CHAPTER 1 Lecture 1

This lecture will be a summary of the material to be covered in the next few lectures; many details will be left until then.

1. Galois Modules

Let K be a field, and fix a separable closure K_{s}. Set

G_{K} = Gal(K_{s}/K) = \underleftarrow{lim}_{K⊆K′⊆K_{s}} Gal(K′/K)

where the projective limit is over all finite Galois extensions K′ of K contained in K_{s}. We give G_{K} its usual profinite (Krull) topology.

Let M be a G_{K}-module; that is, M is an abelian group with a left G_{K}-action. We will say that M is a G_{K}-Galois module if this action is continuous with respect to the profinite topology on G_{K} and the discrete topology on M. (These are called topological G_{K}-modules in [Se-LF].) Equivalently, for each m ∈ M there must be an open subgroup H of G_{K} which fixes m; that is,

M = ⋃_{H open subgroup of G_{K}} M^{H}.

(Recall that M^{H} is the subgroup of M of elements invariant under H.)

In particular, if M is a finitely generated G_{K}-Galois module, then there is an open subgroup H of G_{K} which fixes all of M. (H is just the intersection of the open subgroups fixing each of the finitely many generators of M.) Since open subgroups of G are just Gal(K_{s}/K′) for finite extensions K′ of K, replacing K′ by its normal closure we see that the G_{K}-action on M factors through Gal(K′/K) for some finite Galois extension K′ of K.

For future reference we will briefly clarify the relationship between G_{K}-Galois modules and abelian sheaves for the étale topology on Spec K. (See [Mi-EC, p. 53] for more details.) In fact, there is an equivalence of categories between the two, although certain objects often belong more naturally in one category than the other. Specifically, set x = Spec K and \overline{x} = Spec K_{s}. Then given an abelian sheaf F we associate to F the G_{K}-Galois module

M_{F} = F(\overline{x}) = \underrightarrow{lim}_{x′→x}F(x′)

where the direct limit is over x′ = Spec K′ for all finite Galois extensions K′ of K in K_{s}.

0Last modified December 17, 1999

In the other direction, we first recall that for any x-scheme U which is connected and étale we can associate a G_{K}-set (that is, a set with a G_{K}-action) by

U ↦ Hom_{x}(\overline{x}, U),

the K_{s}-valued points of U. Given a G_{K}-Galois module M, we now associate to it the étale sheaf F_{M} such that

F_{M}(U) = Hom_{G_{K}} (Hom_{x}(\overline{x}, U), M)

for any finite étale U over x. We leave it to the reader to check that these associations give the asserted equivalence of categories. (In particular in the first direction one must check that M_{F} really is a discrete G_{K}-module. In the second direction, one must check that F_{M} really is a sheaf and that F_{m}(\overline{x}) = M.)

If M is a G_{K}-Galois module the Galois cohomology groups

H^{q}(G_{K}, M) = \underrightarrow{lim}_{H ◁ G_{K},H open}H^{q}(G_{K}/H, M^{H})

are defined. (Note that if H is open and normal in G_{K} then G_{K}/H is finite. For generalities on the cohomology of finite groups see [Se-LF, Chapters 7 and 8].) In particular,

H^{0}(G_{K}, M) = M^{G_{K}} = Hom_{G_{K}} (ℤ, M),

the G_{K}-invariants of M. More generally, for any q ≥ 0

H^{q}(G_{K}, M) = Ext_{G_{K}}^{q} (ℤ, M).

2. Example : Quasi-Finite Fields

2.1. Cohomology Groups. We now specialize the notions of the previous section to the case of a quasi-finite field k. (See [Se-LF, Chapter 13, Section 2] for the definition of a quasi-finite field.) When working with a quasi-finite field (which for us will almost always actually be finite), we will usually write 𝔤_{k} or even just 𝔤 for its absolute Galois group Gal(k_{s}/k). We then have an isomorphism

𝔤_{k} ≅ ℤ̂,

where ℤ̂ is the profinite completion of ℤ with respect to its subgroups of finite index; that is,

ℤ̂ = \underleftarrow{lim}_{N} ℤ/Nℤ.

(If k is actually finite, then there is a canonical choice for this isomorphism; namely, we want 1 ∈ ℤ̂ to correspond to the Frobenius automorphism. For general quasifinite fields we do not have such a canonical choice.) Fix one such isomorphism and let φ ∈ 𝔤_{k} correspond to 1. In this setting a 𝔤_{k}-Galois module M is simply an abelian group M with an automorphism φ such that every orbit of φ is finite.

