Lecture 24

Setup * : CR, m) is a noetherian local ring , q,is an ideal at . Tg --m ,

M is a finitely generated R-module .

Hilbert - Samuel function of M w- rt. 9, : Hq

. #(n) = Ep ( Mlgnpa ) .

" " " "

: Ppolynomial" "

g. µ,(n) = Hq

,#Cn) for no 0 .

Degree of M : deg Pg ,µ, ; this is independent of g, and coincides

with deg Pm,m

-

Upper bound on deg M : deg M E Up(q) , for any m-primary g, .

In particular , deg M s Me (m) = dim mmlmz .

Last time : If 0 → N→ M → P → o is a s.es. of R - mods

then

Hg, µ,

= Hg, "t Hq

,p- H

where H Cn) = LR ( Nngn9÷M) is polynomial like and

deg H L deg Hg, ,y

.

Since Hg,# , Hyp are non - negative (⇒ leading coefficients of Pg, ,N , Pgp

are positive ) ,we get

deg M = man { dy N, deg P} .

To deg N, deg P E deg M .

Lemma too Under Setup & if a ER is a nonzero divisor on M,then

deg 141am - deg M .

Pf : Consider the s.es .O →ME> M → Mla, → O . Then

deg ( Hg,,µ,t Ha

,na,aµ

- Hq,m) < deg Hq,µ,

= deg M .

-

deg M 1am.

I

Defn : For an R -mod M,the dimension of M ,

denoted dim,zM ,is the Krull dimension of RIAnnp.iq -

sanity check : dim ,zR = dim RIAnnp.pe = dim RIO = dimR-

Example : Let M be a fin . gen . module over a meth . ring R .

Then l ,z(M) < a ⇐ dim ,zM = O.

Proposition 2 : Under setup H ,dim,zME deg M E Up (m) .

Pf : De already know deg M E Mp (m) .To show

dim RM E deg M ,we proceed by induction on deg M .

If deg M = deg Hm, µ,

= 0,then H n >> 0

,l,z(Minn µ ) is

a constant, independent of n . Using the s.es .

o→m÷i÷→i÷. → n'÷. -so ,

it follows that H n >70, LR (mmIµ) = O

NAK

⇒ It nooo, mn M = mnt

' M ⇒ It noo,mhm = O

⇒ F no >> 0 sit . m"E Ann ,zM

⇒ dim ,zM = dim 121pm,zµ,=0 = deg M .

Now suppose d := deg M 7,

I.Then M t O

. Note that

dim ,zM = sup { dim Rtp : p is a minimal prime of Annam} .

Let p be a minimal prime of Ann,zM ⇒ PE Assam because

Supp,zM ,hes ,zM have the lame minimal elements .

Let N E M be a submodule s - t . Nz Rtp .Then

dim ,zN = dim Rtp ,and deg N E deg M .

WANT : dim RIP E deg M =D .

If not,7 a chain of prime ideals

Po =P EP , E - - - ¥ Pati -

Let a E P ,Ip .

Then N E Rip ⇒ a is a nonzero divisor on N

⇒ deg Nyan, L deg N ⇒ dim,zN/aµ E deg Nyan, sd -

Lemma 1 Induction hypothesis

Now, Anna N Ian, = Anne Rtp = Anne R/¢ , a,

= (P , a) .

a CRIP)

To dimr Nyan, = dime Rlqp ,a , 7, d because

Pyg , a , E - i - E PdHkp,a)

is a chain of primes in Rhp , a) of length atleast d .

This contradicts dime Man, < d . By contradiction, for all

minimal primes p of Ann ,zM ,

dim Rip E deg M

⇒ dim Rlann ,zµ, E degM ⇒ dimp ME deg M , as desired .

I

Corollary 30. If Pfm) is a noetherian local ring , then

dim R s dimRim mlmz .

In particular ,a noeth . local ring has finite Krull dimension

.

Pf : Apply Proposition 2 with M = R and use the fact that

Mr (m) = dimRim Mma .

I

Example : KC X, ,Xu, Xs , . . .

)( ×

, ,×, ,×, , . . .)is a local ring of

infinite Krull dimension .

Example : If k is algebraically closed,then

dim kcx , . . . . , xD = n .

Indeed,

dim KEX , .. . .

, xn) z n because

O E (Xi ) ¥44 , Xz) I, - .- E ki , .kz , . . .

. Xn)F

is a chain of primes of length n . Now recall that

climb = sup { dim Rp : PE Speck} = sup{ dim Rm : m is a

man ideal of R}"

Let m be a maximal ideal of KCX , ,. . .

> Xn) . lince k is algebraicallyclosed

,

m = ( X,- a

, , . . .

,Xn - an) for a ; Ek.

Note

dim KCX, , . . . ,Xn] me = dim KCX.su . .

,Xn)# . . . .,×n)

because the K- automorphismkcx , .

. . . ,xn] → KEX , . . . . ,xn]

Xi t Xi -ai

induces an isomorphism of local rings KCX . .. . . ,XnIc×

. .. . . ,×n,Ikki , - . .

.Xin .

To dim kfx , .. . .,XnIm = dim kcxh - -

i.xDµ , . . . ,×n)

Cor . 3

£ chimp @ is . . . . Xn ) KCX ' ' ' - ' ' Xn) (x . . . . . ,xn)

.

(X, , . . . , Xu)" K (X11 - ' ' >

Xn)(x , , . . . ,

Xn)

Lemma 4 : If m is a max ideal of a ring R ,then

mymz I MRI ( as Rm and R - modules)m2Rm

Pf : If a E R-m,then mynaI mlmz is an isomorphism

on myna with inverse given by mime mlmr , where x ER

is such that Xa = I modm .

ooo By the universal property of localization

Fml"

@ma ) = mini .

LocalizationBut

, CR- m)-'

flu) = 42-4-12 = MRL .

is exact

µ-my -1mL m2 Rm

e

Using Lemma 4,

dim kcxi,

- . .

,xD¢ . . . . . ,×n )

E climb 413--13×42 = n .

Hi , . . .

.xn)-

Upshot : ht maximal ideals m of REX , . . . .

.xD

,

dim kcx , , . . . ,xnTm E n -

Thus,n E din thx

, >. . .

. xn) E n .

Next time : Krull 's Haupt ideal satg .

Motivation : Let a be an ideal of a noeth . ring R and

ht a = inf L dimRp :

p is a minimal prime of a} .Is their a way to bound htx using just information about a and

not the prime ideals containing it ?