Krull Dimension Theorem - Stanford University
Transcript of Krull Dimension Theorem - Stanford University
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