lecture 1: module basics - sporadic.stanford.edu

Modules Homomorphisms Categories Short Exact Sequences The isomorphism theorems

Lecture 1: Module basics

Daniel Bump

April 7, 2020


Scope of our reading of Chapter 10

Modules are to rings what vector spaces are to fields. They aretreated in Chapter 10 of Dummit and Foote.

We will be skipping Section 10.5 of Dummit and Foote: thismaterial (important for homological algebra and other reasons)is covered in Math 210A. We will not need it for grouprepresentation theory.

Main topics are:

The isomorphism theoremsFree modulesTensor product


Dummit and Foote Chapter 10: Modules

Let R be a ring (always with unit element 1 = 1R). A leftR-module is an abelian group A together with a multiplicationR× A −→ A that satisfies the distributive laws:

r(m + n) = rm + rn, (r + s)m = rm + sm,

(r, s ∈ R,m, n ∈ M) and the associative law

r(sm) = (rs)m, r, s ∈ R,m ∈ M.

Right modules are defined similarly.

If R is a field these are exactly the axioms for a vector space. AVector space is a module over a field.


Examples of modules

• If R is a ring, then any left ideal is a R-module.

• If R = Z, then a Z-module is just an abelian group. Indeed,the multiplication comes for free:

nx =

x + . . .+ x (n times) if n > 0;−(−nx) if n < 0;0 if n = 0

• If R = Matn(F) with F a field then 1× n column vectors form aleft R-module with matrix multiplication. (Row vectors form aright R-module.)


Modules over a PID

Looking ahead to Chapter 12, there is a structure theorem forfinitely-generated modules of a principal ideal domain. Keyexamples:

If R = Z, this gives information about finite (orfinitely-generated) abelian groups.If R = F[X], also a PID, this has applications to linearalgebra.


The key example for linear algebra

Let V be a finite-dimensional vector space over the field F andlet T : V −→ V be a linear transformation. Then we may makeV into an F[X]-module as follows. If f ∈ R = F[X], v ∈ V let

f (X) =

d∑n=0

anXn

f (X)v =d∑

n=0

anTn(v).

It is easy to check that this is a module. The module structureencodes the linear transformation. We may exploit the structuretheory of modules over a PID to obtain results about lineartransformations. (Chapter 12.)


Homomorphisms

Let M,N be left R-modules. A homomorphism f : M −→ Q is amap that is a homorphism of abelian groups that also respectsthe multiplication, that is,

f (rx) = rf (x), r ∈ R, x ∈ M.

An additive subgroup N of an R-module M is called asubmodule of M if it is closed under multiplication by R. Wemay write RN ⊆ N to express this, that is, rn ∈ N if r ∈ R andn ∈ N. Then N becomes a module, and the inclusion mapN → M is a homomorphism.


Kernel and Image

If f : M −→ N is a homorphism then the kernel

ker(f ) = {x ∈ M|f (x) = 0}

is a submodule of M. Also the image im(f ) = f (M) is asubmodule of N.

Let Q = im(f ). Then we may factor f into first the surjectionf1 : M → Q defined by f1(x) = f (x), that differs from f only in itsrange, followed by the inclusion of Q into N.

f = f2 ◦ f1, M Q Nf1 f2

We have factored the homomorphism into a surjectivehomomorphism followed by an injective one.


Quotients

If N is a submodule of M then the quotient additive group M/Nis naturally an R-module. This is the group of cosets x + Nwhere x ∈ M and we can define r(x + N) = rx + N. Note thatthis is well-defined, independent of the coset representative x.Let us write x = x + N and define p : M −→ M/N by p(x) = x.

Then p : M → M/N is a surjective homomorphism withkernel N.


isomorphisms

Let f : M → N be a homomorphism. If f has an inversehomomorphism N −→ M then it is an isomorphism. In order forthe homomorphism f : M −→ N to be an isomorphism it issufficient that it be bijective.

If there exists an isomorphism M −→ N then we say themodules M,N are isomorphic. This is an equivalence relationon R-modules.


Definition of a Category

Category theory abstracts the notions of homomorphisms andcategories. It is dealt with in Appendix II of [DF]. In a categoryC, there is a class of objects A,B,C, · · · and for every pair A,Bof objects a set HomC(A,B) of morphisms that may becomposed: if f ∈ HomC(A,B) and g ∈ HomC(B,C) then thecomposition g ◦ f ∈ HomC(A,C) is defined. These are subject tosome axioms. Particularly if h ∈ HomC(C,D) then

h ◦ (g ◦ f ) = (h ◦ g) ◦ f .

Also there is a morphism 1A ∈ HomC(A,A) such that iff ∈ HomC(A,B) then

f ◦ 1A = 1B ◦ f = f .


Examples of Categories

The category of sets. Morphisms are mappings.The category of rings. Morphisms are ringhomomorphisms.If R is a ring, left R-modules form a category. Particularly ifF is a field, vector spaces over F form a category.The category of groups. Morphisms are homomorphismsThe category of topological spaces. Morphisms arecontinous maps.


Isomorphism is a categorical notion

If C is any category, a morphism f : A −→ B is an isomorphismif it has an inverse morphism g : B −→ A such that g ◦ f = 1A

and f ◦ g = 1B. Two objects A,B are isomorphic if there exists aisomorphism f : A −→ B. This is an equivalence relation on theobjects of the category.

