Flkeraj - University of Illinois at Chicago
Transcript of Flkeraj - University of Illinois at Chicago
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 .
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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 .
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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