RECALL FROM PREVIOUS Lecture

or 15 PRIMARY IF xyear tea or yheorFOR A SUFFICIENTLY LARGE

THIS IMPLIES Roz is prone

THEOREM IF A IS Noetherian AND 02 IS AN

IDEAL U ADMITS A Decomposer N

or g n nfu gi PRIMARY

nor UNIQUE

PROOF DEFINE AN IDEAL OR TOBE

IRREDUCIBLE If 02 cannot BE EXPRESSED

AS AN INTERSECTION OF STRICTER LARGER

IDEAL A CAN BE EXPRESSED AS A

UNION OF IRREDUCIBLE IDEALS If Not

Since A 15 NOETHERIAN THERE IS A

Primary Decomposition Krull Dimension Theorem

MAXIMAL COUNTEREXAMPLE 02

A NOT IRR

or fine Cbe Lirr Ibe 5 Larger be C Are intersections

of IRREDUCIBLE IDEALS CONTRADICTION

So PROBLEM IS TO SHOW IF A IS

NOETHERIAN 1 IDEAL Cr is PRIMARY

or is primary 0 IS PRIMARY

in Afr

or is Irr 0 IS PRIMARY In Apr

so WE MAY ASSUME 9 Is a

suppose Xy D NEED b O or X o

LET an zeal ZF o V

Un E Guy E

So A Noetherian Un Oh For

some w Yy o consider

uX n b we'LL SHOW THIS IS Zero

ae n ya Aw yetax Xyz o Z D11

An W

so we anti wean Knw o

a o

THIS proves F n y O

But Col is IRREDUCIBLE so

QY o or ryfoX O or y O PROVING THE IRREDKBK

IDEAL IS PRIMARY

THEOREM If A IS NOETHERIAN AND

01 IS ANY IDEAL THERE IS A FINITE

SET OF MINIMAL PRIME IDEALS CONTAINING 9

MORE precisely

if ol Ofn n of n IS A PRIMARY

Decomposition AND of 2M WHEEL IS

PRIME THEN if CONTAINS THE PRIME

IDEAL rffi for some i

INDEED tf off for SOME I If Nnr

LET xegily Then IT X Eng a

BUT Mxify siree y is Prime

CONTRADICTION THEN

y rly refitTHIS PROVES THE THEOREM

REMINDER ABOUT HILBERT SERIES

LET A BE A NOETHERIAN LOCAL RING

WITH MAX L IDEAL M

PROVED LAST TIME THERE IS A

Pawnomial PAY SUCH THAT

lfA mY pal if u

15 SUFFICIENTbe LARGE

LENGTH FUNCTION ON MODULES IS

CHARACTERIZED BY i l amoeef µhad

Ifa n n n a

line evil teen iAND HAHA L if M Is Local

For EXAMPLE f m mh I h

DimfuehluentiT

AS A v 5 over Afm

HOW WE PROVED THIS

DEFINE A GRADED RINGA

Gural airalI A

Gitai mYui

Muto Giving 76mF

IS INDUCED By MULTIPLICATIN

Mix vis mirjNOETHENAN

PROVED if G G IS A GRADED

RING AND GO_M IS A FIELD

G kCX n Xu x egdi

THEN PIG.tl II Drink.lt

gotproved PCG tf p tog r o tdhl

WHERE got IS A POWNOMIAL

APPEINO THIS to Gm.AT

THE GENERATORS ARE Of DFG 1 IN

mini AND FOo

2 smhianitytia gotI

4 tfI

A Driven't

uswa r th 2 faultWE CAN DEDUCE

DmfnE mi IS A POLYNOMIAL

OF DEGREE Eh 1

THENl.LA wEI raw en w k

n Suff LARGE

uh I

2 one.AEmi Ia

ee al

TAM THERE IS A POLYNOMIAL

fufu suck THAT

ltAmY tmklIF W IS SUFF LARGE

Deo Xm DiMCmha of

7 GENERATORS REWED

for ur

THE MAIN Nt REM af DIMENSION

THEORY i DEFINE

