Let R be an entire ring containing a field k as a subring, Suppose that R is a finite

Corollary 7.2. Let E be a finitely generated module and E' a submodule

3. Let R be an entire ring containing a field k as a subring, Suppose that R is a finite

dimensionalvectorspace^overkunder the

ring multiplication,

Show that R isafield, 4. Direct ^sums.

(a) ^{Prove in} detail that the conditions

given

ⁱⁿ

Proposition

^{3,2 for a} sequence ^to

split

^are

equivalent.

^{Show thata} sequence 0^---7> M' M Mil^---7> 0

splits

^if

and

only

if there exists asubmodule N of M such that M is

equal

^{to the direct}

sum1mf^{EB N,} and that if this is the_case,then N is

isomorphic

^toM".

Complete

all thedetails of the

proof

^of

Proposition

^3,2,

(b) ^{Let E and}

Ei(i

⁼ 1".., m) ^{be modules over a}

ring.

Let 'Pi: Ei E and

.pi:

^{E E}ibe

homomorphisms having

^the

following properties:

.11.0{().= id

Y', 'f" ,

.pi

⁰qJj⁼⁰ if_{i :F}j,

m

L

^qJi0

^.pi

^=ide

i⁼t

Showthat_{the map}x^r-+

(.ptJC,..., .pm x)

^isan

isomorphism

^{ofEontothedirect}

product

of the E_i(i⁼ 1,.."

m),

and that the map

(x

t,^...,x_m

)

qJ1X_t + ^...+ qJm^Xm

isan

isomorphism

^{of thisdirect}

product

^ontoE.

Conversely,

^{if E is}

equal

^{to a direct}

product

(or ^direct sum) of submodules E_i(i ⁼ I, ^{, . ,,} m), ^ifwelet 'Pi be the inclusion of EiinE, ^and

.pi

^the

projection

^of

EonE_i, then these maps

satisfy

^the above-mentioned

properties.

5. Let A be^anadditive

subgroup

^ofEuclideanspaceRn,^{andassumethat}in everybounded

region

^ofspace, there is

only

^afinitenumber of elements of A. Showthat A isafree abelian _group on < n generators.

[Hint:

^{Induction on the} maximal number of

linearly independent

^{elementsofA over R. Let Vb..., Vm beamaximal set of such} elements, ^{and let}

Ao

^{be the}

subgroup

^{of A} contained in the

R-space generated by

Vb^..,,vm^-t^.

By

induction,^onemay^assumethat_anyelementof

Ao

^isalinear

integral

combination of _Vb...,V_m-l' Let S be the subset ofelements VEA ofthe form

V⁼ at Vt + ^...+ am^Vmwithrealcoefficients_ai

satisfying

o ^< ai^< 1 o^< _am^< 1.

if i⁼ 1,^{...,m - 1}

If

v

^{isanelementofS withthesmallest}am :F0,^showthat

{Vt,

^...,Vm^-h

v}

^isabasis ofAover

Z,]

Note. Theabove exercise is

applied

ⁱⁿ

algebraic

^number

theory

^toshow that the group of units in the

ring

^of

integers

^ofanumber field modulo torsion is

isomorphic

to alattice in aEuclidean space. See Exercise 4 of

Chapter

^VII.

6. (Artin-Tate). ^{Let G bea}finite group

operating

^{on a finitesetS. Forw E} S, ^denote 1 ^. w

by

[w], ^{sothat we have thedirectsum}

Z(S)

⁼

L Z[w].

WES

Defineanaction of G on

Z(S) by defining

o'[w] ⁼ [o'w] (for ^{w E}S), ^and

extending

0'to

Z(S) by linearity.

^{Let M bea}

subgroup

^of

Z(S)

^ofrank #[S]. Show that M has

a Z-basis

{Yw}wes

such that O'Yw ⁼ Yaw for all ^{w E} S. (Cf, my Algebraic ^Number

Theory, Chapter

IX,

4,

^Theorem I.)

