DYER-LASHOF OPERATIONS ON TATE COHOMOLOGY OF FINITE GROUPS

MARTIN LANGER

Abstract. Let k = 𝔽_{p} be the field with p > 0 elements, and let G be a finite group. By exhibiting an E_{∞}-operad action on Hom(P, k) for a complete projective resolution P of the trivial kG-module k, we obtain power operations of Dyer-Lashof type on Tate cohomology Ĥ^{∗}(G; k). Our operations agree with the usual Steenrod operations on ordinary cohomology H^{∗}(G). We show that they are compatible (in a suitable sense) with products of groups, and (in certain cases) with the Evens norm map. These theorems provide tools for explicit computations of the operations for small groups G. We also show that the operations in negative degree are non-trivial.

As an application, we prove that at the prime 2 these operations can be used to determine whether a Tate cohomology class is productive (in the sense of Carlson) or not.

1. Introduction

Let k = 𝔽_{p} be the field with p elements. For every finite group G, let Ĥ^{∗}(G) = Ĥ^{∗}(G, k) denote the graded Tate cohomology algebra of G over k. Then Ĥ^{∗} is functorial with respect to injective group homomorphisms. The starting point of our discussion will be the following Theorem.

Theorem 1.1. There is a family of k-linear operations Q_{s} (and βQ_{s} for p ≥ 3) for all integers s on Tate cohomology Ĥ^{∗}, satisfying the following properties.

(1) The operations Q_{s} are natural with respect to injective group homomorphisms.

(2) The operation Q_{s} lowers the degree by 2s(p − 1) (by s if p = 2), and βQ_{s} lowers the degree by 2s(p − 1) − 1 for p > 2.

(3) Q_{s}(x) = 0 if 2s < −(p − 1)|x| (if s < −|x| for p = 2).

(4) If p > 2, then βQ_{s}(x) = 0 if 2s ≤ −(p − 1)|x|.

(5) Q_{s}(x) = x^{p} if 2s = −(p − 1)|x| (if s = −|x| for p = 2).

(6) Q_{s}(1) = 0 unless s ≠ 0, where 1 ∈ Ĥ^{0}(G) is the unit element.

(7) The internal Cartan formula holds:

Q_{s}(xy) = ∑_{i+j=s} Q_{i}(x)Q_{j}(y),

βQ_{s}(xy) = ∑_{i+j=s} βQ_{i}(x)Q_{j}(y) + (−1)^{|x|}Q_{i}(x)βQ_{j}(y)  for all x, y ∈ Ĥ^{∗}(G).

(8) The Adem relations hold: For r > ps,

Q_{r}Q_{s} = ∑_{i}(−1)^{r+i}(pi − r, r − (p − 1)s − i − 1)Q_{r+s−i}Q_{i}

and for r ≥ ps and p > 2
Q_{r}βQ_{s} = ∑_{i}(−1)^{r+i}(pi − r, r − (p − 1)s − i)βQ_{r+s−i}Q_{i}

− ∑_{i}(−1)^{r+i}(pi − r − 1, r − (p − 1)s − i)Q_{r+s−i}βQ_{i}.

Here the convention is that (a, b) = 0 if a or b is negative, and (a, b) = \binom{a+b}{b} otherwise.

(9) On classes of non-negative degree, the operations agree with the Steenrod operations on H^{∗}(BG; k) = H^{∗}(G). More precisely, for |x| ≥ 0 we have

Q_{−n}(x) = Sq^{n}(x)  for p = 2 and n ≥ 0,
Q_{−n}(x) = P^{n}(x), βQ_{−n}(x) = βP^{n}(x)  for p > 2 and n ≥ 0,
Q_{−n}(x) = 0, βQ_{−n}(x) = 0  for n < 0.

We define the total operation Q = ∑_{i} Q_{i}; then the Cartan formula reads Q(xy) = Q(x)Q(y) for all x, y. We will sometimes use the notation P_{i}(x) = Q_{i−|x|}(x), so that P_{i}(x) = 0 for all i < 0 and P_{0}(x) = x^{p}.

Example 1.2. Let p = 2, and let G = ℤ/2ℤ be the cyclic group of order 2. We can easily compute all the operations on Ĥ^{∗}(G) using the statements of the theorem only. It is known that Ĥ^{∗}(G) ≅ k[s^{±1}] for the unique non-zero class s of degree 1 (see [5], XII.§7). We know that Q(s) = s + s^{2}, so that 1 = Q(1) = Q(s^{−1}s) = Q(s^{−1})(s + s^{2}). Using the fact that Q(s^{−1}) = s^{−2} + (terms of degree less than −2) we obtain

Q(s^{−1}) = s^{−2} + s^{−3} + s^{−4} + ….

More generally we get for all integers i

Q(s^{i}) = (s + s^{2})^{i} = s^{2i}(s^{−1} + 1)^{i} = ∑_{j≥0} \binom{i}{j}s^{2i−j},

so that Q_{j−i}(s^{i}) = \binom{i}{j}s^{2i−j} for all j ≥ 0. Here we use the generalized binomial coefficient

\binom{i}{j} = \frac{i(i − 1) … (i − j + 1)}{j!} for integers i, j with j ≥ 0.

Example 1.3. Slightly more complicated, but still an immediate consequence of the theorem is the case G = ℤ/pℤ for odd primes p. Here Ĥ^{∗}(G) ≅ k[s^{±1}] ⊗ Λ(u), where s is of degree 2 and u is exterior of degree 1. Let us define βQ = ∑_{i} βQ_{i}; then from the topological fact β(u) = s we get for integers i

Q(s^{i}) = ∑_{j≥0} \binom{i}{j}s^{pi−j},  Q(s^{i}u) = Q(s^{i})u,
βQ(s^{i}) = 0,  βQ(s^{i}u) = Q(s^{i})s.

Example 1.4. Let us do an example of a non-commutative group. Let G = Q_{8} be the quaternion group with 8 elements. Then it is known that Ĥ^{∗}(G) ≅ k[s^{±1}, x, y]/(x^{2} + xy + y^{2}, x^{3}) with degrees |x| = |y| = 1 and |s| = 4. We immediately get Q(x) = x + x^{2} and Q(y) = y + y^{2}. Every automorphism of H^{1}(G) is realized by a group automorphism; this implies that Sq^{1}(s) = 0 and Sq^{2}(s) = 0. From the Adem relation Sq^{3}(s) = Sq^{1} Sq^{2}(s) it then follows that Q(s) = s + s^{2}. By the same methods as above, one easily deduces the operations on all of Ĥ^{∗}(G).

Remark 1.5. We will prove Theorem 1.1 by establishing an E_{∞}-operad action on Hom_{kG}(P, k), the cochains of a complete projective resolution P of the trivial kG-module k. There is another way of constructing Dyer-Lashof operations on Tate cohomology, using equivariant stable homotopy theory as follows. In the homotopy category of G-spectra [12] let k_{G} = H𝔽_{p} denote the Eilenberg-MacLane spectrum, regarded as a G-spectrum with ’trivial G-action’. Associated with k_{G} there is a Tate spectrum t = F (EG_{+}, k_{G}) ∧ ẼG (see [9]) with the property that Ĥ^{∗}(G; k) ≅ [S, t]_{G}^{∗} (see [8]). Then [14] shows that t is an E′_{∞}-ring spectrum, that is, we have a non-equivariant operad acting equivariantly, which can be used to define power operations on Ĥ^{∗}(G). The author does not know whether the operations defined in that topological manner agree with the rather algebraically defined operations of this paper.

From now on, assume that the order of G is divisible by p. Let us define a graded submodule M^{∗}(G) of a shift of Ĥ^{∗}(G) as follows:

M^{n}(G) = \begin{cases}Ĥ^{n−1}(G) & if n ≤ 0, \\ 0 & otherwise. \end{cases}

Then M^{∗}(G) inherits the Dyer-Lashof operations from Ĥ^{∗}(G), because classes of negative degrees are mapped to classes of negative degrees (or to 0) by the Q_{i}. Via the identification M^{∗}(G) ≅ (Ĥ^{∗}(G)/H^{∗}(G))[1] we can also view M^{∗}(G) as a left H^{∗}(G)-module. For finite groups G_{1} and G_{2}, we have the Künneth isomorphism H^{∗}(G_{1} × G_{2}) ≅ H^{∗}(G_{1}) ⊗ H^{∗}(G_{2}) which is known to be an isomorphism of modules over the Steenrod algebra. We also have the isomorphism M^{∗}(G_{1} × G_{2}) ≅ M^{∗}(G_{1}) ⊗ M^{∗}(G_{2}) which is an isomorphism of modules over H^{∗}(G_{1} × G_{2}). Even more is true:

Theorem 1.6. For finite groups G_{1} and G_{2}, the Künneth isomorphism M^{∗}(G_{1} × G_{2}) ≅ M^{∗}(G_{1}) ⊗ M^{∗}(G_{2}) is an isomorphism of modules over the Dyer-Lashof algebra. In other words, Q(α ⊗ β) = Q(α) ⊗ Q(β) for all α ∈ M^{∗}(G_{1}) and β ∈ M^{∗}(G_{2}).

Example 1.7. Let us consider the case G = ℤ/2ℤ × ℤ/2ℤ at the prime p = 2. Let φ_{i} be a generator of M^{−i}(ℤ/2ℤ); it corresponds to s^{−i−1} ∈ Ĥ^{∗}(ℤ/2ℤ), but this notation suggests the existence of an internal product which we do not have on M^{∗}. Let us write φ_{ij} ∈ Ĥ^{−i−j−1}(G) for the element φ_{i} ⊗ φ_{j} ∈ M^{∗}(G); then Ĥ^{∗}(G) is the commutative graded algebra generated by polynomial classes x, y of degree 1 (coming from the two factors of G) and the classes φ_{ij} subject to the relations

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

φ_{ij}y = \begin{cases}φ_{i,j−1} & if j ≥ 1, \\ 0 & otherwise, \end{cases}
φ_{ij}φ_{i′j′} = 0.

The total square on φ_{0} is given by Q(φ_{0}) = φ_{1} + φ_{2} + …, see Example 1.2. By the theorem, the total square on φ_{0} ⊗ φ_{0} = φ_{00} ∈ Ĥ^{∗}(G) is given by Q(φ_{00}) = ∑_{i,j≥1} φ_{ij}. More generally we get the formula

Q(φ_{ij}) = ∑_{k,l≥0} \binom{k + i}{k}\binom{l + j}{j}φ_{2i+k+1,2j+l+1}.

In particular P_{0} = 0 and P_{1}(φ_{ij}) = φ_{2i+1,2j+1}.

In the same spirit we can prove:

Corollary 1.8. If a group G is a direct product of r groups of order divisible by p, then P_{j} vanishes on elements of negative degree for all j < \frac{r−1}{2} (for all j < r − 1 if p = 2).

