊗p}; 1−T; M^{⊗p}; N; …; 1−T; M^{⊗p}; φ̃^{⊗p}; k]  (3.58)

Let us denote the bottom row by M. In the following, whenever we write C_{p} we implicitly mean the second of the two factors of G, which is generated by the element h. Then we can consider the kG-modules as kC_{p}-modules. As such M is free of rank 1, generated by the class of [e_{1}], and we will often omit this class and simply write 1 for that generator of M. We can construct another complex

0 → k → M → Λ^{2}M → ⋯ → Λ^{p−1}M → Λ^{p}M → 0,  (3.59)

where the maps are given by m ↦ ∑_{r=0}^{p−1} m ∧ h^{r}. Let K be the cokernel of the first map k → M, and let us splice the complex with the Koszul complex associated to the map φ̃ : M → k (see Lemma 1.6)

0 → Λ^{p}M ⟶^{κφ̃ ; p} Λ^{p−1}M → ⋯ → Λ^{2}M ⟶^{κφ̃ ; 2} M ⟶^{φ̃} k → 0.  (3.60)

Then we get a complex

0 → K → Λ^{2}M → ⋯ → Λ^{p−1}M → Λ^{p−1}M → ⋯ → Λ^{2}M → M → k → 0,

which we denote by L. Finally, we write σ for the composition M^{⊗p} ⟶^{τ} M → K, where the former map is given by

τ : h^{i_{1}} ⊗ h^{i_{2}} ⊗ ⋯ ⊗ h^{i_{p}} ↦ \begin{cases}h^{i_{1}} & if \{i_{1}, …, i_{p}\} = \{0, …, p − 1\} (mod p), \\ 0 & otherwise. \end{cases}  (3.61)

Lemma 3.62. The complex L is exact, and the identity map of k lifts to a map of cochain complexes M → L such that the first map equals −σ : M^{⊗p} → K.

Proof. We take a closer look at the de Rham-complex S^{p}M → M ⊗ S^{p−1}M → Λ^{2}M ⊗ S^{p−2}M → ⋯ → Λ^{p}M → 0. Let us denote by X_{j}^{i} the k-linear subspace of Λ^{i}M ⊗ S^{p−i}M generated by all elements of the form h_{1} ∧ h_{2} ∧ ⋯ ∧ h_{i} ⊗ h_{i+1} … h_{p} where h_{s} ∈ C_{p} for all s = 1, 2, …, p and the set {h_{1}, h_{2}, …, h_{p}} has size j. Then X_{j}^{i} is in fact a kG-module, and we have Λ^{i}M ⊗ S^{p−i}M ≅ ⨁_{j=0}^{p} X_{j}^{i} as kG-modules. Furthermore, the differential of the de Rham-complex respects this decomposition, i.e., we have a direct sum of complexes 0 → X_{j}^{0} → X_{j}^{1} → X_{j}^{2} → ⋯ → X_{j}^{p} → 0. In the case j = p we get the complex (3.59) which is therefore exact, except possibly at the two leftmost entries, where exactness is easily checked (in fact, non-exactness of the de Rham complex completely takes place in the (j = 1)-part of the decomposition above). The Koszul complex (3.60) is also exact by the second sequence of Lemma 1.6, because the dimension of M as a k-vector space equals p, so that Λ^{p+1}M = 0. Altogether we obtain that L is exact. Furthermore, we have a map of cochain complexes from the de Rham-Koszul complex RK of M to L, where the first p − 1 maps Λ^{i}M ⊗ S^{p−i}M → Λ^{i}M are projections onto direct summands, and the last p − 1 maps Λ^{i}M ⊗ S^{p−i}M → Λ^{i}M equal id ⊗S^{p−i}φ̃. Together with Proposition 3.20 we get the claim.

The exact sequence L represents an element γ of Ext_{kG}^{2p−3}(k, K), and Lemma 3.62 combined with diagram (3.58) and Proposition 3.9 says that the product of any element of P_{1}(φ) with γ stably equals the composition

γP_{1}(φ) = α : (Ω^{−2}k)^{⊗p} → M^{⊗p} ⟶^{σ} K.  (3.63)

As a next step, we want to describe γ as a map Ω^{2p−3}k → K more explicitly. Observe that we have a commutative diagram

[figure: L^{2p−2}; L^{2p−3}; …; L^{3}; L^{2}; L; k; M; y; M; y^{p−1}; …; M; y^{p−1}; M; y; M; k]  (3.64)

where the upper row is a continuation of our partially chosen projective resolution, given by matrices of the form

( \begin{matrix}x & y \\ & x^{p−1} & −y^{p−1} \\ & & x & & y \\ & & & . & . & . \\ & & & & & & x^{p−1} & −y^{p−1} \\ & & & & & & & x & y \end{matrix} ) and ( \begin{matrix}x^{p−1} & −y \\ & x & y^{p−1} \\ & & x^{p−1} & −y \\ . & . & . \\ & & & & x^{p−1} & −y \\ & & & & & x & y^{p−1} \end{matrix} ) .

Let us define Ω^{i}k by this projective resolution. The vertical maps are given by projection onto the last factor L^{j} → L followed by the canonical projection map L → L/xL ≅ M.

Furthermore, there is a map of cochain complexes

[figure: M; 1−h; M; (1−h)^{p−1}; …; (1−h)^{p−1}; M; 1−h; M; k; M^{⊗p}; 1−T; M^{⊗p}; N; …; N; M^{⊗p}; 1−T; M^{⊗p}; φ̃^{⊗p}; k]  (3.65)

where each vertical arrow is the kC_{p}-linear map given by sending 1 ∈ M to 1 ⊗ h^{p−1} ⊗ h^{p−2} ⊗ ⋯ ⊗ h ∈ M^{⊗p}; this is in fact a kG-linear map. Together with diagram (3.64) and Lemma 3.62 we obtain a diagram as follows:

[figure: L^{2p−2}; L^{2p−3}; …; L^{2}; L; k; K; Λ^{2}M; …; Λ^{2}M; M; k]

The leftmost vertical map factors as L^{2p−2} ↠ Ω^{2p−3}k → K where the second map represents γ ∈ Ext_{kG}^{2p−3}(k, K). We have therefore shown:

Lemma 3.66. Consider Ω^{2p−3}k as a quotient of L^{2p−2}, and denote the basis elements of the latter by f_{1}, f_{2}, …, f_{2p−2}. Then γ : Ω^{2p−3}k → K is represented by the following unstable map:

[f_{2p−2}] ↦ −σ(1 ⊗ h^{p−1} ⊗ h^{p−2} ⊗ ⋯ ⊗ h),
[f_{j}] ↦ 0  for j ≠ 2p − 2.

At the end of our proof we are going to use the identity

φ_{2p−1,2p−3} · v_{1}^{p−1}u_{1} = φ_{0,2p−3},

so let us now construct unstable representatives of v_{1}^{p−1}u_{1} and φ_{0,2p−3}. Let us define b = [e_{1}] · y^{p−1} ∈ (Ω^{−2}k)^{⊗p}.

Lemma 3.67. The class v_{1}^{p−1}u_{1} ∈ Ĥ^{∗}(G) is represented by the map Ω^{−1}k → (Ω^{−2}k)^{⊗p} which sends [1] ∈ L/x^{p−1}y^{p−1}L = Ω^{−1}k to [e_{1}] ⊗ b^{⊗(p−1)} ∈ Ω^{−2}k. Furthermore, the class φ_{0,2p−3} is represented by the map ϕ : Ω^{−1}k → Ω^{2p−3}k which sends [1] ∈ L/x^{p−1}y^{p−1}L to [f_{2p−2}] ∈ Ω^{2p−3}k.

Proof. We know that v_{1} is represented by the following chain map:

\xymatrix{… & L \ar[l] \ar[d]_{_{( \begin{matrix}y^{p−1} \\ 0 \end{matrix} )}} & L^{2} \ar[l]_{( x y)} \ar[d]^{( y^{p−1} 0 )} & L^{3} \ar[l]_{^{( \begin{matrix}x^{p−1} & −y & 0 \\ 0 & x & y^{p−1} \end{matrix} )}} \ar[d]^{( 1 0 0 )} & … \ar[l] \\ … & L^{2} \ar[l] & L \ar[l]^{_{(\begin{matrix}x \\ y \end{matrix})}} & L \ar[l]^{x^{p−1}y^{p−1}} & … \ar[l]}

In particular, the map sending 1 ∈ k to b ∈ Ω^{−2}k represents v_{1}. Secondly, u_{1} is represented by the chain map

\xymatrix{… & L \ar[l] & L \ar[dd]^{x^{p−2}y^{p−1}} \ar[l]_{x^{p−1}y^{p−1}} & L^{2} \ar[l]_{( x y)} \ar[dd]^{( 1 0 )} & … \ar[l] \\  & _{(\begin{matrix}1 \\ 0 \end{matrix})} \ar[d] \\ … & L^{2} \ar[l] & L \ar[l]^{_{(\begin{matrix}x \\ y \end{matrix})}} & L \ar[l]^{x^{p−1}y^{p−1}} & … \ar[l]}

and therefore the map which sends [1] ∈ L/x^{p−1}y^{p−1}L ≅ Ω^{−1}k to [e_{1}] ∈ Ω^{−2}k represents u_{1}. By forming a suitable tensor product of these two maps we see that v_{1}^{p−1}u_{1} is represented by the map Ω^{−1}k ⊗ k^{⊗(p−1)} → (Ω^{−2}k)^{⊗p} as stated.

For the second part it is enough to show that (v_{2}^{p−2}u_{2}) · ϕ = φ_{0,0} and m · ϕm = 0 for any other monomial m of degree 2p − 3. This follows immediately from the observation that the compositions L^{2p−2} → Ω^{2p−3}k → k (where the second map runs through all monomials of degree 2p − 3) are exactly the projection maps onto the single factors L followed by the augmentation L → k, and v_{2}^{p−2}u_{2} corresponds to projection onto the last factor.

Lemma 3.68. The diagram

\xymatrix{Ω^{−1}k \ar[r]^{φ_{0,2p−3}} \ar[d]_{v_{1}^{p−1}u_{1}} & Ω^{2p−3}k \ar[d]^{γ} \\ (Ω^{−2}k)^{⊗p} \ar[r]_{α} & K}

commutes, and both compositions are stably non-trivial.

Proof. By Lemmas 3.66 and 3.67, the upper-right composition sends [1] ∈ Ω^{−1}k to −σ(1 ⊗ h^{p−1} ⊗ ⋯ ⊗ h) ∈ K. For the other composition note that the quotient map Ω^{−2}k → M sends b to the norm element N = ∑_{j=0}^{p−1} h^{j} ∈ M. By Lemma 3.67, the lower-left composition sends [1] ∈ Ω^{−1}k to σ(1 ⊗ N^{⊗(p−1)}). By definition of σ, it is therefore enough to show that

−τ(1 ⊗ h^{p−1} ⊗ ⋯ ⊗ h) = τ(1 ⊗ N^{⊗(p−1)}) ∈ M,

where τ was constructed in (3.61). But both sides equal −1 = (p − 1)! ∈ M.

If the compositions were stably trivial, they would factor over the inclusion Ω^{−1}k → L^{2}. In particular, the image w of [1] ∈ Ω^{−1}k would lie in K · I, where I = ⟨x, y⟩_{L} is the augmentation ideal of L. But K is isomorphic to k[y]/y^{p−1} (with trivial x-action) and w is a generator of K, so w ∉ K · I.

We are ready to complete the proof of Proposition 3.56. We know that

γ φ_{0,2p−3} = α v_{1}^{p−1}u_{1}  by Lemma 3.68,
= γ P_{1}(φ) v_{1}^{p−1}u_{1}  by (3.63),
= t · γ φ_{0,2p−3}  by Proposition 3.54,

and all these maps in \underline{Hom}_{kG}(Ω^{−1}k, K) are non-trivial by Lemma 3.68. This implies that t = 1.

3.7 Further examples

Let us also consider another family of examples. We take G = (ℤ/pℤ)^{r}, the direct product of r cyclic factors with r ≥ 3. Then it will turn out that P_{1} vanishes on elements of negative degree. Let us briefly recall the multiplicative structure of the graded commutative algebra Ĥ^{∗}(G), see also Proposition 4.12 in [16]. For p = 2 it is generated by elements u_{1}, …, u_{r} of degree 1 and elements φ_{α} of degree −1 − |α| for all multi-indices α ∈ ℕ^{r} subject to the relations

φ_{α}φ_{β} = 0,
φ_{α}u_{i} = \begin{cases}φ_{α−ϵ_{i}} & if α_{i} > 0, \\ 0 & otherwise. \end{cases}

Here ϵ_{i} denotes the multi-index (0, …, 0, 1, 0, …, 0) with 1 sitting in the i-th position. For odd primes p we know that Ĥ^{∗}(G) is generated by exterior classes u_{1}, …, u_{r} of degree 1, polynomial classes v_{1}, …, v_{r} of degree 2, and elements φ_{α} of degree −1 − |α| for all multiindices α ∈ ℕ^{r}, subject to the relations

φ_{α}φ_{β} = 0,
φ_{α}u_{i} = \begin{cases}±φ_{α−ϵ_{i}} & if α_{i} is odd, \\ 0 & otherwise, \end{cases}

φ_{α}v_{i} = \begin{cases}±φ_{α−2ϵ_{i}} & if α_{i} ≥ 2, \\ 0 & otherwise. \end{cases}

Let us write w^{α} for the monomial v_{1}^{c_{1}} ⋯ v_{r}^{c_{r}}u_{1}^{e_{1}} ⋯ u_{r}^{e_{r}} with α_{i} = 2c_{i} + e_{i} and e_{i} ∈ {0, 1}. When α runs through all multi-indices with fixed norm |α|, we obtain a k-linear basis of H^{|α|}(G), and we assume that its dual basis is the set of φ_{α}’s in Ĥ^{−|α|−1}(G).

Proposition 3.69. For G = (ℤ/pℤ)^{r} with r ≥ 3 the power operation P_{1} vanishes on classes of negative degree.

3.7 Further examples

Proof. Let us do the proof for odd primes p, as it is the much more difficult one. As in Lemma 3.48 one sees that P_{1} does not have any indeterminacy. Using the same method as in the proof of Proposition 3.54 we see that it is enough to show that P_{1}(φ_{γ}) = 0 for all φ_{γ} of even degree with γ_{i} ∈ {0, 1} for all i. Suppose that two of the indices, say γ_{1} and γ_{2}, equal zero. Then we have

±P_{1}(φ_{γ}) = P_{1}(φ_{γ+ϵ_{1}+ϵ_{2}} · u_{1}u_{2})

= P_{1}(φ_{γ+ϵ_{1}+ϵ_{2}}) · \underbrace{(u_{1}u_{2})^{p}}_{0} + \underbrace{φ_{γ+ϵ_{1}+ϵ_{2}}^{p}}_{0} ·P_{1}(u_{1}u_{2}) = 0.

It is therefore enough to consider the case γ = (1, 1, 1, …, 1, 1) for r odd and γ = (1, 1, 1, …, 1, 0) for r even. Let us write s = r for r odd and s = r − 1 for r even; then s ≥ 3 is odd, γ_{i} = 1 for all i ≤ s, and γ_{i} = 0 for i > s. The degree of φ_{γ} equals −s − 1. For the proof of P_{1}(φ_{γ}) = 0 we will now use the naturality condition on P_{1} with respect to several automorphisms of G.

Let ψ be the automorphism of G represented by a diagonal matrix with entries a_{1}, a_{2}, …, a_{r} ∈ 𝔽_{p}^{∗}. As in the proof of Proposition 3.49 we get that ψ^{∗}(u_{i}) = a_{i}u_{i}, ψ^{∗}(v_{i}) = a_{i}v_{i}, and therefore ψ^{∗}(w^{α}) = a^{c+e}w^{α}, where α = 2c + e for some multi-indices c and e with e_{i} ∈ {0, 1} for all i. Here we use the notation a^{β} = a_{1}^{β_{1}} … a_{r}^{β_{r}}. By duality we obtain ψ^{∗}(φ_{α}) = a^{c+e}w^{α}. From the naturality of P_{1} we get that

ψ^{∗}(P_{1}(φ_{α})) = P_{1}(ψ^{∗}(φ_{α})) = P_{1}(a^{c+e}φ_{α}) = a^{c+e}P_{1}(φ_{α}).

