ON THE KOLYVAGIN CUP PRODUCT

AMNON BESSER

1. Introduction

In his work on Euler systems, Kolyvagin is using a basic, derivative like, operator to construct cohomology classes from his systems. The operation can be described as follows: let Δ = ⟨τ⟩ be a cyclic group with n elements and let M be a Δ-module. If m ∈ M is such that the norm (or trace) of m with respect to Δ is 0, i.e.,

N m = ∑_{k=0}^{n−1} τ^{k}m = 0,

then the element

Dm = ∑_{k=0}^{n−1} kτ^{k}m (mod nM) ∈ M/nM

is Δ-invariant.

A few cohomological generalizations of this construction have been investigated and used since (Faltings, [Nek92], [Rub93]). Rubin’s construction, which is the most general that we know of, goes as follows: Let H be a normal subgroup of G, and let Δ = G/H. Define the G-module M′ = Hom_{H}(ℤ[G], M) and let M′′ be the kernel of the map

η(f) = ∑_{σ∈Δ} σ^{−1}(f(σ)) from M′ to M.

Now, associated to a homomorphism ψ : Δ → ℤ/n is a G-module map

δ_{ψ} : M′′ → M/n defined by δ_{ψ}(f) = ∑_{σ∈Δ} ψ(σ)σ^{−1}(f(σ)).

The induced maps on cohomology, δ_{ψ}^{∗} : H^{i}(G, M′′) → H^{i}(G, M/n), are the higher dimensional analogies of the Kolyvagin derivatives. The term H^{i}(G, M′′) projects onto the kernel of the corestriction map cor_{H}^{G} : H^{i}(H, M) → H^{i}(G, M) with kernel which is contained in H^{i−1}(G, M) and one hopes that this group is small or that it may be annihilated by a small integer in order to make the operation depend only on the kernel of the corestriction map. When i = 0 this operation precisely corresponds to sending m ∈ Ker N : M^{H} → M^{G} to ∑_{σ∈Δ} ψ(σ)σ^{−1}m. Thus, the Kolyvagin derivative is the special case when i = 0, Δ is a cyclic group of order n and ψ is the isomorphism of Δ onto ℤ/n sending the generator τ to 1.

Our purpose in this work is to give a different, even more general, cohomological extension of the Kolyvagin construction, which coincides with the Rubin construction in the above situation, and also to prove for this construction some of the properties of the Kolyvagin derivative that are used in work on Euler systems. The starting point is to notice that, for i = 0, the operation described by Rubin can be carried out in greater generality when ψ is a 1-cocycle of Δ with values in any Δ-module A. We will call the map

(ψ, m) → ∑_{σ∈Δ} ψ(σ) ⊗ σ(m) ∈ (A ⊗ M)^{G}, (1.1)

the “Kolyvagin operation”. Here, G acts on A through its quotient Δ and A⊗M is given the diagonal action of G. The observation that the above element is G-invariant will be refered to as the “Kolyvagin trick”.

One is tempted to try and interpret (1.1) as a cup product of ψ and m. A simple computation shows that as a function of ψ (1.1) does not factor through H^{1}(Δ, A). However, a cup product interpretation is possible in the following way: Let I be the augmentation ideal in the group ring ℤ[Δ]. the 1-cocycle ψ may be interpreted as a G-homomorphism ψ : I → A (throughout this paper we do not distinguish between the two ways of viewing ψ). The kernel of the norm map on M may be interpreted as Hom_{G}(Ǐ, M) where Ǐ is the quotient of ℤ[Δ] by the ℤ-module generated by the norm element, ∑_{σ∈Δ} σ, and the Kolyvagin operation above amounts then to pulling back ψ ⊗ m ∈ Hom_{G}(I ⊗ Ǐ) by the map γ : ℤ → I ⊗ Ǐ defined by γ(1) = ∑_{σ∈Δ}(σ − 1) ⊗ σ. This operation now carries over almost without any change to the case when m is in Ext_{G}^{i}(Ǐ, M) and gives our generalization to the Kolyvagin derivative, which we call the Kolyvagin cup product (see definition 4.1).

It’s not too hard to see that our Kolyvagin cup product coincides with the operation defined by Rubin in the case that A = ℤ/n. The extra generality gives some interesting applications which we hope to discuss in a future paper.

We prove a few fundamental results about our cup product. Two of these, propositions 5.1 and 5.5, are analogies of the well known rules of composition between the corestriction and restriction. Other results, such as 5.4 and 8.1, allow computations with the Kolyvagin cup product in the case that M is described as the quotient of another G-module C. We should mention that 5.1 and 5.4 are generalizations of results proved by Rubin. The method of proof is different and relies as much as possible on general considerations in homological algebra instead of explicit computations with cocycles. Our hope is that this would allow further generalizations in the future (etale cohomology? K-theory?). In fact, the starting point for this work was to try to find more homological algebra like proofs of Rubin’s results.

The main result, Proposition 8.1, is about the localization of the Kolyvagin cup product. In the applications to Euler systems that we have in mind, it shows that one can carry out the crucial step in the Kolyvagin method of computing the localization of the cohomology classes obtained via the derivative operator by using the Euler system.

We also describe two related operations: One is, in some precise sense (see propositions 7.3 and 7.6), a dual of the Kolyvagin cup product and is also a generalization of an operation described by Rubin in [Rub93]. The other is a higher order derivative which one can define in certain cases. This type of higher derivatives has appeared in dimension 0 in the work of Darmon [Dar92]. This operation can also be seen in special cases as giving a higher dimensional analogue of the more general derivative operator defined by Kolyvagin for the product of cyclic groups. This removes one of the obstacles for getting sharper bounds on Selmer groups in the work of Nekovář on Kuga-Sato varieties ([Nek92], remark on page 124)

Some of this work was done while the author was visiting at MSRI. I would like to thank Jon Rogawski for a number of helpful conversations.

2. Some homological algebra

In this and the next section, we collect together a few results in homological algebra, to be used in later sections.

Definition 2.1. A balanced sequence E is a short exact sequence of chain complexes,

0 ⟶ E^{•}(E) ⟶^{i} F^{•}(E) ⟶^{p} G^{•}(E) ⟶ 0,

together with a map of complexes ε : G^{•}(E) → F^{•}(E) with pε = 0.

As usual, we denote by i_{∗} and p_{∗} the maps induced on homology by i and p respectively.

Definition 2.2. Given a balanced sequence

E = (0 ⟶ E^{•} ⟶^{i} F^{•} ⟶^{p} G^{•} ⟶ 0, ε) ,

we let u_{E} : H(G^{•}) → H(E^{•}) be the map on homology induced from the unique map of complexes ū : G^{•} → E^{•} satisfying iū = ε.

One clearly has i_{∗} ∘ u_{E} = ε_{∗}.

In general, the above definition is not “stable”. For instance, if ε is changed by a homotopic map, the map u_{E} might change. We now describe a special situation where one can “canonify” the map u_{E}:

Suppose we have an exact sequence

0 ⟶ X ⟶^{ĩ} Y ⟶^{p̃} Z ⟶ 0

in some abelian category A, and that we have a map ε̃: Y → X. We then get a map

(2.1)  ũ : Z → Coker ε̃ĩ defined by ũ(z) = ε̃(y) (mod ε̃ĩY), p̃(y) = z.

Let there be given an object M ∈ A. We can construct projective resolutions X^{•}, Y^{•} and Z^{•} of X, Y and Z respectively, so that we may lift the maps ĩ, p̃ and ε̃ to an exact sequence of projective resolutions:

0 ⟶ X^{•} ⟶^{ĩ^{•}} Y^{•} ⟶^{p̃^{•}} Z^{•} ⟶ 0,

with maps ε̃^{•} : Y^{•} → X^{•} and therefore ũ^{•} : Z^{•} → Coker ε̃^{•}ĩ^{•} Assume that we can choose these resolutions and maps in such a way that

• The complex Coker ε̃^{•}ĩ^{•} is a resolution of Coker ε̃ĩ; and

• The map Hom(X^{n}, M) ⟶^{∘(ε̃^{n}ĩ^{n})} Hom(X^{n}, M) is 0. Under all of these assumptions, we see that the exact sequence of complexes

(2.2) 0 ⟶ Hom(Z^{•}, M) ⟶^{i=∘p̃^{•}} Hom(Y^{•}, M) ⟶^{p=∘ĩ^{•}} Hom(X^{•}, M) ⟶ 0

together with the map ε = ∘ε̃^{•}, is a balanced sequence. The next lemma is immediate

Lemma 2.3. The map u_{E} associated with the balanced sequence (2.2) is the composition

Ext^{•}(X, M) \underrightarrow{s_{X,ε̃ĩ}} Ext^{•}(Coker ε̃ĩ, M) \underrightarrow{ũ^{∗}} Ext^{•}(Z, M),

where the map s_{X,ε̃ĩ} is defined as follows:

Definition 2.4. Let X and M be an object of an abelian category A, and let X^{•} be a projective resolution of X. Let f : X → X be an endomorphism of X which can be extended to a map of complexes, f^{•} : X^{•} → X^{•}, in such a way that Coker f^{•} is a resolution of Coker f, and such that all the maps Hom(X^{•}, M) ⟶^{∘f^{•}} Hom(X^{•}, M) are 0. Let Q^{•} be a projective resolutions of Coker f. since Coker f^{•} is a resolution of Coker f, there is a map of complexes h^{•} : Q^{•} → Coker f^{•}. We define

s_{X,f} : Ext^{•}(X, M) → Ext^{•}(Coker f, M)

to be the map induced on the homology from the map

Hom(X^{•}, M) ≅ Hom(Coker f^{•}, M) ⟶^{h^{•}^{∗}} Hom(Q^{•}, M).