Proof. Let G = G_{1}×⋯×G_{r}, and take an element of the form a = a_{1}⊗⋯⊗a_{r} with a_{i} ∈ Ĥ^{|a_{i}|}(G_{i}) and |a_{i}| < 0. Then |a| = |a_{1}| + ⋯ + |a_{r}| + r − 1. Now Q(a_{i}) is a sum of elements of degrees at most p|a_{i}|, and therefore Q(a) can be written as sum of elements b = b_{1} ⊗ ⋯ ⊗ b_{r} with |b_{i}| ≤ p|a_{i}|, so that

|b| = |b_{1}| + ⋯ + |b_{r}| + (r − 1) ≤ p|a_{1}| + ⋯ + p|a_{r}| + (r − 1) = p|a| − (p − 1)(r − 1).

This implies the result. □

Remark 1.9. Notice that, unlike the ordinary Steenrod operations [7], the operations Q_{i} are not compatible with transfers. For instance, if we embed K = ℤ/2ℤ × {0} ⊂ ℤ/2ℤ × ℤ/2ℤ = V, then the diagram

\xymatrix{Ĥ^{−1}(K) \ar[r]^{Q_{1}}_{≅} \ar[d]_{_{tr_{K,V}}} & Ĥ^{−2}(K) \ar[d]_{_{tr_{K,V}}}^{≠0} \\ Ĥ^{−1}(V) \ar[r]^{Q_{1}}_{=0} & Ĥ^{−2}(V)}

cannot commute.

1.1. Notations and conventions. Throughout the paper, p is a prime number and k = 𝔽_{p} is the prime field of characteristic p. Some of the results also hold for arbitrary fields of characteristic p, but then certain k-vector spaces have to be twisted by the Frobenius map. Groups labelled G, K, L are assumed to be finite. All modules are right modules, unless mentioned otherwise. We will mainly work in 𝔪𝔬𝔡-kG, the category of right kG-modules, with its tensor product ⊗ and internal Hom-object Hom_{k} obtained from the Hopf algebra structure on kG. We will use several known results about projective modules without further notice (e.g., projective is the same as injective, the tensor product of a projective and an arbitrary module is projective, and arbitrary products and sums of projectives are projective). The ground field k is considered as an object in 𝔪𝔬𝔡-kG by the trivial G-action. We denote by \underline{𝔪𝔬𝔡}-kG the stable module category, obtained from 𝔪𝔬𝔡-kG by dividing out those morphisms which factor through a projective module. Homomorphisms in \underline{𝔪𝔬𝔡}-kG between modules X, Y are denoted by \underline{Hom}(X, Y). The category \underline{𝔪𝔬𝔡}-kG is a triangulated category with shift functor Σ = Ω^{−1}, and Tate cohomology can be defined as Ĥ^{n}(G) = \underline{Hom}(Ω^{n}k, k), with the composition product as multiplication. A morphism X → Y in 𝔪𝔬𝔡-kG is called a stable equivalence if it induces an isomorphism in the stable category. See [4] for an introduction to the stable module category. Notice that, in this paper, we use the notation ⊗ for the internal tensor product ⊗_{k} of 𝔪𝔬𝔡-kG, but Hom is used for the k-vector space of kG-linear maps, that is, Hom = Hom_{kG}. Furthermore, the symbol ∂ is used for the differential of chain complexes over kG, whereas d often denotes the differential of cochain complexes over k.

1.2. Plan of the paper. In §2, we will construct the E_{∞} operad acting on Hom_{kG}(P, k) for a projective resolution P of the trivial kG-module k. We also compare the Dyer-Lashof operations obtained from that action with the usual Steenrod operations that we have on H^{∗}(G) ≅ H^{∗}(BG), thereby completing the proof of Theorem 1.1. In §3 we prove Theorem 1.6 about products of groups. In §4 we give a description of negative Tate Ext-groups in terms of complexes of projective modules, a tool we need for the proofs in the later sections. The duals of certain operations are shown to commute with the Evens norm map in §5, where we also show that our operations are non-trivial in negative degrees. Finally, in the last section we provide a criterion (Theorem 6.2) for Tate cohomology classes to be productive in the sense of Carlson.

1.3. Acknowledgements. Most of the paper evolved from parts of my PhD thesis written at the University of Bonn under supervision of Stefan Schwede. I would like to thank him for suggesting that project, and for his interest and helpful comments on this paper. Furthermore, I would like to thank Wolfgang Lück for the financial support.

2. The operad

2.1. Resolutions. Let k be a field of characteristic p, and let G be a finite group. Let M be a kG-module. A complete projective resolution of M is a long exact sequence of projective kG-modules

\xymatrix{… & P_{−2} \ar[l] & P_{−1} \ar[l] &  & P_{0} \ar[ll] \ar[dl]^{ε} & P_{1} \ar[l]^{∂_{1}} & P_{2} \ar[l] & … \ar[l] \\  &  &  & M \ar[ul]}

such that ε is the cokernel map of ∂_{1}. The map ε is called augmentation and can be viewed as a chain map ε : P → M, where M is regarded as a complex concentrated in degree 0. If N is another module, then a map ε′ : P_{0} → N (or, equivalently, a chain map ε′ : P → N) will be called quasi-augmentation if there is a stable equivalence f : M → N such that f ∘ ε = ε′. There is a dual notion using injectives. A complete injective resolution of M is a long exact sequence of injective kG-modules

\xymatrix{… & I_{−2} \ar[l] & I_{−1} \ar[l] & I_{0} \ar[l]^{∂_{0}} &  & I_{1} \ar[ll] \ar[dl] & I_{2} \ar[l] & … \ar[l] \\  &  &  &  & M \ar[ul]^{η}}

in which η is the inclusion of the kernel of ∂_{0}. The map η is called coaugmentation and can be viewed as a chain map η : M → I. If N is another module, then a map η′ : N → I will be called quasi-coaugmentation if there is a stable equivalence f : N → M with η ∘ f = η′. Since projectives are the same as injectives, the notions of complete resolutions only differ in the position of the resolved module M. If P is a complete projective resolution of the trivial module k, and N is another kG-module, then the cohomology groups of the complex Hom_{kG}(P, N) define the Tate cohomology of G with coefficients in N, that is, Ĥ^{n}(G; N) ≅ H^{n} Hom_{kG}(P, N).

Proposition 2.1. Let P and Q be complete projective resolutions, and let ϵ : P → M be a quasi-augmentation. If for some chain transformation f : Q → P the composite Q_{0} ⟶^{f_{0}} P_{0} ⟶^{ϵ} M is zero, then f is null-homotopic. The corresponding statement holds for injective resolutions.

We omit the straightforward proof. Whenever C is a cochain complex of kG-modules, we define the dual complex C^{∨} as (C^{∨})_{n} = Hom(C_{−n}, k) with the induced differentials. If P is a complete projective resolution of M with (quasi-)augmentation ϵ, then P^{∨} is a complete injective resolution of M^{∨} with (quasi)- coaugmentation ϵ^{∨}, and the same is true with the roles of projective and injective interchanged. Let k be the trivial kG-module, and choose complete injective resolutions I and I′ of k with coaugmentations η, η′. The tensor product I ⊗ I′ is defined to be the complex with modules (I ⊗ I′)_{n} = ⨁_{i+j=n} I_{i} ⊗ I′_{j} and differential ∂_{I⊗I′} = ∂_{I} ⊗ id + id ⊗∂_{I′} (note here that evaluation of the differential involves the usual sign, i.e., (id ⊗∂)(x ⊗ y) = (−1)^{|x|}x ⊗ ∂y). It is known that the tensor product I ⊗I′ is a complete injective resolution of k with quasi-coaugmentation η ⊗η′ (see [10], §8). Now let P and P′ be complete projective resolutions of k, and assume that all modules P_{i}, P′_{i} are finitely generated. Let us define a new tensor product P ⊠P′ = (P^{∨} ⊗P^{′∨})^{∨}; more explicitly, (P ⊠P′)_{n} = ∏_{i+j=n} P_{i}⊗P′_{j}. By the considerations above, this is a complete projective resolution of k with quasi-augmentation ε ⊠ ε′ : P ⊠ P′ → k. These definitions and observations can be generalized to ⊠-products of finitely many complete projective resolutions.

Remark 2.2. The ⊠-product can be used to define the multiplication on Tate cohomology. By usual homological algebra, the identity map on k can be lifted to a commutative diagram as follows:

[figure: P; Δ; ε; P ⊠ P; ε⊠ε; k; k]

Such a lift is unique up to homotopy. A more explicit construction of Δ is given in the proof of Theorem 4.1 in [5], XII, where it is also shown that Δ induces the Tate cohomology product in the following way: given cycles f, g ∈ Hom_{kG}^{∗}(P, k) we get a cycle (f ⊠ g) ∘ Δ ∈ Hom_{kG}^{∗}(P, k) representing [f] · [g].

2.2. Motivation for the definition of the operad. Let P be a complete projective resolution of k by finitely generated kG-modules. Before we start with the actual construction of an E_{∞}- structure on Hom_{kG}^{∗}(P, k), let us point out a major issue in the construction of power operations which does not turn up in the case of ordinary cohomology H^{∗}(G). For simplicity, let us assume that p = 2 throughout this motivational part. Let us naively transfer to Tate cohomology the construction of Sq_{1} as it is done in ordinary cohomology. We know that the identity map of k can be lifted to a map Δ : P → P^{⊠2} as in Remark 2.2, and any two such liftings are homotopic. Therefore, if T denotes the twist map of P^{⊠2}, then we know that (1 − T) ∘ Δ is the boundary of some map Δ_{1} : P → P^{⊠2} of degree −1. If ζ : P → k is a chain map of degree n representing some cohomology class [ζ] ∈ Ĥ^{n}(G), then we know that ζ^{⊠2} ∘ Δ_{1} is a chain map of degree 2n, and we could define Sq_{1}(ζ) to be the class represented by that map. The problem is here that there is an ambiguity in the choice of the map Δ_{1}, and any two such choices differ by a chain map P → P^{⊠2} of degree −1. Therefore, Sq_{1}(ζ) is only well-defined up to some element in ζ^{2} · Ĥ^{−1}(G). This problem does not occur in ordinary cohomology simply because H^{−1}(G) is zero. We therefore have to rigidify our choice of Δ_{1} in order to get actual operations. To do so, observe that a chain map P → P^{⊠2} of degree −1 certainly represents the zero class if the composite P_{−1} → (P^{⊠2})_{0} ↠ P_{0} ⊗ P_{0} → k vanishes, so that one possibility is to require the map P_{−1} → P_{0} ⊗ P_{0} to be zero. The next step is to elaborate this idea, and because we want an E_{∞}-structure, we need to do so in an ’operadic’ way.

