Dieudonne

Ramification group

Theorem 3.2

Let L/K be a finite separable extension of local fields. Then there exists a unique field L0 such that K subset L0 subset L and L0/K is unramified, L/L0 is totally ramified.

Proof:

We have Hom_K(L0, L) = Hom_kK(kL, kL) 

(question: why?)

and then L0 inj L (why? Hint: consider the id on the right hand side set).

W. Messing

Chapter 1, section 1

Local fields, Serre

1.1 Construction of the complex.

lemma 1.1.17

G_K' subset Intersection_M G_M implies that

(1) K^bar,H subset  K^bar,G_K'=K'

on the other hand,

(2) K'=U K_M subset K^bar,H.

Hence K'=H.

Question:

Suppose that V is a normal variety, and Z is a principal divisor so a closed subvariety of V with codimension 1. Then how do we construct the valuation ring OZ？

Here describe the construction: take an open affine U=Spec R such that U intersection Z is nonempty and indeed a maximal proper closed subset of U, then U int Z corresponds to a minimal prime ideal of R, and so Rp is a normal ring with unique maximal pRp, and it is a DVR. We define OZ to be Rp. More intrinsically we can define OZ to be the set of rational functions on V that are defined over an open U such that U int V is not empty.

Since V is normal, we have R=OX(U)=Γ(U,O)R=OX(U)=Γ(U,O) is integrally closed， but why Rp is a discrete valuation ring. Moreover why is it important that p being a minimal ideal?

Answer:

This is basically pure algebra. By a theorem in chapter 9 of Atiyah Macdonald (I think Proposition 9.2 or 9.3):

 a Noetherian local domain of dimension 1 is a DVR iff it is integrally closed.

So in your case above we are given a minimal prime p (which is necessarily of height 1) and hence Rp is one-dimensional (basically because

 height of a prime = dimension of localization at this prime). Since Rp is integrally closed by the proposition above we have that Rp is a DVR.

Definition 1.2

X/R

Exemple 1.3

Definition 1.4

Remarque

Proposition 1.5

Corollaire 1.6

Let K be a complete discrete valuation field. (not necessarily local fields)

Definition

A  finite separable extension L/K is unramified if 

(1) e(L/K)=1.

(2) kL/kK is separable (which is always the case for local field extension, as residue fields are finite, hence perfect, and finite extension of perfect fields is separable).

Properties:

(1)

(2)

(3) Consider K subset L subset M, then M/K is unramified if and only if L/K and M/L are both unramified.  

Proposition 2.1

Let kL/kK be a separable extension, hence there exists alpha^bar in kL such that kL=kK(alpha^bar).

Let L/K be unramified, and f^bar(X) in kK[X] be the minimal polynomial of alpha^bar. Let f(X) in OK[X] be a lift of f^bar(X) with deg f^bar = deg f. Then 

(1) There exists a unique alpha in OL, such that f(alpha)=0 and alpha^bar=alpha (mod pi_L).

(2) L=K(alpha). 

Proposition 2.2

Let K be a complete discrete valuation field and l/kK be a finite separable extension. Let l=kK(alpha^bar) and f^bar(x) in kK[X] be the minimal polynomial of alpha^bar. Let f(X) in OK[X] be a lift of f^bar such that deg f =deg f^bar. Then 

(1) L=K[X]/f(X) is an unramified extension, with kL=l.

Remark:

We have 

psi:  Hom_K(L, M) --> Hom_{kK}(kL, kM).

Indeed, for any sigma in Hom_K(L, M), it induces  a map on ring of integers (notice that sigma(OL) is integral over K). This induces a map on residue field ((pi_K) subset (pi_L)).

Proposition 2.3

If L/K is unramified, then for any M, psi is a bijective.

Proposition 2.4 (Unramifiedness is preserved under compositum)

If L1/K and L2/K are unramified extensions, so is L1L2/K.

Theorem 2.5

Assume K is a local field, then for any n geq 1, there exists a unique unramified field extension of K of degree n. This field extension is a cyclic Galois extension. (the Galois group is generated by some Frobenius)

Local fields

Let K be a complete discrete valuation field.

Theorem 9.1

Let L/K be a finite separable extension, then

(1) L is (also) a complete discrete valuation field.

(2) O_L coincides with the integral closure of O_K in L.

(3) O_L is a free O_K-module of rank [L:K].

Definition

Let L/K be a finite separable extension of a complete discrete valuation field (hence so is L).

(1) ramification index: pi_K=pi_L^e u; set v_L(pi_L)=1, then e=v_L(pi_K)

(2) residue degree: f=[k_L: k_K].

Theorem 9.2

Let L/K be a finite separable extension of a complete discrete valuation field, then  ef=[L:K].

Proposition 9.3

Let L/K be a finite separable extension of a complete discrete valuation field. Let v_K(pi_K)=v_L(pi_L)=1. Then 

v_L(x)=1/f v_K(N_{L/K}(x)).

Proposition 9.4

Assume K subset L subset M are complete discrete valuation fields, then

f(M/K)=f(M/L)f(L/K)

e(M/K)=e(M/L)e(L/K).