CHAPTER 2 Lecture 2

1. Discrete Valuation Rings

1.1. Definitions. Let A be a commutative ring with unit. We recall six equivalent characterizations of discrete valuation rings. (As a general convention, for any local ring A we let 𝔪_{A} denote its maximal ideal.)

(a) A is a noetherian local ring and 𝔪_{A} is principal, generated by a nonnilpotent.

(b) A is a noetherian, integrally closed integral domain with only one non-zero prime ideal.

(c) A is a principal ideal domain with a unique non-zero prime ideal.

(d) Let A be an integral domain with field of fractions K. Then K possesses a surjective valuation homomorphism

v : K^{∗} ↠ ℤ,

for which, if we set v(0) = +∞, we have

v(x + y) ≥ min{v(x), v(y)}

for all x, y ∈ K, and A can be recovered as

A = {x ∈ K | v(x) ≥ 0}.

(e) A has a non-nilpotent element π such that every non-zero element a ∈ A can be written uniquely as a = uπ^{n} for some u ∈ A^{∗}, n ≥ 0.

(f) A is a discrete valuation ring.

For proofs of the equivalences see [Se-LF, Chapter 1, Section 2]. These proofs are essentially a sort of “resolution of singularities” in Krull dimension 1; that is, the implication (b) implies (a) shows that taking the integral closure of a local domain of dimension 1 gives a regular local ring of dimension 1. Geometrically, this corresponds to replacing a possibly non-regular point with a regular point.

Any element π ∈ A of valuation 1 is called a uniformizer or a local parameter; these are precisely the generators of 𝔪_{A}. This name also has a geometric interpretation: if A is the local ring of a smooth k-rational point x of an algebraic curve X over k, then a “local parameter” in the above sense is simply a rational function on X which provides a local parameter for X about the point x.

If A is a discrete valuation ring, we will write k_{A} (or simply k if it is clear from context) for its residue field A/𝔪_{A}.

0Last modified December 17, 1999

1.2. Completions. There are two different, yet equivalent, points of view on completions. We begin with the algebraic approach. (See [Ma, Chapter 8] for the details.) For any noetherian local ring A, we define its 𝔪_{A}-adic completion (or just completion) to be the ring

Â = \underleftarrow{lim}_{n} A/𝔪_{A}^{n}.

Â is again a local ring, with maximal ideal 𝔪Â = 𝔪_{A}Â. There is a natural injection

A ↪ Â

since by Krull’s theorem ∩_{n}𝔪_{A}^{n} = 0. If A is a discrete valuation ring, then Â is also a discrete valuation ring, and any uniformizer of A is again a uniformizer of Â (although there are of course many uniformizers of Â which do not lie in A). We let K and K̂ be the field of fractions of A and Â respectively; we now have a diagram as below.

\xymatrix{A \ar@{^{(}->}[rr] \ar[dr] &  & Â \ar[dr] \\  & K \ar@{^{(}->}[rr] &  & K̂}

The topological approach begins with the fraction field K; we define a metric on K by

∥ x ∥= e^{−v(x)}

for x ∈ K. (The choice for the base of the exponent is irrelevant for our purposes.) The properties of v make K into a topological field with respect to this metric, and we define K̂ to be the topological completion of K with respect to the induced metric topology. We recover the ring Â simply as

Â = {x ∈ K̂; ∥ x ∥≤ 1}.

For more details on this approach, see [Se-LF, Chapter 2]. For more details on the comparison of the two methods, see [Wes-Ide, Sections 1 and 2].

One easily sees that the residue fields of A and Â coincide; that is, the natural map A ↪ Â induces an isomorphism

k_{A} ⟶^{∼} kÂ.

2. Local Fields

2.1. Definitions. Let A be a complete discrete valuation ring (that is, A = Â) with fraction field K and residue field k. If K has characteristic 0 and k is finite, we will call such a K a local field. (This is not the usual definition of local field, but it will suffice for our present purposes.)

Since A is complete, we have

A = \underleftarrow{lim}_{n} A/𝔪_{A}^{n}.

By assumption A/𝔪_{A} = k is finite, and it follows easily that each A/𝔪_{A}^{n} is finite, since it has finite length over k. Specifically, if k has order q, then A/𝔪_{A}^{n} has order q^{n}. In particular, each A is a profinite topological ring (since it is an inverse limit of finite rings) and thus is compact in its natural profinite (i.e., 𝔪_{A}-adic) topology.

The fraction field K can be written as

K = ⋃_{n∈ℤ} π^{−n}A

for any uniformizer π. Each π^{−n}A is easily seen to be both an open and closed subset of K (since the metric on K is discrete), and each π^{−n}A is a homeomorphic image of a compact set and thus compact. Thus K is a locally compact topological field.

Given only the topology on K, we can recover A and its maximal ideal 𝔪_{A} as

A = {x ∈ K | {x^{ν}}_{ν=1}^{∞} is contained in a compact set}

and

𝔪_{A} = {x ∈ K | lim_{ν→∞} x^{ν} = 0}.