Therefore, P_{1}(φ_{γ}) is a k-linear combination of φ_{α}’s with a^{γ} = a^{c+e} for all a = (a_{1}, …, a_{r}). This can only be true if c_{i} + e_{i} ≡ 1 (mod p − 1) which implies that

α_{i} = 1 or α_{i} = 2 or α_{i} ≥ 2p − 1, for all i = 1, 2, …, s.  (3.70)

Now we take another automorphism as follows: choose indices n, m ≤ s with n ≠ m and let ψ be defined by

ψ^{∗}(u_{m}) = u_{n} + u_{m},  ψ^{∗}(u_{i}) = u_{i} for all i ≠ m.

Then ψ^{∗} acts on ordinary cohomology H^{∗}(G) according to the formula

ψ^{∗}(w^{α}) = \begin{cases}∑_{j=0}^{c_{m}} \binom{c_{m}}{j}(w^{α−2j(ϵ_{m}−ϵ_{n})} + w^{α−(2j+1)(ϵ_{m}−ϵ_{n})}) & if α_{m} is odd and α_{n} is even, \\ ∑_{j=0}^{c_{m}} \binom{c_{m}}{j}w^{α−2j(ϵ_{m}−ϵ_{n})} & otherwise. \end{cases}

for α = 2c + e. By duality we get

ψ^{∗}(φ_{α}) = \begin{cases}∑_{j≥0} \binom{c_{m}+j}{j}(φ_{α+2j(ϵ_{m}−ϵ_{n})} + φ_{α+(2j+1)(ϵ_{m}−ϵ_{n})}) & if α_{m} is even and α_{n} is odd, \\ ∑_{j≥0} \binom{c_{m}+j}{j}φ_{α+2j(ϵ_{m}−ϵ_{n})} & otherwise. \end{cases}

Here we used the convention that φ_{β} = 0 if β_{i} < 0 for some i. Since φ_{γ} is ψ-invariant, we know that P_{1}(φ_{γ}) is in the kernel of ψ^{∗} − id. Let us write A = P_{1}(φ_{γ}) = ∑_{α} f_{α} · φ_{α} with f_{α} ∈ k, and let β be some multi-index with β_{m} = 4 such that |φ_{β}| = |A|. How often does φ_{β} occur in (ψ^{∗} − id)A? The formulae above imply that it occurs with factor \binom{2}{1}f_{β−2(ϵ_{m}−ϵ_{n})} + \binom{2}{2}f_{β−4(ϵ_{m}−ϵ_{n})}. By (3.70) the second summand vanishes, because the m-th entry of the multi-index equals 0. Since (ψ^{∗} − id)A = 0 we get that f_{β−2(ϵ_{m}−ϵ_{n})} = 0. Note that the m-th entry of the multi-index equals 2, and every multi-index α of the right degree with α_{m} = 2 is of that form for a suitable n. Hence A is a linear combination of φ_{α}’s satisfying

α_{i} = 1 or α_{i} ≥ 2p − 1, for all i = 1, 2, …, s.  (3.71)

Now let β be some multi-index with β_{m} = 3 such that |φ_{β}| = |A|. As before we see that φ_{β} occurs in (Ψ^{∗} − id)A with factor

\begin{cases}\binom{1}{0}f_{β−(ϵ_{m}−ϵ_{n})} + \binom{1}{1}f_{β−2(ϵ_{m}−ϵ_{n})} + \binom{1}{1}f_{β−3(ϵ_{m}−ϵ_{n})} & if β_{n} is even, \\ \binom{1}{1}f_{β−2(ϵ_{m}−ϵ_{n})} & otherwise. \end{cases}

In the first case, the first and the last summand vanish due to (3.71). Therefore we get f_{β−2(ϵ_{m}−ϵ_{n})} = 0 in both cases. The m-th entry of the multi-index equals 1 and every multi-index α of the right degree with α_{m} = 1 is of that form. Together with (3.71) we obtain that A is a linear combination of φ_{α}’s satisfying α_{i} ≥ 2p−1 for all i = 1, 2, …, s. In particular, the degree satisfies |φ_{α}| ≤ −s(2p−1)−1. But we know that A = P_{1}(φ_{γ}) lives in degree (−s−1)p−(2p−3) = −(s+3)p+3. So we deduce that −(s+3)p+3 ≤ −s(2p−1)−1, which is equivalent to (s − 3)(p − 1) ≤ −1, a contradiction.

This completes the proof for odd primes p. For p = 2 one only needs to check that P_{1}(φ_{0,0,...,0}) = 0. We know that φ_{0,0,...,0} is invariant under all automorphisms ψ. By applying this fact to permutations of the r factors we get that

P_{1}(φ_{0,0,...,0}) = a · ∑_{i} φ_{2ϵ_{i}} + b · ∑_{i<j} φ_{ϵ_{i}+ϵ_{j}}

for some a, b ∈ k. Then using an automorphism of the form

( \begin{matrix}1 & 1 \\ & 1 \\ & & . & . & . \\ & & & & & 1 \end{matrix} )

one easily deduces that a = b = 0.

Remark 3.72. The following diagram might be conceptually helpful. It gathers information about the action of power operations on Tate cohomology classes of negative degree for elementary abelian p-groups.

ℤ/pℤ  (Z/pℤ)^{×2} (Z/pℤ)^{×3} (Z/pℤ)^{×4} …
P_{0} non-trivial trivial trivial trivial …
P_{1} undetermined non-trivial trivial  trivial …

One might expect that this diagram can be extended downwards by defining higher order power operations, but we do not have any evidence for this.

4. An obstruction for higher order

4 An obstruction for higher order

We are now able to prove that our power operation serves as an obstruction for the order to be one larger. Recall that for every kG-module M we have a graded algebra Êxt_{kG}^{∗}(M, M) (which is not graded commutative in general) and there is a natural morphism of graded algebras Ĥ^{∗}(G) → Êxt_{kG}^{∗}(M, M) given by tensoring with M. We can view Êxt_{kG}^{∗}(M, M) as an Ĥ^{∗}(G)-module via that map.

Theorem 4.1. Let ζ ∈ Ĥ^{n}(G) be a Tate cohomology class, of even degree if p is odd. Assume for some (hence any) element a of P_{1}(ζ) that its image in Êxt_{kG}^{∗}(M, M) is divisible by ζ, that is, a ⊗ M ∈ ζ · Êxt_{kG}^{∗}(M, M). Then ζ-ord(M/ζ) ≥ p − 1.

Corollary 4.2. Let ζ ∈ Ĥ^{n}(G) be a Tate cohomology class, of even degree if p is odd. Assume further that some (hence any) element of P_{1}(ζ) is divisible by ζ, that is, P_{1}(ζ) is contained in ζ · Ĥ^{∗}(G). Then ζ-ord(M/ζ) ≥ p − 1 for every kG-module M, and therefore ζ-ord(\underline{𝔪𝔬𝔡}-kG) ≥ p.

By Corollary 3.46 we also have the following consequence.

Corollary 4.3. Suppose that ζ ∈ H^{n}(G) is an ordinary cohomology class, of even degree if p is odd. Assume further that the Steenrod power βP^{\frac{n}{2} −1}ζ (resp. Sq^{n−1} ζ if p = 2) is divisible by ζ, i.e., it is an element of ζ · H^{∗}(G). Then ζ-ord(M/ζ) ≥ p − 1 for every kG-module M.

In what follows, we will work with the same notation as in §2, that is, the cohomology class [ζ] is represented by a surjective unstable map ζ : Ω^{n}k → k, we have a short exact sequence 0 → L_{ζ} ⟶^{ι} Ω^{n}k ⟶^{ζ} k → 0 and an unstable map η : Ωk → L_{ζ} such that the

triangle Ωk ⟶^{η} L_{ζ} ⟶^{ι} Ω^{n}k ⟶^{ζ} k is exact.

4.1 A commutative square

As a first step in the proof of Theorem 4.1, we will show that there is a commutative diagram in \underline{𝔪𝔬𝔡}-kG

[figure: (Ω^{n}k)^{⊗p}; P_{1}ζ; ≃; Ω^{2p−3}k; Ω^{2p−4}η; Ω^{2p−4}L_{ζ}; ≃; Ω^{n}k ⊗ S^{p−1}Ω^{n}k; γ; Ω^{p−2}Λ^{p}Ω^{n}k; Ω^{p−2}κ_{p}; Ω^{p−2}Λ^{p−1}L_{ζ}; Ω^{p−2}ω_{p−1}]  (4.4)

for every element P_{1}ζ ∈ P_{1}(ζ). The upper row vanishes if and only if P_{1}ζ is divisible by ζ. In a second step we will prove that vanishing of the bottom row implies ζ-ord(X/ζ) ≥ p−1.

Let us define the maps in (4.4). The vertical map on the left-hand side is the canonical morphism (Ω^{n}k)^{⊗p} → Ω^{n}k ⊗ S^{p−1}Ω^{n}k, we will show in Remark 4.11 that this map is a stable isomorphism. For the other vertical map note that by Corollary 1.22 the modules Λ^{i}(Ω^{n}k) are projective for i = 2, 3, …, p − 1, and by Corollary 1.12 the sequence

0 ⟶ Λ^{j}L_{ζ} ↪ Λ^{j}(Ω^{n}k) ⟶^{κ_{j}} Λ^{j−1}(Ω^{n}k) ⟶^{κ_{j−1}} … ⟶^{κ_{3}} Λ^{2}(Ω^{n}k) ⟶^{κ_{2}} L_{ζ} ⟶ 0  (4.5)

is exact for every j = 1, 2, …, p − 1. It therefore represents a stable isomorphism ω_{j} in \underline{Hom}_{kG}(Λ^{j}L_{ζ}, Ω^{j−1}L_{ζ}) whose inverse is given by the same complex viewed as an extension (see Remark 3.10). Note that ω_{1} is simply the identity map of L_{ζ}. Let us also consider the complex

\xymatrix{Ω^{n}k ⊗ S^{p−1}Ω^{n}k \ar[r] & Λ^{2}Ω^{n}k ⊗ S^{p−2}Ω^{n}k \ar[r] & … \ar[r] & Λ^{p−1}Ω^{n}k ⊗ Ω^{n}k \ar[r] & Λ^{p}Ω^{n}k;}

again we use that the modules Λ^{i}Ω^{n}k are projective for i = 2, 3, …, p − 1, so we get a class γ in \underline{Hom}_{kG}(Ω^{n}k ⊗ S^{p−1}Ω^{n}k, Ω^{p−2}Λ^{p}Ω^{n}k). Remark 4.6. Before we start proving that the diagram commutes, let us consider the case p = 2 and draw some analogies to the topological world. The diagram takes the following form:

[figure: Ω^{n}k ⊗ Ω^{n}k; P_{1}ζ; Ωk; η; L_{ζ}; Ω^{n}k ⊗ Ω^{n}k; ·∧·; Λ^{2}Ω^{n}k; κ_{2}; L_{ζ}]

This enables us to prove Theorem 4.1 in the case when X = k by considering the commutative diagram on the left-hand side:

\xymatrix{Ω^{n}k ⊗ L_{ζ} \ar[dd] \ar[rr]^{ζ⊗id} &  & L_{ζ} & S/2 \ar[dd]_{pinch} \ar[r]^{2·} & S/2 \\  & Λ^{2}Ω^{n}k \ar@{-->}[ur]_{κ_{2}} \\ Ω^{n}k ⊗ Ω^{n}k \ar[rr]_{Sq_{1} ζ} \ar@{-->}[ur]^{·∧·} &  & k \ar[uu]_{η} & ΣS \ar[r]_{η=Sq_{1}(2)} & S \ar[uu]_{incl}}

Note the similarity to the topological situation on the right-hand side, where η denotes the Hopf map (compare §5 in [23]).

Proposition 4.7. The diagram (4.4) commutes in \underline{𝔪𝔬𝔡}-kG.

Proof. We postpone the case p = 2 to the end of the proof and assume p ≥ 3. As in Lemma 3.29 denote by K the kernel of the surjective map ζ^{⊗p} : (Ω^{n}k)^{⊗p} → k. Then we get the dashed arrows making the following diagram commute in 𝔪𝔬𝔡-kG:

[figure: Λ^{2}Ω^{n}k ⊗ S^{p−2}Ω^{n}k; λ; Ω^{n}k ⊗ S^{p−1}Ω^{n}k; S^{p}Ω^{n}k; S^{p}ζ; K; ρ; (Ω^{n}k)^{⊗p}; ζ^{⊗p}; id ⊗ζ^{⊗(p−1)}; k; L_{ζ}; ι; Ω^{n}k; ζ; k]  (4.8)

4.1 A commutative square

On the other hand the diagram

\xymatrix{Λ^{2}Ω^{n}k ⊗ S^{p−2}Ω^{n}k \ar[r] \ar[d]_{κ_{2}⊗S^{p−2}ζ} & Ω^{n}k ⊗ S^{p−1}Ω^{n}k \ar[d]^{id ⊗S^{p−1}ζ} \\ L_{ζ} \ar[r]_{ι} & Ω^{n}k}

commutes because both compositions are given by

x_{1} ∧ x_{2} ⊗ x_{3} … x_{p} ↦ (ζ(x_{1})x_{2} − ζ(x_{2})x_{1})ζ(x_{3}) … ζ(x_{p}).

From the two diagrams we learn that ρλ = κ_{2} ⊗ S^{p−2}ζ. Therefore, the leftmost square in the diagram

[figure: L_{ζ}; Λ^{2}Ω^{n}k; κ_{2}; Λ^{3}Ω^{n}k; κ_{3}; …; κ_{4}; Λ^{p−1}Ω^{n}k; κ_{p−1}; Λ^{p}Ω^{n}k; κ_{p}; L_{ζ}; Λ^{2}Ω^{n}k ⊗ S^{p−2}Ω^{n}k; ρλ; id ⊗S^{p−2}ζ; Λ^{3}Ω^{n}k ⊗ S^{p−3}Ω^{n}k; id ⊗S^{p−3}ζ; …; Λ^{p−1}Ω^{n}k ⊗ Ω^{n}k; id ⊗ζ; Λ^{p}Ω^{n}k]

commutes. When we apply Example 1.17 to the map ζ : Ω^{n}k → k we get that the remaining squares commute. The upper row (and therefore also the lower row) represents the composition Λ^{p}Ω^{n}k ⟶^{κ_{p}} Λ^{p−1}L_{ζ} ⟶^{ω_{p−1}} Ω^{p−2}L_{ζ} due to Remark 3.13.

We can now splice the lower row of the previous diagram with the complex representing γ and obtain a complex

Ω^{n}k ⊗ S^{p−1}Ω^{n}k → Λ^{2}Ω^{n}k ⊗ S^{p−2}Ω^{n}k → ⋯ → Λ^{2}Ω^{n}k ⊗ S^{p−2}Ω^{n}k → L_{ζ},  (4.9)

representing the composition

Ω^{n}k ⊗ S^{p−1}Ω^{n}k ⟶^{γ} Ω^{p−2}Λ^{p}Ω^{n}k ⟶^{Ω^{p−2}κ_{p}} Ω^{p−2}Λ^{p−1}L_{ζ} ⟶^{Ω^{p−2}ω_{p−1}} Ω^{2p−4}L_{ζ}.

On the other hand, the complex (4.9) represents the composition

Ω^{n}k ⊗ S^{p−1}Ω^{n}k ≅ (Ω^{n}k)^{⊗p} ⟶^{P_{1}ζ} Ω^{2p−3}k ⟶^{Ω^{2p−4}υ} Ω^{2p−4}K ⟶^{Ω^{2p−4}ρ} Ω^{2p−4}L_{ζ}

in \underline{𝔪𝔬𝔡}-kG by Lemma 3.29, so we will be done as soon as we show ρ ∘ υ = η. But η and υ are represented by the bottom and middle row of (4.8), respectively. By Remark 3.15 the diagram shows ρυ = η.