In a general category, objects may not be sets, and morphismsmay not be mappings. It is a special property of the categoriesof sets and modules that a morphism is an isomorphism if andonly if it is a bijection.


Initial objects

Let C be a category, and I an object. We say that I is an initialobject if for any other object A there is a unique morphismI −→ A.

PropositionAny two initial objects are isomorphic.

Indeed, if I and J are initial objects, there is a unique morphismf : I −→ J (since I is initial) and similarly a unique morphismg : J −→ I. We claim g ◦ f = 1I. Indeed, g ◦ f and 1I are bothmorphisms I −→ I, but there is a unique such morphism since Iis initial, so g ◦ f = 1I. Similarly f ◦ g = 1J and so I and J areisomorphic.


Initial and terminal objects in the Set category

We say an object T is a terminal object if for any object A thereis a unique morphism A −→ T. Similarly any two terminalobjects are isomorphic.

For example in the category of sets, the empty set ∅ is an initialobject, and a set 1 with one element is a terminal object. Thiscategory has both an initial and a terminal object, but in thecategory of sets the initial and terminal object are notisomorphic.

We say “the” initial object and “the” terminal object since theyare unique up to isomorphism.


Initial and terminal objects in the module category

In the category of R-modules, let 0 = {0} be the module with asingle element 0. If M is any R-module, there is a uniquehomomorphism 0 −→ M and a unique morphism M −→ 0.Thus 0 is both an initial and a terminal object. Like the categoryof sets, the category of R-modules has both an initial and aterminal object, but unlike the category of sets in the modulecategory the the initial and terminal object are the same.

Of course some categories may not have an initial object (or aterminal object) though if these objects exist they are unique upto isomorphism.


Exactness

Let A, B be R-modules. Then the set HomR(A,B) of R-modulehomorphisms is an abelian group. It has an identity element0 : A→ B, the unique map that sends every element to 0.!Let f : A→ B and g : B→ C be homomorphisms of R-modules.Then im(f ) and ker(g) are both submodules of B. A necessaryand sufficient condition for g ◦ f = 0 is that im(f ) ⊆ ker(g).

If im(f ) = ker(g) we say the sequence

A B Cf g

is exact at B.


Exactness and injectivity

Consider a morphism f : A→ B of R-modules. We ask if thesequence

0 A Bf

is injective. We do not have to specify the first map since thereis only one. A necessary and condition is that ker(f ) = 0, i.e.this sequence is exact if and only if f is injective.

Similarly

A B 0f

is exact if and only if f is surjective.


Short Exact Sequences

Now consider a sequence:

0 A B C 0f g

This is called a short exact sequence if it is exact at A, B and C.Exactness at A means f is injective; exactness at C means g issurjective; and exactness at B means im(f ) = ker(g).

For example let N be a submodule of M. Let i : N → M be theinclusion map and p : M → M/N the quotient map. Then

0 N M M/N 0i p

is a short exact sequence.


Equivalent sequences

We say two sequences

0 A B C 0f g

0 A′ B′ C′ 0f ′ g′

are equivalent if there are isomorphisms α : A→ A′, β : B→ B′

and γ : C→ C′ such that

0 A B C 0

0 A′ B′ C′ 0

f

α

g

β γ

f ′ g′

commutes. This means βf = f ′α and γg = g′β.


First isomorphism theorem

Some theorems in group theory have exact analogs inmodules. These include the isomorphism theorems. The groupversions are in [DF] Section 3.3; the module versions are in[DF] Section 10.2.

PropositionLet G, Q be groups and f : G −→ Q a surjective homomorphism.Then N := ker(f ) is a normal subgroup and Q ∼= G/N.

PropositionLet M,Q be R-modules and let f : M −→ Q be a surjectivehomorphism. Then N = ker(f ) is a submodule of M andQ ∼= M/N.


First isomorphism theorem (continued)

Let us prove the module version. To repeat:

PropositionLet M,Q be R-modules and let f : M −→ Q be a surjectivehomorphism. Then N = ker(f ) is a submodule of M andQ ∼= M/N.

If x ∈ M let x = x + N be the corresponding coset in M/N.Define a homomorphism ϕ : M/N −→ Q by ϕ (x) = f (x). Wemust check that this is well defined and bijective.

x = y ⇐⇒ x− y = 0 ⇐⇒ x− y ∈ N

⇐⇒ f (x− y) = 0 ⇐⇒ f (x) = f (y)

This implies that the definition ϕ (x) = f (x) is well-defined, andthat it defines an injective map M/N −→ Q. It is surjective sincef is surjective. Hence ϕ is an isomorphism.


Application to Short Exact Sequences

Consider a short exact sequence:

0 A B C 0f g

We will show this is equivalent to one of the form

0 N M M/N 0i p

where N is a subodule of M. We may take M = B andN = im(f ) ⊆ B. Since f is injective, A ∼= im(f ) = N; let α be thisisomorphism. If i : N → M is the inclusion:

0 N M

0 A B C 0

i

α

f g


Application to Short Exact Sequences (continued)

0 N M

0 A B C 0

i

α

f g

Now g is surjective so by the first isomorphism theorem,C ∼= B/ ker(g). By exactness at B, ker(g) = im(f ) = N. ThusC ∼= M/N. Let γ : M/N → C be this isomorphism. We have acommutative diagram:

0 N M M/N 0

0 A B C 0

i

α

p

γ

f g

lecture 1: module basics - sporadic.stanford.edu

Documents