CHAPTER 3 Lecture 3

1. Ramification Groups

1.1. Definitions. In an effort to further understand the ramification in Galois extensions of local fields we will now define a certain filtration of the Galois group. As before let L/K be a finite Galois extension of local fields with ring of integers B and A respectively. Let ℓ and k be their residue fields, and let p be the residue characteristic. For any i ≥ 0 and s ∈ G, consider the following two equivalent conditions:

(a) s is the identity on B/𝔪_{B}^{i+1};

(b) v_{L}(s(b) − b) ≥ i + 1 for all b ∈ B.

Here v_{L} is the valuation on L, normalized so that

v_{L} : L^{∗} ↠ ℤ.

If in addition we have that B = A[β] for some β ∈ B, then we have a third condition equivalent to the previous two.

(c) v_{L}(s(β) − β) ≥ i + 1.

The proof of the equivalence is easy; see [Se-LF, Chapter 4, Section 1, Lemma 1]. We define G_{i} to be the subgroup of G of elements s satisfying any of the above conditions. G_{i} is called the i^{th} ramification group of the extension L/K. It follows easily from condition (a) that each G_{i} is normal in G. By (b) we also see that for sufficiently large i we must have G_{i} = {1}. Thus we have a filtration

{1} ◁ ⋯ ◁ G_{i} ◁ G_{i−1} ◁ ⋯ ◁ G_{1} ◁ G_{0} ◁ G.

We note that G_{0} = I by definition, where I is the inertia group for the extension L/K. This means that the rest of the group G is irrelevant for the computation of the ramification groups; in particular, we may as well replace K by K^{ur} (recall that this is the maximal unramified extension of K in L; that is, K^{ur} = L^{I}) and thus assume that L/K is totally ramified. We make this assumption for the remainder of our discussion of ramification groups.

We now investigate the quotients G_{i}/G_{i+1}. We begin with the special case of i = 0. By definition G_{0} stabilizes 𝔪_{B} and G_{1} stabilizes 𝔪_{B}^{2}. Thus the natural action of G_{0} on 𝔪_{B}/𝔪_{B}^{2} factors through G_{0}/G_{1}; in fact, G_{0}/G_{1} acts faithfully on 𝔪_{B}/𝔪_{B}^{2}, since any element of G_{0} which fixes all of 𝔪_{B}/𝔪_{B}^{2} actually lies in G_{1}. Since 𝔪_{B} is principal, 𝔪_{B}/𝔪_{B}^{2} is free of rank 1 over B/𝔪_{B} = ℓ. Further, the action of G_{0}/G_{1} on 𝔪_{B}/𝔪_{B}^{2} is l-linear; this is easy to check using the fact that if b ∈ B lifts an element of ℓ then s(b) ≡ b (mod 𝔪_{B}) for any s ∈ G_{0}. Combining all of this we have the following proposition.

0Last modified December 17, 1999

Proposition 1.1. There is a natural injection

G_{0}/G_{1} ↪ Aut_{l}(𝔪_{B}/𝔪_{B}^{2}) ≅ ℓ^{∗}.

Corollary 1.2. G_{0}/G_{1} is cyclic of order dividing q_{B} − 1, where q_{B} is the order of ℓ; in particular, it is prime to p.

We now go off on a complete digression before considering the higher quotients.

1.2. Digression : Filtrations of Matrix Groups. Let R be an arbitrary commutative, noetherian, complete local ring with finite residue field k and maximal ideal 𝔪_{R}. Thus

R = \underleftarrow{lim}_{n} R/𝔪_{R}^{n}

and we have an exact sequence

0 → 𝔪_{R} → R → k → 0.

R is profinite, and thus a compact topological ring. There is also a multiplicative version of the above exact sequence:

1 → 1 + 𝔪_{R} → R^{∗} → k^{∗} → 1.

This is essentially the exact sequence comparing GL_{1}(R) with GL_{1}(k), where for any ring R, GL_{N}(R) is the group of invertible N by N matrices with entries in R. (Recall that a necessary and sufficient condition for such a matrix to be invertible is that its determinant lies in R^{∗}.)

Fix some positive integer N. Consider the group GL_{N}(R). It follows from the completeness of R that we have

GL_{N}(R) = \underleftarrow{lim}_{n} GL_{N}(R/𝔪_{R}^{n});

thus GL_{N}(R) is a compact topological group. For any i ≥ 1, we consider the surjection

GL_{N}(R) ↠ GL_{N}(R/𝔪_{R}^{i}).

