Lecture 41

Recall that H j> I we have projections Tj : III → My,zjµ,onto the j - th coordinate

, whose kernel is I ITII

.

Proposition I 8 Let I be a finitely generated ideal of R .

① It jai , her Ty.= It Fit

,ie .

.I't III = Etat

.

Thus,

IIIJ I E MID'M .

② The map M/Ijµ → # III's I induced by the canonical

map M → III is an iso .

③ III is I - adically complete .

Pf : ① I = fin . gen .⇒ It is fin . gen .

, sayIt = ( f , . . . .

. fr) .

gµ,Oth→ IJMwe get a surjection

µ , . . . . .mn ) 1-7 f,mi t - - - tfkm

I qISince It preserves surjection ,

we get a surjection Moth → It ?This gives a map at

g(PII )

Oth µ¥I → Etna I - Kerry. ↳ III ,=

which is checked to map

(III ) kz (a , , - → an) I i fix , too . tfpxh .The image of this

mapis Ker Ltj) and also It III

.

Since Tej is surjective , k¥1,=j pie I Mitra .

② The composition M→ III M/Ijµ, is the canonical

projection whose kernel is It'M.

ooo we get a map

MIIIM → F'Flkeraj =FiIi I

whose composition with the iso. FIFI; I =→M/Ijµ, is the identity .

Upshot : M/Ijµ, → ¥I/IjpqI is the inverse of an iso.

,so is

also an iso.

⑤ We get a commutative diagram

ooo → My,=j+, µ, → M/Ijµ, → o . . → M/Iµ,

~ - I= =

x x x

°o° → M±/IjtiµI → ¥TIjµiI → oo . → FIFI , I

where the vertical maps are isomorphisms by ⑦ . Taking limit

of the rows then gives an iso

ETI is fixity at

which can be verified to be the canonical map .

I

More details : The iso III It"' induced by the above diagram sends

(mnt Inn ) n> , ↳ ( (mnt IJM) j, + In Hm, ,

-

since @wt Id'M )j, , = fmjt IJM ) j, , mod Ker Tun-

- I" I

,we get

( (mnt It'

Mlj, + In Itn> ,

= ( (mjtI.im/jy,tInKit-)ny , ,ie .

III → It "'

is indeed the canonical map .

Observation : If R→S is a ring map and I is an ideal

of R , then

⑤I = IT

where T = the expansion IS of I to S.

This followsbecause

Sting = Fests = Stsns -

Corollary 2 °

.Let I be a finitely generated ideal of a ring R .

① It is III - adically complete .

③ IE E Jac ( II) .

③ The canonical projection I , : II → RII induces a bijection ,via contraction of ideals, between the maximal ideals ofRII and the maximal ideals of III .

④ If R is local with maximal ideal me that is finitely generated ,then Im is local with maximal ideal m Im .

Pf : ① III = expansion of I to It under R → It.

8 . It is I - adically complete ⇒ It is III -adically

complete .

But It is I- adically complete by Prop . I.

② Since It is III - adically complete by ①,

III E Jac (II)

by Leo . 40,Lem . I ④ .

③ Prop .

I ⇒ here,= III

.

I. The iso .

'

R' 'EI → RII

shows contraction induces a bijection{ max ideals of RII } ← Imax ideals of It containing ITE'T

1120

{ max ideals of It }

④ Applying ③ to m= I , we get £" is local because Ryn is.

Now, (O) is the max ideal of Rlm and the contraction of Co)

( contraction is just inverse image) under the mapI

,: Im → Rim

Prop 1is Kera

,= men .

I. m#

m

is the UNIQUE Max'd ideal of Em .

I

Exercise % Let I be a finitely generated ideal of a ring , then

for all m> n > O,

InEI

FEI± Item .

Hint : Use the isomorphism R =→ Rhem .

IMEI

Examples : ① If p EI> o is a prime number,

then It" is called

the p-adic integers .

② Consider the polynomial ring REX, ,

. . .

.Xn) = :RCI] and the ideal

I = (Xi , . ..

,Xn) = : (E) .

Hj 7,1 , a ERCEII# j , 7 ! fa E REE) sit . deg fa f j - I .

and a = fat (E)I

o! An element of RTI) # = lim RCE),

#j is UNIQUELY

ja

given by ( fjt tlj , , where

fj is a polynomial of degree E j - I

and

fjt , = fj t monomials of degree j , for all j > I .

ooo f = fo t ( f,

- fo) t ( fa - f , ) t H - fz ) t . . . gives a

power series in

Rffx, ,

. ..

, Xn ) ) -

④Upshot : Get a set map RTI] I RCCX , ,

. . .

,xn))

.

Conversely , H j > I , RAI gj = REED,µj compatible with

RCI ,#it → REY i .

EsUniversal property of lim gives a ring map RAID → RTII#

.

One

can verify q = inverse of Es .

← ⇐ ,so, REET = RCCX , . . .

.,XnD .

Black box Theorem : R is noetherian ⇒ REX , .. . .

,Xin is

noetherian for all n > O .

Proposition 3 : If R is noetherian,then for ANY ideal I of R ,

AIR

is noetherian .

Pf : Suppose I = ( in . . .

, in ) . Define

y : REX , . . . .

,xn) →R .

Rar t r

Xj t ij

Viewing R as a REX, ,

- ..

,Xn) - module

, if (E) = Hi , . . .

,Xn)

,then

we get a surjection of REX , . . . .,xn] -modules

X''#

# 9 a#RCI) -4 R

,

a#Check : y is a ring map .

The expansion of (E) to R is just I .

A# A #Rsince R = R = It

,we see It is a quotient of the

A #noeth . ring RCE ) = Rdx

, ,. . .

,xn )) .

I