The cohomology theory in this case is quite simple, at least when M is torsion. It is determined by the exact sequence

0 ⟶ M^{𝔤_{k}} ⟶ M ⟶^{1−φ} M ⟶ M_{𝔤_{k}} ⟶ 0

where M_{𝔤_{k}} is the 𝔤_{k}-coinvariants M/(1 − φ)M. (This is the largest quotient of M on which 𝔤_{k} acts trivially, just as M^{𝔤_{k}} is the largest submodule of M on which 𝔤_{k} acts trivially.) In fact, these give the only non-zero cohomology groups.

Theorem 2.1. If k is a quasi-finite field and M is a torsion 𝔤_{k}-Galois module, then there are natural isomorphisms

H^{q}(𝔤_{k}, M) = \begin{cases}M^{𝔤_{k}} & q = 0; \\ M_{𝔤_{k}} & q = 1; \\ 0 & q ≥ 2. \end{cases}

In particular, given a 𝔤_{k}-module map f : M → N the following diagrams commute:

H^{0}(𝔤_{k}, M) ⟶^{f} H^{0}(𝔤_{k}, N)
≅↓  ↓≅
M^{𝔤_{k}}  ⟶^{f} N^{𝔤_{k}}
H^{1}(𝔤_{k}, M) ⟶^{f} H^{1}(𝔤_{k}, N)
≅↓  ↓≅
M_{𝔤_{k}}  ⟶^{f} N_{𝔤_{k}}

where the horizontal maps in each case are the natural maps induced by f.

Proof. See [Se-LF, Chapter 13, Section 1].

In fact, somewhat more is true: suppose we are given an exact sequence

0 → M → N → P → 0

of 𝔤_{k}-Galois modules. Consider the exact commutative diagram

\xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & M^{𝔤_{k}} \ar[r] \ar[d] & N^{𝔤_{k}} \ar[r] \ar[d] & P^{𝔤_{k}} \ar[d] \\ 0 \ar[r] & M \ar[r] \ar[d]^{1−φ} & N \ar[r] \ar[d]^{1−φ} & P \ar[r] \ar[d]^{1−φ} & 0 \\ 0 \ar[r] & M \ar[r] \ar[d] & N \ar[r] \ar[d] & P \ar[r] \ar[d] & 0 \\  & M_{𝔤_{k}} \ar[r] \ar[d] & N_{𝔤_{k}} \ar[r] \ar[d] & P_{𝔤_{k}} \ar[r] \ar[d] & 0. \\  & 0 & 0 & 0}

The horizontal arrows in the first and last rows are either obvious or come from functoriality, depending on which you prefer. The snake lemma now yields an exact sequence

0 → M^{𝔤_{k}} → N^{𝔤_{k}} → P^{𝔤_{k}} → M_{𝔤_{k}} → N_{𝔤_{k}} → P_{𝔤_{k}} → 0.

In particular, we obtain a boundary map P^{𝔤_{k}} → M_{𝔤_{k}}.

Proposition 2.2. In the situation above, the diagram

H^{0}(𝔤_{k}, P) ⟶ H^{1}(𝔤_{k}, M)
≅↓  ↓≅
P^{𝔤_{k}} ⟶ M_{𝔤_{k}}

commutes, where the top and bottom maps are the boundary map in cohomology and the map constructed above respectively.

Proof. This follows easily from the explicit isomorphisms of [Se-LF, Chapter 13, Section 1].

2.2. Cup Products. We continue to assume that k is a quasi-finite field (with a fixed topological generator φ of 𝔤_{k}) and that M is a torsion 𝔤_{k}-module. Since most of the cohomology groups of M are trivial, the cup product structure must be particularly simple. (Recall that for any profinite group G and discrete G-modules M and N, the cup product is a pairing

H^{i}(G, M) ⊗ H^{j}(G, N) → H^{i+j}(G, M ⊗ N)

for any i, j ≥ 0. See [AW, Section 7]. Recall that if M and N are G-modules, then M ⊗ N is made into a G-module by setting g(m ⊗ n) = gm ⊗ gn.) Using Theorem 2.1 it is easy to describe the non-trivial cup product pairings for 𝔤_{k}.

Proposition 2.3. For i = j = 0 the cup product is simply the natural map

M^{𝔤_{k}} ⊗ N^{𝔤_{k}} → (M ⊗ N)^{𝔤_{k}}.

Similarly, for i = 1, j = 0 the cup product is the natural map

M_{𝔤_{k}} ⊗ N^{𝔤_{k}} → (M ⊗ N)_{𝔤_{k}}.

Proof. This follows easily from the definition of cup product and the usual isomorphisms.

We can use the above proposition to determine the Pontrjagin dual of M. Recall that this is defined to be

M^{∨} = Hom(M, ℚ/ℤ).

(As always, if G is a group and M and N are G-modules, Hom(M, N) is given a G-module structure by (gf)(m) = gf(g^{−1}m), f ∈ Hom(M, N), g ∈ G, m ∈ M. In particular, the G-invariants of Hom(M, N) are precisely the G-equivariant homomorphisms.)