7, Let M be a

finitely generated

abelian group,

By

^{aseminorm onM we mean areal-}

valued functionv

I

^v

I satisfying

^the

following properties:

I

^v

I

^{> 0 for allv EM;}

Invl

⁼

In I I vi

^{forn EZ;}

I

^{v +}

wi

^<:

I

^v

I

⁺

I

^W

I

^{for all v, WEM.}

By

^thekernel of the seminormwe meanthe subset of elements^vsuch that

I

^v

I

^{= 0,}

(a) Let

Mo

be the kernel. Show that

Mo

^{is a}

subgroup.

^If

Mo

⁼

{O},

^{then the}

seminorm is calleda norm.

(b) Assume that M has rankr. Let VI' ^{,.., vrE}M be

linearly independent

^over

Z mod

Mo.

Prove that there exists abasis

{W.,.,

^{., wr}

}

^of

M/Mo

^{such that}

i

Iwil

^<:

L Ivjl.

j=1

[H^{int: An}

explicit

version of the

proof

of Theorem 7.8

gives

^{the result.}

Withoutloss of

generality,

^{we canasume}

Mo

⁼

{O}.

^{Let M}I

=

(V.,

^{,.., vr}

),

Letd be the exponent^{of M}

/

^MI. ThendM has afinite index in

MI.

^Let nj,j be the smallest

positive integer

^{such that}there exist

integers

nj,^I'..., nj,j_1

satisfying

nj,IV^{I + ... +} nj,jv^{j = dWjforsome wjE M.}

Without loss of

generality

^wemay^assume0^<:nj,k^<:d- 1. Then the elements WI'^{.., , W}rform the desiredbasis.]

8, Consider the

multiplicative

group

Q*

^ofnon-zerorational numbers. For anon-zero rational numberx ⁼

a/b

^{witha, bE Z and}(a, b) ⁼ 1, define the

height

h(x) ⁼

log

^max(

I

^a

I, I

^b

I).

(a) ^{Show that h defines aseminorm on}

Q*,

^whose kernel consists of ⁺1 (the torsion

group).

(b) ^{Let M}Ibea

finitely generated subgroup

^of

Q*

^,

generated by

rational numbers XI'^{, ..,x}m^' Let M be the

subgroup

^of

Q* consisting

of those elementsXsuch thatX^S EM_I forsome

positive integer

^s, Show that M is

finitely generated,

and

using

^{Exercise 7, finda}bound for the seminorm ofasetofgenerators of M interms of the seminormsofx

I'^,..,X_m.

Note. The above two exercises are

applied

ⁱⁿ

questions

^of

diophantine approximation,

See my

Diophantine approximation

^ontoruses, Am. J. Math.

86 (1964), pp. 521-533, and the discussion and references I

give

ⁱⁿEncy-

clopedia

ofMathematicalSciences,^Number

Theory

III,

Springer Verlag,

^1991,

pp. 240-243,

Localization

9, (a) ^{Let A bea}commutative

ring

^{andlet M bean}A-module. Let S bea

multiplicative

subset of A. DefineS-I M ina manner

analogous

^{to the one weused to define} S-I A,and show thatS-IMisanS-IA-module.

(b) ^{If 0 M' M M" 0 is an exact} sequence, show that the sequence o S-IM' S-IM S-IM" 0 isexact.

10. (a) If p is ^a

prime

^{ideal, and S =A - p is the}

complement

^{ofpin the}

ring

A, ^then S-IM is denoted

by

^Mp. Show that the natural map

M

n Mp

ofamodule M into the direct

product

of all localizations M_pwhere_pranges^over all maximalideals, ^is

injective.

(b) ^{Show thata}sequence 0 M' M M" 0 isexactif and

only

if the sequence o

M Mp M"p

^{0 isexactforall}

^primes

^p.

(c) Let Abe ^anentire

ring

and let M be atorsion-free module. For each

prime

^pof

Ashow that the natural map M

Mp

^is

^injective.

^In

^particular

^A

^Ap

^is

^injective,

but you^{can see}that

directly

^{from the}

imbedding

of A in its

quotient

^{field K,}

Projective

^{modules overDedekind}

rings

Forthe nextexercise we assumeyou have done the exercises ^on Dedekind

rings

ⁱⁿ

the

preceding chapter

^.Weshallseethat for such

rings,

^somepartsof their module

theory

canbe reducedtothecaseof

principal rings by

localization. We let0beaDedekind

ring

and Kits

quotient

^field.

11. Let M bea

finitely generated

torsion-free moduleover o. Provethat M is

projective.