It is immediately verified that s_{X,f} is a section of the canonical map Ext^{•}(Coker f, M) → Ext^{•}(X, M) coming from the projection X → Coker f.

Remark 2.5. We have suppressed the dependence on the choice of the lifts f^{•} since in applications we’ll be even more specific: Our category A will be the category of G-modules for some group G, the modules X, Y and Z will be ℤ free and the map ε̃ĩ will be just multiplication by some integer n, which in turn annihilates the module M. It is not hard to see in this case that we may choose our resolutions and maps in such a way that ε̃^{•}ĩ^{•} is again multiplication by n and that all the other assumptions that we have made above hold. This choice canonifies s_{X,f}.

Cup products

We will need to recall a few well known facts about cup products:

Definition 2.6. Let X, Y, A and M be ℤ[G]-modules, for some group G. Let X^{•} and Y^{•} be ℤ[G] free resolutions of X and Y respectively. If we assume that either X or Y are ℤ free, then X^{•} ⊗Y^{•} is a resolution of X ⊗ Y. given [f] ∈ Ext_{G}^{i}(X, A) and [g] ∈ Ext_{G}^{j}(Y, M), we may define f ∪ g ∈ Ext_{G}^{i+j}(X ⊗ Y, A ⊗ M) as follows: Take representatives f ∈ Hom_{G}(X^{i}, A) and g ∈ Hom_{G}(Y^{i}, M) for [f] and [g] respectively. We may extend f and g to maps from X^{•} to A and Y^{•} to M by making them 0 in all the other degrees. Now we may define [f] ∪ [g] to be represented by the homomorphism f ⊗ g : X^{•} ⊗ Y^{•} → A ⊗ M.

The next lemma is then obvious:

Lemma 2.7. If ϕ : X → X′ and ψ : Y → Y′ are two maps of G-modules, then the following diagram commutes

Ext_{G}^{i}(X′, A) × Ext_{G}^{j}(Y′, M) ⟶^{ϕ^{∗}×ψ^{∗}} Ext_{G}^{i}(X, A) × Ext_{G}^{j}(Y, M)
∪↓  ^{∪}↓
Ext_{G}^{i+j}(X′ ⊗ Y′, A ⊗ M) ⟶^{(ϕ⊗ψ)^{∗}} Ext_{G}^{i+j}(X ⊗ Y, A ⊗ M)

We now show a connection between the cup product and the map of definition 2.4.

Lemma 2.8. Let X and Y be G-modules which are ℤ-free, and let A and M be G-modules killed by an integer n. Let s_{Y,n} : Ext_{G}^{i}(Y, M) → Ext_{G}^{i}(Y/n, M) be the canonical map constructed in 2.4, where n denotes multiplication by n on Y. Let r_{?,n} :? →?/n be the reduction map for ? = X, Y. Then, the following diagram commutes

Ext_{G}^{j}(X/n, A) × Ext_{G}^{i}(Y, M) \underrightarrow{r_{X,n}^{∗} × s_{Y,n}} Ext_{G}^{j}(X, A) × Ext_{G}^{i}(Y/n, M)
∪↓  ^{∪}↓
Ext_{G}^{i+j}(X/n ⊗ Y, A ⊗ M)  ⟶^{≅}  Ext_{G}^{i+j}(X ⊗ Y/n, A ⊗ M)

where the isomorphism in the bottom row is induced by the canonical isomorphism X ⊗ Y/n ≅ X/n ⊗ Y.

Proof. This is an immediate consequence of the fact that s_{Y,n} is a section of r_{Y,n}^{∗} and of lemma 2.7 with the maps ϕ and ψ being the reduction maps r_{X,n} and r_{Y,n}.

3. An ugly lemma

The purpose of this section is to prove lemma 3.1. The proof is done using a rather nasty diagram chase. It is possible that the lemma admits a simpler and more conceptual proof (especially the first part) but we couldn’t find one.

To make it easier to follow the computation, we will use a different notational convention then the one used in section 2:

• Chain complexes are denoted by bold capital letters;

• The differential of all complexes is denoted d and is raising degrees. It is assumed to commute with all the other maps whenever that makes sense;

• If C denotes a complex, C[h] denotes the same complex with degrees shifted by h;

• elements in a chain complex are denoted by bold letters matching the complex from which they are taken, e.g., b^{x}, b_{0}^{x} ∈ B^{X};

• The kernel of d on the complex C is denoted by Z(C) and the homology of the complex by H(C);

• The homology class of c ∈ C is denoted by [c];

• Maps between complexes are always denoted by either i for injection or p for projection, with an upper indexing.

Lemma 3.1. Suppose we are given the following commuting diagram of chain complexes, with exact rows and columns:

0  0  0
↓  ↓  ↓
0 ⟶ B^{Z} ⟶^{i^{z}} C^{Z} ⟶^{p^{z}} M^{Z} ⟶ 0
i^{b}↓  _{i^{c}}↓  _{i^{m}}↓
0 ⟶ B^{Y} ⟶^{i^{y}} C^{Y} ⟶^{p^{y}} M^{Y} ⟶ 0
p^{b}↓  ^{p^{c}}↓  ^{p^{m}}↓
0 ⟶ B^{X} ⟶^{i^{x}} C^{X} ⟶^{p^{x}} M^{X} ⟶ 0
↓  ↓  ↓
0  0  0

(3.1)

which gives rise to the following doubly long exact sequence of homology groups

⋮  ⋮  ⋮  ⋮  ⋮
⋯ H (M^{Y}[−2]) ⟶^{δ^{y}} H (B^{Y}[−1]) ⟶^{i_{∗}^{y}} H (C^{Y}[−1]) ⟶^{p_{∗}^{y}} H (M^{Y}[−1]) ⟶^{δ^{y}} H (B^{Y}) ⋯
p_{∗}^{m}↓  ^{p_{∗}^{b}}↓  ^{p_{∗}^{c}}↓  ^{p_{∗}^{m}}↓  ^{p_{∗}^{b}}↓
⋯ H (M^{X}[−2]) ⟶^{δ^{x}} H (B^{X}[−1]) ⟶^{i_{∗}^{x}} H (C^{X}[−1]) ⟶^{p_{∗}^{x}} H (M^{X}[−1]) ⟶^{δ^{x}} H (B^{X}) ⋯
δ^{m}↓  _{δ^{b}}↓  _{δ^{c}}↓  _{δ^{m}}↓  _{δ^{b}}↓
⋯ H (M^{Z}[−1]) ⟶^{δ^{z}} H (B^{Z}) ⟶^{i_{∗}^{z}} H (C^{Z}) ⟶^{p_{∗}^{z}} H (M^{Z}) ⟶^{δ^{z}} H (B^{Z}[+1]) ⋯
i_{∗}^{m}↓  ^{i_{∗}^{b}}↓  ^{i_{∗}^{c}}↓  ^{i_{∗}^{m}}↓  ^{i_{∗}^{b}}↓
⋯ H (M^{Y}[−1]) ⟶^{δ^{y}} H (B^{Y}) ⟶^{i_{∗}^{y}} H (C^{Y}) ⟶^{p_{∗}^{y}} H (M^{Y}) ⟶^{δ^{y}} H (B^{Y}[+1]) ⋯
p_{∗}^{m}↓  ^{p_{∗}^{b}}↓  ^{p_{∗}^{c}}↓  ^{p_{∗}^{m}}↓  ^{p_{∗}^{b}}↓
⋯ H (M^{X}[−1]) ⟶^{δ^{x}} H (B^{X}) ⟶^{i_{∗}^{x}} H (C^{X}) ⟶^{p_{∗}^{x}} H (M^{X}) ⟶^{δ^{x}} H (B^{X}[+1]) ⋯
δ^{m}↓  _{δ^{b}}↓  _{δ^{c}}↓  _{δ^{m}}↓  _{δ^{b}}↓
⋯ H (M^{Z}) ⟶^{δ^{z}} H (B^{Z}[+1]) ⟶^{i_{∗}^{z}} H (C^{Z}[+1]) ⟶^{p_{∗}^{z}} H (M^{Z}[+1]) ⟶^{δ^{z}} H (B^{Z}[+2]) ⋯
⋮  ⋮  ⋮  ⋮  ⋮

(1) Assume that we have [m^{z}] ∈ H (M^{Z}) and [c^{y}] ∈ H (C^{Y}) s.t.

i_{∗}^{m} [m^{z}] = p_{∗}^{y} [c^{y}] ∈ H (M^{Y}) .

