This document discusses the application of Kähler differentials to the study of ramification in algebraic number theory. It defines Kähler differentials and constructs the module of relative differentials. Properties like exact sequences are proven. The concept of the different ideal is introduced, which encodes ramification data in field extensions. It is shown that the different ideal is the annihilator of the module of Kähler differentials, providing a geometric characterization of ramification.

1. K¨ahler Differential and Application to
Ramification
Ryan Lok-Wing Pang
lwpang@ust.hk
May 18, 2015
Contents
1 Introduction 1
2 Construction of K¨ahler Differentials 1
3 Properties of K¨ahler 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 using
the notion of K¨ahler differential and hence giving a way for it to fit into
higher dimensional algebraic geometry.
2 Construction of K¨ahler Differentials
The notion of K¨ahler differential is a very general way to encode a notion of
differential form.
Let A be a commutative ring with unity, B an A-algebra, and let M be
a B-module.
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2. 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 B
over A to be a B-module Ω1
B/A (its elements are called K ahler differentials),
together with an A-derivation d : B → Ω1
B/A, which satisfies the universal
property: for any B-module M, and for any A-derivation d : B → M, there
exists a unique B-module homomorphism f : Ω1
B/A → M such that d = f ◦d,
i.e. the following diagram is commutative:
B M
Ω1
B/A
d
d
∃!f
We first show the existence and uniqueness:
Theorem 2.3. The module of relative differential forms Ω1
B/A, d exists and
unique up to unique isomorphism.
Proof. The uniqueness follows from the definition. To show existence, Let
F be the free B-module generated by the symbols {db|b ∈ B}. Let E be
the 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 → Ω1
B/A by sending b to db, It is clear that
Ω1
B/A, d has the required properties.
There is a more concrete way to construct Ω1
B/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 → bb .
Let I = ker(f). Consider B⊗A B 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
by
db = 1⊗b−b⊗1 (mod I2
). Then I/I2
, d is a module of relative differentials
for B/A.
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3. Proof. [2].
Example 2.5. Let B = A[x1, · · · , xn] be the polynomial ring over A, then
Ω1
B/A is the free B-module of rank n generated by dx1, · · · , dxn.
3 Properties of K¨ahler 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
Ω1
B /A
∼= Ω1
B/A ⊗B B .
Furthermore, if S is a multiplicative system in B, then
Ω1
S−1B/A
∼= S−1
Ω1
B/A.
Proof. [2].
Theorem 3.2 (First Exact Sequence). Let A −→ B −→ C be rings and
homomorphisms. Then there is a natural exact sequence of C-modules
Ω1
B/A ⊗B C → Ω1
C/A → Ω1
C/B → 0.
Proof. Define f : Ω1
B/A ⊗B C → Ω1
C/A by f(db ⊗ c) = cdb, and let g : Ω1
C/A →
Ω1
C/B be defined as g(dc) = dc. First, note that Ω1
B/A⊗B C is by definiton a C-
module and hence the above map is indeed a map of C-module. Surjectivity
of g is clear, since g maps generators of Ω1
C/A onto the generators of Ω1
C/B.
The only difference is that Ω1
C/B has more relations; namely that we must
ensure db = 0 for all b ∈ B and this does not affect the generating set of
Ω1
C/B. Finally, observe that the element db ⊗ 1 generates Ω1
B/A ⊗B C as a
C-module. But then f(db ⊗ 1) = db for all b ∈ B and these are precisely the
elements in ker(g).
Alternatively, it suffices to prove that for any C-module N, the dual
sequence
0 → HomC(Ω1
C/B, N) → HomC(Ω1
C/A, N) → HomC(Ω1
B/A ⊗B C, N)
is exact. For details, see [2]
Theorem 3.3 (Second Exact Sequence). Let B be an A-algebra, let I be
an ideal of B and let C = B/I. Then there is a natural exact sequence of
C-modules
I/I2
→ Ω1
B/A ⊗B C → Ω1
C/A → 0.
Proof. [2].
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4. 4 Application to Algebraic Number Theory
The study of different ideal provides information on ramified primes, and also
gives asort of duality which will plays a role in both the algebraic study of
ramification and on analytic duality. First, we define the notion of different
ideal, then we will how K¨ahler differential can be applied to the study of
algebraic number theory.
Let L/K be a finite separable field extension, A ⊆ K a Dedekind domain
with 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., the
trace 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 duality
is justified by the isomorphism
I∗ ∼
−→ HomA(I, A)
x → (y → 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. Its
inverse, DB/A = C−1
B/A is called the different ideal of B/A.
The name different is explained by the following description, which was
Dedekind’s original way to define it. Let α ∈ B and let f(x) ∈ A[x] be the
minimal polynomial of α. We define the different of the element α by
δL/K(α) =
f (α) if L = K(α),
0 if L = 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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5. Proof. [1] or [3].
The link of K¨ahler differential with the different is now this:
Theorem 4.3. The different ideal DB/A is the annihilator of the B-module
Ω1
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 preserved
under localization and completion. Hence without loss of generality, we may
assume 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, then
Ω1
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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