PROJECTIVE SYSTEMS ON TREES AND THE RIBENBOIM APPROXIMATION THEOREM

Universitetet i Oslo November 1964

PROJECTIVE

SYST~~

ON TREES AND THE RIBENBOIM APPROXIMATION THEOREM

by

Olav Arnfinn Laudal

- 1 -

INTRODUCTION

The first part of this paper is devoted to the study of the functor on the category of projective systems of modules on a tree ~ We show,

p r 0 p 0 ~ i t i 0 n 1 , that if

r

satisfies a mild condition we will have lim(p) ^{= 0}

~ for p~ 2 •

This was proved independently by the author, ((1)) , and by Nobeling, ((2)) • The second part of P r o p o s i t i o n 1 was first proved by Nobeling, · ( (2)) , but the proof established here is somewhat shorter. It should, however, be mentioned that NobelingVs results are slightly more general.

The second part contains a new proof of the Ribenboim approximation the- orem based on the machinery developed in the first part. Note that we con- sider only valuations of finite rank.

We recall the Ribenboim approximation theorem (around

0),

((3)) •

T h e o r e m . If v. , i=1 , ^{ooo ,} r are valuations on a field

K,

l

and if for i=1 , ••• , r , 0(. is an element of the value group

l of v.

l

such that for each couple i, j the image of ⁰⁽ i and 0( j group of V.A V.

l J coincides, then there exists an element

v.(x)

=

^0(.

l l for all i = 1 , ••• r

in the value xtK such that:

Remark. The proof presented below works for valuations of rank

~x if x is an ordinal such that each element in a representation of x admits an immediate successor.

§ 1

Let L be a unitary ring and let

r

be a partially ordered set. If .6 is a subset of

r

then we write:

/\

6

t=

hdoE f\ s< 01~.1-1]

₆ ^v _{= {}

t J ~t r)

^1/

> o;.

^E

⁶ ~

and in particular:

Denote by C

.. ^~~

the category of all projective systems F =

1

^{F J' I}

"Ji lor ^r

of L-modules F l' on

r ,

and of all homomorphisms between such • We know, ((1)) , that there is a~ isomorphism of functors on C :

Ext(p)(L,F)

c

where 1 denotes the constant projective system on ~ defined by the ring

L •

Now, if {

F

₀

f

6 ^E f"'" is a family of L-modules we may construct a prp- jective system F

= {

F0,

YJ }'}

^on

~-,.

by putting

the homomorphism

).;'I

'1~/ associated with a relation

6

< 2(¹ in

r

^be:,Lng

induced by the inclusion F is then said to be

ll

-co- flabby, or the

i l

-coflabby proi§.9tive system generated by the family

• If for every ~f

r

the L-module Fx _{_,} is projective, we say that F is

l.L

-projectiveo We shall use the easily proved fact that any Jj_ -projective object of

C

is projectiveo

A

^I~

D e f i n i t i o n o partially ordered set is a t r e e if for all

of ^{I' '}

^~ is totally orderedo

We shall suppose that all trees appearing in this note satisfy the fol- lowing condition.

- 3 -

(*)

Given two elements ()'1

< '6

₂ of

r ,

then the set is finite.

This condition implies the e:xistehce for every

'0!? r

of

r

with the following properties:

1) If

'6

^~ t-

R;r

^then

{} ) ;:)

^'

'6

^y

f:. 't

2)

If

01 '02E

^{R0 then} ({ 1

f ' ^0'

^{2 s () 2}

~ ⁰¹

of a subset R ~

3)

If

~1E G

then there exists an

"t'

f R ;:{ such that

~

01 >

~

P r o p o s i t i o n 1 • Let

f'

be a tree satisfying the condition (3E) , then we may conclude

(i)

for p~2

, where

C(>

is the homomorphism

--~

TT

Fmin(~,~·~

?f~

r

(('~ R .>;

defined by

'7~ f~' v' ^•

P r o o f ^o For every ()E

r

^{put L't =} L an.d let P⁰ be the

11.

-coflabby projective syretem generated by the family

t L~ J

^{()tF o Note}

that P⁰ is

LL

-projective~ thus projective. Let L denote the constant projective system on ~ defined by the ring L , and consider the homo- morphism (, of P⁰ onto L induced by the identities

L 0

^{'----j L ~ = L}

for

?/

E-

T;:

We contend that ker [, is

iL

-projective. To see this, define the projective system P1 by

P~ =

the homo-

~

r·

morphisms associated with a rela-c.ion ((1

<

2( ₂ in being induced by the inclusion I~ ₂ ~

G-

1 , and look at the homomorphism d: 1

P ~~ker £_

defined by: d~(1

'6')

⁼ 1~- 1o-' where 1 ~ is the identity element of

1?$'.

