CHAPTER 6 Lecture 6

1. Duality Preliminaries

1.1. Pontrjagin Duality. As always, let K be a local field with residue field k of characteristic p. Let G_{K} and 𝔤_{k} be the absolute Galois groups Gal(K̄/K) and Gal(\overline{k}/k) respectively. We let M be a finite G_{K}-module (where the G_{K}-action is continuous with respect to the discrete topology on M, of course).

We define two duals of M. The first is the Pontrjagin dual

M^{∨} = Hom_{ℤ}(M, ℚ/ℤ),

the Hom being as abelian groups. M^{∨} receives the usual adjoint G_{K}-action; in this case, since G_{K} acts trivially on ℚ/ℤ, this means that for any f : M → ℚ/ℤ and σ ∈ G_{K}, σf : M → ℚ/ℤ is given by

(σf)(m) = f(σ^{−1}m)

for all m ∈ M. There is a natural perfect pairing of abelian groups

M ⊗_{ℤ} M^{∨} → ℚ/ℤ,

given by

m ⊗ f ↦ f(m)

for m ∈ M and f ∈ M^{∨}. (Recall that for a pairing M ⊗_{A} N → P to be a perfect pairing of A-modules means that the induced maps

M → Hom_{A}(N, P)

and

N → Hom_{A}(M, P)

are isomorphisms.) The pairing is perfect in this case because M is a finite abelian group, and thus (loosely speaking) every element of M can find something in ℚ/ℤ to map to. In fact, if M ⊗ M^{∨} is given the usual induced G_{K}-action, one easily checks that this pairing, and thus the induced maps as well, are G_{K}-equivariant. Note also that (M^{∨})^{∨} is canonically isomorphic to M as G_{K}-modules.

Now, let H be an arbitrary subgroup of G_{K}. Then M is an H-module in the obvious way, and one easily checks directly from the definitions that (M^{H})^{∨} = (M^{∨})_{H} and (M_{H})^{∨} = (M^{∨})^{H}, where M^{H} and M_{H} are the H-invariants and coinvariants respectively. (These two equalities are actually formally equivalent; simply replace M by M^{∨} in either to get the other one.) Put differently, our above perfect pairing “restricts” to perfect pairings

M^{H} ⊗ (M^{∨})_{H} → ℚ/ℤ

0Last modified February 23, 1998

and

M_{H} ⊗ (M^{∨})^{H} → ℚ/ℤ.

1.2. Cartier Duality. The Cartier dual of M is a G_{K}-module M^{∗} which is isomorphic to M^{∨} as abelian groups, but with a different G_{K}-action. Precisely, we define

M^{∗} = Hom_{ℤ}(M, μ(K̄)),

where μ(K̄) is the group of roots of unity in K̄. The fact that M^{∨} ≅ M^{∗} (noncanonically) as abelian groups follows from the fact that there are non-canonical isomorphims ℚ/ℤ ≅ μ(K̄). The difference is that μ(K̄) is a non-trivial G_{K}-module, so the induced adjoint G_{K}-action on M^{∗} is different from that on M^{∨}: for any f : M → μ(K̄) and σ ∈ G_{K}, we have

(σf)(m) = σf(σ^{−1}m)

for all m ∈ M, the outside action being that of G_{K} on μ(K̄). We still have (M^{∗})^{∗} = M as G_{K}-modules.

Just as in the Pontrjagin case, we have a canonical perfect pairing

M ⊗_{ℤ} M^{∗} → μ(K̄)

given by evaluation. It is again easily checked to be G_{K}-equivariant. In this case, however, the pairing does not always restrict to subgroups. Specifically, let H be any subgroup of G_{K}. Then the direct argument from the definitions shows only that (M^{H})^{∗} = (M^{∗})_{H} and (M_{H})^{∗} = (M^{∗})^{H} if H acts trivially on μ(K̄). However, with a little more work we can recover these results in the following special case.