We are left with the slightly different case p = 2. We get the dashed arrows making the following diagram commute in 𝔪𝔬𝔡-kG:

[figure: Ω^{n}k ⊗ Ω^{n}k; 1+T; λ; Ω^{n}k ⊗ Ω^{n}k; S^{2}Ω^{n}k; S^{2}ζ; K; ρ; (Ω^{n}k)^{⊗2}; ζ^{⊗2}; id ⊗ζ; k; L_{ζ}; ι; Ω^{n}k; ζ; k]  (4.10)

As before we get ρυ = η in the stable category. Recall that γ : Ω^{n}k ⊗ Ω^{n}k → Λ^{2}Ω^{n}k is the canonical projection map; therefore, the diagram

\xymatrix{Ω^{n}k ⊗ Ω^{n}k \ar[r]^{1+T} \ar[d]_{κ_{2}∘γ} & Ω^{n}k ⊗ Ω^{n}k \ar[d]^{id ⊗ζ} \\ L_{ζ} \ar[r]_{ι} & Ω^{n}k}

commutes. Together with (4.10) we get that ρλ = κ_{2}γ. By Lemma 3.29, λ = υ ∘ P_{1}ζ for every element P_{1}ζ ∈ P_{1}(ζ). Gathering all the results, we have

ηP_{1}ζ = ρυP_{1}ζ = ρλ = κ_{2}γ.

Remark 4.11. Let us prove that the canonical map (Ω^{n}k)^{⊗j} → S^{j}Ω^{n}k is a stable isomorphism for j < p. To do so, it is enough to check that the map Ω^{n}k⊗S^{j−1}Ω^{n}k → S^{j}Ω^{n}k is a stable isomorphism; but this map sits in an exact sequence

0 → Λ^{j}Ω^{n}k → Λ^{j−1}Ω^{n}k ⊗ Ω^{n}k → ⋯ → Ω^{n}k ⊗ S^{j−1}Ω^{n}k → S^{j}Ω^{n}k → 0 in which all the Λ^{i}Ω^{n}k are projective for i = 2, …, j.

4.2 Completion of the proof

Now we are ready to complete the proof of Theorem 4.1. We need a new kind of objects which store more information than s-coherent modules do. Definition 4.12. An (s, t)-coherent module consists of

• an s-coherent module X,

• a graded left S^{∗}L_{ζ}-comodule Y_{∗} which vanishes outside degrees 0, 1, …, t, and

• a morphism σ : X → F (Y_{t}) of s-coherent modules such that the sequence

0 → X_{i} ⟶^{σ_{i}} Λ^{i}L_{ζ} ⊗ Y_{t} → Λ^{i+1}L_{ζ} ⊗ Y_{t−1} → ⋯ → Λ^{i+t}L_{ζ} ⊗ Y_{0} → 0  (4.13)

is exact for all i = 0, 1, …, s, the maps being the same as in the chain complex associated to (Λ^{∗}L_{ζ}, Y_{∗}).

We call X_{1} the underlying object of the (s, t)-coherent module. Lemma 4.14. Suppose that (X, Y) is an (s, t)-coherent module. Then the stable iso-

morphisms Ωk ⊗ X_{i} ⟶^{η⊗id} L_{ζ} ⊗ X_{i} → X_{1+i} and

Ωk ⊗ Λ^{i}L_{ζ} ⊗ Y_{0} ⟶^{η⊗id} L_{ζ} ⊗ Λ^{i}L_{ζ} ⊗ Y_{0} → Λ^{1+i}L_{ζ} ⊗ Y_{0}

4.2 Completion of the proof

induce isomorphisms of the groups Êxt_{kG}^{t}(Λ^{i+t}L_{ζ} ⊗ Y_{0}, X_{i}) for all i = 1, 2, …, s. Up to these isomorphisms, the exact sequences

0 → X_{i} ⟶^{σ_{i}} Λ^{i}L_{ζ} ⊗ Y_{t} → Λ^{i+1}L_{ζ} ⊗ Y_{t−1} → ⋯ → Λ^{i+t}L_{ζ} ⊗ Y_{0} → 0

represent the same element for all i = 1, 2, …, s. In particular, the coherent module induces an element in Êxt_{kG}^{t}(Λ^{t+1}L_{ζ} ⊗ Y_{0}, X_{1}) ≅ \underline{Hom}_{kG}(Ω^{t}Λ^{t+1}L_{ζ}, X_{1}).

Proof. We have a commutative diagram as follows:

[figure: 0; Ωk ⊗ X_{i}; η⊗id; Ωk ⊗ Λ^{i}L_{ζ} ⊗ Y_{t}; …; Ωk ⊗ Λ^{i+t}L_{ζ} ⊗ Y_{0}; η⊗id; 0; 0; L_{ζ} ⊗ X_{i}; L_{ζ} ⊗ Λ^{i}L_{ζ} ⊗ Y_{t}; …; L_{ζ} ⊗ Λ^{i+t}L_{ζ} ⊗ Y_{0}; 0; 0; X_{1+i}; (a); Λ^{1+i}L_{ζ} ⊗ Y_{t}; (b); …; Λ^{1+i+t}L_{ζ} ⊗ Y_{0}; 0]

Here (a) commutes because σ : X → F (Y_{t}) is a map of s-coherent modules, and (b) commutes because it is the map of complexes associated to the map of Λ^{∗}L_{ζ}-modules L_{ζ} ⊗ Λ^{∗}L_{ζ} → Λ^{1+∗}L_{ζ} and the S^{∗}L_{ζ}-comodule Y_{∗}.

Definition 4.15. We denote by Φ(X, Y) the stable map Ω^{t}Λ^{t+1}L_{ζ} ⊗Y_{0} → X_{1} constructed in Lemma 4.14.

The next lemma allows us to construct coherent modules inductively.

Lemma 4.16. Suppose that (X, Y) is an (s, t)-coherent module with t < p − 1, and let φ : K → X_{1} be any map of kG-modules. Let us write ν for the S^{∗}L_{ζ}-coaction map of Y. Then there exists a unique graded left S^{∗}L_{ζ}-comodule Z with coaction map ν′ satisfying the following conditions:

(i) Z_{u} = Y_{u} for all u = 0, 1, 2, …, t, Z_{t+1} = K, and Z_{u} = 0 for u outside {0, 1, …, t+1},

(ii) ν′_{r,u} = ν_{r,u} for all 0 ≤ r ≤ r + u ≤ t, and

(iii) ν′_{1,t} : Z_{t+1} → L_{ζ} ⊗ Z_{t} agrees with the composition K ⟶^{φ} X_{1} ⟶^{σ_{1}} L_{ζ} ⊗ Y_{t}.

Proof. The first step is to define the remaining coaction maps ν′_{r,t+1−r}. Consider the following diagram:

\xymatrix{ & Z_{t+1} \ar[r]^{ν′_{1,t}} \ar@{-->}[d] & L_{ζ} ⊗ Z_{t} \ar[r] \ar[d] & Λ^{2}L_{ζ} ⊗ Z_{t−1} \ar[d] \\ 0 \ar[r] & S^{r}L_{ζ} ⊗ Z_{t+1−r} \ar[r] & L_{ζ} ⊗ S^{r−1}L_{ζ} ⊗ Z_{t+1−r} \ar[r] & Λ^{2}L_{ζ} ⊗ S^{r−2}L_{ζ} ⊗ Z_{t+1−r}}

The upper composition is zero because the sequence (4.13) is a complex (for i = 1). Now one has to show that the square on the right hand side commutes — either by an easy verification, or by using the fact that ν : Y_{∗} → S^{∗}L_{ζ} ⊗ Y_{t+1−r} is a map of comodules and passing to suitable associated chain complexes. Since the bottom row is exact by Corollary 1.9, we get the existence of the dashed arrow which we call ν′_{r,t+1−r}. To check that the maps ν′ define a comodule structure on Z_{∗}, consider the diagram

[figure: S^{1+i+j}L_{ζ} ⊗ Z_{l}; L_{ζ} ⊗ S^{i+j}L_{ζ} ⊗ Z_{l}; L_{ζ} ⊗ S^{i}L_{ζ} ⊗ S^{j}L_{ζ} ⊗ Z_{l}; L_{ζ} ⊗ Z_{i+j+l}; L_{ζ} ⊗ S^{i}L_{ζ} ⊗ Z_{j+l}; Z_{1+i+j+l}; S^{1+i}L_{ζ} ⊗ Z_{j+l}; S^{1+i}L_{ζ} ⊗ S^{j}L_{ζ} ⊗ Z_{l}; (b); (a); (c); (c); ν′_{1+i+j,l}; ν′_{1+i,j+l}; ∗]

for i, j, l ≥ 0 with i + j + l = t. We claim that the exterior square commutes. The square (a) commutes because Y_{∗} is a comodule, (b) commutes because S^{∗}L_{ζ} is a coalgebra, and the squares (c) commute by definition of ν′. Injectivity of the map ∗ proves the claim.

Proposition 4.17. Let X_{1} be the underlying object of some (s, t)-coherent module (X, Y) with t < p−1, and let f : K → X_{1} be a surjective map. Then the kernel of the composition

L_{ζ} ⊗ K ⟶^{id ⊗f} L_{ζ} ⊗ X_{1} ⟶^{μ_{1,1}} X_{2}

is the underlying object of some (s − 1, t + 1)-coherent module.

Proof. Let (X, Y, σ) be an (s, t)-coherent module and denote by ν : Y_{∗} → S^{∗}L_{ζ} ⊗ Y_{∗} the structure map of Y_{∗}. We want to define an (s − 1, t + 1)-coherent module (C, Z, σ′) with ν′ : Z_{∗} → S^{∗}L_{ζ} ⊗ Z_{∗} being the structure map of Z_{∗}. As in the proof of Proposition 2.6, define C to be the kernel of the composite

F (K) ⟶^{F (f)} F (X_{1}) → X[1].

Then C is an (s − 1)-coherent module by Lemma 2.4. Furthermore, define the S^{∗}L_{ζ}- comodule Z_{∗} to be the one defined in Lemma 4.16 out of Y_{∗} and φ = f : K → X_{1}. The inclusion of C into F (K) then defines the injective map σ′ : C → F (Z_{t+1}). Now we have defined all the data - it remains to show that

0 → C_{i−1} ⟶^{σ′_{i−1}} Λ^{i−1}L_{ζ} ⊗ Z_{t+1} → Λ^{i}L_{ζ} ⊗ Z_{t} → ⋯ → Λ^{i+t}L_{ζ} ⊗ Z_{0} → 0  (4.18)

is an exact sequence for all i = 1, … s. By definition, the sequence

\xymatrix{0 \ar[r] & C_{i−1} \ar[r]^{σ′_{i−1}} & Λ^{i−1}L_{ζ} ⊗ Z_{t+1} \ar[rr] \ar@{-->}[dr]_{id ⊗f} &  & X_{i} \ar[r] & 0 \\  &  &  & Λ^{i−1}L_{ζ} ⊗ X_{1} \ar@{-->}[ur]_{μ_{i−1,1}}}

is exact. The Yoneda splice of this with (4.13) gives an exact sequence, which agrees with (4.18) due to the following commutative diagram:

\xymatrix{Λ^{i−1}L_{ζ} ⊗ Z_{t+1} \ar[r]^{id ⊗ν′_{1,t}} \ar[dr]_{id ⊗f} & Λ^{i−1}L_{ζ} ⊗ L_{ζ} ⊗ Z_{t} \ar[r]^{μ_{i−1,1}⊗id} & Λ^{i}L_{ζ} ⊗ Z_{t} \\  & Λ^{i−1}L_{ζ} ⊗ X_{1} \ar[r]_{μ_{i−1,1}} \ar[u]_{id ⊗σ_{1}} & X_{i} \ar[u]_{σ_{i}}}

Suppose that (X, Y) is an (s, t)-coherent module with t < p − 1. By applying Lemma 4.16 to Y_{∗} and the identity map φ = id : X_{1} → X_{1}, we can view (Y_{0}, Y_{1}, …, Y_{t}, X_{1}) as an S^{∗}L_{ζ}-comodule.

Definition 4.19. We denote by λ(X, Y) : X_{1} → S^{t+1}Ω^{n}k ⊗ Y_{0} the composition of the coaction map X_{1} → S^{t+1}L_{ζ} ⊗ Y_{0} and the canonical inclusion S^{t+1}L_{ζ} ↪ S^{t+1}Ω^{n}k tensored with Y_{0}.

By Remark 1.14 the exact sequence 0 → L_{ζ} → Ω^{n}k → k → 0 induces a map of graded right Λ^{∗}L_{ζ}-modules κ : Λ^{∗+1}Ω^{n}k → Λ^{∗}L_{ζ}; we therefore get a morphism of the associated chain complexes:

\xymatrix{ & Ω^{n}k ⊗ X_{1} \ar[r] \ar[d]^{ζ⊗id} & Λ^{2}Ω^{n}k ⊗ Y_{t} \ar[r] \ar[d] & Λ^{3}Ω^{n}k ⊗ Y_{t−1} \ar[r] \ar[d] & … \ar[r] & Λ^{2+t}Ω^{n}k ⊗ Y_{0} \ar[d] \\ 0 \ar[r] & X_{1} \ar[r] & L_{ζ} ⊗ Y_{t} \ar[r] & Λ^{2}L_{ζ} ⊗ Y_{t−1} \ar[r] & … \ar[r] & Λ^{1+t}L_{ζ} ⊗ Y_{0} \ar[r] & 0} (4.20)

On the other hand, the coaction map is a map of comodules Y_{∗} ⟶^{ν} S^{∗}L_{ζ} ⊗Y_{0} and therefore induces a morphism of the associated chain complexes:

[figure: Ω^{n}k ⊗ S^{t+1}L_{ζ} ⊗ Y_{0}; Λ^{2}Ω^{n}k ⊗ S^{t}L_{ζ} ⊗ Y_{0}; …; Λ^{2+t}Ω^{n}k ⊗ Y_{0}; Ω^{n}k ⊗ X_{1}; Λ^{2}Ω^{n}k ⊗ Y_{t}; …; Λ^{2+t}Ω^{n}k ⊗ Y_{0}]  (4.21)

Lemma 4.22. Suppose that X_{1} is the underlying object of some (1, p−2)-coherent module (X, Y). Then ζ_{X_{1}} : Ω^{n}k ⊗ X_{1} → X_{1} stably factors as the composition

Ω^{n}k ⊗ X_{1} ⟶^{id ⊗λ(X,Y)} Ω^{n}k ⊗ S^{p−1}Ω^{n}k ⊗ Y_{0}

⟶^{γ⊗id} Ω^{p−2}Λ^{p}Ω^{n}k ⊗ Y_{0} ⟶^{Ω^{p−2}κ_{p}⊗id} Ω^{p−2}Λ^{p−1}L_{ζ} ⊗ Y_{0} ⟶^{Φ(X,Y)} X_{1}. (4.23)

Proof. As in (4.20) we get a diagram as follows:

\xymatrix{ & Ω^{n}k ⊗ X_{1} \ar[r] \ar[d]^{ζ⊗id} & Λ^{2}Ω^{n}k ⊗ Y_{p−2} \ar[r] \ar[d] & Λ^{3}Ω^{n}k ⊗ Y_{p−3} \ar[r] \ar[d] & … \ar[r] & Λ^{p}Ω^{n}k ⊗ Y_{0} \ar[d]_{κ_{p}⊗id} \\ 0 \ar[r] & X_{1} \ar[r] & L_{ζ} ⊗ Y_{p−2} \ar[r] & Λ^{2}L_{ζ} ⊗ Y_{p−3} \ar[r] & … \ar[r] & Λ^{p−1}L_{ζ} ⊗ Y_{0} \ar[r] & 0}

5. Toda brackets

The exact lower row represents the element φ = Φ(X, Y) in Êxt_{kG}^{p−2}(Λ^{p−1}L_{ζ} ⊗ Y_{0}, X_{1}). Since the modules Λ^{i}Ω^{n}k are projective for i = 2, …, p−1, the upper row represents some element ψ ∈ Êxt_{kG}^{−p+2}(Ω^{n}k ⊗ X_{1}, Λ^{p}Ω^{n}k ⊗ Y_{0}). Commutativity of the diagram tells us that ζ_{X_{1}} = φκ_{p}ψ by Proposition 3.9. Using diagram (4.21) and the inclusions S^{i}L_{ζ} ⊂ S^{i}Ω^{n}k, we get a diagram

[figure: Ω^{n}k ⊗ S^{p−1}Ω^{n}k ⊗ Y_{0}; Λ^{2}Ω^{n}k ⊗ S^{p−2}Ω^{n}k ⊗ Y_{0}; …; Λ^{p}Ω^{n}k ⊗ Y_{0}; Ω^{n}k ⊗ X_{1}; λ=id ⊗λ(X,Y); Λ^{2}Ω^{n}k ⊗ Y_{p−2}; …; Λ^{p}Ω^{n}k ⊗ Y_{0}]

where the upper row represents γ ⊗ id_{Y_{0}} and the lower row represents ψ. Hence, ψ = γλ. Therefore, ζ_{X_{1}} = φκ_{p}ψ = φκ_{p}γλ.