Proposition 2.4. Let M and N be finite 𝔤_{k}-Galois modules. Suppose that there is a perfect pairing

M ⊗ N → ℚ/ℤ.

(This is equivalent to N ≅ M^{∨}.) Then the induced cup product pairings

H^{i}(𝔤_{k}, M) ⊗ H^{1−i}(𝔤_{k}, N) → H^{1}(𝔤_{k}, M ⊗ N) → H^{1}(𝔤_{k}, ℚ/ℤ) = ℚ/ℤ

are also perfect pairings for i = 0, 1.

Proof. This follows easily from Proposition 2.3.

2.3. Finite Fields. We now specialize to the case where k is a finite field. Set p = char k and q = p^{f} = |k|. Since k is finite it is perfect and we have k_{s} = \overline{k}, the algebraic closure of k. Furthermore, in this case we have a canonical isomorphism

𝔤_{k} → ℤ̂

given by sending the Frobenius automorphism φ to 1. (We normalize φ so that it sends x to x^{q}.) For every n ≥ 1 there is a unique extension k_{n} of k of degree n; we will write 𝔤_{n} for the Galois group Gal(k_{n}/k). It is the unique (cyclic) quotient of 𝔤_{k} of order n.

We have |k_{n}| = q^{n}; set w_{n} = |k_{n}^{∗}| = q^{n} −1. For any m dividing n we have norm maps

k_{n}^{∗} ⟶^{N} k_{m}^{∗};

these are in fact surjective. (See [Fr, Section 7, Corollary to Proposition 3].) Define Δ_{k} to be the projective limit of the k_{n}^{∗} under these maps. Δ_{k} is a 𝔤_{k}-module, although it is not a 𝔤_{k}-Galois module in the sense of Section 1.

Proposition 2.5. Δ_{k} is (very non-canonically) isomorphic to ∏_{l≠p} ℤ_{l}, where the product is over all primes l not equal to p.

Proof. First note that the Chinese remainder theorem shows that the above product is simply the projective limit

\underleftarrow{lim}_{N,(N,p)=1}ℤ/Nℤ,

the limit taken over only those N relatively prime to p. Further, note that for any N relatively prime to p we can find a w_{n} with N dividing w_{n}. The proposition now follows easily from the fact that Δ_{k} is a projective limit of cyclic groups of all orders prime to p by surjective maps.

The proposition implies that for any finite extension k′ of k, Δ_{k′} and Δ_{k} are abstractly isomorphic. In fact, a canonical isomorphism is given by the norm map. (This is one case where it is much more natural to think of things in terms of étale abelian sheaves.) They are even isomorphic as 𝔤_{k′}-modules; since 𝔤_{k} doesn’t act on Δ_{k′}, this is the best we could hope for. In any event, this leads one to consider Δ_{𝔽_{p}}. Here one is able to use roots of unity to construct a canonical isomorphism

Δ_{𝔽_{p}} ≅ ∏_{l≠p} ℤ_{l}(1),

where ℤ_{l}(1) is the Tate module of the l-power roots of unity. That is,

ℤ_{l}(1) = \underleftarrow{lim}_{n} μ_{l^{n}},

the maps

μ_{l^{n+1}} → μ_{l^{n}}

being the l^{th}-power map. Note that the ℤ_{l}(1) notation is slightly misleading; ℤ_{l}(1) is not canonically isomorphic to ℤ_{l} as an abelian group. ℤ_{l}(1) is called the Tate twist of ℤ_{l}; 𝔤_{k} acts on it simply by the cyclotomic action.

3. Local Fields

We continue with the notation of the previous section: k is a finite field of characteristic p and order q. Recall that to k one can associate the ring of Witt vectors

W (k) = {(a_{0}, a_{1}, a_{2}, …) | a_{i} ∈ k}

which is made into a ring using certain universal polynomials. (See [Se-LF, Chapter 2, Section 6]. With some effort they can be reconstructed by attempting to model the operations in ℤ_{p}.) W (k) has two natural operators F and V given by

\begin{aligned}F & : & (a_{0}, a_{1}, a_{2}, …) & ↦ & (a_{0}^{p}, a_{1}^{p}, a_{2}^{p}, …); \\ V & : & (a_{0}, a_{1}, a_{2}, …) & ↦ & (0, a_{0}, a_{1}, a_{2}, …). \end{aligned}

Clearly F and V commute, and in fact V F = F V is simply multiplication by p in W (k). It turns out that W (k) is a complete, absolutely unramified (that is, with uniformizer p) discrete valuation ring with residue field k.