Lemma 1.1. Suppose that M is a finite G_{K}-module of prime to p order, where p is the residue characteristic of k. Then

(M^{I_{K}})^{∗} = (M^{∗})_{I_{K}}

and

(M_{I_{K}})^{∗} = (M^{∗})^{I_{K}} ,

where I_{K} is the inertia subgroup of G_{K}.

Proof. Let μ′(K̄) be the subgroup of μ(K̄) of roots of unity of prime to p order. Using the decomposition

μ(K̄) = μ′(K̄) × μ_{p^{∞}}(K̄)

we get a decomposition

M^{∗} = Hom(M, μ(K̄)) = Hom(M, μ′(K̄)) × Hom(M, μ_{p^{∞}}(K̄)),

and the last of these groups is zero, since M has prime to p order. Thus

M^{∗} = Hom(M, μ′(K̄)).

Now, since K(μ′(K̄)) is unramified over K, I_{K} acts trivially on μ′(K̄). The usual argument now extends to this case to finish the proof.

1.3. Tate Modules. In order to clarify the relationship between M^{∨} and M^{∗} we will now present formally a subject which we have touched on before. Let l be a prime number (possibly equal to p), and let W be a l-divisible, l-torsion abelian group. Thus

W = ⋃_{n} W [l^{n}],

where W [l^{n}] is the l^{n}-torsion in W. We define a functor Ta_{l} from the category of such groups to the category of ℤ_{l}-modules by defining

Ta_{l}(W) = \underleftarrow{lim}_{n} W [l^{n}],

the inverse limit being with respect to the maps

W [l^{n+1}] ⟶^{l} W [l^{n}]

given by multiplication by l. These are surjective since W is l-divisible. Ta_{l}(W) is actually a ℤ_{l}-module since each W [l^{n}] is a ℤ/l^{n}ℤ-module. In fact, if W [l] is finite, then it follows easily that Ta_{l}(W) is a ℤ_{l}-module of finite rank equal to |W [l]|/l. (The relevant structure of W is actually pretty much determined by W [l], as can be seen just by staring at it for a moment.) It is also easy to see that if W is a G_{K}-module, then Ta_{l}(W) is also a G_{K}-module. However, it is not true that Ta_{l}(W) need be discrete, even if W was.

The canonical example is, of course,

Ta_{l}(ℚ_{l}/ℤ_{l}) = ℤ_{l};

it follows immediately that

Ta_{l}((ℚ_{l}/ℤ_{l})^{r}) = ℤ_{l}^{r},

the free ℤ_{l}-module of rank r. Since μ_{l^{∞}}(K̄) is isomorphic to ℚ_{l}/ℤ_{l} as abelian groups, Ta_{l}(μ_{l^{∞}}(K̄)) must be a free ℤ_{l}-module of rank 1. In fact, one checks easily (since all of the G_{K}-actions involved essentially come from μ_{l^{∞}}(K̄)) that

Ta_{l}(μ_{l^{∞}}(K̄)) ≅ ℤ_{l}(1).

(See Lecture 3, Section 3.2.) In the case l = p we use this to define the module ℤ_{p}(1), which we had not previously defined.

If M is any pro-l G_{K}-module, we define its Tate twist by

M(1) = M ⊗_{ℤ_{l}} ℤ_{l}(1).

More generally, we define inductively

M(n) = M(n − 1) ⊗_{ℤ_{l}} ℤ_{l}(1)

for any positive integer n. We also define

ℤ_{l}(−1) = Hom_{ℤ_{l}}(ℤ_{l}(1), ℤ_{l}),

where the ℤ_{l} in the range has trivial G_{K}-action (this makes sense if one considers the induced G_{K}-action on ℤ_{l}(−1)), and correspondingly

M(−1) = M ⊗_{ℤ_{l}} ℤ_{l}(−1) = Hom_{ℤ_{l}}(ℤ_{l}(1), M)

and

M(n) = M(n + 1) ⊗_{ℤ_{l}} ℤ_{l}(−1)

for any negative integer n.