Then there exist [b^{x}] ∈ H (B^{X}) which satisfies the two equalities: (a) i_{∗}^{x} [b^{x}] = p_{∗}^{c} [c^{y}] ∈ H (C^{X}); and

(b) δ^{b} [b^{x}] = −δ^{z} [m^{z}].

(2) Assume, in addition, that there are maps of complexes

ε^{?} :?^{X} ⟶?^{Y}, ? = B, C, M,

commuting with the vertical maps between the complexes in the obvious way, that there is an endomorphism π of all the complexes, commuting with all the other maps, and such that

p^{?} ∘ ε^{?} = π|_{?^{X}}, ? = b, c, m,

and

π|_{M^{X}} = 0.

Finally, assume that there exist some [c_{0}^{x}] ∈ H (C^{X}) s.t. [c^{y}] = ε_{∗}^{c} [c_{0}^{x}]. Let u_{M} : H (M^{X}) → H (M^{Z}) be the map u_{E} corresponding the balanced sequence E formed of the rightmost column of the diagram (3.1) and the map ε^{m}. Then, there exist [b_{0}^{y}] ∈ H (B^{Y}) and [m_{0}^{x}] ∈ H (M^{X}[−1]) s.t.

(a) i^{y} [b_{0}^{y}] = π_{∗} [c^{y}];

(b) δ^{y}ε_{∗}^{m} [m_{0}^{x}] = ε_{∗}^{b} [b_{0}^{x}] − [b_{0}^{y}]; and

(c) δ^{m} [m_{0}^{x}] + u_{M}p_{∗}^{x} [c_{0}^{x}] = [m^{z}].

Proof. (1) We begin by choosing representatives m^{z} ∈ Z (M^{Z}) and c^{y} ∈ Z (C^{Y}) for [m^{z}] and [c^{y}] respectively. The condition i_{∗}^{m} [m^{z}] = p_{∗}^{y} [c^{y}] implies

i^{m}m^{z} = p^{y}c^{y} + dm^{y} for some m^{y} ∈ M^{Y}[−1].

We can choose c̄^{y} ∈ C^{Y}[−1] s.t. p^{y}c̄^{y} = m^{y}. Replacing c^{y} by c^{y} + dc̄^{y}, which still represents [c^{y}], we may assume

i^{m}m^{z} = p^{y}c^{y}.

The class p_{∗}^{c} [c^{y}] is represented by p^{c}c^{y} ∈ Z (C^{X}).

p^{x}p^{c}c^{y} = p^{m}p^{y}c^{y} = p^{m}i^{m}m^{z} = 0 ⇒ p^{c}c^{y} = i^{x}b^{x} for some b^{x} ∈ Z (B^{X}) .

We choose [b^{x}] to be the cohomology class represented by b^{x}. To find δ^{b} [b^{x}] we choose b^{y} ∈ B^{Y} s.t. p^{b}b^{y} = b^{x}. Then

p^{b}db^{y} = db^{x} = 0 ⇒ db^{y} = i^{b}b^{z} with b^{z} ∈ Z (B^{Z}[1]) .

and we have δ^{b} [b^{x}] = [b^{z}]. We find

p^{c}i^{y}b^{y} = i^{x}p^{b}b^{y} = i^{x}b^{x} = p^{c}c^{y} ⇒ p^{c}(c^{y} − i^{y}b^{y}) = 0.

Therefore,

c^{y} − i^{y}b^{y} = i^{c}c^{z} with c^{z} ∈ C^{Z},

and

i^{m}p^{z}c^{z} = p^{y}i^{c}c^{z} = p^{y}(c^{y} − i^{y}b^{y}) = p^{y}c^{y} = i^{m}m^{z} ⇒ p^{z}c^{z} = m^{z}.

We may therefore use c^{z} to compute δ^{z} [m^{z}]: we have

dc^{z} = i^{z}b̄^{z} with b̄^{z} ∈ Z (B^{Z}[1]) ,

and the class of b̄^{z} is

\left[{ \atop b̄^{z}}\right] = δ^{z} [m^{z}] .

We are done if we show [b^{z}] = − \left[{ \atop b̄^{z}}\right]. In fact

i^{c}i^{z}b̄^{z} = i^{c}dc^{z} = di^{c}c^{z} = d(c^{y} − i^{y}b^{y}) = −di^{y}b^{y} = −i^{y}db^{y} = −i^{y}i^{b}b^{z} = −i^{c}i^{z}b^{z}

and therefore b^{z} = −b̄^{z}.

(2)

p^{y}πc^{y} = πp^{y}c^{y} = 0 ⇒ πc^{y} = i^{y}b_{0}^{y} with b_{0}^{y} ∈ Z (B^{Y}) .

We set [b_{0}^{y}] be the homology class of b_{0}^{y}. The first property is obviously satisfied. The assumption [c^{y}] = ε_{∗}^{c} [c_{0}^{x}] implies that

c^{y} = ε^{c}c_{0}^{x} + dc̃^{y} with c̃^{y} ∈ C^{Y}[−1].

Therefore

i^{x}b^{x} = p^{c}c^{y} = p^{c}(ε^{c}c_{0}^{x} + dc̃^{y}) = πc_{0}^{x} + dp^{c}c̃^{y},

and

i^{y}b_{0}^{y} = πc^{y} = πε^{c}c_{0}^{x} + dπc̃^{y}.

It follows that

i^{y}(ε^{b}b^{x} − b_{0}^{y}) = ε^{c}i^{x}b^{x} − i^{y}b_{0}^{y} = (ε^{c}πc_{0}^{x} + dε^{c}p^{c}c̃^{y}) − (πε^{c}c_{0}^{x} + dπc̃^{y}) = d(ε^{c}p^{c} − π)c̃^{y},

and that therefore a preimage of ε_{∗}^{b} [b^{x}] − [b_{0}^{y}] is given by the class of

p^{y}(ε^{c}p^{c} − π)c̃^{y} = p^{y}ε^{c}p^{c}c̃^{y} = ε^{m}p^{m}p^{y}c̃^{y}.

It is easy to check that dp^{m}p^{y}c̃^{y} = 0. Therefore, we may take m_{0}^{x} = p^{m}p^{y}c̃^{y} to represent the cohomology class [m_{0}^{x}] and the second property holds. Finally, we compute δ^{m} [m_{0}^{x}]. It is represented by m_{0}^{z} ∈ Z (M^{Z}) which satisfies

i^{m}m_{0}^{z} = dp^{y}c̃^{y} = p^{y}dc̃^{y} = p^{y}(c^{y} − ε^{c}c_{0}^{x}) = i^{m}m^{z} − ε^{m}p^{x}c_{0}^{x}.

Thus i^{m}(m^{z} − m_{0}^{z}) = ε^{m}p^{x}c_{0}^{x} so by the definition of u_{M} we have

[m^{z}] − [m_{0}^{z}] = u_{M}p_{∗}^{x} [c_{0}^{x}] .