2.3. Definition of the operad. As before, let P be a complete projective resolution of k by finitely generated kG-modules. We are now going to define an acyclic operad which acts on Hom_{kG}^{∗}(P, k). To do so, we will work in the category of (increasing degree) differential graded modules over k (or, equivalently, the category of cochain complexes of k-vector spaces) with its symmetric monoidal tensor product ⊗. Recall that if X and Y are chain complexes of kG-modules with differential ∂, then we get such a differential graded module Hom^{∗}(X, Y) by defining

Hom^{n}(X, Y) = ∏_{j∈ℤ} Hom_{kG}(X_{n+j}, Y_{j})

with differential d(f) = ∂f − (−1)^{n}f∂. Let us recall some basics about operads; see, e.g., [11] for an introduction. A symmetric operad C is given by a differential graded module C(j) for every integer j ≥ 0 together with a Σ_{j}-action, equivariant structure maps

C(j) ⊗ C(i_{1}) ⊗ … C(i_{j}) → C(i_{1} + ⋯ + i_{j})

for all j, i_{1}, …, i_{j}, and a unit map k → C(j) for each j; all these maps have to satisfy certain coherence diagrams. A typical example of such an operad is the so-called coendomorphism-operad F(j) = Hom^{∗}(P, P^{⊠j}) for j ≥ 0, whose structure maps are given by

Hom(P, P^{⊠j}) ⊗ Hom(P, P^{⊠i_{1}}) ⊗ ⋯ ⊗ Hom(P, P^{⊠i_{j}}) → Hom(P, P^{⊠(i_{1}+···+i_{j})})
g ⊗ f_{1} ⊗ ⋯ ⊗ f_{j} ↦ (f_{1} ⊠ ⋯ ⊠ f_{j}) • g.

We have written • here because we want to stress that the Koszul sign rule also applies to this situation; whenever a and b are composable maps of certain degrees, we write a • b for (−1)^{|a||b|} · a ∘ b, so that expressions like b ⊗ a ↦ a • b indeed yield maps of chain complexes. The symmetric group Σ_{j} acts on P^{⊠j} by permutation of the factors (note that this also involves the usual signs), and we therefore get an action of Σ_{j} on Hom(P, P^{⊠j}). The unit map k → Hom(P, P) is given by the identity of P. The operad we are up to will be a sub-operad of the coendomorphism-operad F. An operad C is called unital if C(0) = k. In that case, the C(j) have augmentations coming from the operad structure maps

C(j) ≅ C(j) ⊗ C(0)^{j} → C(0) = k.

The operad is called acyclic if the augmentations are quasi-isomorphisms of chain complexes. An operad C is called an E_{∞}-operad if it is acyclic and for every j, C(j) is free as a kΣ_{j}-module. A differential graded module A is called a C-algebra if there are structure maps

C(j) ⊗ A^{j} → A

for every j ≥ 0 which are associative, unital and equivariant (see [11], §2 for details). Our goal is to define an acyclic operad C (and later an E_{∞}-operad) and a C-algebra structure on A = Hom^{∗}(P, k). This structure can then be used to define the operations Q_{i} on H^{∗}A ≅ Ĥ^{∗}(G), and also for proving most of Theorem 1.1. Let us begin with the definition of C. For every non-negative integer j, we define a differential graded submodule C(j) of Hom^{∗}(P, P^{⊠j}) as follows:

C(j)^{m} = 0  for m > 0,
C(j)^{0} = {f ∈ Hom^{0}(P, P^{⊠j}) | df = 0}
C(j)^{m} = \{f ∈ Hom^{m}(P, P^{⊠j}) | \begin{matrix}P_{i} ⟶^{proj∘f} P_{s_{1}} ⊗ P_{s_{2}} ⊗ ⋯ ⊗ P_{s_{j}} vanishes \\ for all i < 0 and all s_{1}, …, s_{j} ≥ 0 \end{matrix} \} for m < 0.

In order to check that C(j) is indeed a differential graded submodule, we have to prove dC(j)^{m} ⊆ C(j)^{m+1}. This is clear for m ≥ −1, and in case m < −1, the map P_{i} ⟶^{proj∘df} P_{s_{1}} ⊗⋯⊗P_{s_{j}} is the

sum of P_{i} ⟶^{∂} P_{i−1} ⟶^{proj∘f} P_{s_{1}} ⊗⋯⊗P_{s_{j}} and maps P_{i} ⟶^{proj∘f} P_{s_{1}} ⊗⋯⊗P_{s_{t}+1}⊗⋯⊗P_{s_{j}} ⟶^{id ⊗∂⊗id}

P_{s_{1}} ⊗ ⋯ ⊗ P_{s_{j}}, all of which are zero by assumption. Next we show that C is a sub-operad of the co-endomorphism operad F. In order to do so, we only need to show that it is closed under the structure maps, the Σ-action, and the unit. The latter two are immediate consequences of the definition, so let us take g ∈ C(j), f_{i} ∈ C(j_{i}) for i = 1, …, j and prove that (f_{1} ⊠ ⋯ ⊠ f_{j}) • g ∈ C(j_{1} + ⋯ + j_{j}). If one of the chosen elements is of positive degree, then the composition is zero. If all the chosen elements are of degree zero, then they are chain transformations and so is the composition. Now we can assume that the composition is of negative degree, and we have to show that the composite

P_{i} ⟶^{g} P_{s_{1}} ⊗ ⋯ ⊗ P_{s_{j}} ⟶^{f_{1}⊠···⊠f_{j}} P_{t_{1,1}} ⊗ ⋯ ⊗ P_{t_{1,j_{1}}} ⊗ ⋯ ⊗ P_{t_{j,1}} ⊗ ⋯ ⊗ P_{t_{j,j_{j}}}

is zero for all i < 0 and t_{l,n} ≥ 0. If s_{l} is negative, then P_{s_{l}} ⟶^{f_{l}} P_{t_{l,1}} ⊗ ⋯ ⊗ P_{t_{l,j_{l}}} vanishes and so does the composition. But if all the s_{l}’s are non-negative, then g is zero, so we are done. The operad C is unital, that is, C(0) is isomorphic to k concentrated in degree 0. Here we use the convention P^{⊠0} = k; then C(0)^{m} = 0 unless m = 0, in which case

C(0)^{0} = {f ∈ Hom^{0}(P, k) | df = 0} ≅ k ⟨ε⟩ .

So we get augmentations C(j) ≅ C(j) ⊗ C(0)^{j} → C(0) ≅ k given by postcomposition with ε^{⊠j}.

2.4. Acyclicity of the operad. We are now going to show that the augmentations C(j) → k are quasi-isomorphisms. To do so, let us consider another complete projective resolution Q of k, constructed as follows. Let us define P^{+} to be the non-negative part of P, that is P_{n}^{+} = P_{n} for n ≥ 0 with the induced differentials. Then k ⟵^{ε} P^{+} is an acyclic augmented complex, and by the Künneth theorem

k ⟵^{ε^{⊗j}} (P^{+})^{⊗j} is also acyclic. Next, we define a complex ⋯ ← R_{−2} ← R_{−1} ← R_{0} by setting R_{n} = P_{n} for n < 0 and R_{0} = k, the differential R_{−1} ← R_{0} being the coaugmentation of P. Then R is acyclic, and by the Künneth theorem R^{⊗j} is also acyclic. Note that (R^{⊗j})_{0} = k, so we can splice the complexes R^{⊗j} and k ← (P^{+})^{⊗j} to get a complex Q, which then is a complete projective resolution of k. There is a chain map Φ : P^{⊠j} → Q which in non-negative degrees is given by projections, and in negative degrees the maps

P_{s_{1}} ⊗ P_{s_{2}} ⊗ ⋯ ⊗ P_{s_{j}} → R_{s_{1}} ⊗ R_{s_{2}} ⊗ ⋯ ⊗ R_{s_{j}}

are zero unless all the s_{i}’s are non-positive, in which case the map is the tensor product of identity maps and the augmentation ε : P_{0} → R_{0} = k. Since the composition P^{⊠j} ⟶^{Φ} Q ⟶^{ε} k equals the quasi-augmentation ε^{⊠j}, we get that Φ is a chain homotopy equivalence. Therefore, the induced map λ : Hom^{∗}(P, P^{⊠j}) → Hom^{∗}(P, Q) is a quasi-isomorphism. Moreover, λ is surjective because Φ is levelwise onto.

Proposition 2.3. Suppose that λ : A → B is a surjective quasi-isomorphism of differential graded modules, and let C ⊆ B be a differential graded submodule of B. Then the restriction λ : λ^{−1}(C) → C is a quasi-isomorphism as well.

Proof. Let us denote by K the kernel of λ. Since λ is a quasi-isomorphism, the long exact sequence in homology implies that H^{∗}(K) = 0. Since K is also the kernel of λ |_{λ^{−1}(C)}, using the long exact sequence in homology again we get that the restriction of λ is a quasi-isomorphism. □

Now the idea is to choose a dg submodule C of Hom^{∗}(P, Q) quasi-isomorphic to k, and such that λ^{−1}(C) is (close to) our C(j). Define:

C^{m} = 0  for m > 0,
C^{0} = {f ∈ Hom^{0}(P, Q) | df = 0}
C^{m} = {f ∈ Hom^{m}(P, Q) | P_{i} ⟶^{f} Q_{j} is zero for all i < 0 ≤ j}  for m < 0.

Then C is indeed a dg submodule of Hom^{∗}(P, Q).

Proposition 2.4. We have H^{∗}(C) ≅ k.

Proof. Clearly, H^{m}(C) = 0 for m > 0. Let m < 0, and let f ∈ C^{m} be a cocycle. Define g : P_{i+m−1} → Q_{i} to be zero for all i = 0, 1, …, −m. By common homological algebra we can extend g to a chain null-homotopy for f (the conditions needed for the inductive construction of g is that ∂g∂ = f at the two boundary points of the domain on which g has been defined, and this condition is clearly satisfied). Then dg = f with g ∈ C, and hence H^{m}(C) = 0 for m > 0. Finally, we claim that the image of d : C^{−1} → C^{0} is the same as the image of d : Hom^{−1}(P, Q) → C^{0} (then it follows that H^{0}(C) ≅ H^{0}(G) ≅ k). Let f ∈ Hom^{−1}(P, Q); then the bottom row in the diagram

\xymatrix{P_{0} \ar[rr]^{∂} \ar[dr]_{ε} &  & P_{1} \ar[r]^{f} & Q_{0} \ar[dr]_{ε^{⊗j}} \ar[rr]^{∂} &  & Q_{−1} \\  & k \ar[rrr] \ar[ur]_{η} &  &  & k \ar[ur]}

is stably trivial and therefore the zero map (we assume here that |G| is divisible by p, which is the only interesting case). Therefore the upper row vanishes, and by usual homological algebra there is a cocycle g ∈ Hom^{−1}(P, Q) with f_{0} = g_{0} : P_{−1} → Q_{0}. Then f − g ∈ C^{−1} and d(f − g) = df, so we are done. □

We finally use a method of chopping off the positive part of a dg module. Given a dg module A, define F (A) to be the dg submodule given by

F (A)^{m} = \begin{cases}0 & if m > 0, \\ cycles of A^{0} & if m = 0, \\ A^{m} & if m < 0. \end{cases}

(This can be viewed as the (co)connected cover of A.) Then the inclusion F (A) ⊆ A induces an isomorphism H^{∗}(F (A)) ≅ H^{∗}(A) in non-positive degrees.