This entire theory generalizes to the case where one considers all primes l at once. Specifically, we define a functor Ta from torsion, divisible groups to ℤ̂-modules (which are just profinite abelian groups) by

Ta(W) = \underleftarrow{lim}_{N} W [N],

where the maps of the inverse system are the multiplication by m maps

W [mN] ⟶^{m} W [N];

these are surjective since W is divisible. As above, Ta(W) inherits a G_{K}-action from W if W is a G_{K}-module.

As an example, we find

Ta(ℚ/ℤ) = ℤ̂

canonically, and

Ta(μ(K̄)) = ∏_{l prime} ℤ_{l}(1)

which we will denote simply ℤ̂(1). Suppressing the p-part, we find that we have essentially recovered the Galois group Δ_{k}.

If M is any profinite G_{K}-module, we define its Tate twist to be

M(1) = M ⊗ℤ̂ ℤ̂(1).

Also defining

M(−1) = Homℤ̂(ℤ̂(1), M),

we proceed as above to get G_{K}-modules M(n) for all integers n. In the case that M is pro-l it is easy to check that these definitions agree with our earlier ones.

It is now immediate from these definitions that if M is any finite G_{K}-module, the Pontrjagin and Cartier duals are related by

M^{∗} = M^{∨}(1).

1.4. Cohomology of Topologically Cyclic Profinite Groups. Let C be a profinite group which is (not necessarily canonically) isomorphic to

∏_{l∈L} ℤ_{l}

for some set of prime numbers L. Let M be a finite discrete C-module, with continuous C-action. If C were actually canonically isomorphic to the product of ℤ_{l}’s, meaning that it has a chosen generator γ, then there is a canonical isomorphism

H^{1}(C, M) = M_{C} = M/(γ − 1)M,

given by sending a cocycle φ : C → M to φ(γ). (This adapts easily from [Se-LF, Chapter 13, Section 1].) In the case where C has no chosen generator, we can still obtain a canonical isomorphism via the map

C ⊗ℤ̂ H^{1}(C, M) → M_{C}

given by sending a pair c ⊗ φ of an element of C and a cocycle to the class of φ(c) in M_{C}. That this is an isomorphism follows easily either from the fact that the non-canonical version is, or else it can also be seen easily from the finite cyclic case.

One can actually “solve” for H^{1}(C, M) in the above equation in the case where M has order divisible only by primes in L. Simply take the tensor product of both sides with Homℤ̂(C, ∏_{l∈L} ℤ_{l}). This yields an isomorphism

H^{1}(C, M) = Homℤ̂(C, M_{C}),

since Hom_{Zhat}(C, ∏_{l∈L} ℤ_{l}) ⊗ ℤ̂C = ∏_{l∈L} ℤ_{l}.

2. Tate Local Duality

2.1. An Example : Local Class Field Theory. We keep the notation of the last section. Recall that Tate local duality states that the cohomology groups H^{1}(G_{K}, M) and H^{1}(G_{K}, M^{∗}) are finite, and that there is a perfect pairing

H^{1}(G_{K}, M) ⊗ H^{1}(G_{K}, M^{∗}) → ℚ/ℤ

given by cup product, Cartier duality and the local invariant map. (See Lecture 5, Section 3.)

We begin by working through the implications in an example. Let N be a positive integer and set M = ℤ/Nℤ, with trivial G_{K}-action. Then

M^{∗} = Hom(ℤ/Nℤ, μ(K̄)) = Hom(ℤ/Nℤ, μ_{N}(K̄)) = μ_{N}(K̄),

where μ_{N}(K̄) is the group of N^{th} roots of unity in K̄. Thus in this case Tate local duality gives a perfect pairing

H^{1}(G_{K}, ℤ/Nℤ) ⊗ H^{1}(G_{K}, μ_{N}(K̄)) → ℚ/ℤ;

in fact, the image must land in \frac{1}{N} ℤ/ℤ, since both groups are N-torsion. Since ℤ/Nℤ has trivial G_{K}-action, the first of these cohomology groups is given by

H^{1}(G_{K}, ℤ/Nℤ) = Hom(G_{K}, ℤ/Nℤ) = Hom(G_{K}^{ab}, ℤ/Nℤ).