4. Definition of the cup product

Let G be a group and H a normal subgroup with a finite index, so that we have the short exact sequence

1 ⟶ H ⟶ G ⟶ Δ ⟶ 1, (4.1)

with Δ a finite group. Let ℤ[Δ] be the group ring of Δ, with the obvious (left) G action. The standard augmentation map, ε : ℤ[Δ] → ℤ, sits inside a short exact sequence,

0 ⟶ I ⟶^{i} ℤ[Δ] ⟶^{ε} ℤ ⟶ 0, (4.2)

in which I is the augmentation ideal. The dual exact sequence is

0 ⟶ ℤ ⟶^{N} ℤ[Δ] ⟶^{p} Ǐ ⟶ 0. (4.3)

Here, N is the norm (or trace) map that sends 1 ∈ ℤ to ∑_{g∈Δ} g, and Ǐ = ℤ[Δ]/ℤ ∑_{g∈Δ} g. The duality is given by the ℤ[Δ] map γ̌ : ℤ[Δ] ⊗ ℤ[Δ] → ℤ defined by the pairing

⟨∑_{σ∈Δ} n_{σ}σ, ∑_{σ∈Δ} m_{σ}σ⟩ = ∑_{σ∈Δ} m_{σ}n_{σ}, (4.4)

under which ℤ ∑_{g∈Δ} g and Ǐ are exact annihilators. We define γ : ℤ → I ⊗ Ǐ to be the dual map to the map induced from γ̌ on Ǐ⊗ I. A simple computation shows that γ is the unique map satisfying

γ(1) = ∑_{σ∈Δ}(σ − 1) ⊗ p(σ). (4.5)

For future use we record the following two commutative diagrams with exact rows. The first is

\begin{aligned}0 & ⟶ & I ⊗ Ǐ & ⟶^{i⊗id} & ℤ[Δ] ⊗ Ǐ & ⟶^{ε⊗id} & Ǐ & ⟶ & 0 \\ & & ^{γ}↑ & & ^{β}↑ & & ∥ \\ 0 & ⟶ & ℤ & ⟶^{N} & ℤ[Δ] & ⟶^{p} & Ǐ & ⟶ & 0. \end{aligned} (4.6)

The first row is the sequence (4.2) tensored with Ǐ, the second is just (4.3) and the map β is defined by

β(σ) = σ ⊗ p(σ).

In the second diagram the first raw is (4.3) tensored with I while the second row is (4.2):

\begin{aligned}0 & ⟶ & I & ⟶^{id ⊗ N} & I ⊗ ℤ[Δ] & ⟶^{id ⊗p} & I ⊗ Ǐ & ⟶ & 0 \\ & & ∥ & & ^{β′}↑ & & ^{γ}↑ \\ 0 & ⟶ & I & ⟶^{i} & ℤ[Δ] & ⟶^{ε} & ℤ & ⟶ & 0. \end{aligned} (4.7)

Here

β′(σ) = σ (∑_{τ∈Δ}(τ − 1) ⊗ τ) .

Let M and A be G-modules.

Definition 4.1. The Kolyvagin cup product is the map

∪ k : Ext_{G}^{j}(I, A) × Ext_{G}^{i}(Ǐ, M) ⟶ H^{i+j}(G, M ⊗ A)

which is the composition

Ext_{G}^{j}(I, A) × Ext_{G}^{i}(Ǐ, M) ⟶^{∪} Ext_{G}^{i+j}(I ⊗ Ǐ, M ⊗ A) ⟶^{γ^{∗}} Ext_{G}^{i+j}(ℤ, M ⊗ A) ≅ H^{i}(G, M ⊗ A)

We now give a more concrete interpretation for the domain of the Kolyvagin cup product.

Lemma 4.2. In the situation described above we have the following long exact sequences

⋯ ⟶ H^{i−1}(G, M) ⟶^{δ_{M}^{(4.3)}} Ext_{G}^{i}(Ǐ, M) ⟶^{p^{∗}} H^{i}(H, M) ⟶^{cor_{H}^{G}} H^{i}(G, M) ⟶ ⋯ (4.8)

Here we use the notation δ_{M}^{(4.3)} to denote the boundary map on the long exact sequence of Ext(•, M) associated to the short exact sequence (4.3), and we use similar notation for other long exact sequences.

Proof. Since Ext_{G}^{i}(ℤ[Δ], M) ≅ H^{i}(H, M), (4.8) is just the long exact sequence associated with the short exact sequence (4.3).

Remark 4.3. If H^{i−1}(G, M) = 0 (in particular if i = 0), the sequence (4.8) implies

Ext_{G}^{i}(Ǐ, M) ≅ Ker H^{i}(H, M) ⟶^{cor_{H}^{G}} H^{i}(G, M).

We will often assume that this is the situation. In case it is not, we may define a “Kolyvagin cup product” for

ψ × m ∈ Ext_{G}^{j}(I, A) × (Ker H^{i}(H, M) ⟶^{cor_{H}^{G}} H^{i}(G, M))

by using the following procedure: We choose a surjection pr : M ↠ M′ in such a way that pr_{∗} kills H^{i−1}(G, M) (for instance we can choose M′ = M/M^{G} if i = 1). It is readily seen that for any z ∈ Ext_{G}^{i}(Ǐ, M) mapping to m,

(pr ⊗ id)_{∗}(ψ ∪ k z) ∈ H^{i+j}(G, A ⊗ M′)

will only depend on ψ and m. All of the results which are stated here under the assumption H^{i−1}(G, M) = 0 can also be applied in this more general case after the above modification.

Remark 4.4. With the exception of 7.3, we will always assume that H acts trivially on A. This allows us to identify Hom_{G}(I, A) with the group of cocycles of Δ with values in A.

5. Basic properties of the Kolyvagin cup product

In the situation (4.1), the well known action of Δ on H^{i}(H, M) may be described as follows: If σ ∈ Δ and m ∈ H^{i}(H, M) ≅ Ext_{G}^{i}(ℤ[Δ], M), then σ(m) is the pullback of m under the automorphism of ℤ[Δ] given by τ → τσ. With the assumption that A = H^{0}(H, A), we also have a cup product A × H^{i}(H, M) → H^{i}(H, A ⊗ M) which may be described by sending a × m ∈ Ext_{G}^{0}(ℤ[Δ], A) × Ext_{G}^{i}(ℤ[Δ], M) to the pullback of a ∪ m ∈ Ext_{G}^{i}(ℤ[Δ] ⊗ ℤ[Δ], A ⊗ M) by the map ℤ[Δ] → ℤ[Δ] ⊗ ℤ[Δ] which sends τ to τ ⊗ τ.

Proposition 5.1. Let ψ : Δ → A be a one-cocycle, viewed as an element of Ext_{G}^{0}(I, A). Then the following diagram commutes:

Ext_{G}^{i}(Ǐ, M) ⟶^{ψ ∪ k •} H^{i}(G, A ⊗ M)
p^{∗}↓  ^{res}_{H}^{G} ↓
H^{i}(H, M) ⟶^{(1)} H^{i}(H, A ⊗ M),

where the map (1) is

x ∈ H^{i}(H, M) ⟶^{(1)} ∑_{σ∈Δ} ψ(σ) ∪ σ(x).

Proof. We start with ψ ∈ Hom_{G}(I, A) and m ∈ Ext_{G}^{i}(Ǐ, M), and form

ψ ∪ m ∈ Ext_{G}^{i}(I ⊗ Ǐ, A ⊗ M).

Going along the upper and right sides of the diagram,

res_{H}^{G}(ψ ∪ k m) ∈ H^{i}(H, A ⊗ M) ≅ Ext_{G}^{i}(ℤ[Δ], A ⊗ M)

is the pullback of ψ ∪ m by the map

ℤ[Δ] ⟶^{ε} ℤ ⟶^{γ} I ⊗ Ǐ. (5.1)

Going the other way, for each σ ∈ Δ, ψ(σ) ∈ H^{0}(H, A) is the pullback of ψ by f_{σ} : ℤ[Δ] → I defined by f_{σ}(τ) = t(σ − 1). Therefore, (1)(p^{∗}(m)) is the pullback of ψ ∪ m along the map ℤ[Δ] ⟶ ℤ[Δ] ⊗ ℤ[Δ] ⟶ I ⊗ ℤ[Δ] ⟶ I ⊗ Ǐ given by

τ ⟶ τ ⊗ τ ⟶ ∑_{σ∈Δ} τ(σ − 1) ⊗ τσ ⟶ ∑_{σ∈Δ} τ(σ − 1) ⊗ p(τσ) (5.2)

To see that (5.1) and (5.2) are the same it is enough to check τ = 1 where the equality is clear.

Remark 5.2. If i = 0, then the restriction map above is just the injection (A⊗M)^{G} ↪ (A⊗M)^{H}. It follows that in this case the Kolyvagin cup product is the same as the operation (1.1).