From this we also recover A^{∗} = A − 𝔪_{A} and the valuation homomorphism, since we know 𝔪_{A}. Again, for more details on the topology of local fields see [Wes-Ide].

2.2. Construction of Local Fields. Let F be a number field with ring of integers O_{F}. (Recall that this means that F is a finite extension of ℚ and O_{F} is the integral closure of ℤ in F.) Let 𝔭 be a prime ideal of O_{F}, and consider

O_{F,𝔭} = \underleftarrow{lim}_{n} O_{F} /𝔭^{n}.

Then O_{F,𝔭} is a complete discrete valuation ring (see [Se-LF, Chapter 1, Section 3]). In fact, it can be shown that any local field (as we have defined them) arises in this way.

There is another approach which is in some sense more self-contained. The basic result is the following theorem.

Theorem 2.1. Let K be a local field and let L be a finite extension of K. Then there is a unique local field structure on L extending that of K. (That is, there is a unique topology on L giving it the structure of a local field which restrictson K to the given topology.)

Proof. (Sketch. For details, see [Se-LF, Chapter 2, Section 2].) Let B be the integral closure of A in L.

\xymatrix{B \ar@{^{(}->}[r] & L \\ A \ar[u] \ar@{^{(}->}[r] & K \ar[u]}

L is a finite dimensional vector space over K, so there is only one compatible topology on L. (See [Cas, Section 10].) Now let 𝔭 be a prime ideal of B. Suppose 𝔭 ∩ A = 0. Then one can use the integrality of B over A to show that 𝔭 must be 0. Thus, for any non-zero prime 𝔭 of B we must have 𝔭 ∩ A = 𝔪_{A}. Again, integrality insures that some such prime must exist. Furthermore, 𝔭 is unique; for if it weren’t, each 𝔭 could be used to define a different topology on L compatible with that on K (since 𝔭 ∩ A = 𝔪_{A}). It then follows from the equivalent characterizations of discrete valuation rings that B is a discrete valuation ring, and thus that L is a local field.

Now, choose a rational prime p. We start with the field ℚ_{p}, which is a local field with ring of integers ℤ_{p}. Take K to be any finite extension of ℚ_{p}; by the theorem it is a local field in a unique way, with ring of integers A the integral closure of ℤ_{p} in K. Again, it can be shown that these are all of the possible local fields (as we have defined them) with residue field of characteristic p.

3. The Galois Theory of Local Fields

Let L/K be a finite Galois extension of local fields. Set G = Gal(L/K). Note first that G stabilizes B; that is, if s ∈ G, then s(B) = B. This is essentially because B is integral over A and s fixes A and thus integral equations over A as well. Further, G stabilizes 𝔪_{B}. (For this use the topological characterization of 𝔪_{B} and the fact that elements of G are homeomorphisms.)

It follows that G induces an action on B/𝔪_{B} = k_{B}. To unify notation, let ℓ = k_{B} for the remainder of this section. Since ℓ/k is a finite extension of finite fields it is automatically Galois. Set 𝔤 = Gal(ℓ/k). We now have a diagram

[figure: 𝔪_{B}; G; B; G; L; G; 𝔪_{A}; A; K; ℓ; 𝔤; B/𝔪_{B}; 𝔤; k; A/𝔪_{A}]

Since we have shown that G acts on ℓ, and since it obviously fixes k, we have constructed a natural map

G → 𝔤

which is obviously a group homomorphism. We define the inertia group I to be the kernel of this map; it is a subgroup of G.

Theorem 3.1. The natural map G → 𝔤 is surjective; thus we have an exact sequence

0 → I → G → 𝔤 → 0.

Proof. k is finite and thus perfect, so the primitive element theorem implies that we can write ℓ = k[b̄] for some b̄ ∈ ℓ. Consider the set of k-conjugates of b̄ in ℓ. 𝔤 acts on this set, by definition, and it acts simply transitively since b̄ is a primitive element. Now lift b̄ to an element b of B. Let P_{b}(T) be the minimal polynomial of b over A; it has the form

P_{b}(T) = ∏_{s∈S⊆G}(T − s(b)) ∈ A[T]

where S is a subset of G such that S acts simply transitively on the K-conjugates of b. Consider the reduction of P_{b}(T), say P̄_{b}(T) ∈ k[T]. By construction b̄ is a root of P̄_{b}(T). Since P̄_{b}(T) has coefficients in k, it follows that any k-conjugate of b̄ is also a root of P̄_{b}(T); that is, for any σ ∈ 𝔤 there exists an s ∈ G such that σ(b̄) = \overline{s(b)}. Since b̄ generates the entire extension ℓ over k, it follows that s maps to σ under the natural map G → 𝔤, and thus that this map is surjective.

Let n, f and e be the number of elements in G, 𝔤 and I respectively. n is simply the degree of L/K; we call f the residue degree and e the inertial degree or ramification index of the extension L/K. By Theorem 3.1 we have n = ef. If we set (temporarily) K^{ur} = L^{I}, the subfield of L fixed by I, we have a diagram

[figure: L; G; K^{ur}; I; totally ramified; K; 𝔤; unramified]