We can compute H^{1}(G_{K}, μ_{N}(K̄)) using Kummer theory. We begin with the exact sequence of G_{K}-modules

0 → μ_{N}(K̄) → K̄^{∗} →^{N} K̄^{∗} → 0.

The long exact sequence of cohomology gives us an exact sequence

0 → K^{∗}/(K^{∗})^{N} → H^{1}(G_{K}, μ_{N}(K̄)) → H^{1}(G_{K}, K̄^{∗})[N] → 0.

But H^{1}(G_{K}, K̄^{∗}) = 0 by Hilbert’s Theorem 90. (See [Se-LF, Chapter 10, Section 1, Proposition 2]. Alternatively, if one thinks of H^{1}(G_{K}, K̄^{∗}) as classifying isomorphism classes of one-dimensional K-vector spaces which become isomorphic over K̄, the result is clear, since isomorphism classes of vector spaces are completely determined by dimension.) Thus

K^{∗}/(K^{∗})^{N} ⟶^{∼} H^{1}(G_{K}, μ_{N}‘(K̄^{∗}).

Combining these results, we have a perfect pairing

Hom(G_{K}^{ab}, ℤ/Nℤ) ⊗ K^{∗}/(K^{∗})^{N} → ℚ/ℤ.

Equivalently, we have an isomorphism

Hom(G_{K}^{ab}, ℤ/Nℤ) ≅ Hom(K^{∗}/(K^{∗})^{N}, ℚ/ℤ).

Taking the direct limit over N yields

Hom(G_{K}^{ab}, ℚ/ℤ) ≅ Hom(K̂^{∗}, ℚ/ℤ),

where K̂^{∗} is the profinite completion of K^{∗}. (K^{∗} is not already complete; it is isomorphic to U × ℤ, where U is the group of units. U is complete, but ℤ is not.) Taking Pontrjagin duals finally yields the isomorphism

G_{K}^{ab} ≅ K̂^{∗}.

Thus we have obtained the main isomorphism of local class field theory from Tate local duality. However, there are two flaws with this approach. First, local class field theory is an essential ingredient in most proofs of Tate local duality, so the argument is circular. (In fact, in spirit a good deal of the content of the proof of Tate local duality consists in compiling the known reciprocity isomorphisms of local class field theory.) Second, it would require a lot of effort to relate the above computation to the usual reciprocity isomorphism, which is a key ingredient of the theory.

2.2. Restrictions of Tate Local Duality : First Attempt. We now make two assumptions on the finite G_{K}-module M. First, we assume that it is of prime to p order. Second, we assume that the G_{K}-action on M is at worst tamely ramified; that is, we assume that the wild ramification group P_{K} acts trivially on M. Thus G_{K} acts on M through G_{K}/P_{K} = T_{K}, in our usual notation.

We will refer to the inflation-restriction exact sequence

0 → H^{1}(𝔤_{k}, M^{I_{K}}) → H^{1}(G_{K}, M) → H^{1}(I_{K}, M)^{𝔤_{k}} → 0

as the basic exact sequence. (It is exact on the right since H^{2}(𝔤_{k}, M^{I_{K}}) = 0, as M is torsion.) We are interested in relating this exact sequence with Tate local duality.

Recall that we have an exact sequence

1 → Δ_{k} → T_{K} → 𝔤_{k} → 1,

where Δ_{k} is the image of I_{K} in T_{K}. We claim that we have an isomorphism

H^{1}(Δ_{k}, M) ≅ H^{1}(I_{K}, M).

To see this, we simply write down a portion of the inflation-restriction sequence associated to the exact sequence

1 → P_{K} → I_{K} → Δ_{k} → 1;

specifically, consider

0 → H^{1}(Δ_{k}, M^{P_{K}}) → H^{1}(I_{K}, M) → H^{1}(P_{K}, M).