Proposition 4.24. Let X_{1} be the underlying object of some (s, t)-coherent module (X, Y) with s + t = p − 1. If the composition

Ω^{n}k ⊗ S^{p−1}Ω^{n}k ⟶^{γ} Ω^{p−2}Λ^{p}Ω^{n}k ⟶^{Ω^{p−2}κ_{p}} Ω^{p−2}Λ^{p−1}L_{ζ}

is stably zero when we tensor it with Y_{0}, then ζ-ord(X_{1}) ≥ s.

Proof. We do this by induction on s, starting with s = 1. By Lemma 4.22 and our assumptions, ζ_{X_{1}} = 0 or equivalently ζ-ord(X_{1}) ≥ 1 = s. The inductive step is done exactly the same way as in the proof of Proposition 2.6 by using Proposition 4.17. Notice that the object Y_{0} does not change during the induction.

Proof of Theorem 4.1. Notice that L_{ζ} ⊗ M is the underlying object of some (p − 1, 0)- coherent module as follows: the (p − 1)-coherent module is X = F (M), and Y_{0} = M. The morphism σ : X → F (M) is simply the identity map, and the sequence (4.13) reduces to

0 → X_{i} ⟶^{σ_{i}} Λ^{i}L_{ζ} ⊗ M → 0.

Our assumptions together with Proposition 4.7 tensored by M imply that the conditions of Proposition 4.24 are satisfied, so we get ζ-ord(M/ζ) = ζ-ord(M⊗L_{ζ}) ≥ p−1.

5 Toda brackets

So far we only have considered lower bounds on the order. In this section, we will introduce techniques which enable us to prove upper bounds on the order. The main ingredient will be (2p−1)-fold Massey products. As a first step, we will recall the definition of these higher products and then show the connection to Toda brackets, an analog which can be defined in any triangulated category. In a second step we show that certain Toda brackets can be used to build up a criterion for the ζ-order to be bounded from above. More explicitly, we show that if certain (2s − 1)-fold Toda brackets of the form ⟨ζ, α_{1}, ζ, …, α_{s}, ζ⟩ do not contain zero, then ζ-ord(k/ζ) ≤ s−1. We will then try to find examples of such Toda brackets, and to do so, we will have to compute single elements of them, which is usually a difficult task. In our case we can use the relation α_{1}α_{2} … α_{p−1}P_{1}(ζ) ⊆ ⟨ζ, α_{1}, ζ, …, α_{p−1}, ζ⟩, which we will prove in the third subsection using a connection between Toda brackets and coherent modules. Finally we are able to give our explicit example in the fourth part of this section.

5.1 Toda brackets and Massey products

5.1 Toda brackets and Massey products

We are now going to prove that Massey products in Ĥ^{∗}(G) agree with certain Toda brackets in \underline{𝔪𝔬𝔡}-kG.

Proposition 5.1. Suppose we are given cohomology classes a_{1}, a_{2}, …, a_{n} ∈ Ĥ^{∗}(G). We will also view these elements as stable morphisms Ω^{|a_{j}|}k → k. The Massey product ⟨a_{1}, a_{2}, …, a_{n}⟩ and the Toda bracket ⟨a_{1}, a_{2}, …, a_{n}⟩ associated to the sequence of maps Ω^{?}k ⟶^{a_{n}} Ω^{?}k ⟶^{a_{n−1}} … ⟶^{a_{1}} k (where the Ω^{?}k are suitable shifts of k) define the same subset of Ĥ^{|a_{1}|+···+|a_{n}|−(n−2)}(G).

Let us recall the definitions first. Fix a complete projective resolution P of the trivial kG-module k, and let ∂ be the differential of P. Denote by A = Hom_{kG}^{∗}(P_{∗}, P_{∗}) the endomorphism dga associated to P. More explicitly, A is the differential graded algebra whose degree n-part is given by

A^{(n)} = ∏_{j∈ℤ}Hom_{kG}(P_{n+j}, P_{j}),

and the differential is df = ∂f − (−1)^{n}f∂. Then degree n-cocycles in A are chain transformations P → P [n], and two of those differ by a coboundary if and only if the chain transformations are homotopic. Therefore, H^{∗}A ≅ Ĥ^{∗}(G).

An n-fold defining system is a collection {b_{ij}} of elements of A for all pairs (i, j) with 1 ≤ i ≤ j ≤ n and (i, j) ≠ (1, n) satisfying

(−1)^{g_{i−1}}db_{ij} = ∑_{r=i}^{j−1} b_{i,r}b_{r+1,j}

for all 1 ≤ i ≤ j ≤ n with (i, j) ≠ (1, n), where g_{j} = 1 + ∑_{r=1}^{j}(|b_{rr}| − 1). In particular, the b_{ii}’s are cocycles. Every defining system gives rise to a cocycle

c = ∑_{r=1}^{n−1} b_{1,r}b_{r+1,n}.

For fixed cocycles b_{11}, …, b_{nn}, the set of cohomology classes represented by all cocycles arising from defining systems will be denoted by ⟨b_{11}, …, b_{nn}⟩ and is called n-fold Massey product. It is well-known that this set only depends on the cohomology classes represented by b_{11}, …, b_{nn}, and we will also write ⟨b_{11}, …, b_{nn}⟩ = ⟨c_{1}, …, c_{n}⟩ where c_{i} is the cohomology class represented by b_{ii}. Remark 5.2. It is worth noting that there are several different definitions of Massey products in the literature (see [12], [13], [20]), which differ by certain signs. A quick check of signs asserts that our definition leads to (−1)^{n} times the set given in [12]. We have chosen this one because, as we will prove now, it agrees with the definition via certain Toda brackets which does not involve any choice of signs. Also it has the nice property that the two-fold Massey product is just the usual product (which in the other definitions is not always the case).

Remark 5.3. We also use a slightly different language than previous definitions of Massey products. Usually one says that a Massey product is defined if there is a defining system for it, and then the product is the set of cocycles associated to its defining systems. In the following, we will adopt the notion from Toda brackets and simply say that every Massey product is defined but might be the empty set.

For the definition of Toda brackets we take the definition of filtered objects from [26], Definition A.1. Let us work in a triangulated category T with shift functor [1]. Suppose

that X_{n−1} ⟶^{λ_{n−1}} X_{n−2} ⟶^{λ_{n−2}} … ⟶^{λ_{2}} X_{1} is a sequence of n − 2 composable maps in T . An (n − 1)-filtered object X ∈ {λ_{2}, …, λ_{n−1}} is a sequence of maps ∗ = F_{0}X ⟶^{i_{0}} F_{1}X ⟶^{i_{1}} … ⟶^{i_{n−1}} F_{n−1}X = X together with choices of exact triangles

F_{j}X ⟶^{i_{j}} F_{j+1}X ⟶^{p_{j+1}} X_{j+1}[j] ⟶^{d_{j}} F_{j}X[1]

such that p_{j}[1] ∘ d_{j} = λ_{j+1}[j]. The maps X_{1} ≅ F_{1}X → F_{n−1}X and F_{n−1}X → X_{n−1}[n − 2] are denoted by σ_{X} and σ′_{X}, respectively. The filtered object can be visualized as follows:

[figure: ∗ = F_{0}X; F_{1}X; F_{2}X; …; F_{n−2}X; F_{n−1}X; X_{1}; o; X_{2}[1]; o; o; X_{3}[2]; o; o; …; o; X_{n−1}[n − 2]; o; o]

Here \xymatrix{X \ar[r]|{o} & Y} denotes a map of degree 1, that is, a map X → Y [1]. Also, the diagram is commutative-exact in the following sense:

Definition 5.4. A diagram is called commutative-exact if every triangle with exactly one map [figure: o]of degree one is an exact triangle, and all other triangles in the diagram are commutative.

To be more precise, the first condition means that every triangle of the form

[figure: X; a; Y; b; Z; c; o]

represents an exact triangle X ⟶^{a} Y ⟶^{b} Z ⟶^{c} X[1].

In the definition of a filtered object, suppose that the given set of composable maps λ_{n−1}, …, λ_{2} lives in a strictly full triangulated subcategory S. Then every (n − 1)-filtered object X and its filtrations F_{0}X, F_{1}X, …, F_{n−1}X also lie in S. Recall that S is a strictly full subcategory if it is a full subcategory and for every isomorphism B ≅ B′ in T for which B belongs to S we also have that B′ belongs to S.

Definition 5.5 ([26], Definition A.2). Suppose that X_{n} ⟶^{λ_{n}} X_{n−1} ⟶^{λ_{n−1}} … ⟶^{λ_{1}} X_{0} is a sequence of n composable maps in a triangulated category T . We say that a map γ ∈ T (X_{n}[n − 2], X_{0}) belongs to the n-fold Toda bracket ⟨λ_{1}, …, λ_{n}⟩ if and only if there is an (n − 1)-filtered object X ∈ {λ_{2}, …, λ_{n−1}} and a commutative diagram

\xymatrix{ & X_{1} \ar[dr]^{λ_{1}} \ar[d]_{σ_{X}} \\ X_{n}[n − 2] \ar[r]^{γ_{n}} \ar[dr]_{λ_{n}[n−2]} & X \ar[r]_{γ_{0}} \ar[d]^{σ′_{X}} & X_{0} \\  & X_{n−1}[n − 2]}  (5.6)

such that γ = γ_{0}γ_{n}.

Let us note here that everything takes place in S if the n maps we started with belong to S. It also follows from the definition that Toda brackets are compatible with exact equivalences of triangulated categories.

We will apply this to the case when T = \underline{𝔪𝔬𝔡}-kG and S is the strictly full triangulated subcategory generated by the trivial kG-module k. There is an equivalence of triangulated categories (see, e.g., [14], proof of Proposition 7.13 and Example 7.16)

Z : K_{ac}(inj−kG) ⟶^{≅} \underline{𝔪𝔬𝔡}-kG  (5.7)

where K_{ac}(inj−kG) denotes the homotopy category of (unbounded) acyclic chain complexes of (finitely generated) injective kG-modules. Let M be the strictly full triangulated subcategory generated by P in K_{ac}(inj−kG). Since Z(P) ≅ k we get that M agrees to S under the equivalence Z, so we are going to describe M more explicitly.

Consider the set C of all chain complexes which arise from the following construction. Choose a non-negative integer r and integers n_{1}, n_{2}, …, n_{r}; then consider the complex (n_{1}, n_{2}, …, n_{r}, D) whose modules are the same as in P [n_{1}] ⊕ P [n_{2}] ⊕ ⋯ ⊕ P [n_{r}], but the differential is given by a matrix of the form

D = (\begin{matrix}(−1)^{n_{1}}∂ & a_{11} & a_{12} & a_{13} & … & a_{1,r} \\ & (−1)^{n_{2}}∂ & a_{22} & a_{23} & … & a_{2,r} \\ & & (−1)^{n_{3}}∂ & a_{33} & … & a_{3,r} \\ & & & ... & ... & ⋮ \\ & & & & & (−1)^{n_{r}}∂ \end{matrix}) .

Here ∂ is the differential of P, so that (−1)^{n_{i}}∂ is the differential of P [n_{i}]. The a_{ij}’s are elements of the endomorphism dga A, and we assume that they are of suitable degrees and satisfy certain relations so that D^{2} = 0. We will sometimes write [D] for this complex.

Lemma 5.8. The elements of C belong to M, and every object in M is isomorphic (in M) to an element of C.

Proof. Recall that one possibility of constructing the mapping cone of a map of chain complexes f : D → E is given as follows: take the same modules as in E ⊕ D[1], but with differential \binom{∂_{E} f}{−∂_{D}}. If we have two objects (n_{1}, …, n_{r}, D) and (m_{1}, …, m_{s}, E) in C then a morphism [D] → [E] is a matrix F with entries in A satisfying EF = F D. Therefore, its mapping cone (m_{1}, …, m_{s}, n_{1} + 1, …, n_{r} + 1, \binom{E F}{−D}) is again an object in C.

Let us prove by induction on r that (n_{1}, …, n_{r}, D) ∈ C belongs to M. For r = 0 this says that the trivial complex is in M; for r = 1 we need to show that shifts of P lie in M, which is true by construction. Suppose r ≥ 2, then we know that the object in question is the mapping cone of a morphism between two objects of C with smaller r. By induction hypothesis, this morphism belongs to M, and so does its cone.

Now consider the strictly full subcategory N of M consisting of all objects which are isomorphic to objects in C. Then N is closed under shifts and mapping cones (because the set C is), and P is contained in N. This proves the second statement.

The following simple fact is a useful tool for lifting homotopy-commutative diagrams to strictly-commuting ones.

Lemma 5.9. Suppose that h : P → Q is a chain map between acyclic complexes of injective modules, and that R is also an acyclic complex of injectives.

(i) Assume further that g : P → R is a chain map and f : Q → R is a morphism such that fh = g in K_{ac}(inj−kG). If h is levelwise an inclusion then there is a lift of f to a chain map f̂ such that f̂h = g as chain maps, not only in K_{ac}(inj−kG).

(ii) Assume that g : R → Q is a chain map and f : R → P is a morphism such that hf = g in K_{ac}(inj−kG). If h is levelwise a surjection then there is a lift of f to a chain map f̂ such that hf̂ = g as chain maps, not only in K_{ac}(inj−kG).

\xymatrix{(i) & P \ar[r]^{h} \ar[d]_{g} & Q \ar@{-->}[dl]^{f} & (ii) & P \ar[r]^{h} & Q \\  & R &  &  &  & R \ar[u]_{g} \ar@{-->}[ul]^{f}}

Proof. Let us prove (i) only. We know that fh = g up to some homotopy H : P → R[−1], that is, fh − g = ∂_{R}H + H∂_{P}. Since h is levelwise an inclusion of a direct summand, we can lift H to a map H′ : Q → R[−1] satisfying H = H′h. Then f̂ = f − ∂_{R}H′ − H′∂_{Q} satisfies f̂h = g.

Proposition 5.10. Suppose we are given a sequence of composable maps

P [−m_{1}] = X_{1} ⟵^{a_{2}} X_{2} ⟵^{a_{3}} X_{3} ← … ⟵^{a_{n−1}} X_{n−1}

in K_{ac}(inj−kG), where each a_{i} is a cocycle in A of degree m_{i}, m_{1} is some integer, and the X_{i}’s are suitable shifts of the complex P. Let us write g_{i} = 1 + ∑_{j=1}^{i}(m_{i} − 1). Then every (n − 1)-filtered object in {a_{2}, …, a_{n−1}} is isomorphic to an (n − 1)-filtered object of the following form:

F_{j}X = (−g_{1}, −g_{2}, …, −g_{j}, ( \begin{matrix}(−1)^{g_{1}}∂ & a_{22} & a_{23} & ... & a_{2,j} \\ & (−1)^{g_{2}}∂ & a_{33} & ... & a_{3,j} \\ & & ... & ... & ⋮ \\ & & & (−1)^{g_{j−1}}∂ & a_{j,j} \\ & & & & (−1)^{g_{j}} ∂ \end{matrix} ))

and i_{j} : F_{j}X → F_{j+1}X is the inclusion into the first summands, p_{j+1} : F_{j+1}X → X_{j+1}[j] is the projection onto the last summand, d_{j} : X_{j+1}[j] → F_{j}X[1] is the map − (a_{2,j+1} a_{3,j+1} … a_{j+1,j+1})^{T}, and a_{jj} = −a_{j} for all j.