[Hint:^{Given a}

prime

^{ideal p,the} localized module

Mp

^is

finitely generated

^torsion-

freeover0p, which ^is

principal.

^Then

Mp

^is

projective,

^so if F is finite freeover 0, andf: ^{F M is a}

surjective homomorphism,

^then

fp: Fp Mp

^{has a}

splitting

gp: ^Mp

Fp,

^{such thatf}p

0 gp ^{= id}Mp^. There exists c

p E0 such that cp

ft

p and

cpgp(M)

^{CF. The}

family {c

p

}

generates ^{the unit ideal0}

(why?),

^{sothere is afinite}

number of elements c

p,and elements X; ^E0 such that

2:

x;cp,

= 1. Let 9 ⁼

2:

x;cp,gp,.

Then show that_g: M F

gives

^a

homomorphism

^{such that} fog ^{= id}M,]

12. (a) ^Leta,b be ideals. Show that there isan

isomorphism

^of0-modules

aEBboEBab

[Hint: First do thiswhen_a,b are

relatively prime.

Consider the

homomorphism

a EBb a +b, ^{and use} Exercise 10. Reduce the

general

^{case tothe}

relatively prime

^case

by using

^{Exercise 19 of}

Chapter

II.]

(b) ^Leta,b be fractional ideals, ^and

letf:

^{a b bean}

isomorphism

(of o-modules, ofcourse).

Thenfhas

^{anextensiontoaK-linear}

mapfK:

^{K K, Letc =}

fK(I).

Show that b⁼ caand thatf^is

given by

^the

mapping

me:^{X cx}

(multiplication by

^c).

(c) ^{Let a bea}fractional ideal. For each b E a-I

the map mb: ^{a 0} is an element of the dual aV. Show that a-I = a^{v =} Hom_o(a, 0) under this map, and ^so

a^{VV =} a.

13. (a) ^{Let M be a}

projective

^{finite moduleoverthe Dedekind}

ring

^o. Show that there exist free modules F and F' such that F ::) M ::) F', ^and F, F' have the same

rank, which is called the rank of M.

(b) Prove that there exists abasis

{e.,

^...,

en}

of F and ideals a.,^{, .,,a}n suchthat M ⁼ aiel ^{+ ... +}anen^,orin other words,^{M =} EB_a;.

(c) Prove that M ⁼ on-I EB a for some ideala, and that the association M ^a induces an

isomorphism

^of

Ko(

0) with the group of ideal classes Pic( 0). (The group

Ko(o)

is the group of

equivalence

^{classes of}

projective

modules definedat theend of

4.)

A few snakes

14. Consideracommutative

diagram

of R-modules and

homomorphisms

such that each

rowisexact:

) M

qj

)0 M'

Ij

)M"

o ⁾N' ⁾ N ⁾

hj

N"

Prove:

(a)

Iff,

^hare

monomorphisms

then g is^a

monomorphism, (b) Iff,

^hare

surjective,

then g^IS

surjective.

(c) ^Assumein addition that 0^--+M'^--+Misexact and that N^--+N"^--+0 isexact.

Prove that if any^two

off,

g, h^are

Isomorphisms,

^{thenso ISthethud.}

[Hint:-

Use thesnake

lemma,]

15. The five lemma. Consider acommutative

diagram

^{ofR-modulesand}

homomorph-

ismssuch that eachrowisexact:

Mt

'.j

)M₂

f,j

) M4

14j

)

M3

1.j

)

Ms

Nt ^{) N}2 ⁾ N₃ ) N₄ ⁾

1,j

N_s Prove:

(a) ^If

11

^is

surjective

^and

12,14

^are

monomorphisms, then/

^{3IS a}

monomorphism, (b)

^If

Is

^isa

monomorphism

^and

12,14

^are

surjective,

^then

13

^is

surjective, [Hint:

Usethe snake

lemma,]

Inverse limits

16. Prove that the inverse limit ofasystem^of

simple

groups in which the

homomorphisms

are

surjective

is either the trivial group, ^{or a}

simple

group.