Proposition 2.5. The augmentation C(j) → k is a quasi-isomorphism. Thus, the operad C is acyclic.

Proof. Note that C(j) = F (λ^{−1}(C)), so that H^{∗}(C(j)) ≅ H^{∗}(C) ≅ k by Proposition 2.4. Since

there is a cocycle f ∈ C(j)^{0} such that P ⟶^{f} P^{⊠j} ⟶^{ε^{⊠j}} k equals the augmentation ε, the map C(j) → k is onto in H^{0} and therefore a quasi-isomorphism. □

For every operad A, the module A(0) is an algebra over A via the action map A(j) ⊗ A(0)^{⊗j} → A(0). In particular, Hom^{∗}(P, k) is an algebra over the co-endomorphism operad Hom^{∗}(P, P^{⊗j}), and we can restrict the operad action to the sub-operad C. Hence, Hom^{∗}(P, k) is a C-algebra.

Proposition 2.6. The operad C induces an E_{∞}-structure on Hom^{∗}(P, k) in such a way that the product on H^{∗} Hom^{∗}(P, k) agrees with the composition product of the Tate cohomology ring Ĥ^{∗}(G).

Proof. The operad C might itself not be Σ-free, so we have to choose an approximation of C by an E_{∞}-operad. One possible way of doing so is to choose an arbitrary E_{∞}-operad E′ and tensor its augmentation E′ → k with C. Then E = C ⊗ E′ is an E_{∞}-operad acting on Hom^{∗}(P, k) via the action of C pulled back along the morphism of operads E → C. The statement about the product follows from the fact that the element Δ ∈ C(2)^{0} ⊆ Hom^{0}(P, P ⊠P) given in Remark 2.2 generates the cohomology H^{0}(C(2)) and induces the right product on H^{∗} Hom^{∗}(P, k). □

2.5. Comparison with Steenrod reduced powers. For the proof of part (9) of Theorem 1.1 we need to recall the construction of Steenrod operations in the cohomology of cocommutative Hopf algebras. Let P̃ be an ordinary projective resolution of k, viewed as a complex ⋯ ← 0 ← P_{0} ← P_{1} ← . . .. Then P̃^{⊗j} is a projective resolution of k for all j. Consider the suboperad A(j) = F (Hom(P̃, P̃^{⊗j})) of the coendomorphism-operad Hom(P̃, P̃^{⊗j}). Then A is acyclic, and Hom(P̃, k) is an A-algebra in the obvious way. Using an E_{∞}-approximation of A, this operad action defines the Steenrod operations on H^{∗} Hom^{∗}(P̃, k) ≅ H^{∗}(G). Extend P̃ to a complete projective resolution P of k. We are now going to write down a quasiisomorphism of unital operads C → A. Let us begin with a function Ψ which maps an element f ∈ Hom^{∗}(P, P^{⊠j}) to the element in Hom^{∗}(P̃, P̃^{⊗j}) given by the composition

P̃ ⟶^{ι} P ⟶^{f} P^{⊠j} ⟶^{π^{⊠j}} P̃^{⊗j}.

Notice here that the inclusion map ι is not quite a chain map; its differential dι in Hom(P̃, P) is zero everywhere except for P̃_{0} → P_{−1}. On the other hand, the projection map π is a chain map, and therefore

d(πfι) = πd(f)ι ± πfd(ι)

in Hom^{∗}(P̃, P̃^{⊗j}). Now assume that f ∈ C(j); then either f is of non-negative degree, in which case πfd(ι) is zero (because π vanishes in negative degrees), or f is of negative degree, but then f is zero as maps P_{−1} → P_{s_{1}} ⊗P_{s_{2}} ⊗⋯⊗P_{s_{j}} for all s_{i} ≥ 0, and π is zero on all other factors of P^{⊠j} of interest. Hence d(Ψ(f)) = Ψ(df), so that Ψ restricted to C(j) is indeed a map of dg modules. We get a map Ψ : C → A of unital operads, and we need to show that Ψ commutes with the augmentations of C(j) and A(j). This follows from the following commutative diagrams:

\xymatrix{P̃ \ar[r]^{ι} \ar[dr]_{ε} & P \ar[d]^{ε} & P^{⊠j} \ar[dr]_{ε^{⊠j}} \ar[r]^{π^{⊠j}} & P̃^{⊗j} \ar[d]^{ε^{⊗j}} \\  & k &  & k}

Proof of Theorem 1.1. Everything except part (9) is a consequence of Proposition 2.6 and the fact that E_{∞}-structures can be used to construct power operations with the desired properties; see, e.g., I.§7 in [11], I.§1 in [6], and [13]. For part (9), note that by construction of the operations Q_{i} via C and the Steenrod operations via A we get the desired statement for n ≥ 0. To prove Q_{−n}(x) = 0 and βQ_{−n}(x) = 0 for n < 0 it is enough to notice that for elements f in C(p) we have that f : P_{neg} → P_{|x|}^{⊗p} vanishes. □

3. Products of groups

This section is devoted to the proof of Theorem 1.6. Let G be any finite group whose order is divisible by p. As a first step, we shall define a new operad action defining some power operations on M^{∗}(G). In the second step we prove that these operations agree with the Dyer-Lashof operations coming from Ĥ^{∗}(G). Let P be a complete projective resolution of k as a kG-module. We denote by \overline{P} the complex ⋯ ← P_{−2} ← P_{−1} ← 0 ← 0 ← …, with the P_{−1} sitting in degree 0 and with differential ∂\overline{P} = −∂_{P}, and let η : k → \overline{P} be the coaugmentation. For j ≥ 1 define the differential graded module B(j) = F (Hom^{∗}(\overline{P} , \overline{P}^{⊗j})). Also put B(0) = k; we want to turn B into a unital operad, so we need to define the structure maps

γ : B(j) ⊗ B(i_{1}) ⊗ ⋯ ⊗ B(i_{j}) → B(i_{1} + ⋯ + i_{j}).

As long as all i_{s}’s are positive, we simply take the usual structure maps of the coendomorphismoperad Hom^{∗}(\overline{P} , \overline{P}^{⊗j}). If one of the i_{s}’s is zero, then we put γ = 0 unless i_{1} = i_{2} = ⋯ = i_{j} = 0, in which case

γ : B(j) ⊗ B(0)^{⊗j} = B(j) → Hom_{kG}(k, k) = k = B(0) (3.1)

sends a chain map \overline{P} → \overline{P}^{⊗j} in B(j) to the induced map k → k^{⊗j} = k on zero-cycles. It is now straightforward to check that B is indeed a unital symmetric operad. Also, B is acyclic because by usual homological algebra the augmentations B(j) → k are quasi-isomorphisms. Now Hom^{∗}(\overline{P} , k) is a B-algebra, so we get Dyer-Lashof operations on H^{∗} Hom^{∗}(\overline{P} , k) ≅ M^{∗}(G) which we are now going to compare with those obtained from C. Let ι ∈ Hom^{1}(\overline{P} , P) be the inclusion, and let π ∈ Hom^{−1}(P, \overline{P}) be the projection map. Then dι = 0, but dπ ≠ 0. Let K be the cochain complex of k-vector spaces generated by an element x of degree −1 which is mapped by the differential to a non-trivial element y in degree 0:

[figure: …; 0; k ⟨y⟩; k ⟨x⟩; 0; …; y; x]

Let Y = K^{⊗p}, and then define the augmented cochain complex X by the formula X_{i} = Y_{i−1} for all i ≤ 0 with augmentation X_{0} → Y_{0} = k ⟨y^{p}⟩. Then X is an acyclic augmented complex of kΣ_{p}-modules. The map of cochain complexes K → Hom^{∗}(P, \overline{P}) given by x ↦ π induces a map of cochain complexes φ : Y → Hom^{∗}(P^{⊠p}, \overline{P}^{⊗p}). Let us define σ : X ⊗ C(p) → B(p) by the formula σ(v ⊗ f) = (−1)^{|f|}φ(v) ∘ f ∘ ι (the sign coming from shifting Y to X).

Proposition 3.2. The map σ : X ⊗ C(p) → B(p) enjoys the following properties:

(a) it is a Σ_{p}-equivariant cochain map lifting the identity of k,

(b) for every f ∈ C(p) we have σ(x^{p} ⊗ f) = (−1)^{|f|}π^{⊠p} ∘ f ∘ ι, and

(c) for every element w ∈ X ⊗ C(p) of bidegree (m, n) with m > −p + 1, and for every cocycle a ∈ Hom^{∗}(\overline{P} , k), we have γ(σ(w) ⊗ a^{⊗p}) = 0, where γ is the operad action of B on Hom^{∗}(\overline{P} , k).

Proof. (b) follows directly from the definition. To show (c), let w = w_{1} ⊗⋯⊗w_{p} ⊗f ∈ X ⊗C(p) with w_{i} ∈ {x, y} for all i. Up to a sign, γ(σ(w) ⊗ a^{⊗p}) is given by the composition

\overline{P} ⟶^{ι} P ⟶^{f} P^{⊠p} ⟶^{u_{1}⊠···⊠u_{p}} \overline{P}^{⊗p} ⟶^{a^{⊗p}} k,

where u_{i} = π if w_{i} = x and u_{i} = dπ if w_{i} = y. From the condition on the bidegree of w we know that at least one of the u_{i}’s equals dπ, so that a ∘ u_{i} = 0, which implies (c).

For (a), let f ∈ C(p) be a cocycle in degree 0 mapping to 1 under the augmentation C(p) → k; then consider the following diagram

\xymatrix{k \ar[rrr]^{η} \ar[ddd]_{η^{⊗p}} &  &  & P_{−1} \ar[ddd]^{f} \\  &  & P_{0} \ar[ull]_{ε} \ar[ur]^{∂} \ar[dl]^{f} \\  & (P^{⊠p})_{0} \ar[uul]_{ε^{⊠p}} \ar[dl]^{−(dπ)^{⊠p}} \ar[drr]^{∂} \\ (\overline{P}^{⊗p})_{0} &  &  & (P^{⊠p})_{−1} \ar[lll]^{π⊠(dπ)^{⊠(p−1)}}}

All smaller parts commute, and since P_{0} ⟶^{ε} k is surjective we can deduce that the exterior square commutes. Therefore, σ indeed lifts the identity of k. Also, σ is Σ_{p}-equivariant and is a cochain map because dι = 0, so (a) is proved. □

Proposition 3.3. The operad action of B on Hom^{∗}(\overline{P} , k) and the action of C on Hom^{∗}(P, k) define the same operations on M^{∗}(G).