Here we used the notation M^{T} for the transpose of the matrix M.

Proof. As a first step we show that the given object is indeed an (n − 1)-filtered object,

i.e., X_{j+1}[j − 1] ⟶^{−d_{j}[−1]} F_{j}X ⟶^{i_{j}} F_{j+1}X ⟶^{p_{j+1}} X_{j+1}[j] is an exact triangle and p_{j}[1] ∘ d_{j} =

−a_{j+1,j+1} = a_{j+1}. The latter actually holds strictly (not only up to homotopy), and the former follows from the fact that F_{j+1}X is the mapping cone (as constructed in the proof of Lemma 5.8) of −d_{j}[−1], with i_{j} and p_{j+1} being the associated maps.

Now we argue by induction and consider a diagram

[figure: ∗ = F_{0}X; F_{1}X; F_{2}X; …; F_{j−1}X; (∗); F′_{j}X; X_{1}; o; X_{2}[1]; o; o; X_{3}[2]; o; o; …; o; X_{j}[j − 1]; o; o]

in which we assume that the boxed region is already of the form we want it to be. Since F_{j−1}X → X_{j−1}[j − 2] is levelwise a surjection, we can assume by Lemma 5.9 that the map d_{j−1} : X_{j}[j − 1] → F_{j−1}X is chosen in such a way that the triangle (∗) commutes strictly. Then d_{j−1} is of the form − (a_{2j} a_{3j} … a_{j,j})^{T} with a_{j,j} = −a_{j}. Now let F_{j}X be constructed as in the statement of the proposition; then the triangles

X_{j}[j − 2] ⟶^{−d_{j}[−1]} F_{j−1}X → F′_{j}X → X_{j}[j − 1]

and

X_{j}[j − 2] ⟶^{−d_{j}[−1]} F_{j−1}X → F_{j}X → X_{j}[j − 1]

are both exact and hence isomorphic.

Proof of Proposition 5.1. Due to the equivalence (5.7) we can work with Toda brackets in M instead of \underline{𝔪𝔬𝔡}-kG, so we have a sequence of composable maps P = X_{0} ⟵^{a_{1}} X_{1} ⟵^{a_{2}} … ⟵^{a_{n−1}} X_{n−1} ⟵^{a_{n}} X_{n} with a_{i} ∈ A and the X_{i}’s are suitable shifts of P. Let us start with an (n − 1)-filtered object X and maps γ_{n}, γ_{0} such that γ_{0}γ_{n} defines an element of the Toda bracket as in Definition 5.5. We can assume that X is of the form described in Proposition 5.10, where m_{1} is the degree of a_{1}. Then σ_{X} : P [−m_{1}] → X is the inclusion of the first summand, and σ′_{X} : X → P [−g_{n−1}] is the projection onto the last summand. Using Lemma 5.9 we can assume that the diagram of chain complexes and chain maps

\xymatrix{ & X_{1} \ar[dr]^{a_{1}} \ar[d]_{σ_{X}} \\ X_{n}[n − 2] \ar[r]^{γ_{n}} \ar[dr]_{a_{n}[n−2]} & X \ar[r]_{γ_{0}} \ar[d]^{σ′_{X}} & X_{0} \\  & X_{n−1}[n − 2]}

commutes strictly, which means that γ_{n} is of the form − (a_{2,n} … a_{n−1,n} a_{n,n})^{T} with a_{nn} = −a_{n}, and γ_{0} is of the form − (a_{11} a_{12} … a_{1,n−1}) with a_{11} = −a_{1}. The fact that γ_{0} and γ_{n} are chain maps can be expressed by saying that

( \begin{matrix}(−1)^{g_{0}}∂ & a_{11} & a_{12} & a_{13} & ... & a_{1,n−1} & 0 \\ & (−1)^{g_{1}}∂ & a_{22} & a_{23} & ... & a_{2,n−1} & −a_{2,n} \\ & & (−1)^{g_{2}}∂ & a_{33} & ... & a_{3,n−1} & −a_{3,n} \\ & & & (−1)^{g_{3}}∂ & ... & a_{4,n−1} & −a_{4,n} \\ & & & & ... & ⋮ & ⋮ \\ & & & & & (−1)^{g_{n−1}}∂ & −a_{n,n} \\ & & & & & & (−1)^{gn}∂ \end{matrix} )^{2} = (\begin{matrix}0 & 0 & … & −c \\ & 0 & ... & 0 \\ & & 0 & 0 \\ & & & 0 \end{matrix}) ,

where c is some cocycle representing the map γ = γ_{0}γ_{n}. By putting b_{ij} = −a_{ij} (and in particular b_{ii} = −a_{ii} = a_{i}) we have therefore found a defining system whose associated cocycle is the element of the Toda bracket we started with. Going backwards through the arguments we get that every defining system yields an element of the Toda bracket.

In view of Proposition 5.1 we will often not distinguish between Massey products and its corresponding Toda brackets. Still, one has to be careful with the signs: whenever we write a Toda bracket ⟨a_{1}, …, a_{n}⟩ with a_{i} ∈ Ĥ^{∗}(G), we always mean the Toda bracket associated to the sequence of composable maps

Ω^{?}k ⟶^{a_{n}} Ω^{?}k ⟶^{a_{n−1}} … ⟶^{a_{2}} Ω^{?}k ⟶^{a_{1}} k,

where the last object is k (not some shift of k).

Definition 5.11. We say that a Toda bracket ⟨λ_{1}, λ_{2}, …, λ_{n}⟩ is strictly defined if for each pair 1 ≤ i < j ≤ n with (i, j) ≠ (1, n) we have ⟨λ_{i}, …, λ_{j}⟩ = {0}.

Corollary 5.12. Let c_{1}, c_{2}, …, c_{n} ∈ Ĥ^{∗}(G). The Toda bracket ⟨c_{1}, …, c_{n}⟩ in \underline{𝔪𝔬𝔡}-kG is strictly defined if and only if the corresponding Massey product is (in the sense of May, [20]).

Strictly defined Toda brackets (and Massey products) are much easier to deal with than arbitrary ones, mainly because every partially defined filtered object (and defining system) can be extended to a completely defined one. We will use this fact in the following version later.

Lemma 5.13. Suppose that X_{n} ⟶^{λ_{n}} X_{n−1} ⟶^{λ_{n−1}} … ⟶^{λ_{1}} X_{0} is a sequence of composable maps such that the Toda bracket ⟨λ_{1}, …, λ_{n}⟩ is strictly defined. Then for every (n − 1)- filtered object X ∈ {λ_{2}, …, λ_{n−1}} there are maps γ_{n}, γ_{0} such that λ_{n}[n − 2] = σ′_{X}γ_{n} and λ_{1} = γ_{0}σ_{X}, that is, the diagram (5.6) commutes.

Proof. Let us show the existence of γ_{n} only. For every pair 0 ≤ i < j < n choose exact triangles F_{i}X → F_{j}X → F_{j}/F_{i}X. Notice that the exact triangles F_{j}X → F_{j+1}X → X_{j+1}[j] induce exact triangles F_{j}/F_{i}X → F_{j+1}/F_{i}X → X_{j+1}[j]. Now we claim that for every i = 0, 1, …, n − 2 there is a map β_{i} : X_{n}[n − 2] → F_{n−1}/F_{i}X such that the composition X_{n}[n − 2] ⟶^{β_{i}} F_{n−1}/F_{i}X → X_{n−1}[n − 2] equals λ_{n}[n − 2]. For i = n − 2 we know that the second map in the latter composition is an isomorphism. Assume we have constructed β_{i} and let us construct β_{i−1}. We have an (n − 1 − i)-filtered object ∗ = F_{i}/F_{i}X → F_{i+1}/F_{i}X → ⋯ → F_{n−1}/F_{i}X which lies in {λ_{i+2}, …, λ_{n−1}}, up to shifts. There is a commutative diagram

\xymatrix{ & F_{i+1}/F_{i}X ≅ X_{i+1}[i] \ar[dr]^{λ_{i+1}[i]} \ar[d] \\ X_{n}[n − 2] \ar[r]^{β_{i}} \ar@{-->}[dr]_{λ_{n}[n−2]} & F_{n−1}/F_{i}X \ar[d] \ar[r]_{τ} & F_{i}/F_{i−1}X[1] ≅ X_{i}[i] \\  & X_{n−1}[n − 2]}

where the map τ is the composite F_{n−1}/F_{i}X → F_{i}X[1] → F_{i}/F_{i−1}X[1]. Since τβ_{i} is an element of the Toda bracket ⟨λ_{i+1}, …, λ_{n}⟩ = {0} we know that τβ_{i} = 0, which means

that β_{i} can be factorized as X_{n}[n − 2] ⟶^{β_{i−1}} F_{n−1}/F_{i−1}X → F_{n−1}/F_{i}X. This completes the inductive step, and putting γ_{n} = β_{0} we get the result.

We end this section with two technical lemmas. In the first one we basically recall the octahedral axiom in the form we will use it later, and we draw an immediate conclusion.

Lemma 5.14. Suppose we are given the left-hand side diagram

[figure: D; b; o; v; o; B; x; α; F; o; w; a; C; z; u; A; β; γ; E; o; y; c; E[−1]; y[−1]; c[−1]; A; γ; D[−1]; v[−1]; B]

in a triangulated category T , without the dashed arrows, and assume further that it is commutative-exact. Then the dashed arrows exist in such a way that the diagram is still commutative-exact and γ[1]y = vc and uα = ax. Furthermore, E[−1] is a weak pullback in the right-hand side diagram.

Proof. The first statement is the octahedral axiom. For the second statement let T ⟶^{λ} A and T ⟶^{κ} D[−1] be any two maps with γλ = v[−1]∘κ. Then βλ = αγλ = α∘v[−1]∘κ = 0,

so that λ factors as T ⟶^{e} E[−1] ⟶^{y[−1]} A for some map e. Then v[−1] ∘ (c[−1] ∘ e − κ) =

γ ∘ y[−1] ∘ e − γλ = 0. Therefore, c[−1] ∘ e − κ factors as T ⟶^{τ} C[−1] ⟶^{z[−1]} D[−1]. Define f = e − u[−1]τ : T → E[−1]. Then c[−1]f = κ and y[−1]f = λ, as desired.

Lemma 5.15. Suppose that X_{n} ⟶^{λ_{n}} X_{n−1} ⟶^{λ_{n−1}} … ⟶^{λ_{2}} X_{1} is a sequence of n − 1 composable maps in a triangulated category T , and let X ∈ {λ_{2}, …, λ_{n−1}} be an (n − 1)-

filtered object. Assume that the composition X_{n}[n − 3] ⟶^{λ_{n}[n−3]} X_{n−1}[n − 3] ⟶^{d_{n−2}} F_{n−2}X can also be written as a composite X_{n}[n − 3] ⟶^{ϕ} X_{1} ≅ F_{1}X → F_{n−2}X. Then (−1)^{n−1}ϕ is an element of the (n − 1)-fold Toda bracket ⟨λ_{2}, …, λ_{n}⟩.

Proof. Choose exact triangles F_{j}X[−1] ⟶^{π_{j}} G_{j}X ⟶^{w_{j}} F_{1}X → F_{j}X for all j = 1, …, n − 1. For all j = 1, 2, …, n − 2 we obtain a diagram

[figure: X_{j+1}[j − 1]; d̃_{j}; o; −d_{j}[−1]; o; F_{j}X[−1]; −π_{j}; i_{j}[−1]; G_{j}X; o; w_{j}; −ĩ_{j}; F_{j+1}X[−1]; p_{j+1}[−1]; π_{j+1}; F_{1}X[−1]; G_{j+1}X; o; −w_{j+1}; p̃_{j+1}]

(the signs coming from shifting of triangles), and the octahedral axiom assures that the dashed maps exist and form an exact triangle. Also, p̃_{j}[1] ∘ d̃_{j} = p̃_{j}[1]π_{j}[1]d_{j}[−1] =

p_{j} ∘ d_{j}[−1] = λ_{j+1}[j − 1]. Therefore ∗ ≅ G_{1}X ⟶^{−ĩ_{1}} G_{2}X ⟶^{−ĩ_{2}} … ⟶^{−ĩ_{n−2}} G_{n−1}X is an

(n − 2)-filtered object in {λ_{3}, …, λ_{n−1}}. Its associated maps are σ̃′_{X} = p̃_{n−1} : G_{n−1}X → X_{n−1}[n − 3] and σ̃_{X} = (−1)^{n−1}ĩ_{n−2} … ĩ_{2}p̃_{2}^{−1} : X_{2} → G_{n−1}X. Our goal is to choose maps γ_{n}, γ_{1} such that the diagram

\xymatrix{ & X_{2} \ar[d]_{σ̃_{X}} \ar[dr]^{λ_{2}} \\ X_{n}[n − 3] \ar[r]^{γ_{n}} \ar[dr]_{λ_{n}[n−3]} & G_{n−1}X \ar[r]^{(a)}_{(b)}_{γ_{1}} \ar[d]^{σ̃′_{X}} & X_{1} \\  & X_{n−1}[n − 3]}

5.2 Toda brackets and order

commutes and γ_{1}γ_{n} = (−1)^{n−1}ϕ. We choose γ_{1} = (−1)^{n−1}p_{1}w_{n−1}, then (a) commutes due to the relations w_{j} = w_{j+1}ĩ_{j} and λ_{2}p̃_{2} = p_{1} ∘ d_{1}[−1] ∘ p̃_{2} = p_{1}w_{2}.

Let us write i : F_{1}X → F_{n−2}X. By Lemma 5.14 and our assumptions, we have a weak pullback diagram

\xymatrix{X_{n}[n − 3] \ar[ddr]_{λ_{n}[n−3]} \ar[drr]^{−p_{1}^{−1}ϕ} \ar@{-->}[dr]_{γ_{n}} \\  & G_{n−1}X \ar[r]^{−w_{n−1}} \ar[d]^{p̃_{n−1}} & F_{1}X \ar[d]^{i} \\  & X_{n−1}[n − 3] \ar[r]_{−d_{n−2}} & F_{n−2}X}

which implies the existence of the dashed arrow γ_{n}. Then (b) commutes by definition, and γ_{1}γ_{n} = (−1)^{n−1}ϕ is an element of the Toda bracket.

5.2 Toda brackets and order

From now on, we fix a Tate cohomology class ζ of even degree n.

Lemma 5.16. Suppose that t < p − 1 and α_{1}, …, α_{t+1} ∈ Ĥ^{∗}(G) are Tate cohomology classes of even degree such that the (2t + 3)-fold Toda bracket ⟨ζ, α_{1}, ζ, …, ζ, α_{t+1}, ζ⟩ is strictly defined. Then there is a (2t + 2)-filtered object ∗ = F_{0}Z → F_{1}Z → ⋯ → F_{2t+2}Z

in {ζ, α_{1}, ζ, …, α_{t}, ζ}, associated to the maps k = Z_{1} ⟵^{ζ} Z_{2} ⟵^{α_{1}} Z_{3} ← … ⟵^{ζ} Z_{2t+2} (where

the Z_{i}’s are suitable shifts of k), with the following properties:

(i) ζ-ord(F_{2t+2}Z) ≥ ζ-ord(k/ζ) − t.