(This is easily seen to be a surjection; since R is local, any lifting to R of the entries of a matrix in GL_{N}(R/𝔪_{R}^{i}) will work.) Let Γ_{i} be the kernel; it is a normal subgroup of GL_{N}(R) which is closed and of finite index, and thus open. Thus there is an exact sequence

1 → Γ_{i} → GL_{N}(R) → GL_{N}(R/𝔪_{R}^{i}) → 1,

and we have defined a filtration

⋯ Γ_{i+1} ◁ Γ_{i} ◁ ⋯ ◁ Γ_{1} ◁ GL_{N}(R)

of GL_{N}(R).

We now consider the quotients Γ_{i}/Γ_{i+1}. The first one, GL_{N}(R)/Γ_{1}, is just GL_{N}(k), which is an interesting group. Next consider the higher quotients. It is easy to write down Γ_{i} explicitly; it is just the set of matrices of the form 1 + (A_{αβ}), with each entry A_{αβ} ∈ 𝔪_{R}^{i}. (Here by 1 we mean the N × N identity matrix and by (A_{αβ}) we mean the N × N matrix with (α, β) entry A_{αβ}.) Now, define a map

Γ_{i}/Γ_{i+1} → {(Ā_{αβ}) | Ā_{αβ} ∈ 𝔪_{R}^{i}/𝔪_{R}^{i+1}}

by sending 1 + (A_{αβ}) to the reduction of (A_{αβ}) modulo 𝔪_{R}^{i+1}. (Here the second set is made into a group by addition of matrices.) The fact that this is a group homomorphism comes from the calculation

(1 + (A_{αβ}))(1 + (B_{αβ})) = 1 + (A_{αβ} + B_{αβ}) + (A_{αβ})(B_{αβ}) = 1 + (A_{αβ} + B_{αβ})

since (A_{αβ})(B_{αβ}) has all entries in 𝔪_{R}^{2i}, which vanishes modulo 𝔪_{R}^{i+1}. It is easy to see that this map is actually an isomorphism. The group structure of the second group is quite simple; it is just isomorphic to

\overbrace{𝔪_{R}^{i}/𝔪_{R}^{i+1} × ⋯ × 𝔪_{R}^{i}/𝔪_{R}^{i+1}}^{n^{2}},

n^{2} copies of the additive group 𝔪_{R}^{i}/𝔪_{R}^{i+1}. This in turn is an abelian group of exponent p.

We now look at the simplest case of this; take N = 1. Our Γ_{i}’s are thus defined by the exact sequences

1 → Γ_{i} → R^{∗} → (R/𝔪_{R}^{i})^{∗} → 1.

If we apply our above results with the ring B from earlier in this section, we obtain a filtration of the topological group B^{∗}. The usual notation for the Γ_{i} in this situation is U_{i}; B^{∗} is denoted by U. We thus have a filtration

⋯ ⊆ U_{i+1} ⊆ U_{i} ⊆ ⋯ ⊆ U_{1} ⊆ U.

Our above results show that each U_{i}/U_{i+1} is a finite p-group, and thus that U_{1} is a pro-p group; that is, U_{1} is an inverse limit of finite p-groups.

1.3. The Higher Ramification Groups. We now compare the two filtrations

{1} ◁ ⋯ ◁ G_{i} ◁ G_{i−1} ◁ ⋯ ◁ G_{1} ◁ G_{0}

and

⋯ ⊆ U_{i+1} ⊆ U_{i} ⊆ ⋯ ⊆ U_{1} ⊆ U

that we have constructed. As before, we assume that L/K is totally ramified, so that G_{0} = I = Gal(L/K). Choose a uniformizer π = π_{B} for B. One can easily check that B = A[π]. (See [Se-LF, Chapter 1, Section 6].) So in this case for any s ∈ G_{0} we have s ∈ G_{i} if and only if

s(π) ≡ π (mod 𝔪_{B}^{i+1}).

This is in turn is equivalent to

\frac{s(π)}{π} ≡ 1 (mod 𝔪_{B}^{i}),

since π has exact valuation 1. That is, s ∈ G_{i} if and only if

\frac{s(π)}{π} ∈ U_{i}.

Now consider the map

θ_{i} : G_{i} → U_{i}/U_{i+1}

defined by

s ↦ \frac{s(π)}{π}.