By Lecture 5, Lemma 2.4, H^{1}(P_{K}, M) = 0 since P_{K} is pro-p and M has prime to p order. Since by hypothesis P_{K} acts trivially on M, we obtain the desired isomorphism.

Thus we can rewrite our basic exact sequence as

0 → H^{1}(𝔤_{k}, M^{Δ_{k}}) → H^{1}(G_{K}, M) → H^{1}(Δ_{k}, M)^{𝔤_{k}} → 0.

We now use the results of Section 1.4 to compute the outside terms. Since 𝔤_{k} ≅ ℤ̂ canonically, we have a canonical isomorphism

H^{1}(𝔤_{k}, M^{Δ_{k}}) ≅ (M^{Δ_{k}})_{𝔤_{k}}.

Since Δ_{k} ≅ ∏_{l≠p} ℤ_{l}(1) and M has prime to p order, we have a canonical isomorphism

H^{1}(Δ_{k}, M)^{𝔤_{k}} ≅ Hom(Δ_{k}, M_{Δ_{k}})^{𝔤_{k}} ≅ Hom (∏_{l≠p} ℤ_{l}(1), M_{Δ_{k}} )^{𝔤_{k}} ≅ (M_{Δ_{k}}(−1))^{𝔤_{k}}.

Our basic exact sequence now takes the form

0 → (M^{Δ_{k}})_{𝔤_{k}} → H^{1}(G_{K}, M) → (M_{Δ_{k}}(−1))^{𝔤_{k}} → 0.

We now rewrite the Pontrjagin dual of our basic exact sequence together with the basic exact sequence for M^{∗}.

\xymatrix{0 \ar[r] & ((M_{Δ_{k}}(−1))^{𝔤_{k}})^{∨} \ar[r] & H^{1}(G_{K}, M)^{∨} \ar[r] & ((M^{Δ_{k}})_{𝔤_{k}})^{∨} \ar[r] & 0 \\ 0 \ar[r] & ((M^{∗})^{Δ_{k}})_{𝔤_{k}} \ar[r] & H^{1}(G_{K}, M^{∗}) \ar[r] \ar[u]^{≅} & ((M^{∗})_{Δ_{k}}(−1))^{𝔤_{k}} \ar[r] & 0}

The vertical isomorphism is induced by Tate duality. Now, we claim that the other two vertical pairs of groups are isomorphic. For example,

((M^{∗})^{Δ_{k}})_{𝔤_{k}} = ((M^{∨}(1))^{Δ_{k}})_{𝔤_{k}}
= (((M(−1))^{∨})^{Δ_{k}})_{𝔤_{k}}
= ((M(−1)_{Δ_{k}})^{∨})_{𝔤_{k}}
= ((M(−1)_{Δ_{k}})^{𝔤_{k}})^{∨}
= ((M_{Δ_{k}}(−1))^{𝔤_{k}})^{∨}.

Here we have used some easy isomorphisms together with some already mentioned in Section 1.1. This shows that the first two terms are isomorphic, and the last two are done in a similar way.

One should note, however, that there is absolutely no apparent reason why these isomorphisms should make the above diagram commutative. We will state what will be useful to us as a proposition.

Proposition 2.1. With K, k and M as above, the groups H^{1}(𝔤_{k}, M^{I_{K}}) and H^{1}(I_{K}, M^{∗})^{𝔤_{k}} have the same order. Similarly, H^{1}(I_{K}, M)^{𝔤_{k}} and H^{1}(𝔤_{k}, (M^{∗})^{I_{K}}) have the same order.

Proof. This follows immediately from the isomorphisms constructed above, after translating back into the language of cohomology. Note that the fact that these groups are all finite follows from the statement of Tate duality.