Proposition 5.3. The following diagram commutes:

Ext_{G}^{j}(I, A) × H^{i−1}(G, M) ⟶^{id ×δ_{M}^{(4.3)}} Ext_{G}^{j}(I, A) × Ext_{G}^{i}(Ǐ, M)
δ_{A}^{(4.2)}×id↓  ∪ k↓
H^{j+1}(G, A) × H^{i−1}(G, M) ⟶^{∪}  H^{i+j}(G, A ⊗ M)

Proof. The commutativity will follow if we show the commutativity of the following two diagrams:

Ext_{G}^{j}(I, A) × H^{i−1}(G, M) ⟶^{∪} Ext^{i+j−1}(I, A ⊗ M)
δ_{A}^{(4.2)}×id↓  ^{δ_{A⊗M}^{(4.2)}} ↓
H^{j+1}(G, A) × H^{i−1}(G, M) ⟶^{∪} H^{i+j}(G, A ⊗ M)

and

Ext_{G}^{j}(I, A) × H^{i−1}(G, M) ⟶^{id ×δ_{M}^{(4.3)}} Ext_{G}^{j}(I, A) × Ext_{G}^{i}(Ǐ, M)
∪↓  ^{∪}↓
Ext_{G}^{i+j−1}(I, A ⊗ M)  ⟶^{δ^{(4.7)}} Ext_{G}^{i+j}(Ǐ⊗ I, A ⊗ M)
∥  ^{γ^{∗}}↓
Ext_{G}^{i+j−1}(I, A ⊗ M)  ⟶^{δ_{A⊗M}^{(4.2)}}  H^{i+j}(G, A ⊗ M)

since the composition of the two maps on the right hand side is ∪ k. The map δ^{(4.7)} is the boundary map derived from the first row of (4.7). The commutativity of the first diagram and the top square of the second diagram follows from a standard argument in homological algebra. The commutativity of the bottom square follows from the commutativity of (4.7).

Let

0 ⟶ B ⟶^{i′} C ⟶^{p′} M ⟶ 0 (5.3)

be a short exact sequence of G-modules. Let B′ be the kernel of A⊗C → A⊗M, so that there is a surjection pr : A ⊗ B ↠ B′ and a short exact sequence,

0 ⟶ B′ ⟶ A ⊗ C ⟶ A ⊗ M ⟶ 0, (5.4)

and therefore a long exact sequence with a boundary map

H^{i+j}(G, A ⊗ M) ⟶^{δ^{(5.4)}} H^{i+j+1}(G, B′). (5.5)

Let m ∈ Ker H^{i}(H, M) ⟶^{cor_{H}^{G}} H^{i}(G, M) and let z ∈ Ext_{G}^{i}(Ǐ, M) satisfy p^{∗}(z) = m. Assume there is c ∈ H^{i}(H, C) s.t. p′_{∗}(c) = m. It follows that there is b ∈ H^{i}(G, B) s.t. i′_{∗}(b) = cor_{H}^{G}(c).

Proposition 5.4. In the situation described above, we may choose the class b ∈ H^{i}(G, B) such that the following holds: For any ψ ∈ Ext_{G}^{j}(I, A),

δ^{(5.4)}(ψ ∪ k z) = − pr_{∗}(δ_{A}^{(4.2)}(ψ) ∪ b)

Proof. Choose projective resolutions X^{•}, Y^{•} and Z^{•} of ℤ, ℤ[Δ] and Ǐ respectively in such a way that we may lift the short exact sequence (4.3) to a short exact sequence of complexes

0 ⟶ X^{•} ⟶^{i} Y^{•} ⟶^{p} Z^{•} ⟶ 0.

If we apply part 1 of lemma 3.1 to the doubly short exact sequence of complexes

0  0  0
↓  ↓  ↓
0 ⟶ Hom_{G}(Z^{•}, B) ⟶^{i′∘} Hom_{G}(Z^{•}, C) ⟶^{p′∘} Hom_{G}(Z^{•}, M) ⟶ 0
∘p↓  ^{∘p}↓  ^{∘p}↓
0 ⟶ Hom_{G}(Y^{•}, B) ⟶^{i′∘} Hom_{G}(Y^{•}, C) ⟶^{p′∘} Hom_{G}(Y^{•}, M) ⟶ 0
∘i↓  ^{∘i}↓  ^{∘i}↓
0 ⟶ Hom_{G}(X^{•}, B) ⟶^{i′∘} Hom_{G}(X^{•}, C) ⟶^{p′∘} Hom_{G}(X^{•}, M) ⟶ 0
↓  ↓  ↓
0  0  0

we find that we may choose b ∈ H^{i}(G, B) = H(Hom_{G}(X^{•}, B)) in such a way that

δ_{B}^{(4.3)}b = −δ^{(5.3)}z in Ext_{G}^{i+1}(Ǐ, B).

Therefore,

−ψ ∪ k δ^{(5.3)}z = ψ ∪ k δ_{B}^{(4.3)}b = δ_{A}^{(4.2)}(ψ) ∪ b,

where the second equality follows from proposition 5.3. A general argument about cup products gives

δ^{(5.4)}(ψ ∪ k z) = pr_{∗}(ψ ∪ k δ^{(5.3)}z)

and the result follows.

The next result is an analogue of the known formula for the composition of corestriction and restriction: cor_{H}^{G} ∘ res_{H}^{G}(x) = |G/H|x. Let n = |Δ| and assume that n annihilates both M and A. Let ψ : Δ → A be a 1-cocycle. Since nA = 0, the Kolyvagin trick shows that the element

a(ψ) := ∑_{σ∈Δ} ψ(σ)

is G-invariant. Recall that we have a map p^{∗} : Ext_{G}^{i}(Ǐ, M) → H^{i}(H, M) induced from the map p : ℤ[Δ] → Ǐ of (4.3).

Proposition 5.5. In the above situation, there is a map

u_{M,G} : H^{i}(G, M) → Ext_{G}^{i}(Ǐ, M)

satisfying

(1) p^{∗} ∘ u_{M,G} = res_{H}^{G} as maps from H^{i}(G, M) to H^{i}(H, M);

(2) The following diagram commutes:

Hom_{G}(I, A) × H^{i}(G, M) ⟶^{id ×u_{M,G}} Hom_{G}(I, A) × Ext_{G}^{i}(Ǐ, M)
a×id↓  ∪ k↓
H^{0}(G, A) × H^{i}(G, M) ⟶^{∪}  H^{i}(G, A ⊗ M).

Proof. Let u_{M,G} be the map u_{E} with the balanced sequence E associated to the short exact sequence (4.3) and the usual augmentation map as described before lemma 2.3. For this short exact sequence we denote the map ũ of (2.1) by ε̄. The first assertion is then clear from the remark at the end of definition 2.2. Since A is killed by n, the reduction map r_{I,n} : I → I/n induces an isomorphism r_{I,n}^{∗} : Hom_{G}(I/n, A) → Hom_{G}(I, A). If ϕ is in Hom_{G}(I/n, A), it is easy to see that a(r_{I,n}(ϕ)) ∈ H^{0}(G, A) is the pullback of ϕ under the map

w : ℤ → I/n defined by w(1) = ∑_{σ∈Δ}(σ − 1)

(notice that this element is invariant in I/n). According to lemma 2.3, u_{M,G} = ε̄^{∗} ∘ s_{ℤ,n}. We may therefore reformulate the proposition as saying that the diagram

[figure: Hom_{G}(I/n, A) × Ext_{G}^{i}(ℤ, M); Hom_{G}(I, A) × Ext_{G}^{i}(ℤ/n, M); Hom_{G}(ℤ, A) × Ext_{G}^{i}(ℤ, M); Hom_{G}(I, A) × Ext_{G}^{i}(Ǐ, M); Ext_{G}^{i}(ℤ ⊗ ℤ, A ⊗ M); Ext_{G}^{i}(I ⊗ Ǐ, A ⊗ M); Ext_{G}^{i}(ℤ, A ⊗ M); r_{I,n}^{∗}×s_{ℤ,n}; w^{∗}×id; id ×ε̄^{∗}; ∪; ∪; ≅; γ^{∗}]

commutes. In view of lemma 2.8, both ways of getting to Ext_{G}^{i}(ℤ, A ⊗ M) involve pulling back the same element via the two maps

ℤ ⟶^{≅} ℤ ⊗ ℤ ⟶^{w⊗id} I/n ⊗ ℤ and ℤ ⟶^{γ} I ⊗ Ǐ ⟶^{id ⊗ε̄} I ⊗ ℤ/n.

One checks directly that these maps are identical hence the result.

6. Higher derivatives

In this section we define, under certain restrictions, a notion of a derivative of order k,

\frac{∂^{k}}{∂f} m ∈ H^{i}(G, M), m ∈ Ker cor_{H}^{G} : H^{i}(H, M) → H^{i}(G, M),

for the cohomology class m with respect to “a function f of order k on Δ” in such a way that the Kolyvagin cup product defined in section 4 may be identified with the derivative \frac{∂^{1}}{∂ψ} m w.r.t. the cocycle ψ. Things become more complicated as more and more choices are involved. It is for this reason that we restrict to the following situation:

• G and H are the same as in section 4;

• R is a ring with a trivial G action;

• M is an R[G]-module;

• We fix i ≥ 0 and assume that for any finite R-module L with a trivial G action, Ext_{R[G]}^{i−1}(L, M) = 0 (in the most interesting case, in which i = 1, this condition amounts to M^{G} = 0).