17. (a) ^{Let n} range ^over the

positive integers

and let p be ^a

prime

^{number, Show that}

the abelian groups

An

⁼

Z/pnz

^{form a}

projective

system under the canonical

homomorphism

^{ifn > m, Let}

Zp

^be its inverse limit. Show that

Zp

^{maps sur-}

jectively

^oneach

Z/pnz;

^that

Zp

^hasnodivisors of0,^{and hasa}

unique

^maximal

ideal

generated by

p. Show that

Zp

^{is factorial, with}

^only

^one

^prime, namely

^p

itself.

(b) Next consider all ideals of Zas

forming

^adirected system,

by divisibility.

^Prove

that

!!!!! ^Z/(a)

⁼

n _Zp,

(a) ^p

where the limit is taken ^over all ideals (a), ^{and the}

product

^{is taken over all}

pnmes p.

18. (a) ^Let

{An}

^{be an}

^inversely

directed sequence of commutative

rings,

^{and let}

{M

ⁿ

}

bean

inversely

^directedsequence ofmodules, ^Mn

being

^amoduleover

An

^such

that the

following diagram

^iscommutative:

An+1

^{x M}n +1 M_{n +1}

An

^{X Mn Mn}

The vertical maps ^are the

homomorphisms

of the directed sequence, and the horizontal maps

give

^the

operation

^{of the}

ring

^onthe module. Show

that!!!!!

^Mn

is amodule

over!!!!! ^An.

(b) ^{Let M bea}

p-divisible

group. Show that

Tp(A)

^{is amoduleover}

Zp.

(c) ^{LetM, N be}

p-divisible

groups, Show that

Tp(M

^{EB N) =}

Tp(M)

^EB

Tp(N),

^as

modulesover

Zp.

Directlimits

19. Let

(A;,f)

^beadirected

family

^{ofmodules. Let akE}Ak^forsomek,and suppose that the

image

of ak in the directlimitAiso. Show that thereexistssomeindexj^{> ksuch} that

f(ak)

^{=O. In other words whether some element in some group Ai vanishes}

In thedirect limitcan

already

^{beseenwithin the}

original

^data, One way^toseethis isto usetheconstruction of Theorem 10.1.

20. Let I, J be twodirected sets, and

give

^the

product

^{I x J}the obvious

ordering

^that

(i,j)

^<

(i',j')

^{if i < i' and}j ^<

j'.

^{Let A}ij be a

family

^ofabelian groups, with homo-

morphisms

^indexed

by

^{I x} J,^and

forming

^{a directed}

family,

Show that the direct limits

lim limAij

^and

^lim limAij

i j j ⁱ

exist andare

isomorphic

^inanaturalway, State and prove the^sameresultfor inverse limits.

21. Let

(M,f), ^(M;, g)

^bedirectedsystems of modules^{over a}

ring. By

^a

homomorphism (M;) (M;)

one means a

family

^of

homomorphisms

^Ui:

M;

^Mifor each i whichcommutewith the

f, g. ^Suppose

^{we are}

^given

^{anexactsequence}

o

(MD

(Mi)

(M')

⁰ of directed systems,

meaning

^{that foreach}i,the sequence

o M_& M._I^-+ M'_I 0

isexact. Show that thedIrectlimit preservesexactness,that is o hmM

hill

M;

h111 M;'

⁰

ISexact.

22. (a) ^Let

{M;}

^bea

family

^{ofmodulesover a}flng. For any module N show that

Hom(ffi ^M;,

^{N) =}

n

^Hom(Mi,N) (b) ^{Show that}

Hom(N,

n

^M;)=

n

^Hom(N,

^M;).

23, Let

{M

i

}

^beadirected

family

^{of modulesover a}

ring.

For any module N show that

11m

Hom(N,

M;)

⁼ Hom(N,

Jim M;)

24. Showthatany module is^adirectlimit of

finitely generated

submodules.

A module Mis called

finitely presented

if there isanexactsequence

FlFoMO

where F_0,

Flare

^{freewithfinitebases. The}

image

^{of F1in F0is saidtobethesubmodule}

ofrelations,among thefreebasis elements of F₀^.

25. Show that any module is ^adirect limit of

finitely presented

^modules(not

necessarily submodules).

^Inotherwords,

given

M,there existsadirectedsystem

{M;, fJ}

^{with Mi}

finitely presented

^{for all isuch that}

M

lim Mi.

