Lecture 19

Artin-Rees Lemma 8 Let R be a noeth .

ring , I be an ideal,M be

a fin . gen . R-mod and N be a clubmodule .

Then F 127,0 sit . H c > 0 ,

N n Ik#M = Ic ( N n Ik M) .

MAIN COROLLARY : Under the hypothesis of the Artin- Rees Lemma , ifI,= topology on N induced by the I- adic filtration { INN}n>o

11 " ' l U l l

T,

=

"

filtration { INMAN} n>o ,

then I,= Ez .

Pf : I" N E INMAN ⇒ e

,2 Ez .

Conversely , H n> 0 , choosing k as in Artin -Rea,we get

Intkpan N = In ( IkMn N) I I"N

.

To Zz Z T ,-

II

Another famous consequence of Artin- Rees is

Krutls Intersection Theorem : Let R be a meth . ring , I an ideal

and M a fin . gen . R-mod .

If I E Jac CRI,then

f) In M = O.

h 7,0

In particular , if § ,m) is a noeth

. local ring , then nm" = O-

M, 0

Topological interpretation : The I- adic filtration on M ( resp . m - adic filtrationon µ , m) ) induces a Hausdorff top . on M ( resp , a Hausdorff filtration on R) .

Pf of Krull n : The second assertion follows from the first upontaking M =D

.

Let

N o.= NI" M

.

N7,0

Note N is a fin . gen . R- mod since M is noeth .

Now, by Artin - Rees

,I k > o s - t .

Ikt 'M AN = I ( Ik MAN) .

But,

Itt ' M ? N ⇒ Ikt 'M NN =N. Similarly , Ik MnN=N .

Nakayama's

o! N = IN ⇒ N = O

.

Lemma

II

Defn : Let (R ,{ In}n

> o) be a filtered ring .

A filtered R -mod

(M , LMnln.cz )is I - stable ( or I- good ) if F KEI sit . Hn > 0 , InMk = Mean .

(usually kno)

Note that since Mr F M necessarily , I - stable filtration don't necessarily coincide with

the I- adic filtration on M for ur> 0 .

Example : The I - adic filtration on M is I - stable .

Defh o

.

Let R be a ring and I be an ideal . The

Rees - algebra of I is the ring

⑤ Inn7,0

Where multiplication on the components is defined by the action I"

X Im→ In"

( s 't ) ↳ ij-

Often to avoid confusion between In occurring as a summand,

and I"

sitting inside the summand Im for men , the Rees algebra of Iis identified as the subring

RCIEI := Ot Int"

n> 0

of the polynomial ring RCTI.

Lemma l : If R is a meth . ring , then the Rees- algebra

④ I"

h70

is a meth -

ring .

Pf : Let I = ( i , . . . . , in) .

We then get a surjective ring map

REX , . . . .

. xD → ⑤ In.

N? O

Xj t ij E I

'

Rar t r E IO -

-R.

By Hilbert's Basis Thm, REX , ,

ooo , Xk] is meth .

,hence so is a

quotient of REX , ) .. .

, XD .

a

Note that if (M , Lianne ) is a filtered (R . {In} n >o ) module,then

⑦ Mn and Ot MwNEE h7,0

are modules over the Rees - algebra ④ In,where the action on

n 7,0

the components is given by I"

x Mm → I"

Mme Mmtn(i , a) t ia

The Rees- algebra characterizes I-stable filtration .

Proposition 2 : Let R be a meth . ring ,and M be a fin . gen .

R-mod .

Suppose (M , {Mn}n⇐ ) is a filtered (R, {In}n> o) - mod .

The filtration is I - stable ⇒ ftp.Mw is a fin . gen . not.

In -module.

Pf : 1¥ Choose 127,0 sit - In Mr = Mun for all n> 0 .

Then not. .

Mn is generated as a Qt,

I"

by the subgroupke

Ot Mw.

h-

- o

b

Note that Ot Mn is a fin . gen .R-mode because M is noeth .

neo

k

Exercise : Any generating set of ⑤ Mn as an R- mod also generatesUI O

n,④,o Mn as a ④ In -module.

M, O

1¥ Now suppose Mn is a fin . generated ¥.

In - mode .

Then one can choose a finite set of generators X, ,

. . .

, Xm

S - t . X ;E Mn

;,ie

.

, Xi lives in degree ni , for i =L, .

. .

,m .

Now take 127,max In , , .

. .

, nm } .

It follows that InMn = Mktn

for all n >,0 by degree considerations .

I

Proof of Artin- Rees : The I- adic filtration on M is I - stable

⇒ ¥.

TIM is a fin . gen . II. In - mod .

However, if N is a submodule of M , then ⑦ NAIM is an

M, 0

not, .

In - submodule of ftp..IM .Since In is a meth . ring

( here you use that 12 is noeth) by Lemma I,it follows that

Ot Nn InnaNT

, O

is a fin . gen . Ot In - module . Then Prop . I ⇒ the filtrationM, O

{NAI" M}nyo

on N is I - stable.

But this means I k> O s-t.

It n >, O ,I" ( NnIkM) = Nn IntkM

.

I