We may replace ℤ by R in the setup of section 4. In particular, we redefine I (resp. Ǐ) to be the kernel (resp. the cokernel) of the obvious map ε : R[Δ] → R (resp. N : R → R[Δ]). Since

H^{•}(G, M) = Ext_{R[G]}^{•}(R, M) and H^{•}(H, M) = Ext_{R[H]}^{•}(R, M) ≅ Ext_{R[G]}^{•}(R[Δ], M),

The long exact sequence (4.8) is isomorphic to the sequence

… ⟶ H^{i−1}(G, M) ⟶ Ext_{R[G]}^{i}(Ǐ, M) ⟶ H^{i}(H, M) ⟶^{cor_{H}^{G}} H^{i}(G, M) ⟶ … (6.1)

We will need two different multiplications of R[Δ]. If we think of R[Δ] as the set of function f : Δ → R, the first is the convolution product, which is the one used to define the group ring structure on R[Δ], and will be denoted by ∗. The second is pointwise multiplication of functions. As before, we have a pairing

R[Δ] × R[Δ] ⟶ R, ⟨f, g⟩ = ∑_{σ∈Δ} f(σ)g(σ).

Definition 6.1. We define an increasing filtration

0 = F^{−∞} = ⋯ = F^{−1} ⊂ F^{0} ⊂ ⋯ ⊂ F^{∞} := ⋃_{r∈ℕ} F^{i}

on R[Δ] by setting, for r ≥ 1, F^{r−1}R[Δ] to be the annihilator of the ideal I^{∗r} (i.e. the r-th power, with respect to the usual product in the group ring, of the augmentation ideal)

Lemma 6.2. With respect to the filtration above:

(1) The function f ∈ R[Δ] is in F^{r+1}R[Δ] if and only if its image in R[Δ]/F^{r} is Δ-invariant.

(2) With respect to the pointwise product, F^{r}F^{s} ⊂ F^{r+s}.

Proof. (1) Suppose f ∈ F^{r+1} and let g ∈ I^{∗r}. Then

⟨σ(f) − f, g⟩ = ⟨σ(f), g⟩ − ⟨f, g⟩ = ⟨f, σ^{−1}(g)⟩ − ⟨f, g⟩ = ⟨f, (σ^{−1} − 1) ∗ g⟩ = 0,

the last equality because (σ^{−1} − 1) ∗ g ∈ I^{∗r+1}. Hence σ(f) − f ∈ F^{r}. Conversely, if σ(f) − f ∈ F^{r}, then the same equation shows that f annihilates every element of the form (σ^{−1} − 1) ∗ g with g ∈ I^{∗r}. Since I^{∗r+1} is generated by these elements the result follows.

(2) The case r = s = 0 is easy as F^{0} is just the set of constant functions. Given f ∈ F^{r} and g ∈ F^{s}, we find

σ(fg) − fg = σ(f)σ(g) − fg = (σ(f) − f)σ(g) + (σ(g) − g)f.

The result therefore follows by induction on r + s and part 1.

For any f ∈ F^{k} we associate the left ideal R[Δ] ∗ f filtered by

F^{r}(R[Δ] ∗ f) = F^{r} ∩ R[Δ] ∗ f

with r ≤ k. For such an r we denote the inclusion F^{r}(R[Δ] ∗ f) ⊂ R[Δ] by N_{f,r}. We then have a pullback map

H^{i}(H, M) ≅ Ext_{R[G]}^{i}(R[Δ], M) ⟶^{N_{f,r}^{∗}} Ext_{R[G]}^{i}(F^{r}(R[Δ] ∗ f), M)

We now define the higher derivatives. We first remark that by our assumptions it follows from part 1 of lemma 6.2 that Ext_{R[G]}^{i−1} (F^{k}(R[Δ] ∗ f), M) = 0 for all k.

Definition 6.3. Let m ∈ H^{i}(H, M) and let f ∈ F^{k}. Assume that N_{f,k−1}^{∗}(m) = 0. By the last remark, N_{f,k}^{∗}(m) pulls back uniquely to an element

z ∈ Ext_{R[G]}^{i−1} (F^{k}(R[Δ] ∗ f)/F^{k−1}(R[Δ] ∗ f), M) .

We define

\frac{∂^{k}}{∂f} m ∈ Ext_{R[G]}^{i}(R, M) ≅ H^{i}(G, M)

to be the pullback of z by the canonical G-map R → F^{k}(R[Δ] ∗ f)/F^{k−1}(R[Δ] ∗ f) sending 1 to f (mod F^{k−1}(R[Δ] ∗ f))

We leave it for the reader to verify the following two propositions:

Proposition 6.4. There is an inclusion H^{1}(Δ, R) ≅ Hom(Δ, R) ↪ F^{1} which induces F^{1}/F^{0} ≅ H^{1}(Δ, R). If z ∈ Ext_{G}^{i}(Ǐ, M) is the unique pullback of m ∈ H^{i}(G, M) and ψ ∈ Hom(Δ, R), then \frac{∂^{1}}{∂ψ} m is the push forward of ψ ∪ k z ∈ H^{i}(G, R ⊗ M) by the canonical map R ⊗ M → M.

Proposition 6.5. We have

res_{H}^{G} \frac{∂^{k}}{∂f} m = ∑_{σ∈Δ} f(σ)σ(m),

where δ acts on H^{i}(H, M) in the usual way.

It is possible to check the conditions for defining the k-th derivative by verifying the vanishing of all lower order derivatives. To be precise, we have

Proposition 6.6. If N_{f,k−1}^{∗}(m) = 0, then for any r < k and any g ∈ F^{r}(R[Δ] ∗ f) we have

\frac{∂^{r}}{∂g} m = 0.

Conversely, let, for r = 0, … k − 1,

f_{r,s} ∈ F^{r}(R[Δ] ∗ f), s = 1, … n(s),

be functions whose images generate F^{r}(R[Δ] ∗ f)/F^{k−1}(R[Δ] ∗ f). If \frac{∂^{r}}{∂f_{r,s}} m = 0 for all r and s, then N_{f,k−1}^{∗}(m) = 0.

We now discuss a few special cases:

1. Let f_{1}, …, f_{k} ∈ Hom(Δ, R). The R-module generated by all the monomials in f_{1}, …, f_{k} of degree ≤ k contains R[Δ] ∗ f_{1} · f_{2} ⋯ f_{k} and the r-th step in its filtration is generated by all the monomials of degree ≤ r. Thus, we may define inductively the derivative \frac{∂^{k}}{∂f_{1}·f_{2}···f_{k}} m if all the lower order derivatives \frac{∂^{r}}{∂f_{i_{1}} ···f_{ir}} m vanish for r < k.

2. A particular case of the above is when

• Δ = Δ_{1} × ⋯ Δ_{k} with projections π_{r} : Δ → Δ_{r} and Π_{r} : Δ → Δ̃_{r} := Δ_{1} × ⋯ × Δ̂_{r} × ⋯ × Δ_{k}. We let G_{r} be the preimage of Δ_{r} under the projection G → Δ;

• we have g_{r} ∈ Hom(Δ_{r}, R) and f_{r} = g_{r} ∘ π_{r};

• m ∈ H^{i}(H, M) is killed by all the partial corestriction, i.e., cor_{G_{r}}^{H} m = 0. In this situation, all the lower order derivatives vanish. Indeed, the partial corestrictions cor_{G_{r}}^{H} m may be described as pulling m ∈ Ext_{R[G]}^{i}(R[Δ], M) by the map Π_{r}^{∗} : R[Δ̃_{r}] → R[Δ] and the inclusion of the Rmodule generated by f_{1}, …, f̂_{s}, …, f_{k} in R[Δ] may be factored through Π_{s}. The derivative operator in this case is a higher dimensional analogue of the general Kolyvagin operator which is the composition of the derivative operators for all the G_{r}.

7. The Kolyvagin cap product

In this section we construct a new operation which is dual to the Kolyvagin cup product we have constructed in section 4. The setup is the same as that of section 4.

Definition 7.1. The Kolyvagin cap product is the cup product

Ext_{G}^{j}(I, A) × H^{i}(G, M) ⟶ Ext_{G}^{i+j}(I, A ⊗ M).

We denote this product by ψ × x ⟶ ψ ∩ k x.

As usual, we’ll be mostly interested in the case j = 0 where ψ is just a one-cocycle with values in A. The short exact sequence (4.2) gives rise to

(7.1) 0 ⟶ Coker [H^{i}(G, A ⊗ M) ⟶^{res_{H}^{G}} H^{i}(H, A ⊗ M)] ⟶^{i^{∗}} Ext_{G}^{i}(I, A ⊗ M)

⟶^{δ_{A⊗M}^{(4.2)}} Ker [H^{i+1}(G, A ⊗ M) ⟶^{res_{H}^{G}} H^{i+1}(H, A ⊗ M)] ⟶ 0.

