Download Finitely generated abelian groups, abelian categories

The fundamental theorem about finitely
generated abelian groups
Theorem: Suppose A is a finitely generated
abelian group. Then there are integers r, s ≥ 0
and n1, . . . , ns ≥ 1 such that
1. ni divides ni+1 for all i.
2. A is isomorphic to the direct sum

Zr
M
s
M


Z/ni
i=1
3. The integers r, s, n1, . . . , ns both determine
and are determined by the isomorphism class
of A.
Terminology: The ideals niZ are called the
invariant factors of A.
1
Example: The isomorphism classes of abelian
groups A of order n correspond bijectively to
those choices of (r, s, n1, . . . , ns) such that r =
Qs
0 and n = i=1 ni.
I’ll postpone the proof of the theorem till we
consider its generalization to modules over a
principal ideal domain.
Note that the subgroup Ator of elements of
finite order in A is just
Ator =
s
M
Z/ni.
(1)
i=1
2
Since Ator is a finite abelian group, its nilpotent
and hence the product of its p-Sylows. (One
can also show this directly from (1). Applying
the theorem to each of the p-Sylows of Ator
gives a direct sum decomposition
M
bp
M
prime p|#Ator
i=1
Ator =
Z/pap,i
(2)
for some integers ap,i. The ideal pap,i Z are
called the elementary divisors of Ator .
To find the elementary divisors from the invariant factor description
Ator =
s
M
Z/ni
i=1
one writes
Z/ni =
M
Z/pbi,p
prime p|ni
where pbi,p ||ni.
3
To find the invariant factors ni from the elementary divisor description, note the largest
ns is the product over all p|#G of the largest
power of p appearing in the elementary divisor
description
Ator =
M
bp
M
prime p|#Ator
i=1
Z/pap,i
One then cancels out these summands and
continues.
Random abelian groups: Statistical questions about randomly chosen abelian groups
have recently become relevant to number theory due to work of Cohen and Lenstra ( “Heuristics on class groups of number fields. Number theory,” Lecture Notes in Math., 1068.)
For example, one can consider the asymptotic
number of isomorphism classes of abelian groups
of order n as n → ∞. Sometimes it is more natural to weight the isomorphism class of A by
1/#Aut(A).
4
Direct limits in categories
Def: An index set I is directed if if has a partial
order ≤ on I such that for each i, j ∈ I there
is a k ∈ I such that i ≤ k and j ≤ k.
Example:
Z with the usual ordering.
Def: A directed family of morphisms in a category C is given by a collection {Ai}i∈I of objects
in C together with morphisms fi,j : Ai → Aj
for i ≤ j which respect composition. So, if
i ≤ j ≤ k then
fi,k = fj,k ◦ fi,j
5
Example 1: Let I be the positive integers.
1 · Z, and let
Define Ai = i!
1
1
· Z → Aj = · Z
i!
j!
be the inclusion homomorphism.
fi,j : Ai =
Example 2: Again let I be the positive integers. Define Ai to be the free group on generators {x1, . . . , xi}, and let Ai → Aj be the natural
inclusion.
6
Def: An object D in C together with a family of
morphisms d = {di : Ai → D}i∈I in C is a direct
limit for the family {Ai, fi,j } if the following is
true:
1. The homomorphism di respect the maps
fi,j : Ai → Aj for i ≤ j. In other words,
di = dj ◦ fi,j .
2. The pair (D, d) is universal with respect to
pairs satisfying condition (1). So if (D0, d0)
is any other such pair, there is a unique
h : D → D0 in C such that
d0i = h ◦ di
for all i.
Notation: The direct limit is
D = lim Ai
−→
i
7
Example: Suppose as before that
1
Ai = Z
i!
for all i, with Ai → Aj being inclusion if i ≤ j.
Lets check that D = Q with the inclusion maps
di : Ai → D is a direct limit lim Ai.
−→
i
The di certainly satisfy the consistency condition (1).
For condition (2), we need to define h : D →
D0. But each element of D arises as di(α) for
some sufficiently large i and some α ∈ Ai. So
we define h(di(α)) = d0i(α). This is consistent
(i.e. independent of the choice of i) because
of the consistency condition for (D0, {d0i}i).
Exercise: What happens if Ai = Z for all i
and fi,j : Ai → Aj is multiplication by j!/i! for
i ≤ j?
8
Theorem: Direct limit exist in the category
of abelian groups.
Proof: Let {Ai}i∈I be a directed family of abelian
groups, with transition homomorphisms
fi,j : Ai → Aj
for i ≤ j. Define
A=
M
Ai
i∈I
to be the direct sum of the Ai. Let B be the
submodule of A0 generated by all elements of
A0 of the form
(0, 0, 0, . . . , ai, 0, . . . , 0, −fi,j (ai), 0, . . .)
where i ≤ j and ai ∈ Ai.