Proof. Let ϵ : X → k[1 − p] be the kΣ_{p}-linear chain map given by x^{p} ↦ 1, and define τ = ϵ ⊗ id_{C} : X ⊗ C(p) → C(p)[1 − p]. By suitably shifting the action of C(p) on the negative part of Hom^{∗}(P, k) we get a map defined by

γ_{C} : C(p)[1 − p] ⊗ Hom^{∗}(\overline{P} , k)^{⊗p} → Hom^{∗}(\overline{P} , k)
f ⊗ w ↦ (−1)^{|f|·(|w|+1)}w ∘ π^{⊠p} ∘ f ∘ ι

for all f ∈ C(p)[1 − p] and w ∈ Hom^{∗}(\overline{P} , k)^{⊗p}. The sign is due to the Koszul sign rule, and the check that this is indeed a map of chain complexes uses the fact that a ∘ (dπ) = 0 for all a ∈ Hom(P, k). Now γ_{C} can be used to construct the power operations on M^{∗}(G) as follows. Let W be the standard free resolution of the trivial kC_{p}-module k (where C_{p} denotes the cyclic group of order p), so that W_{i} is generated by a single element e_{i}. Since X ⊗ C(p) is an acyclic augmented complex of kC_{p}-modules, we can lift the identity of k to a C_{p}-equivariant chain map ϑ : W → X ⊗ C(p). We then have a diagram like this:

\xymatrix{ &  & B(p) \\ W \ar[r]^{ϑ} & X ⊗ C(p) \ar@{-->}[ur]^{σ} \ar@{-->}[dr]_{τ} \\  &  & C(p)[1 − p]}

For cocycles a ∈ Hom^{∗}(\overline{P} , k), define D_{i}^{C}(a) to be the cohomology class of the cocycle γ_{C}(τϑ(e_{i})⊗ a^{p}) ∈ Hom^{∗}(\overline{P} , k), and define D_{i}^{B}(a) to be the class of γ_{B}(σϑ(e_{i}) ⊗ a^{p}) ∈ Hom^{∗}(\overline{P} , k). We need to show that D_{i}^{C} = D_{i}^{B}, and for this it suffices to prove the identity

γ_{B}(σ(w) ⊗ a^{p}) = γ_{C}(τ(w) ⊗ a^{p})

for all w ∈ X ⊗ C(p) and a ∈ Hom^{∗}(\overline{P} , k). We can write w = x^{p} ⊗ f + ∑_{i} u_{i} ⊗ f_{i} with u_{i} ∈ X of degree |u_{i}| > 1 − p, and f, f_{i} ∈ C(p). By Proposition 3.2 we have that

γ_{B}(σ(w) ⊗ a^{p}) = γ_{B}(σ(x^{p} ⊗ f) ⊗ a^{p}) = (−1)^{|f|}γ_{B}((π^{⊠p} ∘ f ∘ ι) ⊗ a^{p})

On the other hand, γ_{C}(τ(w) ⊗ a^{p}) = γ_{C}(f ⊗ a^{p}), and all these expressions equal (−1)^{|f|(|a^{p}|+1)} times the composition

\overline{P} ⟶^{ι} P ⟶^{f} P^{⊠p} ⟶^{π^{⊠p}} \overline{P}^{⊗p} ⟶^{a^{⊗p}} k.  □

Proof of Theorem 1.6. Let us write G = G_{1} ×G_{2}. Choose complete projective resolutions P and Q for k as trivial kG_{1}- and kG_{2}-module, respectively. Then k → \overline{P} ⊗ \overline{Q} is the negative part of a projective resolution of k as kG-module. We denote by B^{G_{1}}, B^{G_{2}} and B^{G} the operads constructed above using these resolutions; then we get a quasi-isomorphism of unital operads B^{G_{1}}⊗B^{G_{2}} → B^{G} by tensoring morphisms. Let us denote by A_{1} = Hom_{kG_{1}}^{∗}(\overline{P} , k), A_{2} = Hom_{kG_{2}}^{∗}(\overline{Q}, k) and A = Hom_{kG}^{∗}(\overline{P} ⊗ \overline{Q}, k) the corresponding B-algebras, then the commutative diagram

\xymatrix{B^{G_{1}}(p) ⊗ A_{1}^{p} ⊗ B^{G_{2}}(p) ⊗ A_{2}^{p} \ar[r] \ar[d] & A_{1} ⊗ A_{2} \ar[d] \\ B^{G}(p) ⊗ A^{p} \ar[r] & A}

implies the desired result. □

4. An alternative description of 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 B by A. We will now give a similar description of Êxt_{kG}^{−n}(A, B) ≅ \underline{Hom}_{kG}(A, Ω^{n}B), which will be used throughout the next two sections. 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). Let us fix a projective resolution of B:

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

Theorem 4.2. 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) which is natural in G.

To prove this, we need the following proposition.

Proposition 4.3. 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.

The proof is standard homological algebra, and we omit it.

Proof of Theorem 4.2: 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}∂ \\ 0 \end{matrix} )}; _{( \begin{matrix}id \\ ∂∘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 (4.1) 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 Proposition 4.3, 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. □

Example 4.4. 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 4.2, 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.

Proposition 4.5. 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 Proposition 4.3. □

Remark 4.6. 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).

Proposition 4.7. The composition products on K_{∗} and Êxt_{kG}^{−∗} coincide under the bijection of Theorem 4.2.

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 4.5, the left diagram shows that γ = βα. After splicing the two diagrams the result follows from Proposition 4.3. □

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:

Proposition 4.8. 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 4.5. □

Proposition 4.9. 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 4.5. □

Remark 4.10. There is a similar statement for extensions. Suppose that we have a diagram as in Proposition 4.9, 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.

Remark 4.11. If k → I_{0} → I_{1} → ⋯ → I_{j} is an injective resolution and λ : I_{j} → k represents some cohomology class in Ĥ^{−1−j}(G) ≅ H^{j} Hom(I, k), then the complex k → I_{0} → ⋯ → I_{j} ⟶^{λ} k represents the same class.

5. The Evens norm map and the dual operations

In this part we are going to show that some of the dual operations Q_{i}^{∗} on ordinary group cohomology are compatible with the Evens norm map in certain cases. For simplicity we restrict to the case p = 2. Recall (see e.g. [1], § 4.1) that the Evens norm map is a function

norm_{K,G} : H^{i}(K) → H^{ni}(G)

for all i ≥ 0, where G is a finite group and K ≤ G is a subgroup of index n. It can be defined as follows: let x ∈ H^{i}(K) = Ext_{kK}^{i}(k, k) be represented by an exact sequence k = E_{i} → E_{i−1} → ⋯ → E_{0} → k, which we think of as an augmented complex E → k. Then define x^{⊗n} to be the augmented complex E^{⊗n} → k, which is an exact sequence of k(Σ_{n} ≀ K)-modules. It therefore represents some class in H^{ni}(Σ_{n} ≀ K). We then choose a suitable inclusion ι : G ↪ Σ_{n} ≀ K and define norm_{K,G}(x) = ι^{∗}(x^{⊗n}). In the following, we will often implicitly identify H^{∗}(G) with the dual of Ĥ^{−1−∗}(G) by the use of Tate duality. In particular, we have dual operations Q_{i}^{∗} : H^{i+j}(G) → H^{j}(G).

Theorem 5.1. Let k = 𝔽_{2}, and let K be a subgroup of index n of a finite group G. (1) For all i ≥ 0 the diagram

\xymatrix{H^{i}(K) \ar[r]^{Q_{i}^{∗}} \ar[d]_{(−)^{⊗n}} & H^{0}(K) \ar[d]^{(−)^{⊗n}} \\ H^{ni}(Σ_{n} ≀ K) \ar[r]^{Q_{ni}^{∗}} & H^{0}(Σ_{n} ≀ K)}

commutes.

(2) If K is a central factor of G (e.g., a central subgroup or a direct factor), then for x ∈ H^{i}(K) we have that norm_{K,G} Q_{i}^{∗}(x) = Q_{ni}^{∗}(norm_{K,G} x).

(3) For x ∈ H^{i}(K) we have that Q_{i}^{∗}(x) = Q_{ni}^{∗}(x^{n}) ∈ H^{0}(K).

Remark 5.2. Recall that K is a central factor of G if and only if the product of K with its centralizer is the whole group G. The condition we really need for the proof of part (2) is that we can choose coset representatives for K in G which commute with all elements of order 2 in K. This is true if K is a central factor, but it is also true in other cases like ℤ/4ℤ ⊆ Q_{8}. The condition is not satisfied for ℤ/2ℤ × ℤ/2ℤ ⊂ D_{8} (the dihedral group with 8 elements), and we will see in Remark 5.18 that the conclusion fails in that case.

Corollary 5.3. If the order of the finite group G equals an odd multiple of 2^{i} with i ≥ 1, then the operation Q_{n} : Ĥ^{−1}(G) → Ĥ^{−1−n}(G) is non-trivial whenever n is divisible by 2^{i}.

Proof. Let P ≤ G be a 2-Sylow subgroup, which is of order 2^{i}. The commutative diagram

\xymatrix{Ĥ^{−1}(G) \ar[d]^{_{res_{P,G}}}_{≅} \ar[r]^{Q_{n}} & Ĥ^{−1−n}(G) \ar[d]^{_{res_{P,G}}} \\ Ĥ^{−1}(P) \ar[r]_{Q_{n}} & Ĥ^{−1−n}(P)}

shows that it is enough to consider the case of a 2-group P. Let K ≤ P be a central cyclic subgroup of order 2; then the commutative diagram

\xymatrix{H^{2^{i}j}(P) \ar[r]^{Q_{2^{i}j}^{∗}} & H^{0}(P) \\ H^{j}(K) \ar[u]^{norm} \ar[r]_{Q_{j}^{∗}}^{≅} & H^{0}(K) \ar[u]^{norm}_{≅}}

proves the claim. □

For the proof of Theorem 5.1 we use the following reinterpretation of Q_{i} : Ĥ^{−1}(G) → Ĥ^{−1−i}(G).

Proposition 5.4. Let G be a finite group of order divisible by p = 2, and let φ ∈ Ĥ^{−1}(G) be the canonical generator. For every i ≥ 0, the complex

k ⟶^{N^{⊗2}} kG^{⊗2} ⟶^{1+T} kG^{⊗2} ⟶^{1+T} … ⟶^{1+T} kG^{⊗2} ⟶^{ε^{⊗2}} k

with i + 1 projective modules kG^{⊗2} represents the element Q_{i}(φ) ∈ Ĥ^{−1−i}(G). Here, T denotes the twist map interchanging the two factors of kG^{⊗2}, and ε is the augmentation map.

Proof. We use the definition of Q_{i} using the operad B from §3. Let P be a complete projective resolution of k as trivial kG-module, where we assume that P_{−1} = kG and k ↪ P_{−1} is the norm map N. Let W be the standard free resolution of k as trivial kΣ_{2}-module, with one generator e_{j} in degree j for every j ≥ 0. Choose a Σ_{2}-equivariant chain map Ψ : W → B(2) lifting the identity of k, and define α_{j} : P_{−1−j} → P_{−1}^{⊗2} to be the degree 0-part of the map Ψ(e_{j}) ∈ B(2) ⊆ Hom^{∗}(\overline{P} , \overline{P}^{⊗2}). We get a commutative diagram

[figure: k; N; P_{−1}; α_{0}; P_{−2}; α_{1}; P_{−3}; α_{2}; …; P_{−i}; α_{i−1}; λ; k; k; N^{⊗2}; P_{−1}^{⊗2}; 1+T; P_{−1}^{⊗2}; 1+T; P_{−1}^{⊗2}; 1+T; …; 1+T; P_{−1}^{⊗2}; ε^{⊗2}; k]

By definition, the class of λ in H^{i} Hom(\overline{P} , k) ≅ Ĥ^{−1−i}(G) represents Q_{i}(φ). The commutative diagram then shows the claim by Remark 4.11. □

Proposition 5.5. Let G be a finite group of order divisible by p = 2, and let ν : k → kG be

the norm map. If α : kG^{⊗2} → k is a map for which k ⟶^{ν^{⊗2}} kG^{⊗2} ⟶^{α} k is a complex, then that complex represents ∑_{g∈G} α(1 ⊗ g) ∈ k = Ĥ^{−1}(G).