We now show that the operation just defined coincides with the one defined by Rubin in [Rub93, §3] when i = 0 as follows: Let x ∈ M^{G} = H^{0}(G, M) and let ψ ∈ Ext_{G}^{0}(I, A) be a one cocycle ψ : Δ → A. Assume there is y ∈ M^{H} satisfying N_{G/H} y = x. One can now form

ψ ∩ k x = ∑_{σ∈Δ} ψ(σ) ⊗ σ(y) (mod (A ⊗ M)^{G}) ∈ (A ⊗ M)^{H}/(A ⊗ M)^{G}.

if N_{G/H} y = 0, then the basic Kolyvagin trick shows that ∑_{σ∈Δ} ψ(σ) ⊗ σ(y) is G-invariant, hence the operation is well defined.

The next proposition shows that the above operation, when defined, is given by the Kolyvagin cap product.

Proposition 7.2. The following diagram commutes:

H^{i}(H, M) ⟶^{cor_{H}^{G}} H^{i}(G, M)
(1)↓  _{ψ ∩ k •}↓
H^{i}(H, A ⊗ M) ⟶^{i^{∗}} Ext_{G}^{i}(I, A ⊗ M)

where (1) is the map

y ⟶ − ∑_{σ∈Δ} ψ(σ) ∪ σ(y)

Proof. As in the proof of Proposition 5.1, we start with m ∈ Ext_{G}^{i}( ℤ[Δ], M) and form ψ ⊗ m ∈ Ext_{G}^{i}(I ⊗ ℤ[Δ], A⊗M). Going along the upper and right side of the diagram (resp. the left and lower sides) corresponds to pulling back ψ ∪ m by

\begin{matrix}I & ⟶^{≅} & I ⊗ ℤ & ⟶^{id ⊗ N} & I ⊗ ℤ[Δ] \\ τ − 1 & ⟶ & (τ − 1) ⊗ 1 & ⟶ & (τ − 1) ⊗ (∑_{σ∈Δ} σ) \end{matrix}

(resp.

\begin{aligned}I & ⟶^{i} & ℤ[Δ] & ⟶ & I ⊗ ℤ[Δ] \\ & & τ & ⟶ & − ∑_{σ∈Δ} τ(σ − 1) ⊗ τσ). \end{aligned}

It thus remains to observe that

(τ − 1) ⊗ (∑_{σ∈Δ} σ) = − ∑_{σ∈Δ} τ(σ − 1) ⊗ τσ

We now show that the Kolyvagin cap product is dual, in a sense we shall define soon, to the Kolyvagin cup product defined in section 4. To motivate what follows, recall from [Mil86, ch. 1] that Tate duality is defined by the pairing

⟨ , ⟩ : H^{k}(G, M) × H^{i}(G, M̂(1)) ⟶^{∪} H^{2}(G, μ_{p^{∞}}) ⟶^{inv_{G}} ℚ_{p}/ℤ_{p}. (7.2)

Here, G is the Galois group of K̄/K, where K is a nonarchimedean local field (we’ll refer to this type of groups as local Galois groups), k and i are nonnegative integers summing to 2, μ_{p^{∞}} is the G-module of p-power roots of unity in K, M̂(1) := Hom(M, μ_{p^{∞}}) is the Kummer dual of M and inv_{G} is the invariant map of local class field theory. When H is a normal group of finite index in G, it is also a local Galois group, so the same duality holds for H. It is known that the invariant commutes with corestriction, i.e., if x ∈ H^{2}(H, μ_{p^{∞}}), then

inv_{G}(cor_{H}^{G}x) = inv_{H}(x).

Our next result shows that the Kolyvagin cup product may be considered as giving a new pairing which is closely connected to the pairing in the Tate duality.

Proposition 7.3. Let A and M be G-modules (we are not assuming now that A has a trivial H action!). Let p^{∗} : Ext_{G}^{i}(Ǐ, M) → H^{i}(H, M) and i^{∗} : H^{k}(H, A) → Ext_{G}^{k}(I, A) be the maps induced on cohomology from the natural maps p and i in (4.2) and (4.3). Then, p^{∗} and i^{∗} are “adjoint” with respect to the usual cup product and the Kolyvagin cup product, i.e., for x ∈ Ext_{G}^{i}(Ǐ, M) and y ∈ H^{k}(H, A) one has

i^{∗}(y) ∪ k x = cor_{H}^{G}(y ∪ p^{∗}(x)).

Proof. The result follows from the commutativity of the diagram:

[figure: ℤ; ℤ[Δ]; ℤ[Δ] ⊗ ℤ[Δ]; I ⊗ Ǐ; ℤ[Δ] ⊗ Ǐ; N; γ; (1); id ⊗p; i⊗id]

where the map (1) sends σ ∈ ℤ[Δ] to σ ⊗ σ.

Definition 7.4. suppose that G is a local Galois group and that M is a G-module. Define the Kolyvagin pairing,

⟨ , ⟩_{K} : Ext_{G}^{i}(I, M̂(1)) × Ext_{G}^{k}(Ǐ, M) ⟶ ℚ_{p}/ℤ_{p},

with i + k = 2, to be the composition of the Kolyvagin cup product with the invariant map.

Corollary 7.5. If, in the situation of Proposition 7.3, G is a local Galois group, i + k = 2 and A = M̂(1), then one has

⟨i^{∗}(y), x⟩_{K} = ⟨y, p^{∗}(x)⟩.

We can now state the adjointness of the Kolyvagin cup and cap products:

Proposition 7.6. Let M, N and A be G-modules and let ψ ∈ Ext_{G}^{j}(I, A). Then the following diagram commutes:

H^{k}(G, N) × Ext_{G}^{i}(Ǐ, M)  ⟶^{id ×(ψ ∪ k •)} H^{k}(G, N) × H^{i+j}(G, A ⊗ M)
(ψ ∩ k •)×id↓  ^{∪}↓
Ext_{G}^{k+j}(I, A ⊗ N) × Ext_{G}^{i}(Ǐ, M) \underrightarrow{∪ k}  H^{i+j+k}(G, A ⊗ N ⊗ M).

Proof. Follows from the commutativity of

ℤ ⊗ Ǐ⊗ I ⟵^{id ⊗γ} ℤ ⊗ ℤ
↑  ∥
Ǐ⊗ I ⟵^{γ} ℤ

Corollary 7.7. Let G be a local Galois group and let A and M be G-modules with a trivial action of H on A. Let i and k be two nonnegative integers with i + k = 2. Let ψ be a one-cocycle of Δ with values in A. Let x ∈ H^{k}(G, { \atop A ⊗ M}(1)) and y ∈ Ext_{G}^{i}(Ǐ, M). Then,

(1) We have

⟨x, ψ ∪ k y⟩ = ⟨ψ ∩ k x, y⟩_{K},

where we used the identification A⊗{ \atop (A ⊗ M)}(1) ≅ M̂(1) to view ψ ∩ k x as an element of Ext_{G}^{i}(I, M̂(1)).

(2) If ψ ∩ k x = i^{∗}(z) with z ∈ H^{i}(H, M̂(1)), then

⟨x, ψ ∪ k y⟩ = ⟨z, p^{∗}(y)⟩.

(3) In particular, if x = cor_{H}^{G}(x′), with x′ ∈ H^{k}(H, { \atop A ⊗ M}(1)), then

⟨x, ψ ∪ k y⟩ = −⟨∑_{σ∈Δ} ψ(σ) ⊗ σ(x′), p^{∗}(y)⟩.

Proof. All the results are immediate consequences of Propositions 7.6, 7.3 and 7.2.

8. Localization of the Kolyvagin cup product

In this section, we describe a method which, under certain conditions, allows one to compute the localization of the Kolyvagin cup product of given cohomology classes. The result is inspired by the computations of Nekovář in [Nek92].