We claim first that θ_{i} is independent of the choice of π. To see this, let π′ be another uniformizer for B; then π′ = uπ for some u ∈ U. We compute

\frac{s(π′)}{π′} = \frac{s(uπ)}{uπ} = \frac{s(u)}{u} \frac{s(π)}{π} = \frac{u + m_{i+1}}{u} \frac{s(π)}{π}

for some m_{i+1} ∈ 𝔪_{B}^{i+1}, since s ∈ G_{i}. But

\frac{u + m_{i+1}}{u} \frac{s(π)}{π} ≡ \frac{s(π)}{π} (mod U_{i+1})

since u is a unit. This shows that θ_{i} is well-defined independent of π. In fact, it also shows that θ_{i} is a group homomorphism. To see this, let s, t ∈ G_{i}. Note that π′ = t(π) is also a uniformizer. Thus

\frac{st(π)}{π} = \frac{st(π)}{t(π)} \frac{t(π)}{π} = \frac{s(π′)}{π′} \frac{t(π)}{π},

which shows that θ_{i} is a homomorphism, since in U_{i}/U_{i+1} we have s(π′)/π′ = s(π)/π.

θ_{i} clearly vanishes on G_{i+1}, so we obtain a map

G_{i}/G_{i+1} → U_{i}/U_{i+1}

which we shall hereafter refer to as θ_{i}. It is easy to see that θ_{i} is now injective, using criterion (c) for lying in G_{i+1}. Thus, by the determination of U_{i}/U_{i+1} earlier we now see that G_{i}/G_{i+1} is abelian of exponent p, for i ≥ 1. This yields two important corollaries.

Corollary 1.3. The group G_{1} is a p-group.

Corollary 1.4. If L/K is tamely ramified, then G_{1} = {1}. (Recall that an extension L/K is tamely ramified if its ramification index e is prime to p, the residue characteristic.)

We now summarize our investigation of ramification. So assume L/K is any finite Galois extension of local fields (in particular, we no longer assume that it is totally ramified) with Galois group G and residue Galois group 𝔤. Set P = G_{1} and Δ = G_{0}/G_{1}. Then P is a p-group (the wildly ramified inertia subgroup), Δ is cyclic of order prime to p, and we have an exact sequence

1 → P → I → Δ → 1.

Next, if we let T be the Galois group of the maximal tamely ramified extension of K in L (that is, the extension L^{P}), then we have exact sequences

1 → P → G → T → 1

and

1 → Δ → T → 𝔤 → 1.

Δ was previously identified with a subgroup of ℓ^{∗}, and one can check that the conjugation action of 𝔤 on Δ is just the usual action of 𝔤 on ℓ^{∗}.

1.4. The Upper Indexing. To this point we have been using the lower indexing of ramification groups. This is the simplest to define, but it unfortunately behaves very poorly with respect to extensions. To correct for this, one is led to introduce the upper indexing as follows. (For more details, see [Se-LF, Chapter 4].) First, we extend the definition of G_{i} to real indices by saying that s ∈ G_{u} if and only if v_{L}(s(b) − b) ≥ u + 1 for all b ∈ B. Now define

φ(u) = ∫_{0}^{u} \frac{dt}{(G_{0} : G_{t})}.

(The integrand is piecewise constant, so this is really just a sum; the integral notation is more convenient.) This can be checked to give a bijection from the positive reals to itself. We define the upper indexing by

G_{u} = G^{φ(u)}.

The inverse function of φ can be determined to be

ψ(v) = ∫_{0}^{v}(G^{0} : G^{w})dw.

Thus

G^{v} = G_{ψ(v)}.

It turns out that this upper indexing is compatible with extensions, in the sense that if we have a diagram

[figure: M; G_{M/K}; L; K; G_{L/K}]

then under the natural surjection

G_{M/K} ↠ G_{L/K}

each G_{M/K}^{v} maps onto G_{L/K}^{v}.

2. Witt Vectors

Let R be a perfect ring of characteristic p. We define the Witt vectors over R to be the set

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

which is turned into a ring using certain universal polynomials. (See [Se-LF, Chapter 2, Section 6] for all references for this section.) The fundamental example is W (𝔽_{p}) = ℤ_{p}, which is really used to define the univeral polynomials in general. In this case the Witt vector of α ∈ ℤ_{p} is just the p-adic expansion of α; that is,

(a_{0}, a_{1}, ⋯) ↔ α = a_{0} + a_{1}p + a_{2}p^{2} + · · · .

In general the relationship is not this simple; it involves twisting by Frobenius.

Now let k be a finite field of characteristic p. Then W (k) is is a complete discrete valuation ring with residue field k. Further, W (k) is absolutely unramified; that is, it has p as a uniformizer. The construction of Witt vectors is also functorial, so we get a map

ℤ_{p} = W (𝔽_{p}) ↪ W (k).

The Galois group 𝔤 = Gal(k/𝔽_{p}) acts on W (k), and one has a canonical isomorphism

𝔤 ≅ Gal(K/ℚ_{p}),

where K is the fraction field of W (k). More generally, we have the following proposition.

