Setup - Rankeya Datta 520, Lecture 24.pdfit follows that H n >70 LR (mmIµ) = O NAK 㱺 It nooo mn M...
Transcript of Setup - Rankeya Datta 520, Lecture 24.pdfit follows that H n >70 LR (mmIµ) = O NAK 㱺 It nooo mn M...
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 ?