M-.O is exact, we say that u is an epimorphism

EXERC1SES

N-. M-.O is exact, we say that u is an epimorphism

M'

M'/!(M)

^{rather than the module} itself. The context should make clear which is meant. Thus the cokernel is ^a factor module ofM'^.

Canonical

homomorphisms

discussed in

Chapter ^{I, 3} apply

^{to modules}

mutatis mutandis. For the convenience of the

reader,

^{we summarise these}

homomorphisms:

Let

N,

^{N' be two submodules}

of

^{a module M. Then N + N' is also asub-}

module,

^andwehavean

isomorphism

Nj(N

ⁿ

N') (N

⁺

N')jN'.

If

^{M ::J M' ::J Milare}

^modules,

^then

(MjM")j(M'jM") MjM'.

Iff:

^{M-. M' isa}

module-homomorphism,

^{and N' isasubmodule}

of M',

^then

f

-

l(N')

^isasubmodule

of

^{Mandwehaveacanonical}

injective homomorphism J:Mjf-l(N')

^-.

M'jN'.

Iff

^is

surjective,

^then

J

^isa

module-isomorphism.

The

proofs

^{are obtained}

by verifying

^{that all}

homomorphisms

^which ap-

peared

^when

dealing

with abelian _groups ^{are now}

A-homomorphisms

^of

modules. We leave the verification ^tothe reader.

Aswith_groups,weobserve that^a

module-homomorphism

^whichis

bijective

is a

module-isomorphism.

^Here

again,

^the

proof

^{is the same as} for groups,

adding only

the observation that the inverse _map,whichweknow is agroup-

isomorphism, actually

^{is a}

module-isomorphism. Again,

^{weleavethe verifica-}

tion to the reader.

Aswith abelian_groups,^wedefineasequenceof

module-homomorphisms

M'

1.

M Mil

to be exact if

Imf

⁼ Ker g. We have an exact sequence associated with a

submoduleN ofamodule

M, namely

o^-. N^-. M^-. M

j

^N-.

0,

the _map ofN into M

being

^the

^inclusion,

^{and the}

subsequent

map

being

^the

canonical_map. The notion ofexactnessis dueto

Eilenberg-Steenrod.

Ifa

homomorphism

^{u :N -.M} is such that

O-.NM

is exact, then^wealso _say thatu is a

monomorphism

^{or an}

embedding. Dually,

if

u

N-.M-.O

Algebras

There are some

things

ⁱⁿ mathematics which

satisfy

all the axioms ofa

ring except

for the existence of a unit element. We gave the

example

^of

L} (R)

ⁱⁿ

Chapter ^II,

^{1. There are also some}

things

^{which do not}

satisfy associativity,

but

satisfy distributivity.

^For instance let R be a

ring,

^{and forx, y E R define}

the bracket

product

[x, y]

⁼ xy

-

y^x.

Then this bracket

product

^{is not} associative in most cases when R is not com-

mutative,

^but it satisfies the distributive law.

Examples.

^A

typical example

^{is the}

ring

of-differentialoperators^with Coo

coefficients, operating

^{on the}

ring

^of Coo functions on an open ^set in R n. The bracket

product

[D},

^D2

]

⁼

D}

^{0 D}2

- D₂ ⁰

D}

oftwodifferential

operators

^is

again

^adifferentialoperator.^{In the}

theory

^{of Lie}

groups, the

tangent

space ^at the

origin

also has such a bracket

product.

Such considerations leadusto define amore

general

^notionthana

ring.

^Let

A be a commutative

ring.

^Let

^E,

^{F be modules.}

By

^a bilinear map g: E ^x E F

we mean a map such that

given

^{x E}

^E,

the map y ^.-..+

g(x, y)

^is

^A-linear,

^and

given

^{y E}

^E,

the map ^x

g(x, y)

is A-linear.

By

^an

A-algebra

^{we mean a}

module

together

^{with a}bilinear map g: E ^x E E. We view such amap ^{as a} law of

composition

^{on E.} But in this

book,

unless otherwise

specified,

^{we shall}

assume that our

algebras

^are associative and have a unit element.

Aside from the

examples already mentioned,

^{we note} that the group

ring A[G] (or

^monoid

ring

^{when G isa}

monoid)

^isan

A-algebra,

also called the group

(or monoid) algebra. Actually

the group

algebra

^{can be viewed as a}

special

case of the

following

^situation.

Let

f:

^{A B be a}

ring-homomorphism

^{such that}

f(A)

is contained in the centerof

B, i.e.,f(a)

^commuteswith every element of B for every^a EA. Then

we may view B ^{as an}

A-module, defining

^the

operation

^{of A on B}

by

the map

(a, b) f(a)b

foralla EA and b EB. The axioms for amodule are

trivially satisfied,

^{and the}

multiplicative

^{law of}

composition

^{B xB Bis}

clearly

^bilinear

(i.e., A-bilinear).

In this

book,

unless otherwise

specified, by

^an

algebra

^over

^A,

^weshall

always

mean a

ring-homomorphism

^as above. We say that the

algebra

^is

finitely

gen- erated if B is

finitely generated

^{as a}

ring

^over

f(A).

Several

examples

^{ofmodules over a}

polynomial algebra

^{or a} group

algebra

will be

given

^{in the next}

^section,

^{where we also} establish the

language

^of

representations.