The setup is as follows:

• We have the short exact sequence of groups (4.1): 1 ⟶ H ⟶ G ⟶ Δ ⟶ 1 with Δ of order n;

• G_{l} (l for local) is subgroup of G. We let H_{l} = H ∩ G_{l} and demand that G_{l}/H_{l} ≅ Δ;

• A and M are G-modules killed by n, and A has a trivial H action;

• M sits in a short exact sequence of G-modules 0 ⟶ B ⟶^{i′} C ⟶^{p′} M ⟶ 0;

• We are given i > 0, and cohomology classes m ∈ H^{i}(H, M), c ∈ H^{i}(H, C) and c_{l} ∈ H^{i}(G_{l}, C) s.t.

m = p′_{∗}c and res_{H_{l}}^{H}c = res_{H_{l}}^{G_{l}}c_{l};

• We have a 1-cocycle ψ : Δ → A. As in the setup of lemma 5.4, we can choose z ∈ Ext_{G}^{i}(Ǐ, M) s.t. p^{∗}(z) = m, and we can form the Kolyvagin cup product ψ ∪ k z ∈ H^{i}(G, A ⊗ M). Our aim in this section is to compute res_{G_{l}}^{G}(ψ ∪ k z)

As in lemma 5.4, we find a class b ∈ H^{i}(G, B), with i′_{∗}(b) = cor_{H}^{G}(c). In addition, since p′_{∗}(nc) = nm = 0, we can also find b′ ∈ H^{i}(H, B) such that i′_{∗}(b′) = nc. It follows from the assumption G/H ≅ G_{l}/G_{l} that res_{G_{l}}^{G} ∘ cor_{H}^{G} = cor_{H_{l}}^{G_{l}} ∘ res_{H_{l}}^{H}, and therefore

res_{H_{l}}^{G}cor_{H}^{G}c = res_{H_{l}}^{G_{l}}res_{G_{l}}^{G}cor_{H}^{G}c = res_{H_{l}}^{G_{l}}cor_{H_{l}}^{G_{l}}res_{H_{l}}^{H}c
= res_{H_{l}}^{G_{l}}cor_{H_{l}}^{G_{l}}res_{H_{l}}^{G_{l}}c_{l} = res_{H_{l}}^{G_{l}}nc_{l} = nres_{H_{l}}^{H}c.

Let b_{1} = res_{H}^{G}b − b′ ∈ H^{i}(H, B). The above computation shows that

i′_{∗}(res_{H_{l}}^{H}b_{1}) = res_{H_{l}}^{H}(res_{H}^{G}cor_{H}^{G}c − nc) = 0,

and so

res_{H_{l}}^{H}b_{1} = δ^{(5.3)}m_{1} for some m_{1} ∈ H^{i−1}(H_{l}, M).

Proposition 8.1. There exist a choice of b and b′ above such that the following holds: There exist m_{2} ∈ H^{i−1}(G_{l}, M) with

m_{1} = res_{H_{l}}^{G_{l}}m_{2} and res_{G_{l}}^{G}(ψ ∪ k z) = a(ψ) ∪ p′_{∗}(c_{l}) + m_{2} ∪ [ψ],

where a(ψ) = ∑_{σ∈Δ} ψ(σ) and [ψ] denotes as usual the cohomology class of ψ.

Proof. The condition G/H ≅ G_{l}/G_{l} shows that when viewing (4.3) as a sequence of G_{l}-modules it will give the long exact sequence (4.8) with G and H replaced by G_{l} and H_{l}. We take the projective resolutions X^{•}, Y^{•} and Z^{•} that were used in the proof of proposition 5.4, and form the same doubly exact sequence of complexes as in that proof but replacing Hom_{G} by Hom_{G_{l}}. According to remark 2.5, we can lift the augmentation map ε : ℤ[Δ] → ℤ to a map of complexes

ε : Y^{•} → X^{•} such that ε ∘ i = multiplication by n.

It is now easy to check that we may apply part 2 of lemma 3.1 with the map π being multiplication by n. The lemma gives the existence of m_{2} such that

m_{1} = res_{H_{l}}^{G_{l}}m_{2} and z = δ_{M}^{(4.3)}m_{2} + u_{M,G_{l}}p′_{∗}c_{l}.

with u_{M,G_{l}} being the map defined in proposition 5.5. Propositions 5.3 and 5.5 now give the required result.

To make a more explicit computation we now add further assumptions:

(1) a(ψ) = 0 (this is for instance true if Δ or A have no 2-torsion and A has a trivial G action);

(2) res_{H_{l}}^{G_{l}} is an isomorphism on H^{i}(G_{l}, C) and H^{i}(G_{l}, B);

(3) There is an operator T, acting in a compatible way on B, C and M, T M = 0, and we have x ∈ H^{i}(G, C) with cor_{H}^{G}c = T x;

(4) There is an operator Fr(l), acting compatibly on the H_{l} cohomology with coefficients in C, B and M, and res_{H_{l}}^{H}c = Fr(l)res_{H_{l}}^{G}x;

(5) The cohomology groups H^{i−1}(G, M) and H^{i−1}(G_{l}, B) vanish.

Remark 8.2. The reader should have in mind the following situation: K/F is some finite extension of number fields G = Gal(ℚ̄/F), H = Gal(ℚ̄/K). l is some prime of F which is totally ramified in K and G_{l} is the decomposition group of l (notice that G/H ≅ G_{l}/H_{l}). C is a ℤ_{p}-free G-module which has pure weight different from 0 and with good reduction at l. p is assumed not to lie below l. B = p^{k}C. i = 1. Condition (2) is then satisfied because the weight assumptions imply, according to [Nek92, Lemma 4.1], that H^{1}(G_{l}, C) ≅ H^{1}(G_{l}/I_{l}, C), with I_{l} the inertia at l (one should probably be a little more careful and replace C with its quotient by a sufficiently large power of p), and the same is true with G_{l} replaced by H_{l} and C replaced by B, and because K_{l}/F_{l} is totally ramified. The conditions (3) and (4) are typical conditions for Euler systems of elliptic type (see for instance [Nek92, Proposition 6.1]).

Because of the condition (2) we can define Fr(l) on H^{i}(G_{l}, C) and H^{i}(G_{l}, B). Denote by \frac{T}{i′} : C → B the composition (i′)^{−1} ∘ T which is well defined since by assumptions T has values in the image of i′. Define \frac{n}{i′} similarly. Since H^{i−1}(G_{l}, B) = 0, the boundary map δ^{(5.3)} : H^{i−1}(G_{l}, M) → H^{i}(G_{l}, C) is injective and we denote by δ^{−1} its inverse when defined. Define χ_{ψ} from a subspace of H^{i}(G_{l}, C) to H^{i}(G_{l}, A ⊗ M) by

χ_{ψ}(x_{l}) = ψ ∪ δ^{−1} ((\frac{T}{i′} )_{∗} − (\frac{n}{i′} )_{∗} Fr(l)) x_{l} whenever it’s defined ,

i.e., whenever (\frac{T}{i′} )_{∗} x_{l} − (\frac{n}{i′} )_{∗} Fr(l)x_{l} is in the image of δ^{(5.3)}. As usual, we can choose z ∈ Ext_{G}^{i}(Ǐ, M)

such that p^{∗}(z) = m. Since H^{i−1}(G, M) = 0, z is in fact unique.

Corollary 8.3. In the situation described above, χ_{ψ} is defined on res_{G_{l}}^{G}x and we have

res_{G_{l}}^{G}(ψ ∪ k z) = χ_{ψ}(res_{G_{l}}^{G}x).

Proof. The assumption H^{i−1}(G, M) = 0 implies that the choice of b and b′ in proposition 8.1 is unique. Therefore,

b = (\frac{T}{i′} )_{∗} x and b′ = (\frac{n}{i′} )_{∗} c.

From (4) we deduce

res_{H_{l}}^{H}b′ = res_{H_{l}}^{H} (\frac{n}{i′} )_{∗} c = Fr(l) (\frac{n}{i′} )_{∗} res_{H_{l}}^{G}x

and therefore

δ^{(5.3)}res_{H_{l}}^{G_{l}}m_{2} = res_{H_{l}}^{G}b − res_{H_{l}}^{H}b′ = (\frac{T}{i′} )_{∗} res_{H_{l}}^{G}x − Fr(l) (\frac{n}{i′} )_{∗} res_{H_{l}}^{G}x

Since res_{H_{l}}^{G_{l}} is an isomorphism, we have

δ^{(5.3)}m_{2} = ((\frac{T}{i′} )_{∗} − (\frac{n}{i′} )_{∗} Fr(l)) res_{G_{l}}^{G}x

and the result is clear.

References

[Dar92] H. Darmon, A refined conjecture of Mazur-Tate type for Heegner points. Invent. math. 110 (1992), 123–146.

[Mil86] J.S. Milne, Arithmetic duality theorems. Perspectives in Mathematics. Academic Press, 1986.

[Nek92] J. Nekovář, Kolyvagin’s method for Chow groups of Kuga-Sato varieties. Invent. math. 107 (1992), 99–125.

[Rub93] K. Rubin, Abelian varieties, p-adic heights and derivatives. preprint 033-93, MSRI, January 1993.

University of California, Department of Mathematics, 405 Hilgard Avenue, Los Angeles, California 90024-1555

   E-mail address: besser@math.ucla.edu