Lecture t

1 PID's2 MainThm on meduly PID's3 Proofof the main ThmRef Dammit Foote Chapter42

BONUS Finitedimensionalmodules overClay

11 PID definition examples A is a commutative ring

Definition Anideal ICA isprincipal if I a forsomeAEA

Say A is PID if A is a domain every idealinA isprincipal

Examples TL FEx F isfield are PID's everyEuclidean domain is a PIDNonexamples I 55 TLCx F xy are not PID

2,1 55 Tx Yay netprincipal

12 Uniquefactorization a b EA PID ideal s6 CAF Le Al la6 dd divides both a6 ble a be dI dividesboth a6 ddividesLfxaty6

T

So d GCD 961 moreover d x aty6 for some xyEAClassical application uniquefactorization holdsfor ARecall that bydefin p EA is p p is aprimeidealUF property Hae A decomposes as aproductofprimeelements in an essentially uniqueway 2 decompositions

are obtainedfrom one another bypermutingfactors

multiplying them by invertible elements

Remark in a PID everyprince idealthe ismaximal if pisprime

then f 3p f dividesp f p or

f APID is Noetherian

2 1 Main theorem A is PIDLet M be afinitelygenerated A module

Thm 1 I KETheprimesPa PeeA da dee74 St

M A O IEA pi2 k is uniquelydeterminedby M pt pete are uniquelydetermined up topermutationExample ATL thisThm classif n offingen'dabeliangripsI

22 Case ofA F x F isalg closedAssume dim M co se k o F is alg closed

primes in F x are X X TEF Cupto invertiblefactor

MainThm I die I die74 s t M É F x x tildiReminder Amodule over F x F vectorspace an operator XFor a fixed F vectorspaceM operators Am X'm M Mgiveisomorphic F x module structures AmX'm are conjugateI F linear4 M M s t 4km4 X'm exercise So theMainThm allows to classify linearoperators up to conjugation

Choose an F basis in FL x Tiki x ti j g di p

X x hit x ex tittytil tdi x til ifjade e

ti x tilt if j di tSo it acts as a Jordanblock

Jai ti fitiMainThm inthis case Jordan NormalForm thm

Let X be a linear operator on a fin dim F vectorspaceM let Fbealg closed Then in some basis X isrepresentedby a Jordanmatrix diag JaAi JdellellCanrecover thepairs hi dede from X will discuss

37in Lec8

3 Proofof Mainthm existence

Part 1 Prove that MIA e E A fi where f fmEAnonzero

Part 2 Prove that for feA a have

A f t.IE A pin wherepeeps arepairwisedistinctprimes f spiepts invertible

Today Part 1 M isfingenerated is aquotientof afree module F t A for some n so have epimorphism

F M K kerst

Since A is Noetherian K is finitelygeneratedSo can choose a basis en en in F

a set ofgeneratorsy yr EkThe crucial claim we can cheese g en kya yr insuch

away that I faitmeA Irmen yi fie

Then M F K EAei EAte Allfiletprecisely claim of Part 1

Now we need toprove the crucial claim We reduce it toa question about matrices w coeff's in A

Yi Isyijej Y yj eMatron A

h Suppose we replace y yr lore en withtheirnondegene

gyratelinear combinations Howdoes this affect Y

A replace Cy yr w ly g R where REMatra Ais invertible Let R EA is invertibleHere Y RYSimilarly Cq en g g N NEMatn A invertible

gives Ya YN

The crucial claim in the language ofmatricesYEMatron A F non degenerate invertibleLet

R E Mat CA Ne Matan A s t RYN Egg ofr fmeA

ProofofthisStepp Spec case 8 2 n tLemma let y geA y GCDly y Then I REMatza AJet R I s t

RIGProof Dividingguy byy can assume GCDlyy t

I a beA stx ay by2 1

Case A is PID Set R Eyby sothatRly e a

Step2 We use thefollowing steps

5T

i Multiply Y w

I r yto kill to 12,1

entry of Y

ii permute rows 2 j j 2By iterating thesesteps arrive at

EtNow multiply on the right by similar matrices permutecolumns

we arrive at finTContinue w this Y arrive at findFinishes Part P s

BONUS Finite dimensional modules ever CxyFix ne ke lur question classify Glxy meduly that have

Limen In the language of Linearalgebra classifypairsofcommuting matrices XY up to simultaneousconjugationFor n large enough there'sno reasonable solution However

various geometric objects related to theproblem are ofgreatimportance and we'll discuss thembelow

qSet C X4 EMatn e XY YX Consider the

subset Cga CC of all pairsfor whichthere is a

cyclicvector ved meaning that V is agenerator of thecorresponding Glxyl module Thegroup 64 E acts

on C by simultaneous conjugation g Xie CgXg gYgExercise Cage is stable under theaction Il thestabilizers for the resulting GlnCE action are trivialPremium exercise the set of Gln E orbits in Cayce isidentifiedwith the set of codim n ideals in laxyIt turns out thatthis set oforbits equivalently thesetofideals has a structure of an algebraic variety Thisvarietyis called the Hilbert scheme of npoints in IGand isdenoted by Hilton ET It is extremelynice veryimportantForexample it is smooth meaning it has no singularitiesOne can split Hilton E into the disjoint union ofaffine

sppacesmeaning 1C Theaffinespaces are labelledbythe

artitions of n ideals in QQy spannedby monomials

for eachpartition we can compute the dimension thusachieving some kindof classification ofpointsOne of the reasons why Hilton El is important is that it

appearsin various developments throughoutMathematics Algebraic

geometry not surprising Representation theory MethPhysicsand even Algebraic Combinatorics k Knottheory 1The structure of theorbitspacefortheaction of 64cal

yenC is FAR more complicatedyet the resultinggeometric

Iobject is still important