Proof. Let b = ∑_{g} α(1 ⊗ g); then the result follows from the commutative diagram

[figure: k; ν^{⊗2}; kG^{⊗2}; α; k; k; ν; b; kG; bε; 1⊗ν; b; k; k; ν; kG; ε; k]

and Example 4.4 and Proposition 4.9. □

Proposition 5.6. Suppose that σ ∈ Σ_{n} satisfies σ^{2} = 1 and σ ≠ 1, and let K be a finite group. Define the map f : ℤ/2ℤ × K → Σ_{n} ≀ K to be (u, g) ↦ (σ^{u}; g, g, …, g). Then there is some m ≥ 1 such that for every x ∈ H^{i}(K) we have f^{∗}(x^{⊗n}) = x^{n−2m}(∑_{r=0}^{|x|} Sq^{r}(x)z^{|x|−r})^{m}, where H^{∗}(ℤ/2ℤ) = k[z].

Proof. We can assume that σ is of the form (1 2) (3 4) … (2m − 1 2m) for some m. For every i and every group L denote by Ψ_{i,L} the map L → Σ_{i} ≀ L given by l ↦ (id; l, l, …, l). Let h be the composition

ℤ/2ℤ × K → Σ_{2} ≀ K ⟶^{Ψ_{m,Σ_{2}≀K}} Σ_{m} ≀ (Σ_{2} ≀ K) ↪ Σ_{2m} ≀ K,

the first map being given by (u; g) ↦ (τ^{u}; g; g), where τ is the generator of Σ_{2}. Also let j be the

composition ℤ/2ℤ × K ⟶^{proj} K ⟶^{Ψ_{n−2m,K}} Σ_{n−2m} ≀ K; then we get a composition

ℤ/2ℤ × K ⟶^{h×j} Σ_{2m} ≀ K × Σ_{n−2m} ≀ K ↪ Σ_{n} ≀ K

which equals f. Now let x ∈ H^{i}(K); then x^{⊗n} ∈ H^{ni}(Σ_{n} ≀K) restricts to (x^{⊗2m})⊗(x^{⊗(n−2m)}) ∈ H^{ni}(Σ_{2m} ≀ K × Σ_{n−2m} ≀ K). Now

h^{∗}(x^{⊗2m}) = res Ψ_{m,Σ_{2}≀K}^{∗}((x^{⊗2})^{⊗m}) = (res x^{⊗2})^{m} = (norm_{K,ℤ/2ℤ×K}(x))^{m}.

On the other hand, j^{∗}(x^{⊗(n−2m)}) = x^{n−2m}, so it remains to show that

norm_{K,ℤ/2ℤ×K}(x) = ∑_{r=0}^{|x|} Sq^{r}(x)z^{|x|−r},

which is done, e.g., in [1], §4.4; note that this can actually be used to define the Steenrod operations on ordinary group cohomology. □

Consider the augmentation kΣ_{n} → k as an augmented chain complex; then the augmented chain complex kΣ_{n} ⊗ kΣ_{n} → k is a chain complex of right Σ_{2} × Σ_{n}-modules, where Σ_{n} acts diagonally and Σ_{2} acts by permuting the factors. Let W be the standard free resolution of k as trivial kΣ_{2}-module. By endowing W with a trivial right Σ_{n}-action, we can consider W → k and hence also W ⊗ kΣ_{n} ⊗ kΣ_{n} → k as augmented chain complexes of right Σ_{2} × Σ_{n}-modules. As such the latter consists entirely of free modules, and we can therefore lift the identity map of k to a map of chain complexes ϑ : W ⊗ kΣ_{n} ⊗ kΣ_{n} → W^{⊗n}, where Σ_{n} acts on W^{⊗n} by permuting the factors and Σ_{2} acts diagonally. Finally note that kΣ_{n} is a right Σ_{n} ≀ K-module via the projection map Σ_{n} ≀ K → Σ_{n}, and kK^{⊗n} is also a right Σ_{n} ≀ K-module. Therefore the tensor product kΣ_{n} ⊗ kK^{⊗n} is a right k(Σ_{n} ≀ K)-module, which is free of rank one. We can now form the following map ξ of augmented chain complexes over k(Σ_{n} ≀ K):

\xymatrix{W ⊗_{Σ_{2}} (k(Σ_{n} ≀ K))^{⊗2} \ar[r]^{≅} \ar[dr]_{ξ} & W ⊗_{Σ_{2}} (kΣ_{n} ⊗ kK^{⊗n})^{⊗2} \ar[r]^{≅} & (W ⊗ kΣ_{n} ⊗ kΣ_{n}) ⊗_{Σ_{2}} (kK^{⊗n})^{⊗2} \ar[d]^{ϑ⊗twist} \\  & (W ⊗_{Σ_{2}} kK^{⊗2})^{⊗n} & W^{⊗n} ⊗_{Σ_{2}} (kK^{⊗2})^{⊗n} \ar[l]^{twist}}

In the following, we consider triples (E, β̄, γ̄) where E → k is an exact sequence k = E_{i} ↪ E_{i−1} → ⋯ → E_{0} ↠ k of kK-modules, β̄ : W ⊗_{Σ_{2}} kK^{⊗2} → E is a map of augmented chain complexes, and γ̄ is defined to be γ̄ = β̄^{⊗n} ∘ ξ. Then γ̄ : W ⊗_{Σ_{2}} (k(Σ_{n} ≀ K))^{⊗2} → E^{⊗n} is a map of augmented chain complexes over k(Σ_{n} ≀ K). Define β to be the composite

kK^{⊗2} ≅ W_{i} ⊗_{Σ_{2}} kK^{⊗2} ⟶^{β̄_{i}} E_{i} = k,

and similarly define γ : k(Σ_{n} ≀K)^{⊗2} → k. Furthermore, for every group L let us define the subset L′ = {l ∈ L | l^{2} = 1} ⊆ L.

Proposition 5.7. If (E, β̄, γ̄) is a triple as above, then ∑_{g∈K} β(1 ⊗ g) = ∑_{g∈K′} β(1 ⊗ g) and ∑_{l∈L} γ(1 ⊗ l) = ∑_{l∈L′} γ(1 ⊗ l).

Proof. The formula

β̄_{i}(w ⊗_{Σ_{2}} (g ⊗ h)) = ε(w)β(g ⊗ h), (5.8)

holds because it is true for w = 1 ∈ Σ_{2} and for w = 1 − τ ∈ kΣ_{2} (where τ is the generator of Σ_{2}) since β̄ is a chain map. The formula implies that β(1 ⊗ g) = β(g ⊗ 1) = β(1 ⊗ g^{−1}), and therefore ∑_{g∈K} β(1 ⊗ g) = ∑_{g∈K′} β(1 ⊗ g). The same proof applies to γ. □

Proposition 5.9. There exist constants c_{n,i,σ} (for all σ ∈ Σ_{n}), not depending on K, with the following property: for all triples (E, β̄, γ̄) as above and all elements g = (σ, k_{1}, k_{2}, …, k_{n}) ∈ Σ_{n} ≀ K we have that

γ(1 ⊗ g) = c_{n,i,σ} · β(1 ⊗ k_{1}) ⋯ β(1 ⊗ k_{n}).

Some of the constants will be determined later in Proposition 5.13.

Proof. We can write ϑ_{ni}(1⊗1⊗σ) = ∑_{s} w_{s,1} ⊗⋯⊗w_{s,n} where w_{s,j} ∈ W. Then γ(1⊗g) equals

γ(1 ⊗ g) = ∑_{s} β̄^{⊗n}((w_{s,1} ⊗_{Σ_{2}} (1 ⊗ k_{1})) ⊗ ⋯ ⊗ (w_{s,n} ⊗_{Σ_{2}} (1 ⊗ k_{n}))).

If the degree of one of the w_{s,j} is bigger than i, then the corresponding s-th summand vanishes because β̄ is the zero map. Therefore, we are only interested in the case where all w_{s,j} are of degree i, in which case we can simplify by (5.8)

γ(1 ⊗ g) = (∑_{s} ε(w_{s,1}) ⋯ ε(w_{s,n})) · β(1 ⊗ k_{1})β(1 ⊗ k_{2}) ⋯ β(1 ⊗ k_{n})
= c_{n,i,σ} · β(1 ⊗ k_{1})β(1 ⊗ k_{2}) ⋯ β(1 ⊗ k_{n}),

where c_{n,i,σ} is some constant in k not depending on the group K. □

Proposition 5.10. Let L be any finite group, and suppose that f : L → Σ_{n} ≀ K is an injective group homomorphism and c ∈ k is some constant. Suppose that for all triples (E, β̄, γ̄) as above we have that c · ∑_{g∈K} β(1 ⊗ g) = ∑_{l∈L} γ(1 ⊗ f(l)). Then for all x ∈ H^{i}(K) the formula Q_{ni}^{∗}f^{∗}(x^{⊗n}) = c · f^{∗}(Q_{i}^{∗}(x)^{⊗n}) ∈ H^{0}(L) holds.

