Kahler Differential and Application toRamification

Ryan Lok-Wing Panglwpang@ust.hk

May 18, 2015

Contents

1 Introduction 1

2 Construction of Kahler Differentials 1

3 Properties of Kahler Differentials 3

4 Application to Algebraic Number Theory 4

1 Introduction

The concept of different ideal is important in algebraic number theory be-cause it encodes the ramification data in extension of algebraic number fields.In this article, we wish to characterize the different ideal geometrically usingthe notion of Kahler differential and hence giving a way for it to fit intohigher dimensional algebraic geometry.

2 Construction of Kahler Differentials

The notion of Kahler differential is a very general way to encode a notion ofdifferential form.

Let A be a commutative ring with unity, B an A-algebra, and let M bea B-module.

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Defintion 2.1 (A-Derivation). An A-derivation of B into M is a map d :B →M such that

(1) d is additive: d(b+ b′) = db+ db′;(2) d satisfies the Leibniz’s rule: d(bb′) = bdb′ + b′db and(3) da = 0 for all a ∈ A.

Defintion 2.2. We define the module of relative differential forms of Bover A to be a B-module Ω1

B/A (its elements are called K ahler differentials),

together with an A-derivation d : B → Ω1B/A, which satisfies the universal

property: for any B-module M , and for any A-derivation d′ : B →M , thereexists a unique B-module homomorphism f : Ω1

B/A →M such that d′ = f d,i.e. the following diagram is commutative:

B M

Ω1B/A

d′

d∃!f

We first show the existence and uniqueness:

Theorem 2.3. The module of relative differential forms 〈Ω1B/A, d〉 exists and

unique up to unique isomorphism.

Proof. The uniqueness follows from the definition. To show existence, LetF be the free B-module generated by the symbols db|b ∈ B. Let E bethe submodule of F generated by all the expressions of the form d(b+ b′)−db− db′, d(bb′)− bdb′ − b′db for b, b′ ∈ B and da for a ∈ A. Set Ω1

B/A = F/E

and define the derivation d : B → Ω1B/A by sending b to db, It is clear that

〈Ω1B/A, d〉 has the required properties.

There is a more concrete way to construct Ω1B/A using the diagonal ho-

momorphism as follows:

Theorem 2.4. Let B be an A-algebra. We consider the diagonal homomor-phism

f : B ⊗A B → B

b⊗ b′ 7→ bb′.

Let I = ker(f). Consider B⊗AB as a B-module by multiplication on the left,then I/I2 inherits a structure of B-module. Define a map d : B → I/I2 bydb = 1⊗b−b⊗1 (mod I2). Then 〈I/I2, d〉 is a module of relative differentialsfor B/A.

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Proof. [2].

Example 2.5. Let B = A[x1, · · · , xn] be the polynomial ring over A, thenΩ1

B/A is the free B-module of rank n generated by dx1, · · · , dxn.

3 Properties of Kahler Differentials

In this section we give some properties of modules of differentials.

Theorem 3.1. If A′ and B are A-algebras, let B′ = B ⊗A A′. Then

Ω1B′/A′

∼= Ω1B/A ⊗B B

′.

Furthermore, if S is a multiplicative system in B, then

Ω1S−1B/A

∼= S−1Ω1B/A.

Proof. [2].

Theorem 3.2 (First Exact Sequence). Let A −→ B −→ C be rings andhomomorphisms. Then there is a natural exact sequence of C-modules

Ω1B/A ⊗B C → Ω1

C/A → Ω1C/B → 0.

Proof. Define f : Ω1B/A⊗B C → Ω1

C/A by f(db⊗ c) = cdb, and let g : Ω1C/A →

Ω1C/B be defined as g(dc) = dc. First, note that Ω1

B/A⊗BC is by definiton a C-module and hence the above map is indeed a map of C-module. Surjectivityof g is clear, since g maps generators of Ω1

C/A onto the generators of Ω1C/B.

The only difference is that Ω1C/B has more relations; namely that we must

ensure db = 0 for all b ∈ B and this does not affect the generating set ofΩ1

C/B. Finally, observe that the element db ⊗ 1 generates Ω1B/A ⊗B C as a

C-module. But then f(db⊗ 1) = db for all b ∈ B and these are precisely theelements in ker(g).

Alternatively, it suffices to prove that for any C-module N , the dualsequence

0→ HomC(Ω1C/B, N)→ HomC(Ω1

C/A, N)→ HomC(Ω1B/A ⊗B C,N)

is exact. For details, see [2]

Theorem 3.3 (Second Exact Sequence). Let B be an A-algebra, let I bean ideal of B and let C = B/I. Then there is a natural exact sequence ofC-modules

I/I2 → Ω1B/A ⊗B C → Ω1

C/A → 0.

Proof. [2].

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4 Application to Algebraic Number Theory

The study of different ideal provides information on ramified primes, and alsogives asort of duality which will plays a role in both the algebraic study oframification and on analytic duality. First, we define the notion of differentideal, then we will how Kahler differential can be applied to the study ofalgebraic number theory.

Let L/K be a finite separable field extension, A ⊆ K a Dedekind domainwith field of fraction K, and let B ⊆ L be its integral closure in L.

The theory of different originates from the fact that we are given a non-degenerate symmetric bilinear form on the the K-vector space L, viz., thetrace form (see [1])

T (x, y) = trL/K(xy).

Then we can associate every fractional ideal I of L to the dual B-module

I∗ = x ∈ L|trL/K(xI) ⊆ A.

It is easy to see that I∗ is again a fractional ideal. The notion of dualityis justified by the isomorphism

I∗∼−→ HomA(I, A)

x 7→ (y 7→ trL/K(xy)).

For a proof, see [3]. We are now ready to define the different of B/A:

Defintion 4.1. The fractional ideal

CB/A = B∗ = x ∈ L|trL/K(xB) ⊆ A

is called a Dedekind’s complementary module, or the inverse different. Itsinverse, DB/A = C−1B/A is called the different ideal of B/A.

The name different is explained by the following description, which wasDedekind’s original way to define it. Let α ∈ B and let f(x) ∈ A[x] be theminimal polynomial of α. We define the different of the element α by

δL/K(α) =

f ′(α) if L = K(α),0 if L 6= K(α)

In the special case where B = A[α] we then obtain

Proposition 4.2. If B = A[α], then the different is the principal ideal

DB/A = (δL/K(α)).

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Proof. [1] or [3].

The link of Kahler differential with the different is now this:

Theorem 4.3. The different ideal DB/A is the annihilator of the B-moduleΩ1

B/A, i.e.

DB/A = x ∈ B|xdy = 0 for all y ∈ B.

Proof. By Theorem 3.1, we see that the module of differential is preservedunder localization and completion. Hence without loss of generality, we mayassume that A is a complete discrete valuation ring . We know that B = A[α]for some α ∈ B, and if f(x) ∈ A[x] is the minimal polynomial of α, thenΩ1

B/A is generated by dα. The annihilator of dα is f ′(α). On the other hand,

by Proposition 4.2, we have DB/A = (f ′(α)). The result follows.

References

[1] S. Lang. Algebraic Number Theory (Graduate Texts in Mathematics).Springer Verlag, 2000.

[2] H. Matsumura. Commutative Ring Theory (Cambridge Studies in Ad-vanced Mathematics). Cambridge University Press, 1989.

[3] N. Neukirch. Algebraic Number Theory (Grundlehren der mathematis-chen Wissenschaften). Springer Verlag, 1999.

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