(ii) If we write Z_{2t+3} = Ω^{|α_{t+1}|}Z_{2t+2}, then the map α_{t+1} : Z_{2t+3} → Z_{2t+2} factors as

Z_{2t+3} \underrightarrow{α̂_{t+1}} Ω^{2t+1}F_{2t+2}Z ⟶^{Ω^{2t+1}σ′_{Z}} Z_{2t+2}.

Proof. Recall that we have the following exact triangle:

Ωk ⟶^{η} L_{ζ} ⟶^{ι} Ω^{n}k ⟶^{ζ} k

We prove the proposition by induction on t, beginning with t = 0. Define the filtered object to be as follows:

[figure: ∗; k; −η[1]; L_{ζ}[1]; ι[1]; k; o; Ω^{n}k[1]; o; ζ[1]; o; ζ[1]]

Then F_{2t+2}Z = L_{ζ}[1] = k/ζ so that (i) holds. The map α̂_{1} exists due to ζα_{1} = 0.

Now let us assume the statement is true for t − 1 and prove it for t. By induction we know that there is a map α̂_{t}[2t] : Z_{2t+1}[2t] → F_{2t}Z[1]. Since

ζ-ord(F_{2t}Z) ≥ ζ-ord(k/ζ) − t + 1 ≥ 1

by induction hypothesis and Corollary 2.8, we know that there exists some extension F : k/ζ ⊗ Z_{2t+1}[2t] → F_{2t}Z[1] of −α̂_{t}[2t] in such a way that there is an exact triangle

F_{2t}Z → F_{2t+2}Z → k/ζ ⊗ Z_{2t+1}[2t] ⟶^{F} F_{2t}Z[1]

with some object F_{2t+2}Z satisfying ζ-ord(F_{2t+2}) ≥ ζ-ord(k/ζ) − t. We get the following diagram without the dashed and dotted arrows:

[figure: F_{2t}Z; F_{2t+1}Z; F_{2t+2}Z; Z_{2t+1}[2t]; α̂_{t}[2t]; o; −η[1]⊗id; Z_{2t+2}[2t + 1]; o; ζ; o; k/ζ ⊗ Z_{2t+1}[2t]; ι[1]; F; o]  (5.17)

If we choose F_{2t+1}Z to be some choice of cone of α̂_{t}[2t], we get the dashed arrows and the octahedral axiom guarantees the existence of the dotted arrows such that the resulting diagram is commutative-exact. We have therefore extended our (2t)-filtered object to a (2t + 2)-filtered object, and (i) is also satisfied.

Now put Z_{2t+3} = Ω^{|α_{t+1}|}Z_{2t+2}. For the existence of α̂_{t+1}, let us choose exact triangles F_{1}Z → F_{i}Z → F_{i}/F_{1}Z ⟶^{π_{i}} F_{1}Z[1] for all i = 1, 2, …, 2t + 2. Then the (2t + 1)-filtered object ∗ = F_{1}/F_{1}Z → F_{2}/F_{1}Z → ⋯ → F_{2t+2}/F_{1}Z is (up to shifts) an element of {α_{1}, ζ, …, α_{t}, ζ}. By Lemma 5.13 there is a map ω yielding a commutative diagram as follows:

\xymatrix{ & Z_{2}[1] \ar[dr]^{ζ} \ar[d] \\ Z_{2t+3}[2t + 1] \ar[dr]_{α_{t+1}} \ar@{.>}[r]^{ω} & F_{2t+2}/F_{1}Z \ar[r]_{π_{2t+2}} \ar[d] & F_{1}Z[1] ≅ Z_{1}[1] \\  & Z_{2t+2}[2t + 1]}

Then π_{2t+2} ω lies in the Toda bracket ⟨ζ, α_{1}, …, ζ, α_{t+1}⟩ = {0}. Therefore, Ω^{2t+1}ω lifts to a map α̂_{t+1} : Z_{2t+3} → Ω^{2t+1}F_{2t+2}Z, proving (ii).

Corollary 5.18. Suppose that s ≤ p − 1, and let α_{1}, …, α_{s} ∈ Ĥ^{∗}(G) be Tate cohomology classes of even degree in such a way that the (2s + 1)-fold Massey product ⟨ζ, α_{1}, ζ, …, ζ, α_{s}, ζ⟩ is strictly defined. If ζ-ord(k/ζ) ≥ s, then 0 is an element of the Massey product.

5.3 Toda brackets and coherent modules

Proof. We put t = s − 1 in Lemma 5.16 and get a 2s-filtered object together with a map α̂_{s} (the dotted arrow) making the diagram commutative-exact, and we know that ζ-ord(F_{2s}Z) ≥ 1. By choosing a cone F_{2s+1}Z of α̂_{s}[2s − 1] we can view the diagram as a (2s + 1)-filtered object. Put Z_{2s+2} = Ω^{n}Z_{2s+1}; then the diagram

[figure: ∗ = F_{0}Z; F_{1}Z; F_{2}Z; …; F_{2s}Z; Z_{1}; o; Z_{2}[1]; o; o; …; o; o; Z_{2s}[2s − 1]; o; Z_{2s+1}[2s]; α̂_{s}[2s]; o; α_{s}[2s]; o]

\xymatrix{Z_{2s+2}[2s − 1] \ar[r]^{ζ} \ar[d]_{Ω^{n}α̂_{s}[2s−1]} & Z_{2s+1}[2s − 1] \ar[d]^{α̂_{s}[2s−1]} \\ Ω^{n}F_{2s}Z \ar[r]^{ζ} & F_{2s}Z}

commutes. Since ζ-ord(F_{2s}Z) ≥ 1, the bottom vertical map vanishes. Hence α̂_{s}ζ = 0, and therefore 0 ∈ ⟨ζ, α_{1}, ζ, …, α_{s}, ζ⟩ due to Lemma 5.15.

5.3 Toda brackets and coherent modules

Proposition 5.19. Suppose that t < p−1 and α_{1}, …, α_{t+1} ∈ Ĥ^{∗}(G) are Tate cohomology classes of even degree such that the (2t + 3)-fold Toda bracket ⟨ζ, α_{1}, ζ, …, ζ, α_{t+1}, ζ⟩ is strictly defined. Then there is a (p − 1 − t, t)-coherent module (X, Y) with Y_{0} = k and a (2t + 2)-filtered object ∗ = F_{0}Z → F_{1}Z → ⋯ → F_{2t+2}Z in {ζ, α_{1}, ζ, …, α_{t}, ζ}, associated to the maps k = Z_{1} ⟵^{ζ} Z_{2} ⟵^{α_{1}} Z_{3} ← … ⟵^{ζ} Z_{2t+2} (where the Z_{i}’s are suitable shifts of k), with the following properties in \underline{𝔪𝔬𝔡}-kG:

(i) F_{2t+2}Z is isomorphic to X_{1}[2t + 1].

(ii) The map σ_{Z} : k → F_{2t+2}Z equals the composition

k ⟶^{−η[1]} L_{ζ}[1] ⟶^{Ω^{−t−1}ω_{t+1}^{−1}} Ω^{t}Λ^{t+1}L_{ζ}[2t + 1] ⟶^{Φ(X,Y)[2t+1]} X_{1}[2t + 1],

where ω_{t+1} and Φ(X, Y) are constructed in (4.5) and Definition 4.15, respectively.

(iii) The composition

Ω^{2t+1}F_{2t+2}Z ⟶^{Ω^{2t+1}σ′_{Z}} Z_{2t+2} = Ω^{|α_{1}|+···+|α_{t}|+(t+1)n}k ⟶^{(−1)^{t}α_{1}...α_{t}} Ω^{(t+1)n}k

equals Ω^{2t+1}F_{2t+2}Z ≅ X_{1} ⟶^{λ(X,Y)} S^{1+t}Ω^{n}k ≅ (Ω^{n}k)^{⊗(1+t)}, where λ(X, Y) is defined

in Definition 4.19.

(iv) Let us write Z_{2t+3} = Ω^{|α_{t+1}|}Z_{2t+2}. The map α_{t+1} : Z_{2t+3} → Z_{2t+2} factors as a

composite Z_{2t+3} \underrightarrow{α̂_{t+1}} X_{1} ≅ Ω^{2t+1}F_{2t+2}Z ⟶^{Ω^{2t+1}σ′_{Z}} Z_{2t+2}.

(v) By (iii) and (iv), the composition

Z_{2t+3} \underrightarrow{α̂_{t+1}} X_{1} ⟶^{λ(X,Y)} S^{1+t}(Ω^{n}k) ≅ (Ω^{n}k)^{⊗(1+t)}

equals (−1)^{t}α_{1}α_{2} … α_{t+1} : Z_{2t+3} = Ω^{|α_{1}|+···+|α_{t+1}|+(t+1)n}k → Ω^{(t+1)n}k.

We will draw a picture visualizing the situation in Remark 5.23, after the proof.

Proof. We do this by induction on t, beginning with t = 0. We have to define a (p − 1, 0)- coherent module, and we take X_{∗} = Λ^{∗}L_{ζ} = F (k), with σ : X → F (Y_{0}) being the identity map. Define the filtered object to be the following:

[figure: ∗; k; −η[1]; L_{ζ}[1]; ι[1]; k; o; Ω^{n}k[1]; o; ζ[1]; o; ζ[1]]

Then X_{1}[1] = L_{ζ}[1] = F_{2t+2}Z so that (i) holds. The map σ_{Z} : k → L_{ζ}[1] equals −η[1] which proves (ii), because Φ(X, Y)[1] ∈ \underline{Hom}_{kG}(L_{ζ}[1], X_{1}[1]) and ω_{1}^{−1} : L_{ζ}[1] → L_{ζ}[1] are the identity maps. The map λ(X, Y) : X_{1} = L_{ζ} → Ω^{n}k equals ι, which proves (iii). The map α̂_{1} exists due to ζα_{1} = 0, thus proving (iv).

Now let us assume the statement is true for t − 1 and prove it for t. By induction we know that there is a map α̂_{t} : Z_{2t+1} → X_{1}, and we let −f be a surjective lift of that stable map to the unstable world. As in Proposition 4.17 we get from f and the (p − t, t − 1)- coherent module (X, Y) a (p − 1 − t, t)-coherent module (X′, Y′) whose underlying object X′_{1} is the kernel of the composition

F : L_{ζ} ⊗ Z_{2t+1} ⟶^{id ⊗f} L_{ζ} ⊗ X_{1} → X_{2}.

We get the following diagram in \underline{𝔪𝔬𝔡}-kG (without the dashed and dotted arrows):

[figure: X_{2}[−1] ≅ X_{1}[−2]; μ; ∗; C; ∗; X′_{1}; ∗; Z_{2t+1}[−1]; −f[−1]; o; −η[1]⊗id; Z_{2t+2}; (♮); o; ζ; o; L_{ζ} ⊗ Z_{2t+1}; ι; ∗; F; o]  (5.20)

We define C to be some fiber of −f and obtain the dashed arrows. The octahedral axiom then tells us that the dotted arrows exist in such a way that the diagram is commutativeexact. Apply the shift functor [2t + 1] to the diagram, and at the same time multiply the ∗-marked arrows by (−1). This assures that the previously exact triangles are still exact. Then we can concatenate the result with the (2t)-filtered object we already have and obtain a (2t + 2)-filtered object with F_{2t+1}Z = C[2t + 1] and F_{2t+2}Z = X′_{1}[2t + 1] which proves (i). To prove (ii), notice that the exact sequence

0 → X′_{i} → Λ^{i}L_{ζ} ⊗ Y′_{t} → Λ^{i+1}L_{ζ} ⊗ Y′_{t−1} → ⋯ → Λ^{i+t}L_{ζ} → 0

is the Yoneda splice of the exact sequences

0 → X_{1+i} → Λ^{1+i}L_{ζ} ⊗ Y_{t−1} → ⋯ → Λ^{i+t}L_{ζ} → 0

and

0 → X′_{i} → Λ^{i}L_{ζ} ⊗ Y′_{t} → X_{1+i} → 0,

as in the proof of Proposition 4.17. Via the isomorphism

Ext^{1}(X_{1+i}, X′_{i}) ≅ \underline{Hom}_{kG}(X_{2}[−1], X′_{1})

the latter represents the map μ of (5.20), and (ii) follows.

In order to prove (iii), we set up two commutative diagrams. The first one uses the induction hypothesis on (v) as follows:

[figure: X′_{1}; Ω^{2t+1}σ′_{Z}; (a); Z_{2t+2}; (v); ≅; (−1)^{t}α_{1}···α_{t}; Ω^{n}k ⊗ S^{t}Ω^{n}k ≅ (Ω^{n}k)^{⊗(t+1)}; L_{ζ} ⊗ Z_{2t+1}; ι⊗id; Ω^{n}k ⊗ Z_{2t+1}; id ⊗f; Ω^{n}k ⊗ X_{1}; id ⊗λ(X,Y)]  (5.21)

The square (a) equals the triangle (♮) in (5.20), so the diagram commutes. The second diagram takes the following form:

[figure: X′_{1}; λ(X′,Y′); (b); S^{1+t}(Ω^{n}k); ≅; (Ω^{n}k)^{⊗(1+t)}; L_{ζ} ⊗ Z_{2t+1}; 1⊗f; L_{ζ} ⊗ X_{1}; ι⊗λ(X,Y); Ω^{n}k ⊗ S^{t}(Ω^{n}k); ≅; (Ω^{n}k)^{⊗(1+t)}]  (5.22)

Here, (b) commutes by the definition of the comodule structure on Y′. Putting the diagrams (5.21) and (5.22) together, we get a proof of (iii).

The existence of α̂_{t+1} in (iv) follows exactly as in the proof of Lemma 5.16. The statement (v) is an immediate consequence of (iii) and (iv), so we are done.

Remark 5.23. The statement of the previous proposition might become clearer by considering the following diagrammatic picture. In the topological world of the stable homotopy category SHC, our trivial kG-module k corresponds to the sphere spectrum S, which can be viewed as a stable 0-cell. So let us consider k as a 0-cell, then Σ^{n}k is an n-cell, Ω^{n}k is a −n-cell, and L_{ζ} has one −1-cell and one −n-cell. We draw pictures of these objects as follows:

[figure: •; −1; •; −1; •; −n; •; −n; Ωk; L_{ζ}; Ω^{n}k; η; ι]

We draw a dot • for every cell, and indicate its dimension on top of it. Note that this does not mean that L_{ζ} decomposes as a direct sum of Ωk and Ω^{n}k, the two cells are ‘linked’ by the map ζ in an appropriate sense, but we do not include this information into the figure. Now we draw a picture visualizing the situation of Proposition 5.19 in the special case t = 2 as follows:

[figure: •; 0; F_{1}; •; −n+1; F_{2}; •; −n−|α_{1}|+2; F_{3}; •; −2n−|α_{1}|+3; F_{4}; •; −2n−|α_{1}|−|α_{2}|+4; F_{5}; •; −3n−|α_{1}|−|α_{2}|+5; F_{6}; •; 0; •; −n+1; •; 0; k; L_{ζ}[1]; X_{1}[2t + 1]; −η[1]; Φ(X,Y); σ_{Z}; •; −3n−|α_{1}|−|α_{2}|−|α_{3}|+5; Z_{2t+3}[2t + 1]; X_{1}[2t + 1]; α̂_{t+1}; •; −3n−|α_{1}|−|α_{2}|+5; •; −3n+5; Z_{2t+2}[2t + 1]; (Ω^{n}k)^{⊗(1+t)}[2t + 1]; σ′_{Z}; α_{1}α_{2}; λ(X,Y)]

Proposition 5.24. Suppose that α_{1}, …, α_{p−1} ∈ Ĥ^{∗}(G) are Tate cohomology classes of even degree such that the (2p − 1)-fold Massey product ⟨ζ, α_{1}, ζ, …, α_{p−1}, ζ⟩ is strictly defined. Then −α_{1} … α_{p−1}P_{1}(ζ) ⊆ ⟨ζ, α_{1}, ζ, …, α_{p−1}, ζ⟩.

Remark 5.25. Notice that the sign occurring in the statement depends on the choice of the definition of Massey products.