Proof. We may assume that the order of both K and L is divisible by p = 2. Then we identify Ĥ^{0}(K) and Ĥ^{0}(L) with k, so that we have to prove

Q_{ni}^{∗}f^{∗}(a^{⊗n}) = c · Q_{i}^{∗}(a)

for all a ∈ H^{i}(K) (recall that k = 𝔽_{2}). Let E be an exact sequence representing a ∈ H^{i}(K). Since the modules of the augmented complex W ⊗ kK^{⊗2} → k are free, we can lift the identity of k to a chain map β̄:

[figure: k; ν^{2}; kK^{⊗2}; 1+T; β; kK^{⊗2}; 1+T; …; 1+T; kK^{⊗2}; ε^{2}; k; k; E_{i−1}; …; E_{0}; k]

Here ν : k → kK is the norm map, and the upper row represents Q_{i}(κ) ∈ Ĥ^{−1−i}(K) for the generator κ ∈ Ĥ^{−1}(K) (by Proposition 5.4). Due to Proposition 4.8 the product Q_{i}(κ)a ∈

Ĥ^{−1}(K) is represented by the complex k ⟶^{ν^{2}} kK^{⊗2} ⟶^{β} k. Therefore, by Proposition 5.5, Q_{i}(κ)a = ∑_{g∈K} β(1 ⊗ g)κ and hence

Q_{i}^{∗}(a) = ∑_{g∈K} β(1 ⊗ g) ∈ k. (5.11)

As before, we get a triple (E, β̄, γ̄) in such a way that the diagram of kL-modules

[figure: k; μ^{2}; kL^{⊗2}; 1+T; (kf)^{⊗2}; kL^{⊗2}; 1+T; …; 1+T; kL^{⊗2}; ε^{2}; k; k(Σ_{n} ≀ K)^{⊗2}; 1+T; γ; k(Σ_{n} ≀ K)^{⊗2}; 1+T; …; 1+T; k(Σ_{n} ≀ K)^{⊗2}; ε^{2}; k; k; (E^{⊗n})_{ni−1}; …; (E^{⊗n})_{0}; k]

commutes, where μ : k → KL is the norm map, such that the upper row represents Q_{ni}(λ) ∈ Ĥ^{−1−ni}(L), where λ ∈ Ĥ^{−1}(L) is the generator. As above, Propositions 4.8 and 5.5 show that Q_{ni}(λ)f^{∗}(a^{⊗n}) = ∑_{l∈L} γ(1 ⊗ f(l))λ, so that

Q_{ni}^{∗}(f^{∗}(a^{⊗n})) = ∑_{l∈L} γ(1 ⊗ f(l)) ∈ k. (5.12)

Combining formulas (5.11) and (5.12) we get the desired result. □

We will now exploit this fact for several maps f.

Proposition 5.13. The constants c_{n,i,σ} satisfy

(5.14)  c_{n,i,id} = 1,
(5.15)  c_{n,i,σ} = 0  if σ^{2} = 1 and σ ≠ id, and
(5.16)  c_{n,i,σ} = c_{n,i,σ^{−1}}  for all σ.

Proof of Proposition 5.13 and Theorem 5.1. As a first step, take L = K and let f : K → Σ_{n} ≀ K be given by g ↦ (id; g, …, g) for all g ∈ K. Then x^{n} = f^{∗}(x^{⊗n}), and the computation

∑_{g∈K} γ(1 ⊗ f(g)) = ∑_{g∈K} γ(1 ⊗ (id; g, …, g)) = c_{n,i,id} ∑_{g∈K} β(1 ⊗ g)^{n} = c_{n,i,id} ∑_{g∈K} β(1 ⊗ g)

shows that Q_{i}^{∗}(x) = c_{n,i,id}Q_{ni}^{∗}(x^{n}) ∈ H^{0}(K). If we put K = ℤ/2ℤ, then the computations in Example 1.2 show that the constant c_{n,i,id} equals 1, so we have proved (5.14) and Theorem 5.1.(3). As a second step, let us take L = ℤ/2 × K and let f : ℤ/2ℤ × K → Σ_{n} ≀ K to be given by (u, g) ↦ (σ^{u}; g, g, …, g), where σ ∈ Σ_{n} is some fixed element of order 2. Then

∑_{g∈ℤ/2ℤ×K} γ(1 ⊗ f(g)) = ∑_{g∈K} γ(1 ⊗ (id; g, …, g)) + γ(1 ⊗ (σ; g, …, g))

= (1 + c_{n,i,σ}) ∑_{g∈K} β(1 ⊗ g).

We take K = ℤ/2ℤ, but we keep the notation K in order to distinguish from the other factor ℤ/2ℤ. We have H^{∗}(K) ≅ k[x] and H^{∗}(ℤ/2ℤ) ≅ k[z] for one-dimensional classes x and z. By Proposition 5.6, we know that f^{∗}((x^{i})^{⊗n}) = f^{∗}(x^{⊗n})^{i} = x^{(n−2m)i}(x^{2} + xz)^{mi}. By the computations in Example 1.7, applying Q_{ni}^{∗} to such a polynomial in x, z equals the sum of the evaluations of that polynomial at (x, z) = (1, 1), (0, 1) and (1, 0); therefore, Q_{ni}^{∗}(f^{∗}((x^{i})^{⊗n})) = 1 ∈ H^{0}(ℤ/2ℤ × K). This implies (5.15). In order to prove (5.16), take a situation in which l ∈ K is of order 2, and β is such that β(1 ⊗ l) ≠ 0. Then put g = (σ; l, l, …, l) and the result follows from γ(1 ⊗ g) = γ(1 ⊗ g^{−1}). Up to this point, we have proved Proposition 5.13 completely. Now we prove Theorem 5.1.(1). Take f : L → Σ_{n} ≀ K to be the identity map of Σ_{n} ≀ K and compute

∑_{g=(σ;k_{1},...,k_{n})∈Σ_{n}≀K} γ(1 ⊗ g) = ∑_{σ} c_{n,i,σ} · ∑_{k_{1},...,k_{n}∈K} β(1 ⊗ k_{1})β(1 ⊗ k_{2}) ⋯ β(1 ⊗ k_{n})

= ∑_{σ} c_{n,i,σ}(∑_{g∈K} β(1 ⊗ g))^{n}.

By Proposition 5.13, ∑_{σ∈Σ_{n}} c_{n,i,σ} = 1, which proves Theorem 5.1.(1). Let π : Σ_{n} ≀ K → Σ_{n} be the projection map. For the proof of Theorem 5.1.(2), choose a set {g_{i}} of coset representatives of K in G with the property that all the g_{i}’s commute with all elements of order 2 in K. Then for each g ∈ G, there are unique elements k_{1}, …, k_{n} ∈ K and σ ∈ Σ_{n} such that gg_{j} = g_{σ(j)}k_{j} for all j, and we get an injection f : G ↪ Σ_{n} ≀ K by g ↦ (σ; k_{1}, …, k_{n}). Then norm_{K,G}(x) = f^{∗}(x^{⊗n}), and we need to investigate

∑_{g∈G} γ(1 ⊗ g) = ∑_{g∈G′} γ(1 ⊗ g)  by Proposition 5.7
= ∑_{\begin{matrix}g∈G′ \\ π(f(g))=id \end{matrix}} γ(1 ⊗ g)  by (5.14) and (5.15).

But if π(f(g)) = id, then gg_{j} = g_{j}k_{j} for all j, which means gg_{j} = k_{j}g_{j} for all j by our condition on K. Therefore, we get g = k_{j} for all j, so that g ∈ K and f(g) = (id; g, g, …, g). Conversely, if g ∈ K, then f(g) = (id; g, g, …, g) by our condition on K. Therefore,

∑_{g∈G} γ(1 ⊗ g) = ∑_{g∈K′} γ(1 ⊗ (id; g, g, …, g)) = ∑_{g∈K′} β(1 ⊗ g)^{n} = ∑_{g∈K′} β(1 ⊗ g).

This proves (2) of Theorem 5.1. □

Example 5.17. Let us work out in detail the operations Q on the generator of Ĥ^{−1}(G) in the case G = D_{8}, the dihedral group with 8 elements. The structure of the cohomology ring H^{∗}(G) is known to be H^{∗}(G) ≅ k[a, b, c]/(ab) where |a| = |b| = 1 and |c| = 2 (see, e.g., [4], Theorem 7.8). From [2], Theorem 3.1 and Lemma 2.1 we get that Ĥ^{−}(G) · Ĥ^{−}(G) = 0 and that Ĥ^{n}(G) · Ĥ^{m}(G) = 0 for n < 0 ≤ n + m. Consider the k-basis {a^{i}c^{j}, b^{i}c^{j}}_{i,j≥0} of H^{∗}(G) and let us define {φ_{a^{i}c^{j}}, φ_{b^{i}c^{j}}} to be the dual basis; in particular, φ_{1} is the canonical generator of Ĥ^{−1}(G). Using Tate duality, one derives the relations

cφ_{a^{i}c^{j}} = \begin{cases}φ_{a^{i}c^{j−1}} & if j > 0, \\ 0 & otherwise, \end{cases}

aφ_{a^{i}c^{j}} = \begin{cases}φ_{a^{i−1}c^{j}} & if i > 0, \\ 0 & otherwise, \end{cases}
aφ_{b^{i}c^{j}} = 0,

and similarly for a and b interchanged. All these facts together completely determine the multiplicative structure of Ĥ^{∗}(G). From Q_{1}(φ_{1}) = φ_{1}^{2} = 0 we get that Q_{1}^{∗} : H^{1}(G) → H^{0}(G) is zero. Therefore, by Theorem 5.1.(3), we get Q_{2}^{∗}(a^{2}) = Q_{2}^{∗}(b^{2}) = 0. Now notice that G ≅ Σ_{2} ≀ ℤ/2ℤ, so that Theorem 5.1 implies that Q_{2}^{∗} : H^{2}(G) → H^{0}(G) is onto and hence Q_{2}^{∗}(c) = 1. We have therefore determined Q_{2}(φ_{1}) = φ_{c}. More generally, note that Q(φ_{1}) = Q(aφ_{a}) = Q(a)Q(φ_{a}) = (a + a^{2})Q(φ_{a}) is divisible by a and, by symmetry, also by b. This fact already implies that Q_{i}(φ_{1}) is a multiple of φ_{c^{j}} for some j. Together with Q_{2i}^{∗}(c^{i}) = Q_{2}^{∗}(c)^{i} = 1 we get

Q_{2i}(φ_{1}) = φ_{c^{i}}  for i ≥ 1,
Q_{2i+1}(φ_{1}) = 0  for i ≥ 0.

Remark 5.18. Let us prove that Theorem 5.1.(2) is not true for arbitrary subgroups K of G. Take K = ℤ/2ℤ × ℤ/2ℤ and G = D_{8}, and let us write norm for norm_{K,G}. We know that H^{∗}(K) ≅ k[x, y] for some one-dimensional classes x, y. Suppose that Theorem 5.1.(2) would hold in that case. Then Q_{2}^{∗}(norm(x)) = norm(Q_{1}^{∗}(x)) = 0, so that norm(x) = αa^{2} + βb^{2} for some α, β ∈ k. Similarly, norm(y) = α′a^{2} + β′b^{2} for some α′, β′ ∈ k. But then norm(xy) = norm(x) norm(y) = αα′a^{4} + ββ′b^{4} and hence

0 = Q_{4}^{∗}(norm(xy)) ≠ norm(Q_{2}^{∗}(xy)) = norm(1),

a contradiction.

6. Productive elements at the prime 2

Let G be a finite group, and let ζ : Ω^{n}k → k be a surjective map representing a Tate cohomology class [ζ] ∈ Ĥ^{n}(G). Define L_{ζ} to be the kernel of ζ; we therefore get an exact triangle

Ωk ⟶^{η} L_{ζ} ⟶^{ι} Ω^{n}k ⟶^{ζ} k. (6.1)

Following Carlson ([4], §9) we call the class [ζ] productive if ζ annihilates the cohomology of L_{ζ}, that is, the map ζ ⊗ id_{L_{ζ}} : Ω^{n}k ⊗ L_{ζ} → L_{ζ} is stably zero. It is known that, for all primes p ≥ 3, a non-zero class [ζ] is productive if and only if its degree n is even (see [3], Theorem 4.1). The case p = 2 is more complicated, and we will show in this section that the operations Q can be used to determine whether a class is productive or not:

Theorem 6.2. Let p = 2, and let G be a finite group. A cohomology class [ζ] ∈ Ĥ^{n}(G) is productive if and only if P_{1}(ζ) is divisible by ζ in Ĥ^{∗}(G).