Now, consider the units W (k)^{∗}. They naturally sit in an exact sequence

0 → 1 + pW (k) → W (k)^{∗} → k^{∗} → 0,

where the map W (k)^{∗} → k^{∗} is reduction modulo p. Since 1 + pW (k) is pro-p and k^{∗} has order prime to p it follows that the sequence splits, so that k^{∗} lifts uniquely to W (k)^{∗} and

W (k)^{∗} ≅ k^{∗} × (1 + pW (k)).

The splitting map k^{∗} → W (k)^{∗} is called the Teichmuller character and denoted ω. (Its existence can also be seen more directly; for example, it follows easily from Hensel’s lemma. (See [Se-LF, Chapter 2, Section 2.2, Lemma] or [Cas, Appendix C].) Alternatively, one can check that k^{∗} is simply the roots of unity μ(W (k)).)

The same construction applies to k_{n}; W (k_{n}) is also an absolutely unramified complete discrete valuation ring. Letting K be the quotient field of W (k), K_{n} the quotient field of W (k_{n}), W (\overline{k}) the union of all of the W (k_{n})’s and K_{∞} its fraction field, we obtain a diagram

[figure: K_{∞}; 𝔤_{k}; W (\overline{k}); K_{n}; 𝔤_{n}; \overline{k}; 𝔤_{k}; W (k_{n}); K; k_{n}; 𝔤_{n}; W (k); k]

This construction can be shown to give all of the unramified extensions of K. (See [Se-LF, Chapter 3, Section 5]. Note that this approach only works for absolutely unramified K.)

The next step from K to K̄ is the tamely ramified piece. For this we use Kummer extensions. We adjoin to the field K_{n} the w_{n}^{th} roots of p. More precisely, set

L_{n} = K_{n}[x]/(x^{w_{n}} − p).

Since K_{n} contains μ_{w_{n}} this is indeed a Kummer extension. Also, x^{w_{n}} − p is an Eisenstein polynomial, so L_{n} is totally ramified of degree w_{n} over K_{n}; since p does not divide w_{n} the ramification is tame. Furthermore, if we let V be the subgroup of K_{n}^{∗}/K_{n}^{∗w_{n}} generated by p, then Kummer theory (as in Lecture 8, Section 3.1) identifies Gal(L_{n}/K_{n}) with Hom(V, μ_{w_{n}}) as Gal(K_{n}/K)-modules. (Recall that Gal(K_{n}/K) acts on Gal(L_{n}/K_{n}) by extending an automorphism of K_{n} to L_{n} and then using it to conjugate Gal(L_{n}/K_{n}); this is well-defined since Gal(L_{n}/K_{n}) is abelian.) In fact, since K_{n} is absolutely unramified and p ∈ K, V is isomorphic to ℤ/w_{n}ℤ with trivial Galois action. Furthermore, we know that μ_{w_{n}} ≅ k_{n}^{∗}, so finally we can identify Gal(L_{n}/K_{n}) with k_{n}^{∗} as Gal(K_{n}/K)-modules. Thus, if we let T_{n} = Gal(L_{n}/K), we see that it sits in an exact sequence

1 → k_{n}^{∗} → T_{n} → 𝔤_{n} → 1.

This sequence does not in general split; however, since p ∈ K we can take a single w_{n}^{th} root of p, say α, and adjoin it to K:

[figure: L_{n}; K(α); 𝔤_{n}; K_{n}; k_{n}^{∗}; K; 𝔤_{n}]

This does allow us to identify a (non-canonical) lifting of 𝔤_{n} to T_{n}, and thus to identify T_{n} with the semi-direct product of k_{n}^{∗} and 𝔤_{n}. We have shown above that the action of 𝔤_{n} for this semi-direct product is simply its natural action on k_{n}^{∗}.

Now, define L_{∞} to be the union of all of the L_{n}, and let T = Gal(L_{∞}/K). That L_{∞} is the maximal tamely ramified extension of K_{∞} can be seen easily from Kummer theory. Gal(L_{∞}/K_{∞}) is the projective limit of the k_{n}^{∗}’s; one can check that the maps (m divides n)

Gal(L_{n}/K_{n}) → Gal(L_{m}/K_{m})

for this projective system are simply the norm maps

k_{n}^{∗} ⟶^{N} k_{m}^{∗}.

Thus Gal(L_{∞}/K_{∞}) ≅ Δ_{k}.

We summarize our constructions to this point in the following diagram.

[figure: K̄; I_{K}; G_{K}; L_{∞}; P; wildly ramified; T; K_{∞}; Δ_{k}; tamely ramified; K; 𝔤_{k}; unramified]

Here I_{K} is the inertia group of K. Also, 𝔤_{k} ≅ ℤ̂ (canonically) and Δ_{k} ≅ ∏_{l≠p} ℤ_{l} (non-canonically). P is a pro-p group which is quite complicated.