Proof of Proposition 5.24. We put t = p − 2 in Proposition 5.19 and see what we get. There is a (1, p − 2)-coherent module (X, Y) and a (2p − 2)-filtered object

[figure: ∗ = F_{0}Z; F_{1}Z; F_{2}Z; …; F_{2p−2}Z; Z_{1}; o; Z_{2}[1]; o; o; …; o; o; Z_{2p−2}[2p − 3]; o; Z_{2p−1}[2p − 2]; α̂_{p−1}[2p−2]; o; α_{p−1}[2p−2]; o]

together with a map α̂_{p−1} (the dotted arrow) making the diagram commutative-exact, and we know that F_{2p−2}Z ≅ X_{1}[2p − 3], and σ_{Z} : k = Z_{1} → F_{2p−2}Z equals the composition

k ⟶^{−η[1]} L_{ζ}[1] ⟶^{Ω^{−(p−1)}ω_{p−1}^{−1}} Ω^{p−2}Λ^{p−1}L_{ζ}[2p − 3] ⟶^{Φ(X,Y)[2p−3]} X_{1}[2p − 3]. By defining

F_{2p−1}Z to be some choice of fibre of α̂_{p−1}[2p − 2] we can also consider the diagram above as a (2p − 1)-filtered object.

Consider the composite Ω^{n}Z_{2p−1} ⟶^{ζ} Z_{2p−1} \underrightarrow{α̂_{p−1}} Ω^{2p−3}F_{2p−2}Z ≅ X_{1}. By naturality of ζ, this equals

Ω^{n}Z_{2p−1} ⟶^{Ω^{n}α̂_{p−1}} Ω^{n}Ω^{2p−3}F_{2p−2}Z ≅ Ω^{n}X_{1} ⟶^{ζ} X_{1}.

Let us choose any element P_{1} ∈ P_{1}(ζ) and consider the following diagram:

[figure: Ω^{n}Z_{2p−1}; Ω^{n}α̂_{p−1}; (−1)^{p−2}α_{1}...α_{p−1}; Ω^{n}k ⊗ X_{1}; id ⊗λ(X,Y); ζ⊗id; (b); (Ω^{n}k)^{⊗p}; P_{1}; (a); Ω^{n}k ⊗ S^{p−1}Ω^{n}k; ≅; γ; Ω^{p−2}Λ^{p}Ω^{n}k; Ω^{p−2}κ_{p}; (c); Ω^{2p−4}L_{ζ}; Ω^{p−2}ω_{p−1}^{−1}; Ω^{2p−4}η; Ω^{p−2}Λ^{p−1}L_{ζ}; Φ(X,Y); Ω^{2p−3}k; −Ω^{2p−3}σ_{Z}; (d); X_{1}]  (5.26)

Here, (d) and (a) commute due to Proposition 5.19 (ii) and (v), (b) commutes because of Lemma 4.22, and (c) is Proposition 4.7.

Now we put Z_{2p} = Ω^{n}Z_{2p−1}. Then we are in the situation of Lemma 5.15 (where the

n in that statement corresponds to 2p in our case), the composable maps being Z_{2p} ⟶^{ζ}

Z_{2p−1} ⟶^{α_{p−1}} … ⟶^{ζ} Z_{1} = k. The filtered object is ∗ = F_{0}Z → F_{1}Z → … F_{2p−2}Z →

F_{2p−1}Z, and the map Z_{1} ≅ F_{1}Z → F_{2p−2}Z equals our σ_{Z}. Furthermore, (5.26) tells us that the diagram

\xymatrix{Z_{2p}[2p − 3] \ar[r]^{ζ} \ar[d]_{α_{1}...α_{p−1}P_{1}} & Z_{2p−1}[2p − 3] \ar[d]^{α̂_{p−1}[2p−3]} \\ Z_{1} = k \ar[r]_{σ_{Z}} & F_{2p−2}Z}

commutes, and by Lemma 5.15 we get that −α_{1} … α_{p−1}P_{1} ∈ ⟨ζ, α_{1}, …, α_{p−1}, ζ⟩.

Remark 5.27. In the case p = 2 we get the result αP_{1}(ζ) ⊆ ⟨ζ, α, ζ⟩, whenever the Massey product is defined. In the case of ordinary cohomology, this is a well-known fact. It is an immediate consequence of a formula relating the ∪_{1}- and the ∪-product in the singular cochain complex of a topological space; see [8]. For G = ℤ/2ℤ×ℤ/2ℤ the formula enables us to compute all Massey products of the form ⟨ζ, α, ζ⟩, and it is consistent with the computation of all Massey products done in [15], §5.

5.4 Example for equality

Now we can prove that Corollary 2.8 is a strong inequality in the sense that for every prime p and every field of characteristic p, there is a group G and a Tate cohomology class ζ of even degree such that ζ-ord(k/ζ) = p − 2. Consider the group

Aff_{1} = {x ↦ ax + b : a ∈ 𝔽_{p}^{∗}, b ∈ 𝔽_{p}} ≅ 𝔽_{p} ⋊ 𝔽_{p}^{∗}

of affine linear transformations of 𝔽_{p}. The group operation is composition of such maps. This group has the normal subgroup of translations

T = {x ↦ x + b : b ∈ 𝔽_{p}},

and the quotient Aff_{1}/T is isomorphic to 𝔽_{p}^{∗}. Furthermore, T is the p-Sylow subgroup of Aff_{1}.

Our example will be the group G = Aff_{1} × Aff_{1}. By the considerations above, G has the normal p-Sylow subgroup N = T × T. It is well-known that (see [4], Theorem 10.1 in Chapter XII) the inclusion i : N → G induces an isomorphism Ĥ^{∗}(G) → Ĥ^{∗}(N)^{G/N} from the Tate cohomology of G to the fixed points in Tate cohomology of N under the action of G/N. We will often suppress this isomorphism from notation. Let us now try to describe the Tate cohomology of G. To do so, we use the description of the Tate cohomology of N given in §3.6. Recall that for a fixed non-negative degree n we have the monomial basis of Ĥ^{n}(N) given by elements v_{1}^{a_{1}}v_{2}^{a_{2}}u_{1}^{ϵ_{1}}u_{2}^{ϵ_{2}} with 2a_{1} + 2a_{2} + ϵ_{1} + ϵ_{2} = n and ϵ_{1}, ϵ_{2} ≤ 1. Using Tate duality, we denoted the dual basis by φ_{2a_{1}+ϵ_{1},2a_{2}+ϵ_{2}}. Let us write D : Ĥ^{n}(N) → Ĥ^{−1−n}(N) for the k-linear map which sends each basis element v_{1}^{a_{1}}v_{2}^{a_{2}}u_{1}^{ϵ_{1}}u_{2}^{ϵ_{2}} to its dual φ_{2a_{1}+ϵ_{1},2a_{2}+ϵ_{2}}. Also, let us write q = 2p − 2.

Lemma 5.28. Suppose that p ≥ 3. The ordinary group cohomology of G is given by H^{∗}(G) = k[b_{1}, b_{2}, e_{1}, e_{2}], the graded commutative algebra generated by two exterior classes e_{1}, e_{2} of degree q − 1 and two polynomial classes b_{1}, b_{2} of degree q. The elements are given by e_{i} = v_{i}^{p−2}u_{i} and b_{i} = v_{i}^{p−1}. The structure of the Tate cohomology of G is obtained from the ordinary cohomology by using the fact that D restricts to an isomorphism D : Ĥ^{n}(G) → Ĥ^{−1−n}(G).

Proof. The computation of H^{∗}(Aff_{1}) is well-known, and H^{∗}(G) is deduced from this using the Künneth theorem. For the last statement it is enough to observe that the elements of G/N ≅ 𝔽_{p}^{∗} × 𝔽_{p}^{∗} act as diagonal matrices with respect to our chosen basis: the element (c_{1}, c_{2}) acts as

(c_{1}, c_{2}) · v_{1}^{a_{1}}v_{2}^{a_{2}}u_{1}^{ϵ_{1}}u_{2}^{ϵ_{2}} = c_{1}^{2a_{1}+ϵ_{1}}c_{2}^{2a_{2}+ϵ_{2}}v_{1}^{a_{1}}v_{2}^{a_{2}}u_{1}^{ϵ_{1}}u_{2}^{ϵ_{2}}.

Therefore, the elements of G/N act as the same diagonal matrices on the dual space Ĥ^{−1−n}(N) with respect to the dual basis φ_{?}.

Remark 5.29. In the case p = 2 the classes u_{1} and u_{2} (and therefore also e_{1} and e_{2}) are polynomial classes with u_{i}^{2} = v_{i}. With this tiny difference in mind, the following goes through even in the case p = 2, because we will never use the fact that the u_{i}’s are exterior classes.

5.4 Example for equality

As an immediate consequence of the preceeding lemma we can describe the structure of Ĥ^{∗}(G) more explicitly in the range we will be interested in.

Corollary 5.30. The following table describes the k-vector spaces Ĥ^{i}(G) in the range −2q + 1 ≤ i ≤ 2q − 2:

deg −2q + 1 −q − 1  −q  −1 0 q − 1  q  2q − 2
basis φ_{2p−3,2p−3} φ_{2p−2,0} φ_{2p−3,0} φ_{00} 1 v_{1}^{p−2}u_{1} v_{1}^{p−1} v_{1}^{p−2}v_{2}^{p−2}u_{1}u_{2}
φ_{0,2p−2} φ_{0,2p−3}  v_{2}^{p−2}u_{2} v_{2}^{p−1}

For all other values of i in the given range we have Ĥ^{i}(G) = 0.

Theorem 5.31. Let ζ = φ_{2p−3,0} ∈ Ĥ^{−q}(G) and α = v_{1}^{p−1} ∈ Ĥ^{q}(G). Then we have the (2p − 1)-fold Massey product ⟨ζ, α, ζ, …, α, ζ⟩ = −φ_{2p−3,2p−3} without indeterminacy. In particular, ζ-ord(k/ζ) = p − 2 by Corollaries 2.8 and 5.18.

One step in the proof is easy: we can compute −α^{p−1}P_{1}(ζ), which is one element of the Massey product due to Proposition 5.24. We know P_{1}(ζ) = φ_{2p^{2}−2p−1,2p−3} from Propositions 3.54 and 3.56, and hence

α^{p−1}P_{1}(ζ) = v_{1}^{p^{2}−2p+1}φ_{2p^{2}−2p−1,2p−3} = φ_{2p−3,2p−3}.

The statement about the indeterminacy is slightly more complicated. Let us recall some facts we will frequently use in the proof.

Lemma 5.32 (May, [20]). Suppose that the Massey product ⟨a_{1}, …, a_{n}⟩ is strictly defined.

(i) For every i = 1, 2, …, n−1 and every x_{i} ∈ Ĥ^{∗}(G) of degree |a_{i}|+|a_{i+1}|−1 the Massey product ⟨a_{1}, …, a_{i−1}, x_{i}, a_{i+2}, …, a_{n}⟩ is strictly defined ([20], Proposition 2.4.(i)).

(ii) If every Massey product as in (i) consists only of the zero element, then ⟨a_{1}, …, a_{n}⟩ has no indeterminacy ([20], Proposition 2.4.(ii)).

(iii) If a_{0} · ⟨a_{1}, …, a_{n}⟩ contains 0, then so does ⟨a_{0}a_{1}, …, a_{n}⟩ (follows from part (ii) of Corollary 3.2 in [20]).

Also, for arbitrary elements a_{1}, …, a_{n}:

(iv) If ⟨a_{1}, …, a_{i−1}, a_{i}a_{i+1}, a_{i+2}, …, a_{n}⟩ = {0} and ⟨a_{1}, …, a_{i}, a_{i+1}a_{i+2}, a_{i+3}, …, a_{n}⟩ is defined, then the latter contains 0 (this follows from Corollary 3.4.(iii) in [20]).

Remark 5.33. Notice that May uses a different sign convention, so we formulated the facts in a sign-independent form we need later.

Let us now take a look at the sequences of the degrees of elements coming up in the Massey products we are interested in. Define S to be the smallest set of integer sequences of finite length with the following properties: the sequence (−q, q, −q, …, q, −q) of length 2p − 1 belongs to S, and whenever a sequence (d_{1}, …, d_{n}) of length n ≥ 3 lies in S then so does (d_{1}, …, d_{i−1}, d_{i} + d_{i+1} − 1, d_{i+2}, …, d_{n}) for every i = 1, …, n − 1. This notion is clearly motivated by Part (i) of Lemma 5.32. For example, in the case p = 3 the set S will consist of the following elements:

[figure: (−4,4,−4,4,−4); (−1,−4,4,−4); (−4,−1,4,−4); (−4,4,−1,−4); (−4,4,−4,−1); (−6,4,−4); (−1,−1,−4); (−1,−4,−1); (−4,2,−4); (−4,−1,−1); (−4,4,−6); (−3,−4); (−6,−1); (−1,−6); (−4,−3)]

The occurring numbers are

q, q − 2, q − 4, …, 2, −1, −3, …, −q + 1, −q, −q − 2, …, −2q + 2.  (5.34)

Let us describe without proof some of the sequences in S more explicitly.

Lemma 5.35. A sequence (d_{1}, …, d_{n}) with 2 ≤ n ≤ 2p − 1 and d_{i} ∈ {q, −1, −q} for all i belongs to S if and only if it is of the form

(\overline{−1}, −q, \overline{−1}, q, \overline{−1}, −q, …, −q, \overline{−1})

where each \overline{−1} stands for an arbitrary long (possibly empty) sequence of (−1)’s, and the total number of (−1)’s is 2p − 1 − n.

We call a Massey product ⟨a_{1}, …, a_{n}⟩ in Ĥ^{∗}(G) admissible if the sequence of the degrees of its elements (|a_{1}|, …, |a_{n}|) belongs to S and for every i = 1, …, n, a_{i} = ζ if |a_{i}| = −q and a_{i} = α if |a_{i}| = q.

Lemma 5.36. Every admissible Massey product is strictly defined.

Proof. By Lemma 5.32.(i) it is sufficient to show that the Massey product ⟨a_{1}, …, a_{2p−1}⟩ with

a_{i} = \begin{cases}ζ & if i is odd, \\ α & if i is even \end{cases}

is strictly defined. The two-fold products ⟨a_{i}, a_{i+1}⟩ = a_{i}a_{i+1} vanish due to the multiplicative relations in Ĥ^{∗}(ℤ/pℤ × ℤ/pℤ). Now we can assume that p ≥ 3. The three-fold products ⟨ζ, α, ζ⟩ and ⟨α, ζ, α⟩ are strictly defined and have no indeterminacy, and by Corollary 5.18 they contain zero.

Suppose we have some product ⟨a_{i}, …, a_{j}⟩ of length n ≥ 4. If n is even, then the product is of degree −n + 2 with 4 ≤ n ≤ 2p − 2 = q; but then Ĥ^{−n+2}(G) = 0 by Corollary 5.30. If n is odd, then the degree of the product is ±q−n+2 with 5 ≤ n ≤ 2p−3, but then Ĥ^{±q−n+2}(G) = 0.

Proposition 5.37. Suppose that ⟨a_{1}, a_{2}, …, a_{n}⟩ is an admissible Massey product. Then it does not have any indeterminacy. If n < 2p − 1, then ⟨a_{1}, …, a_{n}⟩ = {0}.

6. A counterexample

Proof. Let n be the smallest number for which any of the two statement fails. Suppose the first statement fails for that n. Then n ≥ 3 and by minimality of n we know that all products of the form ⟨a_{1}, …, a_{i−1}, x_{i}, a_{i+2}, …, a_{n}⟩ with x_{i} of degree |a_{i}| + |a_{i+1}| − 1 equal {0}, because they are admissible (note here that x_{i} is not of degree ±q). By Lemma 5.32.(ii) we know that ⟨a_{1}, …, a_{n}⟩ does not have any indeterminacy.