Proposition 2.1. Let A be a complete discrete valuation ring with residue field k and fraction field K of characteristic 0. Then there is a unique ring homomorphism

\xymatrix{W (k) \ar@{^{(}->}[rr] \ar[dr] &  & A \ar[dl] \\  & k}

making this diagram commute, where both maps to k are the natural map.

A is in fact a free W (k)-module of rank equal to the absolute ramification index of K. Specifically, one can write A = W (k)[π_{A}], where π_{A} is any uniformizer for A.

3. Projective Limits of Groups of Units of Finite Fields

3.1. Topologically Cyclic Profinite Groups. Let Γ be a profinite group which is topologically cyclic; that is, there is a γ ∈ Γ such that Γ is the closure of the group generated by γ. We further assume that γ has infinite order; this merely excludes the case where Γ is cyclic of finite order. Thus we have a map

γ^{ℤ} ↪ Γ

with dense image. Morally speaking, any such Γ ought to be some sort of profinite completion of the infinite cyclic group ℤ. We will now attempt to make that statement more precise in certain special cases of interest to us.

First of all, we recall the standard examples. We have the full profinite completion of ℤ,

ℤ̂ = \underleftarrow{lim}_{N} ℤ/Nℤ,

the inverse system being defined by divisibilities. We also have

ℤ_{p} = \underleftarrow{lim}_{n} ℤ/p^{n}ℤ = \underleftarrow{lim}_{N,(N,l)=1 for l ≠ p}ℤ/Nℤ.

The first inverse system is with respect to the usual ordering of ℤ, the second with respect to the multiplicative ordering. More generally, if L is any set of prime numbers of ℤ, we have

∏_{l∈L} ℤ_{l} ≅ \underleftarrow{lim}_{N, N has all prime divisors in L}ℤ/Nℤ.

This isomorphism comes from the Chinese remainder theorem. All of these groups are topologically cyclic, with canonical generator the image of 1 ∈ ℤ.

Conversely, suppose that

Γ = \underleftarrow{lim}_{N} C_{N},

where each C_{N} is cyclic of order w_{N} and the maps of the inverse system are surjective. In particular Γ is topologically cyclic, as one can lift generators via the surjective maps. Suppose further that every integer M prime to a fixed prime number p divides some w_{N}, and that all of the w_{N} are prime to p. Then one can easily show that

Γ ≅ ∏_{l≠p} ℤ_{l},

although the isomorphism is horribly non-canonical.

3.2. Example : Units of Finite Fields. Let k be a finite field of order q and characteristic p, let \overline{k} be a fixed algebraic closure of k, and let k_{n} be the unique subfield of \overline{k} of degree n over k.

[figure: \overline{k}; k_{n}; n; k]

Set C_{n} = k_{n}^{∗} and w_{n} = |C_{n}| = q^{n} − 1. This defines an inverse system via norm maps

k_{n′} ⟶^{N} k_{n}

for n dividing n′. These maps can be shown to be surjective (see [Fr, Section 7, Corollary to Proposition 3]), and one easily checks that the w_{n} satisfy the conditions at the end of the previous section. Thus, if we define

Δ_{k} = \underleftarrow{lim}_{n} k_{n}^{∗},

then

Δ_{k} ≅ ∏_{l≠p} ℤ_{l},

although the isomorphism is non-canonical. (Each factor ℤ_{l} is canonical, but the choice of generator and thus the identification with ℤ_{l} is not.) This is actually a 𝔤-module, where 𝔤 = Gal(\overline{k}/k), and the action is continuous when Δ_{k} is given the profinite topology. (It is not continuous if Δ_{k} is given the discrete topology; thus Δ_{k} is not a 𝔤-Galois module.)

Now let k′ be any finite extension of k in \overline{k}. Then

Δ_{k′} ≅ Δ_{k}

since Δ_{k′} is defined to be an inverse limit of a cofinal subsystem of the inverse system defining Δ_{k}. In fact, this is a 𝔤′-isomorphism, where 𝔤′ = Gal(\overline{k}/k′). This suggests that we should look at the Gal(\overline{k}/𝔽_{p})-module Δ_{𝔽_{p}}. We use this module to define Gal(\overline{k}/𝔽_{p}) modules ℤ_{l}(1) by

Δ_{𝔽_{p}} = ∏_{l≠p} ℤ_{l}(1).

This notation is slightly misleading, in that ℤ_{l}(1) is not canonically isomorphic to ℤ_{l}. It can actually be realized as the Tate module of the l-power roots of unity; that is,

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

In fact, Δ_{k} can also be realized as this same product, via our earlier isomorphism. Here 𝔤 acts on each ℤ_{l}(1) simply as it would as a subgroup of Gal(\overline{𝔽}_{p}/𝔽_{p}).