deal sea Ctm8 A MINIMAL NUMBER of

Generators reaureoFor An M PRIMARY IDEAL

Krull DIMENSION OF A 15 THE Lenox d

of tfLONGEST SATURATED CHAIN OF PRIME IDEALS

Yo f f E r E Yd

Theorem Dental dial 8cal

FIRST STEP IS TO GENERALIZE OUR

resuir ABOUT HILBERT Series Suppose

G is M primary Vg m

THEN WE'LL PROVE IF h IS THE

NUMBER OF GENERATORS NEEDED for gTHEN

Deb.fm Eh

SKETCH of Proof

Define GqCal Ggical

Gigi gigiNOW GO Alf Is Not A FIELD

6g WHEREEVERY WE MENTIONED

DIMENSION WE HAVE TO USE LENGTH

THE ARGUMENT GOES THROUGH AND

THERE IS A Polynomial 1g Suck THAT

t.ghy.la qY if

IS SUFFICIENT LARGE AID

SEGAg h of Gers of g

liqig'tLemma i SEG Xp statue f deal

since q is A PRIMARY

M g mn

he Z f z mum

Kamil learn Nahin

WN Elgin E IncunlTHIS IMPLIES fur AND kg HAVETHE SAME DEGREE

therefore doff of crews need

For A Gb PRIMARK

IDEAL

dat Sal

Proof that 8cal E IMAIDIN FAI LENGTH h of A CHAIN OF

PRIME IDEALS

yo f Y E F YnCHOSEN AS LONG AS POSSIBLE

LEMMA SUPPOSE tf Yu Are

PRIME IDEALS 024 Afi THEN

org Uefi

CLEAN AVE If w l

Suppose TVE for U t

we Have kit 0h BUT Not hfjARGUING BY CONTRADICTION

as Uefiso it Must Be Xie YiA tying for j't io

fie a

guyASSUMING

so Xerfn for some h

X IT ti truth tiith II j M

T

Ttfn Ya

so IT dielferith

contradiction since Kieffer fitffBUT Ya is PRIME

h

Tekno to Prove 8cal EDNA

Proposition her A BE A

NOETHERIAN LOCAL RING WITH MAILIDEAL M THERE IS A Seavence

Of ELEMENTS fi ten of W 5041

THAT Ct n Xm is m PRIMARY

AND ANY PRIME IDEAL if

CONTAINING tyre X HAS

HEIGHT N

THE HEIGHT h OF A PRIME YIS THE LONGEST POSSIBLE CHAIN

Yo E Eyu yaf primes So THE HEIGHT of

M IS THE KRUL DIMENSION

I

y07

Ay n

I ixx

ix I

THIS WILL PROVE KRULL DIMENSION

is h Therefore

DIMIAI HEIGHTEN 8cal

SINCE Cte n th will BE A

PRIMARY IDEAL WITH h Generators

8cal smartest number ofGENERATORS of A

m PRIMARY IDEAL

SUPPOSE fi 1 Hi are

constructed If VCA tht M

WE MAY STOP OTHERWISE TIENE

ARE A FINITE NUMBER OF MINMAL

PRIME IDEALS CONTAINING Cte Hml

AND THESE ARE THE MINIMAL

PRIME IDEALS AMONG THE TcfIN A PRIMARY DECOMPOSITION

Cry ionic Aqi

Ayr IM SO THESE MINIMAL

PRIMES Are Nor M BY THE LEMMA

WE CAN AND SOME tin THAT IS

NOT IN ANN OF THESE MINIMAL PRIMES

LET p BE A prime DEAL

CONEAININO

Hi stealTHIS contains Ct r Xml _AgBut If P Is A Prime CONTAINING

fit it CANNOT BE ONE Of THESE

MINIMAL PRIMES SINCE tie 4THESE

P must BE LARGER

Po Ct n Kia p z one ofTHESE MINIMALPRIMES

P ar Aaaaa Igf

DHZine

Hq a Hero T a

THIS proves 8cal E ONAin

RECOMMENSED i CHAPERON

SIMISIMPLE RINGS IN LANG'S ACC