9
We claim that the quotient D = A/B is the
direct limit of the Ai when we let the homomorphisms di : Ai → D be composition of
Ai → A
defined by
ai → (0, . . . , 0, ai, 0, . . .)
with the quotient hom A → D = A/B.
By construction, di(ai) − dj (fi,j )(ai) = 0 in D
for ai ∈ Ai and i ≤ j. So
di = dj ◦ fi,j .
Thus the di have the right property relative to
the fi,j .
10
Suppose D0 and d0i : Ai → D0 for i ∈ I form
another family of homomorphisms respecting
the fi,j .
We get a unique homomorphism
h̃ :
a
i∈I
Ai =
M
Ai = A → D0
i∈I
from the definition of the direct sum as the
coproduct in the category of abelian groups.
This homomomorphism sends B ⊂ A to {0}
since the d0i are compatible with the fi,j . So h̃
factors in a unique way through a homomorphism
h : D = A/B → D0.
This completes the proof.
11
Inverse limits in categories
These result from turning arrows around in the
definition of direct limits.
Suppose I is a directed set of indices.
Def: An inverse system of morphisms in a category C is given by a collection {Ai}i∈I of objects in C together with morphisms fj,i : Aj →
Ai for i ≤ j which respect composition. So, if
i ≤ j ≤ k then
fk,i = fk,j ◦ fj,i
12
Example 1: Suppose I is the set of positive
integers, and that pis a prime. Define
Ai = Z/pi
for all i. If i ≤ j we then have a surjective
homomorphism
fj,i : Aj = Z/pj → Ai = Z/pi
which is reduction mod pi.
Example 2: With I again the set of positive
integers, let Ai be the group R/Z for all i. Define
fj,i : Aj = R/Z → Ai = R/Z
to be multiplication by j!/i! for all i ≤ j.
13
Def: An object D in C together with a family
of morphisms d = {di : D → Ai}i∈I in C is
an inverse limit for the family {Ai, fj,i} if the
following is true:
1. The homomorphism di respect the maps
fj,i : Aj → Ai for j ≤ i. In other words,
dj = di ◦ fj,i.
2. The pair (D, d) is universal with respect to
pairs satisfying condition (1). So if (D0, d0)
is any other such pair, there is a unique
h : D0 → D in C such that
d0i = di ◦ h
for all i.
Notation: The inverse limit is
D = lim Ai
←−
i
14
Theorem: Inverse limits exist in the categories of all groups and the category of abelian
groups.
Construction: Let D be the subgroup of the
product
Y
Ai
i∈I
consisting of all elements
Y
ai
i∈I
such that fj,i(aj ) = ai for i ≤ j. The projection
map of the product to Ai gives the required di :
D → Ai. These are consistent by the definition
of D with the fj,i. One shows they have the
required universal property using the universal
Q
property of the product i∈I Ai.
15
Example: Suppose as in our first example that
fj,i : Aj = Z/pj → Ai = Z/pi
which is reduction mod pi for all integers i ≤ j.
The inverse limit
Zp = lim Ai
←−
i
is called the set of p-adic integers.
One can think of an element {ai}i≥1 in Zp as
the choice of a compatible sequence of residues
ai in Z/pi for all i. Compatibility means that
the residues reduce to previous choices mod
smaller powers of p.
Since addition and multiplication respect reduction mod pi for all i, we can both add and
multiply elements of Zp. This shows Zp the is
a ring.
16
Exercise: Show that we can think of elements
of Zp as infinite formal series
b=
∞
X
bipi
(3)
i∈0
in which each bi is an integer in
{0, 1, 2, . . . , p − 1}
. Associate to b the compatible sequence of
residues {ai}i≥1 defined by
ai =
i−1
X
bipi.
k=0
Show that the addition and multiplication of Zp
corresponds to formally adding and multiplying
series as in (3) and then using the usual rules
about carrying mod higher and higher powers
of p.
17
Exercises about Zp:
1. Show that Zp becomes a topological group
when we say there is a basis of open sets
of the form
a + pj Zp
as a ranges over Zp and j ranges over all
positive integers.
2. Show that Zp is a complete metric space
when we define a metric | |p : Zp → R by
|a|p = p−i0
if a = {ai}i≥1 and i0 is the smallest nonnegative integer for which
ai 6≡ 0
mod
pi0
for i ≥ i0. (Completeness means that Cauchy
sequences with respect to | |p converge to
elements of Zp.)