It is trivially seen that d is an isomorphism of projective systemso

1 ~

11-

It is thus sufficient to show that P is Jj_-projective. Put Q ~= 1~'

't't: R ^2{

and denote by Q1 the JJL-projective system on

f'

generated by the family

['Q

¹

~j

^{(Jf [7 0} Denote by M () the set

1 ( ^{~}' ~vv) I /f'E- rc;)}

^oNE-

Ro-'1

and by N

25

the set

£ 6'

^J

'6

¹

f r,r ) ^ff

^I

tJ-1

⁰ The conditions made on

r

implies that the map

8 :

^M

25

_N3' defined by

e (

^{'t ' y}

^{0 )}

^{Y1 =}

⁰

^{VY is a}

bijection. Using this we find an isomorphism of projective systems

Q

¹

~P

¹ • This implies (i) ^o Finally (ii) results from the relations:

Hom(P⁰,F) =

c ,

Hom(P 1 ,F)

= Tf

F w1i14(v .Y 1)

c ()er

o'"

~'e

R>J

and from the fact that if j is the injection

P

¹

~P

⁰ then the homo- morphism CO

=

^{Hom(j ,F)}

I <:

QED.

Now, if ~- is an ordered set, and if

r

1 is a subset of

r

^{we de-}

note by ~(f .... ,

1",

1 )F the natural homomorphism

Using a general result, ((1)) , we may then prove:

1 e m m a 1 ^o Suppose

r;

^and

^r

2 are subsets of

f'

such that ,...

""'

r;

⁼

f1

^J

r; ⁼ r2 ,

^{and put ro}

⁼ r1 ⁿ G

⁰ Then we have an exact sequence

- 5 -

o--~ lim F/

r

₀ ^{im 6(f1: r)F+im ~(Q ~)F )}

0

L e m m a 2 ^o Suppose ~ is a finite tree, then the following state- ments are equivalent

(i)

( ii) If _o v E:-

r ,

^{R vl.} _o ^{= (} _{(.01 'ei2'} ^v _•oo ^v _Or)

t.

and if

for 1 ~ j ~ r-1 , then

F'/)'

•

P r o o

f •

(i) ~ (ii) • For each

!fc r ,

^{put h(}

^<()

⁼

(_ I

^{:J v' <} < '/ - / 1 B . . ( ') . 1

max

l

n ..J o n -:/: • ^{o o (:.} _{0 0 -} ⁽⁾ 5 • y 1nduct1on on h () , us1ng e m IQ. a 1 we prove that lim ( 1 _{)F =} 0 for all

IS

f

r

By induction on r - j

~ ~~

we prove, again using L emma 1 that

~~( ¹

^)F

=

0 • Put

r ^{= ,6j+}

1 ⁼ 6 j

A /'.. ./.' /"" A

L1jU ?j f"'1 =L:\j

,r

2 =L._j>

r

0 - r

1

^{nl~= (5 o} The conclusion then follows immediately from L e m m a 1 o

(ii)

::;>

^{(i) 0 For each}

?Sf r

put coh(ij) = max [ n

I 3

'!{ =

' ( 0

#

^{o • • ~ ({n}

^?

^o By induction on coh(()) , using 1 e m m a 1 we

prove that lim ( 1 ) F = 0 for every 0 ^E:

r .

The conclusion follows tri vi- +-.A

r: _0'

ally from L e m m a 1 ^o

C o r 0 1 1 a r y o Suppose

r

is a finite treei and let

6

^{be a} subset of

r

'

^{then l:im(1 )F =} 0 implie:J lim( 1)F

=

0

?

^{~ .;6.}

P

r o o f o If (ii) holds for

r'

then it holds for

6

§

2

Let K be a field, and denote by ~ the partially ordered set of all valuations of finite rank on K • We know that

r

is a tree satisfying the condition (K) of §1 •

Given any valuation v E-

-r ,

denote by 111 v the maximal ideal of the valuation ring of v , by V the value group of v , written additively,

v

and by U the group of wlits associated with v , written multiplicative- v

lyo Then we have an exact sequence of Abelian groups

( 1)

i v 1 ----} u · - - - 7

v

v

Kx ----;) v

--4-

o

v

The groups U and V defines obvious projective systems U and V on

v v

r""'

⁰ J.,et

Kx

denote the constant projective system on

I

defined by the

multiplicative group K , then the homomorphisms

x

morphisms of projective systems

i and v define homo- v

Since for every v ~

r'

the sequence (1) is exact we have an exact se- quence of projective systems on

r'

(2)

- 7 -

Suppose the subset

6

of \ has a least element, then vre have

lim(p\± = 0 for p

-~

1 ,

~

Kx

~ ~

• Applying

~

on the sequence (2)

~

£::::.