[Hint: Any finitely generated

submodule is such a direct

limit,

^{since an}

infinitely generated

module of relationscanbeviewedas alimit of

finitely generated

^modulesof relations. Make this

precise

^toget^a

proof.]

26, Let Ebeamoduleover a

ring.

^Let

{M

i

}

^beadirected

family

^{ofmodules. IfEis}

finitely generated,

show that the natural

homomorphism

lim Hom(E, ^Mi

) Hom(E, lim

^Mi

)

IS

Injective.

^{If E is}

finitely presented,

show that this

homomorphism

^isan

isomorphism.

Hint: First _{prove the} statements when E is free with finite basis. Then, say E is

finitely presented by

^anexactsequence F¹ F₀ E O. Consider the

diagram:

o ⁾

li111 Hom(E,

^Mi)

I

) lim

Hom(F

^0,Mi)

I

)limHom(F^l'Mi) o )

Hom(E, liIIl

^Mi) ⁾

Hom(F

0'

lim M;)

⁾

Hom(F

l'

I lim

^Mi)

Graded

Algebras

Let A bean

algebra

^{over a field k.}

By

^a filtration ofA we mean a sequence of k- vectorspacesAi

(i

⁼⁼0,

I,

^..

.)

^{such that}

Ao^c Al ^cA2 ^{c ,., and}

U

^A;==A,

and

A;Aj

^c

A;+j

^{for all i, j >O. In}

particular,

^{A isan}

Ao-algebra.

^{Wethen call A afil-}

tered

algebra,

^{Let R be an}

algebra.

We say that R is

graded

^{if R is a direct sum}

R⁼⁼

EB

^{R; of}

^subspaces

^{such that}

R;Rj

^c

R;+j

^{for alli, j>o.}

27. Let A be a filtered

algebra.

^Define R; ^{for i>0}

by

R;⁼⁼

A;/

A;_I.

By definition,

A_I ⁼⁼

{O}.

^{Let R==}

EB

^{R;, andR;==}

^gr;(A).

^{Defineanatural}

^product

^{on R}

making

Rintoa

graded algebra,

^denoted

by gr(A),

and called the associated

graded algebra.

28. LetA,^Bbefiltered

algebras,

^{A ==}

U

^A;andB==

UBi.

^{Let L: A -+Bbean}

(Ao, Bo)-

linear _map

preserving

^the

filtration,

^{that is}

L( A;)

^{c B; for all i, and}

L( ca)

⁼⁼

L(c)L(a)

^{forcEAoandaEA;for all i.}

(a)

Show that Linduces an

(Ao, Bo)-linear

map

gr;(L): gr;(A)

^-+

gr;(B)

^{forall i,}

(b) Suppose

^that

gr;(L)

^isan

isomorphism

^{for all i. Show thatLisan}

(Ao, Bo)- isomorphism.

29.

Suppose

^{k has}characteristico. Letnbe thesetofall

strictly

upper

triangular

^ma- trices ofa

given

^{sizen x n overk.}

(a)

^Fora

given

^{matrix XEn, let} DI

(X),

^...,

Dn(X)

^{be its}

diagonals,

^soDI ⁼⁼

DI

(X)

^{is the main}

diagonal,

^{and is0}

by

the definition ofn. Let ni be the subsetofn

consisting

of those matrices whose

diagonals

DI,^...,Dn-i^areO.

Thus_no⁼⁼

{O},

^nl consists of all matrices whose components ^{are 0} except

possibly

^{forXnn; n2} consists of allmatrices whosecomponents^{are 0}except

possibly

those in the lasttwo

diagonals;

^{and so}forth. Show that each_{n; is}

an

algebra,

^andits elementsare

nilpotent (in

^{fact the}

(i

^{+ 1}

)-th

powerof its elementsis

0).

(b)

^{Let Ube the set of}elements I + X with XEn. Show that Uis a multi-

plicative

group.

(c)

Let expbe the

exponential

series definedas usual. Show that _expdefines a

polynomial

^{functionon n}

(all

^{butafinitenumberoftermsare0wheneval-} uatedona

nilpotent matrix),

^andestablishes^a

bijection

exp: ^n-+ U.

Show that the inverse is

given by

^thestandard

log

^series.

CHAPTER IV