We can therefore assume that for the minimal n the second statement fails, whereas the first one is true. Suppose that \underline{a} = ⟨a_{1}, …, a_{n}⟩ is admissible; we need to show that 0 ∈ \underline{a}. If there is some index i with |a_{i}| ∉ \{q, −1, −q\} then we can deduce from the list (5.34) and the table in Corollary 5.30 that Ĥ^{|a_{i}|}(G) = 0 and therefore a_{i} = 0. Since \underline{a} is strictly defined, it must contain 0 in this case. We can therefore assume that all Massey products \underline{a} of length n with \underline{a} ≠ \{0\} satisfy a_{i} ∈ \{q, −1, −q\} for all i. For every such Massey product there is a smallest index i with |a_{i}| = −1; denote that index by d(\underline{a}). Let us take one such product with minimal d(\underline{a}) = i.

Case 1: i > 1 and |a_{i−1}| = −q. Then a_{i−1} = ζ. By Tate duality we know that there is some b ∈ Ĥ^{q−1}(G) with a_{i} = bζ. Then, by the description in Lemma 5.35, the Massey product ⟨a_{1}, …, a_{i−2}, a_{i−1}b, ζ, a_{i+1}, …, a_{n}⟩ is admissible, and it equals {0} by minimality of d(\underline{a}). By Lemma 5.32.(iii) \underline{a} contains 0.

Case 2: i > 1 and |a_{i−1}| = q. Then a_{i−1} = α. Again by Tate duality there is some element b ∈ Ĥ^{−q−1}(G) with a_{i} = bα. The product ⟨a_{1}, …, a_{i−2}, a_{i−1}b, α, a_{i+1}, …, a_{n}⟩ is admissible and equals \{0\}. As before we can deduce that \underline{a} contains 0.

Case 3: i = 1. Then we know by Lemma 5.35 that the product ⟨ζ, α, a_{2}, a_{3}, …, a_{n}⟩ is admissible and hence strictly defined. In particular, ⟨α, a_{2}, …, a_{n}⟩ = {0}. By Tate duality we can write a_{1} = bα for some class b ∈ Ĥ^{−q−1}(G). Then b · ⟨α, a_{2}, …, a_{n}⟩ contains 0, and by Lemma 5.32.(iii) we get that 0 ∈ \underline{a}.

6 A counterexample

We shall now show that the lower bound of our Main Theorem (as given in Corollary 2.8) is no longer true in general if we allow ζ to be an arbitrary element of even degree in the graded center of \underline{𝔪𝔬𝔡}-kG. Let p = 3 and G = ℤ/pℤ × ℤ/pℤ. Let k be a field of characteristic 3; then kG ≅ A = k[x, y]/(x^{3}, y^{3}), and from now on we will work with A instead of kG. Since A is commutative, the enveloping algebra of A is simply given by A^{e} = A ⊗ A, which we also denote by B. Define the elements x̄ = 1 ⊗ x − x ⊗ 1 and ȳ = 1 ⊗ y − y ⊗ 1 in B.

Lemma 6.1. The beginning of a projective resolution of A as a right B-module is given by

A ⟵^{ϵ} B ⟵^{( x̄ ȳ)} B^{⊕2} \underleftarrow{(\begin{matrix}x̄^{2} & ȳ \\ & −x̄ & ȳ^{2} \end{matrix} )} B^{⊕3} ← …

where ϵ is the multiplication of A. In particular, we can choose Ω_{B}^{2}A to be the submodule of B^{⊕2} generated by the three elements ( \begin{matrix} & x̄^{2} \\ 0 \end{matrix} ), ( ȳ ; −x̄ ) and ( \begin{matrix}0 \\ & ȳ^{2} \end{matrix} ).

Proof. Consider the algebra L = k[z]/(z^{3}) and let z̄ = 1 ⊗ z − z ⊗ 1 ∈ L^{e}. Simple combinatorial arguments show that

L ⟵^{ϵ} L^{e} ⟵^{z̄} L^{e} ⟵^{z̄^{2}} L^{e} ⟵^{z̄} L^{e} ⟵^{z̄^{2}} …,

with ϵ being the multiplication of L, is a free resolution of L by L^{e}-modules. Tensoring two such complexes and using the Künneth theorem, we get the desired result.

If M is any A-module, we can apply the functor M ⊗_{A} − to the projective resolution of the proposition and obtain a projective resolution of M. In particular, when we put M = k the trivial A-module, we get the resolution

k ⟵^{ϵ} A ⟵^{( x y)} A^{⊕2} \underleftarrow{(\begin{matrix}x^{2} & y \\ & −x & y^{2} \end{matrix} )} A^{⊕3} ← …

Let this resolution define the modules Ω^{n}k for n ≥ 0. For a better understanding, let us draw pictures of these modules as follows: We take a suitable basis of the module as a kvector space (drawn as •) and draw vertical and horizontal lines to indicate multiplication by x and y, respectively:

[figure: •; x; y; •; •; •; •; •; •; •]

This picture represents Ωk = ⟨x, y⟩_{A} ⊂ A. Whenever there is no horizontal line from a • to the right, we mean that the corresponding element vanishes under multiplication by x (and similarly downwards for y). Note that there are A-modules not representable in this form, but all the modules we are interested in will be. Let us draw two more examples:

k as a trivial module: •

Ω^{2}k, considered as a submodule of A^{⊕2}:  (\begin{matrix}x^{2} \\ 0 \end{matrix})_{(\begin{matrix}y \\ −x \end{matrix})} • • • • •_{(\begin{matrix}0 \\ −y^{2} \end{matrix})} • •

Now let ξ : Ω^{2}k → k be the following map:

[figure: •; •; •; •; •; •; •; •; •; •; •]

That is, ( \begin{matrix}y \\ −x \end{matrix} ) ↦ 1 ∈ k and all other drawn k-basis elements map to 0. Let L_{ξ} be the kernel of ξ, which can be drawn like this:

b_{4} b_{3} • b_{2} • • b_{1} • •

It is the A-submodule of A^{⊕2} spanned by the four vectors b_{1} = ( \begin{matrix}0 \\ −y^{2} \end{matrix} ), b_{2} = ( \begin{matrix}y^{2} \\ −xy \end{matrix} ), b_{3} = ( \begin{matrix}xy \\ −x^{2} \end{matrix} ) and b_{4} = ( \begin{matrix}x^{2} \\ 0 \end{matrix} ). The map ξ induces an element − ⊗ ξ of degree 2 in the graded center Z(\underline{𝔪𝔬𝔡}-A).

Let us turn back to the projective resolution of A as a B-module given in Lemma 6.1. Note that the elements x̄ und ȳ belong to the augmentation ideal; we therefore have constructed the beginning of a minimal resolution, and applying the functor Hom_{B}(−, A) to the resolution yields a complex A ⟶^{0} A^{⊕2} ⟶^{0} A^{⊕3} ⟶^{0} …; in particular ĤH^{2}(A) ≅ A^{⊕3}. More explicitly, in order to specify a B-linear map φ : Ω_{B}^{2}A → A we can freely choose the values of φ under the three generators ( \begin{matrix} & x̄^{2} \\ 0 \end{matrix} ), ( ȳ ; −x̄ ) and ( \begin{matrix}0 \\ & ȳ^{2} \end{matrix} ). Let us put

φ : Ω_{B}^{2}A → A
( \begin{matrix} & x̄^{2} \\ 0 \end{matrix} ) ↦ x,
( ȳ ; −x̄ ) ↦ 0,
( \begin{matrix}0 \\  & ȳ^{2} \end{matrix} ) ↦ 0.

This map induces an element − ⊗_{A} φ of degree 2 in the graded center Z(\underline{𝔪𝔬𝔡}-A).

Proposition 6.2. The element ζ = φ + ξ ∈ Z^{2}(\underline{𝔪𝔬𝔡}-A) satisfies ζ-ord(k/ζ) = 0.

Proof. Since x acts trivially on k, we get that φ_{k} = 0, so ζ_{k} = ξ_{k}. Hence k/ζ ≅ k/ξ ≅ Ω^{−1}L_{ξ}. It is enough to show that ζ_{L_{ξ}} ≠ 0. Since ξ comes from a Tate cohomology class, we know that ξ_{L_{ξ}} = 0 by Corollary 2.8, so it remains to prove that φ_{L_{ξ}} ≠ 0.

For this we need two easy but powerful lemmas. Suppose that M is a right A-module and m ∈ M. We say that m is divisible by x if there is some element m′ ∈ M with m = m′ · x, and similarly for y. Define I to be the ideal of A generated by the elements x^{2}, xy and y^{2}.

Lemma 6.3. Let F be a free A-module. Suppose that m ∈ F has the property that m · x is divisible by y and m · y is divisible by x. Then m ∈ F · I ⊂ F.

Proof. One can assume that F is free of rank 1. The set of elements m with m · x being divisible by y is the ideal generated by x^{2} and y; the set of elements m with m · y being divisible by x is the ideal generated by x and y^{2}. The intersection of these two ideals is I.

Lemma 6.4. Suppose that f : M → N is a map of right A-modules which factors through a projective module. Assume further that m ∈ M is such that m · x is divisible by y and m · y is divisible by x. Then f(m) ∈ N · I ⊂ N.

Proof. Because A is the group algebra of a 3-group, all projective modules are free. Write f as a composition M ⟶^{g} F ⟶^{h} N with a free module F. Then g(m) satisfies the conditions of Lemma 6.3; therefore g(m) ∈ F · I and hence f(m) = h(g(m)) ∈ N · I.

We want to apply this to φ acting on L_{ξ}, i.e., the map

L_{ξ} ⊗_{A} Ω_{B}^{2}A → L_{ξ} ⊗_{A} A ≅ L_{ξ}.

Let e_{1}, e_{2} ∈ B^{⊕2} be given by e_{1} = ( \begin{matrix}1 \\ 0 \end{matrix} ) and e_{2} = ( \begin{matrix}0 \\ 1 \end{matrix} ). Define the elements f_{i} = b_{i−1} ⊗ e_{1} − b_{i} ⊗ e_{2} ∈ L_{ξ} ⊗_{A} B^{⊕2} for i = 1, …, 5 (where by convention b_{0} = 0 and b_{5} = 0). When we apply the differential d : L_{ξ} ⊗_{A} B^{⊕2} → L_{ξ} ⊗_{A} B to f_{i} we get

df_{i} = b_{i−1} ⊗ x̄ − b_{i} ⊗ ȳ
= b_{i−1} ⊗ (1 ⊗ x) − b_{i−1}x ⊗ (1 ⊗ 1) − b_{i} ⊗ (1 ⊗ y) + b_{i}y ⊗ (1 ⊗ 1)
= b_{i−1} ⊗ (1 ⊗ x) − b_{i} ⊗ (1 ⊗ y);

here we used b_{i−1}x = b_{i}y.

Now consider b′_{i} = f_{i+1}x^{2} + f_{i+2}xy + f_{i+3}y^{2} (where f_{i} = 0 for i > 5). From the computation above we get db′_{i} = 0, so b′_{i} ∈ L_{ξ} ⊗_{A} Ω_{B}^{2}A, and we have b′_{2}y = b′_{1}x and b′_{2}x = b′_{3}y. Therefore, b′_{2} satisfies the conditions of Lemma 6.4, and it remains to show that the image of b′_{2} does not lie inside L_{ξ} · I. A rather lengthy computation shows that

b′_{2} = b_{4} ⊗ ( ȳ ; −x̄ )y + b_{3} ⊗ ( ȳ ; −x̄ )x + b_{3}x ⊗ ( ȳ ; −x̄ ) + b_{2} ⊗ ( \begin{matrix} & x̄^{2} \\ 0 \end{matrix} ).

Therefore, under φ, b′_{2} is mapped to b_{2} · x ∈ L_{ξ} which is not inside L_{ξ} · I.

Remark 6.5. The last step of the proof might become clearer if we draw the map φ_{L_{ξ}} using the diagrams introduced above. It turns out that, up to a stable isomorphism in the source, the map φ : L_{ξ} ⊗_{A} Ω_{B}^{2}A → L_{ξ} can be viewed as follows:

[figure: •; •; •; ∘; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •; •]

The point marked ∘ satisfies the conditions of Lemma 6.4, thus showing that the indicated map is non-trivial in the stable category.

Remark 6.6. The element ζ of the graded center comes from the Hochschild cohomology of A ≅ kG. Since G is abelian, we know that HH^{∗}(G) ≅ H^{∗}(G) ⊗ A (see [9], or [5] for a more general result). Using the description of H^{∗}G given in §3.6, the element ζ comes from u_{1}u_{2} + v_{1} · x ∈ HH^{∗}(G).

\underline{References}

References

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List of Symbols

Greek letters γ in §4 and §5 a certain map Ω^{n}k ⊗ S^{p−1}Ω^{n}k → Ω^{p−2}Λ^{p}Ω^{n}k, page 56 Δ may refer to

comultiplication maps of the coalgebras S^{∗}X and Λ^{∗}X, page 12 maps Δ_{i} arising in the construction of Steenrod powers, page 39

ζ may refer to an element in the graded center of a triangulated category, page 9 an unstable map Ω^{n}k → k, page 16

η an unstable representative of the stable map Ωk → L_{ζ}, page 16 ι the inclusion L_{ζ} → Ω^{n}k, page 16 κ_{i}^{f} the i-th map in the Koszul complex associated to the map f, page 12 Λ^{n}X the exterior power of a module X, page 12 λ(X, Y) a certain map associated to a coherent module (X, Y), page 61 μ multiplication maps of the algebras S^{∗}X and Λ^{∗}X, page 12 Σ shift functor in \underline{𝔪𝔬𝔡}-kG, page 6 Σ_{n} the symmetric group on n letters, page 11 σ_{X}, σ′_{X} maps associated to a filtered object X, page 64 Φ(X, Y) a certain map associated to a coherent module (X, Y), page 59 Ω inverse of the shift functor of \underline{𝔪𝔬𝔡}-kG, page 6 ω_{j} a certain stable isomorphism Λ^{j}L_{ζ} → Ω^{j−1}L_{ζ}, page 56 Other expressions D_{i} certain predecessor of the power operation, page 30 Êxt Tate cohomology groups, page 7 F (K) the tautological coherent module over L_{ζ} ⊗ K, page 18 G a fixed finite group, page 6 ĤH^{∗} Tate version of Hochschild cohomology, page 9 \underline{Hom} morphisms in the stable category, page 6 Ĥ^{∗}(G) Tate cohomology algebra, page 7 k may refer to

the ground field, of characteristic p, page 6 the trivial kG-module k, unit of ⊗, page 6 K_{ac}(inj−kG) the homotopy category of unbounded acyclic chain complexes of injective

kG-modules, page 65

L_{ζ} the kernel of ζ : Ω^{n}k → k, page 16 𝔪𝔬𝔡-kG the category of finitely generated, right kG-modules, page 6 \underline{𝔪𝔬𝔡}-kG the stable module category, page 6 N may refer to

the sum of cyclic permutations, page 11

the normal p-Sylow subgroup of Aff_{1} × Aff_{1}, page 78

ord the order of an object, page 10 p a prime, the characteristic of the ground field k, page 6 PHom morphisms which factor through a projective, page 6 P_{1} power operation on Tate cohomology, Definition 3.35, properties on page 21 RK de Rham-Koszul complex, page 28 S Steenrod complex, page 28 S^{n}X the symmetric power of a module X, page 12 T may refer to

the cyclic permutation map on an object of the form X^{⊗p}, page 11 the normal cyclic subgroup of translations inside Aff_{1}, page 78

Z(C) graded center of a triangulated category C, page 8 Further symbols

[figure: o] a map of degree 1, page 64 ⟨X⟩_{k} the k-vector space generated by a set X, page 12 ≅_{st} stable isomorphism, page 6 ⟨a_{1}, …, a_{n}⟩ a Massey product (page 63) or a Toda bracket (page 64) X/ζ some cone of the map ζ on X, page 10 X^{♯} the dual of X, page 6

[m] may refer to shift of a cochain complex, page 11 shifted coherent module, page 19 m-fold iterate of the shift functor in a triangulated category, page 64

(−1)^{σ} signum of the permutation σ, page 11 ⊗ = ⊗_{k} the tensor product of modules, page 6 X^{⊗n} the n-fold tensor product of X with itself, page 11