18
Abelian categories
The category A of abelian groups has some
special properties. Namely, for every pair of
objects C, D of A, the morphisms
M orA(C, D) = HomA(C, D)
are an abelian group. Furthermore, each f ∈
HomA(C, D) has a kernel and a cokernel, which
are objects of A. Each such f can be factored
as a composition of an surjective homomorphism followed by an injective homomorphism.
Definition A category C is abelian if it is isomorphic to a subcategory of A which is closed
under finite direct sums and also closed under taking kernels and cokernels of homomorphisms in C.
19
Example 1: Suppose G is a group. A Gmodule is an abelian group M having a (left)
action of G which respects the group law of
M . Thus
g(m1 + m2) = gm1 + gm2
for g ∈ G and m1, m2 ∈ M . A G-module homomorphism M1 → M2 is a homomorphism of
abelian groups which respects the action of G
on M1 and M2. The category M od(G) of all
G-modules is an abelian category.
Example 2: We haven’t defined rings R or
R-modules yet, but the category of R-modules
for any ring R will be an abelian category.
20
Comment: One can give an abstract definition of an abelian category. See §II.1 of
Hartshorne’s “Algebraic Geometry” or ”Homological Algebra” by Hilton and Stammbach.
Our first definition of abelian categories then
becomes a Theorem of Peter Freyd.
One requires first that M orA(C, D) is an abelian
group for all objects C and D of A.
If f ∈ M orA(C, D), the kernel of f can be defined to be a morphism h : K → C in A which
is universally attracting with respect to morphisms such that f ◦ h is the 0 element of
M or(K, D). One can define cokernels similarly.
A morphism is injective if it the kernel of some
morphism from its range, and one can define
surjective morphisms similarly.
21
The axioms for an abelian category A are:
1. M orA(C, D) is an abelian group, and the
composition of morphisms is linear. So h ◦
(f1 + f2) = h ◦ f1 + h ◦ f2 and similarly on
the other side, when these compostions are
defined.
2. coproducts of finitely many objects exist.
3. kernels and cokerns of morphisms exist.
4. If f : K → C is injective, this is the kernel
associated to C → coker(f ). If h : D → H
is surjective, this the cokernel associated
to ker(h) → D.
5. Every morphism can be factored as a composition f ◦ h of a surjection h and an injection f .
22
Presheaves
Suppose X is a topological space. Let UX be
the set of all open subsets of X. We can view
UX as a category, in which the morphisms are
inclusion maps. A presheaf on X is a contravariant functor
F : UX → A = abelian groups
such that F (∅) = {0}. One typically thinks of
F (U ) as a set of functions from U to some fixed
abelian group T . Then if U → V is an inclusion
of sets, F (V ) → F (U ) can be thought of as the
restriction of functions from V to U . Call F (U )
the sections of F over U .
For instance, we could let F (U ) be the additive
group of all continuous functions from U to the
reals R.
23
Fact: We can make the set PX of all presheaves
on X into an abelian category in the following
way.
1. If H : UX → A is another presheaf, a morphism f : F → H is a just a morphism of
functors.
2. We need to define presheaves Kernel(f )
and Cokernel(f ) associated to f . To do
this, we have for each U ∈ UX a group homorphism
f (U ) : F (U ) → H(U )
Define Ker(f ) is the presheaf for which
Ker(f )(U ) = Kernel(f (U ))
and define Coker(f ) similarly.
24
Exercise: Suppose X is the unit interval [0, 1]
on the real line. Define F and H by
F (U ) = {Continuous functionsr : U → Z}
H(U ) = {Continuous functionsr : U → R}
We have a morphism of presheaves f : F → H
in which
f (U ) : F (U ) → H(U )
is induced by Z ⊂ R. What are the presheaf
kernel and the presheaf cokernel of f ?
Exercise: To a presheaf F associate the abelian
group
AF =
Y
F (U ).
open U ⊂X
Show that the association of F with AF makes
the category PX of presheaves on X into a
subcategory of the category of abelian groups
which satisfies the axioms for an abelian category.
25
Sheaves
A presheaf F : UX → A on a topological space
X is a presheaf if it satisfies the following additional properties. (Heuristically, these say that
the sections of F over any open set U ⊂ X are
determined local information.)
1. (Local information determines sections). Suppose {Vi}i is a covering of an open set
U ⊂ X by open sets. If s ∈ F (U ) has image
0 in F (Vi) for all i, then s = 0 in F (U ).
2. (Patching condition). Suppose conversely
that we have sections si ∈ F (Vi) for all i
which are consistent, in the sense that si
and sj ∈ F (Uj ) have the same image in
F (Ui ∩ Uj ) for all i, j. Then there is an
s ∈ F (U ) restricting to si for all i.
26