~

there results an exact sequence of Abelian groups

(3)

(1)

~ {im

u

~Kx--? ~

^v

~

^{p.:m(1 )u ---4 (o)}

~ ~ ~

From this it follovrs easily:

P r o p o s i t i o n

2

The following two statements are equivalent (i)

(ii)

The Ribenboim Approximation theorem holds.

If L\ is finite then lim( 1)u = 0 •

.. 8

Lemma

3

Suppose

6

is a finite subset of

1--.

containing a

? r

v'

^{t Y }}

least valuation v , and put £:::,. =

l.

/v ¹ v -E-

6

Then ~ ^I is a

y

subset of the set of valuations on some field K , and we have

P r o o f •

in K such that

'

v ) v , and that

Denote by I the subgroup of U of those elements x v

1-x E- -"t11v· It is easily seen that Uv /v ~ = U _v ~/I • Thus the sequence

for all

is exact. This sequence may be treated as an exact sequence of projective systems on £:::,. • Since I is constant we have lim(p)I

=

0 for p

>,

1 •

E:b.

By exactness this implies

Q.E.D.

1 e m m a

4 •

Suppose

b :; .6

iS a finite subset of

f"' ,

^{and let}

v E:

6

j

6 j : ;

~ 6v.

1::::1 l

' LJ·

⁼

~v

_j+1

1 ~ j~ r-1 o

an

Then if x E U v

:;::: lim U

y.

J

such that

x

=

x

_{1 •}

x

₂ ^o

it..e pr.,oi cf

a.rtd

P r o o f • By 1 e m m a 2 and 1 e m m a 3 we may assume that v is the trivial valuation. Thus the valuations v.,i = 1 ,

l r , ^~re

-1 ) all of rank 1 • Further, we may aseume v j +₁ (x) ~ 0 , (if not, take x •

Now, let wk , k

=

1, ^{o . • ,} s be the maximal elements of

..6

^{j , and}

let u_{1 , 1} =

1,

o•• , t be the maximal elements of

L. . .

Choose elements J

.~ k , k = 1 , • • • , s , and

Y[·

_{1 , 1} = 1 , o • o , t , such that:

v.r.()Ak)> 0 for i-:lk,

l -

Since for every i = 1, ^ooo r , v. is of rank 1 we may by, eventually,

l

taking big enough powers of

'>]

1 , assume that vi (

ry

_{1 )}

> -

^{vi (x) for}

i

=

1, • • • , j • This implies that wi (

'YJ

_{1 ) )} - w. _l (x) for i = 1, •

o. ,

^{s •}

u1 (x) q; 0 for 1 = 1 ,

' t

⁰ Since vj+1 (x) ~ 0 we have

Note that wk( ~l •

x)) 0

for k

= 1,

^{ooo ,} s Put =

s

t

[_ ftk

+

L_

"]l ^o x • We have

k=1 1=1

Let x 2

=

xX

1 , then u

1

^(~)

=

0 for 1

=

1, ooo t , and the proof is complete.

Q.E.D.

- 9-

C o r o l l a r y o The Ribenboim approximation theorem (arourid 0 ) holds.

P r o o f •

t i o n

2 ^o

Use L e m

m

a

4

^P L e m m a 2 and P r o p o s i -

((1))

((2))

((3))

O.A. Laudal: Sur la limite projective et la theorie de la dimen- sion I

&

II. Seminaire C. Ehresmannj Paris (1962).

G. Nobeling: Uber die Derivierten des inversen und direkten Limes einer Modulfamilie. Topology vol. 1. (1962) pp. 47-62.

P. Ribenboim: Le theoreme dYapproximation pour les valuations de Krullo Math. Zo 68(1957)1-18.

Correctlons to Semtnar report no.13, nov. 1964.

p • 1 ,

l.

4 : l tm ( p )

=

0

9

read : l i..m ( p)

=

0

<::--

r

p.2,

l. 3

from bottom: "· •• all "( e r , r i..s • • • "

read

^{11 •••}

all p.3, formula (ll) prop.1

y e

r

_; t

s .••

_•

l

tm t1)

F =

co er ¢

9

<:--

r

read ~ ltm( 1 )F:= coker ¢

<::--

r

p. 5, l.1

~

read:

Fjtm L\(r ₁ ,r

₀

)F + tm L\(r ₂ ,r

₀

)F -·-;><::lt_!!l(i)F

- - : : >

r