2.3. Restrictions of Tate Duality : Second Attempt. We keep the same assumptions on M. We can now use the results we obtained above to relate Tate local duality to our basic exact sequences

0 → H^{1}(𝔤_{k}, M^{I_{K}}) → H^{1}(G_{K}, M) → H^{1}(I_{K}, M)^{𝔤_{k}} → 0

and

0 → H^{1}(𝔤_{k}, (M^{∗})^{I_{K}}) → H^{1}(G_{K}, M^{∗}) → H^{1}(I_{K}, M^{∗})^{𝔤_{k}} → 0.

We claim that under the Tate pairing

H^{1}(G_{K}, M) ⊗ H^{1}(G_{K}, M^{∗}) → ℚ/ℤ,

H^{1}(𝔤_{k}, M^{I_{K}}) and H^{1}(𝔤_{k}, (M^{∗})^{I_{K}}) are orthogonal. To see this consider the commutative diagram

\xymatrix{H^{1}(𝔤_{k}, M^{I_{K}}) ⊗ H^{1}(𝔤_{k}, (M^{∗})^{I_{K}}) \ar[dd] \ar[r]^{inf ⊗ inf} & H^{1}(G_{K}, M^{I_{K}}) ⊗ H^{1}(G_{K}, (M^{∗})^{I_{K}}) \ar[d] \\  & H^{1}(G_{K}, M) ⊗ H^{1}(G_{K}, M^{∗}) \ar[d] \\ H^{2}(𝔤_{k}, M^{I_{K}} ⊗ (M^{∗})^{I_{K}}) \ar[d] & H^{2}(G_{K}, M ⊗ M^{∗}) \ar[d] \\ H^{2}(𝔤_{k}, μ(\overline{k})) \ar[r]^{inf} & H^{2}(G_{K}, μ(K̄)) \ar[d] \\  & ℚ/ℤ}

where the first vertical map in the second column is induced by the inclusions M^{I_{K}} ↪ M and (M^{∗})^{I_{K}} ↪ M^{∗} and the rest of that column gives the Tate pairing. Note that since M has prime to p order we really do only need to consider μ(\overline{k}). But H^{2}(𝔤_{k}, μ(\overline{k})) = 0, so for H^{1}(𝔤_{k}, M^{I_{K}}) and H^{1}(𝔤_{k}, (M^{∗})^{I_{K}}) the pairing factors through 0; thus they are orthogonal under the Tate pairing, as claimed.

This orthogonality together with Proposition 2.1 implies the following fundamental theorem.

Theorem 2.2. Let K be a local field with residue field k. Let M be a finite tamely ramified G_{K}-module of prime to p order. Then under Tate local duality, H^{1}(𝔤_{k}, M) and H^{1}(I_{K}, M^{∗})^{𝔤_{k}} are orthogonal complements. Similarly, H^{1}(I_{K}, M)^{𝔤_{k}} and H^{1}(𝔤_{k}, M^{∗}) are orthogonal complements. Thus Tate local duality induces perfect pairings

H^{1}(𝔤_{k}, M) ⊗ H^{1}(I_{K}, M^{∗})^{𝔤_{k}} → ℚ/ℤ

and

H^{1}(I_{K}, M)^{𝔤_{k}} ⊗ H^{1}(𝔤_{k}, M^{∗}) → ℚ/ℤ.

In particular, we have an exact commutative diagram

\xymatrix{0 \ar[r] & (H^{1}(I_{K}, M)^{𝔤_{k}})^{∨} \ar[r] & (H^{1}(G_{K}, M))^{∨} \ar[r] & (H^{1}(𝔤_{k}, M))^{∨} \ar[r] & 0 \\ 0 \ar[r] & H^{1}(𝔤_{k}, M^{∗}) \ar[r] \ar[u]^{≅} & H^{1}(G_{K}, M^{∗}) \ar[r] \ar[u]^{≅} & H^{1}(I_{K}, M^{∗})^{𝔤_{k}} \ar[r] \ar[u]^{≅} & 0}

Proof. The injectivity of the first vertical maps and the surjectivity of the second vertical map follow from the the orthogonality constructions above. The fact that they are isomorphisms then follows from Proposition 2.1. The rest of the theorem is just a restatement of the diagram.