2.

^THE

^{GROUP OF} HOMOMORPHISMS

Let A bea

ring,

^{and let}

X,

^X'be A-modules. Wedenote

by HomA(X', X)

the set of

A-homomorphisms

^{ofX' into X. Then}

HomA(X', X)

^{is an abelian}

group,thelaw ofaddition

being

that of addition for

mappings

^{into anabelian}

group.

If A is commutativethen we can make

HomA(X', X)

^{into an}

A-module, by defining affor

^{a EA}

andfE HomA(X', ^X)

^{to be themap such that}

(af)(x)

⁼

af(x).

The verification that the axioms for^anA-modulearesatisfied is trivial.

However,

if Aisnot

commutative,

^thenweview

HomA(X', X) simply

^{as anabelian}group:

Wealso view Hom_A as a functor. It is

actually

^{afunctor oftwo}

variables,

contravariant in the first and covariant in the second.

Indeed,

^{let Y be an}

A-module,

^{and let}

X'

!.

X

bean

A-homomorphism.

^Thenwe

get

^aninduced

homomorphism HomACt: Y): HomA(X, Y)

^-.

HomA(X', Y) (reversing

^the

arrow!) given by

ggof

This is illustrated

by

^the

following

sequence of maps:

X'

!.

_{X Y.}

The fact that

HomACt: Y)

^{is a}

homomorphism

^is

simply

^a

rephrasing

^ofthe

property (g

1 +

g2)

⁰

f

⁼ g1⁰

f

^+g20

f,

^{which is}

trivially

^{verified. If}

f

⁼

id,

then

composition withf

^{actsas an}

identity mapping

^ong, i.e._g⁰id ⁼ _g.

Ifwehaveasequenceof

A-homomorphisms

X' ^-.X ^-.

X",

thenwe

get

^aninduced sequence

HomA(X', Y) HomA(X, Y) HomA(X", Y).

Proposition

^2.1. A sequence

X' X^-. X"^-.0 is exact

if

^and

only if

^thesequence

HomA(X', Y) HomA(X, Y) HomA(X", Y)

⁰

is exact

for

^{all Y.}

Proof

^{This is an}

important fact,

^whose

proof

^is easy. For

instance,

suppose the first _sequence is exact. If_g:X"^-+ Y is an A

-homomorphism,

^its

image

ⁱⁿ

HomA(X, Y)

is obtained

by composing

gwith the

surjective

map of X on X". If this

composition

^is

0,

it follows that _g ⁼ 0 because X ^-+X" is

surjective.

^{As another}

example,

^{consider a}

homomorphism

g: X ^-+ Y such thatthe

composition

X'

!.

X

!!.

Y

is O. Then g vanishes ^on the

image

^{of A.. Hencewe can factor}g

through

^the

factor

module,

X

II

^mA.

X

I

₉ Y

SinceX ^-+X"is

surjective,

^wehavean

isomorphism X/lm

^A.+-+X".

Hencewe can factor_g

through X", thereby showing

^{thatthe kernelof}

HomA(X', Y) HomA(X, Y)

is containedin the

image

^of

HomA(X, Y) HomA(X", Y).

The other conditions neededto

verify

^{exactness areleftto}the reader. Sois the

con verse.

We have a similar situation with

respect

^{to the second}

variable,

^but then the functor is covariant. Thus if X is

fixed,

^{and we have a} sequence of A-

homomorphisms

Y'^-+ Y^-+

Y",

then we

get

^aninduced sequence

HomA(X, Y')

^-+

HomA(X, Y)

^-+

HomA(X, Y").

Proposition

^2.2. A sequence

o ^-+ Y'^-+ Y^-+

Y",

is exact

if

^and

only if

o^-+

HomA(X, Y')

^-+

HomA(X, Y)

^-+

HomA(X, Y")

is exact

for

^allX.

The verification will be lefttothe reader. Itfollowsatoncefromthedefini- tions.

Wenotethatto say that

o Y' Y

is exact meansthat Y' is

embedded

ⁱⁿ

Y,

^{i.e. is}

isomorphic

^{to a}submodule of Y. A

homomorphism

^{into Y' can be viewed as a}

homomorphism

^{into Y ifwe}

have Y' ^c: Y. This

corresponds

^tothe

injection

o

HomA(X, Y') HomA(X, Y).

Let

Mod(A)

^and

Mod(B)

^{be the}

categories

^{of modules over}

rings

^{A and}

B,

and let F:

Mod(A) Mod(B)

^{be a functor. One} says that F is exact if F transforms ^exact sequences into ^exact sequences. We see that the Horn functor in either variable need not be exact if the other variable is

kept

^fixed.

In alater

section,

^wedefine conditions under which exactness is

preserved.

Endomorphisms.

^{Let M be an}A-module. From the relations

(g1

⁺

g2)

⁰

I

⁼ g1 ⁰

I

⁺ g2⁰

I

and its

analogue

^onthe

right, namely

g⁰

(/1

⁺

12)

⁼ g

011

⁺g

0/2,

and the fact that thereisan

identity

^for

composition, namely

^idM^,weconclude that

HomA(M, M)

^{is a}

ring,

^the

multiplication being

^{defined as}

composition

of

mappings.

^{If n is an}

integer

^>

1,

^{we can} write

In

^{to mean the iteration}