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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