Remark 6.3. The ’only if’ part of this theorem has been conjectured and independently proven in the case of ordinary cohomology classes by Ergün Yalçin, using connections to the existence of diagonal approximations of certain chain complexes.

The proof of Theorem 6.2 relies on the following commutative diagram.

Proposition 6.4. Under the conditions of the theorem, the following diagram commutes stably:

\xymatrix{Ω^{n}k ⊗ L_{ζ} \ar[r]^{ζ⊗id} \ar[d]_{id ⊗ι} & L_{ζ} \\ Ω^{n}k ⊗ Ω^{n}k \ar[r]_{P_{1}(ζ)} & Ωk \ar[u]_{η}}

Proof of Theorem 6.2. We assume that [ζ] ≠ 0. If P_{1}(ζ) is divisible by ζ, then there is a map α : Ω^{n}k ⊗ Ω^{n}k → Ω^{n+1}k such that P_{1}(ζ) = ζα. But then η P_{1}(ζ) = ηζα = 0 because ηζ = 0. By Proposition 6.4 we get that ζ is productive. Conversely, suppose that ζ is productive, so that η P_{1}(ζ) · (id ⊗ι) = 0 by Proposition 6.4. Since the triangle

Ω^{n}k ⊗ L_{ζ} ⟶^{id ⊗ι} Ω^{n}k ⊗ Ω^{n}k ⟶^{id ⊗ζ} Ω^{n}k

is exact, we get that η P_{1}(ζ) = λ · (id ⊗ζ) for some map λ : Ω^{n}k → L_{ζ}. When we apply the homological functor \underline{Hom}_{kG}(Ω^{n}k, −) to the triangle (6.1), we get a long exact sequence

\underline{Hom}_{kG}(Ω^{n}k, Ωk) ⟶^{η_{∗}} \underline{Hom}_{kG}(Ω^{n}k, L_{ζ}) ⟶^{ι_{∗}} \underline{Hom}_{kG}(Ω^{n}k, Ω^{n}k) ⟶^{ζ_{∗}} \underline{Hom}_{kG}(Ω^{n}k, k).

Here ζ_{∗} can be viewed as ζ· : Ĥ^{0}(G) → Ĥ^{n}(G) which is injective because the class [ζ] is non-zero. By exactness, ι_{∗} = 0 and η_{∗} is surjective. In particular, λ = ηρ for some map ρ : Ω^{n}k → Ωk. Altogether we have that

η(P_{1}(ζ) − ρ(id ⊗ζ)) = λ(id ⊗ζ) − ηρ(id ⊗ζ) = 0,

and therefore P_{1}(ζ) − ρ(id ⊗ζ) = ζσ for some map σ : Ω^{n}k ⊗ Ω^{n}k → Ω^{n+1}k. But then P_{1}(ζ) = [ρ][ζ] + [ζ][σ], so that P_{1}(ζ) is divisible by [ζ]. □

Remark 6.5. Before we start proving that the diagram commutes, let us draw some analogies to the topological world. Let us define k/ζ to be some choice of cone of ζ : Ω^{n}k → k. Then the commutative square of Proposition 6.4 is a shift of the diagram on the left-hand side:

\xymatrix{Ω^{n}k ⊗ k/ζ \ar[r]^{ζ} \ar[d] & k/ζ & S/2 \ar[r]^{2} \ar[d]_{pinch} & S/2 \\ Ω^{n}k ⊗ Ω^{n−1}k \ar[r]_{Sq_{1}(ζ)} & k \ar[u] & ΣS \ar[r]_{η} & S \ar[u]_{incl}}

Note the similarity to the topological situation on the right-hand side, which takes place in the stable homotopy category. Here, S denotes the sphere spectrum, S/2 is the mod-2-Moore spectrum, a cone of multiplication by 2 on S, and η = Sq_{1}(2) is the Hopf map.

The rest of this section is devoted to the proof of Proposition 6.4. Let p = 2, and let G be a finite group. Let P be a complete projective resolution of the trivial kG-module k, and let this resolution define the modules Ω^{n}k.

Proposition 6.6. There is an unstably commutative diagram

[figure: Ω^{n}k ⊗ Ω^{n}k; Q; k; Ω^{n}k ⊗ Ω^{n}k; 1+T; Ω^{n}k ⊗ Ω^{n}k; ζ^{⊗2}; k]

in which Q is projective, and the upper row is a complex representing P_{1}(ζ) ∈ Ĥ^{2n−1}(G).

Proof. We use the operad C given in §2.3 for the definition of P_{1}(ζ). Choose a morphism of augmented Σ_{2}-chain complexes Δ : W → C(2), where W is the standard free resolution of k as trivial Σ_{2}-module. When we consider Ω^{n}k as a chain complex concentrated in degree 0, then the differential of P induces a chain map π_{n} : P → Ω^{n}k of degree n. Let Δ_{i} = Δ(e_{i}), where e_{i} is the generator of W_{i}. We get a commutative diagram

\xymatrix{P \ar[r]^{Δ_{0}} \ar[d]^{π_{2n}} & P^{⊠2} \ar[d]^{π_{n}^{⊠2}} \ar[r]^{1+T} & P^{⊠2} \ar[d]^{π_{n}^{⊠2}} \\ Ω^{2n}k \ar[r]_{ψ} & (Ω^{n}k)^{⊗2} \ar[r]_{1+T} & (Ω^{n}k)^{⊗2}}

where ψ is a stable equivalence and the upper row equals dΔ_{1}. Since π_{n} is a chain map, this diagram restricts to the following commutative diagram in dimension 2n:

\xymatrix{P_{2n} \ar[r]^{∂} \ar[d]^{π_{2n}} & P_{2n−1} \ar[r]^{Δ_{1}} & (P^{⊠2})_{2n} \ar[d]^{π_{n}^{⊠2}} \\ Ω^{2n}k \ar[r]_{ψ} & (Ω^{n}k)^{⊗2} \ar[r]_{1+T} & (Ω^{n}k)^{⊗2}}

We define λ = π_{n}^{⊠2} ∘ Δ_{1}; then ζ^{⊗2} ∘ λ : P_{2n−1} → k represents P_{1}(ζ) by definition. Since ψ is a stable equivalence, we can choose a map ω : (Ω^{n}k)^{⊗2} → Ω^{2n}k such that ψω − id factors as

ψω − id : (Ω^{n}k)^{⊗2} ⟶^{α} Q ⟶^{β} (Ω^{n}k)^{⊗2}

for some projective module R. Then we have a commutative diagram

[figure: Ω^{2n}k; incl; P_{2n−1}; ζ^{⊗2}∘λ; k; Ω^{n}k ⊗ Ω^{n}k; _{(\begin{matrix}incl∘ω \\ α \end{matrix})}; ω; P_{2n−1} ⊕ R; _{(_{λ, (1+T)∘β})}; _{(_{ζ^{⊗2}∘λ, 0})}; _{(_{id, 0})}; k; Ω^{n}k ⊗ Ω^{n}k; 1+T; Ω^{n}k ⊗ Ω^{n}k; ζ^{⊗2}; k]

proving the claim. □

Proof of Proposition 6.4. Define the map κ : Ω^{n}k ⊗ Ω^{n}k → L_{ζ} by a ⊗ b ↦ ζ(a)b + ζ(b)a, then the upper left triangle in the diagram

\xymatrix{Ω^{n}k ⊗ L_{ζ} \ar[r]^{ζ⊗id} \ar[d]_{id ⊗ι} & L_{ζ} \\ Ω^{n}k ⊗ Ω^{n}k \ar[ur]^{κ} \ar[r]_{P_{1}(ζ)} & Ωk \ar[u]_{η}}

commutes, and we want to show that the bottom right triangle also commutes. To do so, we extend the diagram of Proposition 6.6 as follows:

[figure: Ω^{n}k ⊗ Ω^{n}k; Q; k; Ω^{n}k ⊗ Ω^{n}k; 1+T; κ; Ω^{n}k ⊗ Ω^{n}k; ζ^{⊗2}; ζ⊗id; k; L_{ζ}; ι; Ω^{n}k; ζ; k]

The bottom row is an extension representing [η] ∈ Ext_{kG}^{1}(k, L_{ζ}), so we are done by Proposition 4.5. □

References

[1] D. J. Benson. Representations and cohomology. II, volume 31 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, second edition, 1998. Cohomology of groups and modules.

[2] D. J. Benson and Jon F. Carlson. Products in negative cohomology. J. Pure Appl. Algebra, 82(2):107–129, 1992.

[3] Jon F. Carlson. Products and projective resolutions. In The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986), volume 47 of Proc. Sympos. Pure Math., pages 399–408. Amer. Math. Soc., Providence, RI, 1987.

[4] Jon F. Carlson. Modules and group algebras. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 1996. Notes by Ruedi Suter.

[5] Henri Cartan and Samuel Eilenberg. Homological algebra. Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ, 1999. With an appendix by David A. Buchsbaum, Reprint of the 1956 original.

[6] Frederick R. Cohen, Thomas J. Lada, and J. Peter May. The homology of iterated loop spaces. Lecture Notes in Mathematics, Vol. 533. Springer-Verlag, Berlin, 1976.

[7] Leonard Evens. Steenrod operations and transfer. Proc. Amer. Math. Soc., 19:1387–1388, 1968.

[8] J. P. C. Greenlees. Representing Tate cohomology of G-spaces. Proc. Edinburgh Math. Soc. (2), 30(3):435– 443, 1987.

[9] J. P. C. Greenlees and J. P. May. Generalized Tate cohomology. Mem. Amer. Math. Soc., 113(543):viii+178, 1995.

[10] Henning Krause. The stable derived category of a Noetherian scheme. Compos. Math., 141(5):1128–1162, 2005.

[11] Igor Kříž and J. P. May. Operads, algebras, modules and motives. Astérisque, (233):iv+145pp, 1995.

[12] L. G. Lewis, Jr., J. P. May, M. Steinberger, and J. E. McClure. Equivariant stable homotopy theory, volume 1213 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1986. With contributions by J. E. McClure.

[13] J. Peter May. A general algebraic approach to Steenrod operations. In The Steenrod Algebra and its Applications (Proc. Conf. to Celebrate N. E. Steenrod’s Sixtieth Birthday, Battelle Memorial Inst., Columbus, Ohio, 1970), Lecture Notes in Mathematics, Vol. 168, pages 153–231. Springer, Berlin, 1970.

[14] J. E. McClure. E_{∞}-ring structures for Tate spectra. Proc. Amer. Math. Soc., 124(6):1917–1922, 1996.

E-mail address: martinlanger@uni-muenster.de

Westfälische Wilhelms-Universität Münster, Institut für Mathematik, Einsteinstr. 62, 48149 Münster, Germany