Roots and logarithms of automorphisms of complete local rings

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rings

Citation for published version (APA):

Praagman, C. (1985). Roots and logarithms of automorphisms of complete local rings. (Memorandum COSOR;

Vol. 8511). Technische Hogeschool Eindhoven.

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Published: 01/01/1985

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Memorandum COSOR 85 - 11

Roots and Logarithms of Automorphisms of complete local rings

by

C. Praagman

Eindhoven, The Netherlands June 1985

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INTRODUCTION.

Let k ~,m or

f'

then the following observation can readily be made :

LEMMA 1.

There exist a

X

E

Hom (k,k*)

such that

x

E x(k)

if and only if

for aU

h E :IN

there exists a

z E k

such that

zh

=

x.

Let G be a Lie group,

9

its Lie algebra, and exp :

9

-+ G the exponential map. Then one may pose the question whether an equivalent of LEMMA 1 holds does exp

9

equal the set of elements of G which have roots of arbitrary order

?

In this paper the question is answered in the affirmative for groups of automorphisms of complete local rings with the complex numbers as coeffi-cient field. As an immediate consequence a number of results on iterations of automorphisms, partly well known (LEWIS [3J, REICH [8J, REICH-SCHWAIGER [9J, PRAAGMAN [4,5]), partly new, follows.

§

1

PRELIMINARIES.

NOTATION.

R will denote a complete local (noetherian) ring with maximal ideal m and coefficient field

¢,

i.e. R/m

=

¢

~ R.

R~ = R/m~+l

has finite dimension as f-linear space. Endow

R~

with the ordi-nary topology. Since R

x + x if and only if x

n n

lim R~ this induces a topology on R

~+1

-+ x mod m for all ~

E

:IN •

Let

A

be the group of ¢-automorphisms of R and

V

the Lie-algebra of ¢-derivations of R into

m.

A~ resp. V~ denote the corresponding entities for R~.

C

and C~ denote the group of all t-linear continuous maps of R resp. R~ which map mk into itself for all k.

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c~ has a natural topology induced by the topology on R~. Since

C

this induces a topology on

C,

in which

A

and

V

are closed.

JORDAN DECOMPOSITION.

Every element of A~ or V~ allows a decomposition into a semisimple and a unipotent resp_ nilpotent part (HUMPHREYS [2], THEOREM 15.3). This property transfers to

A

and

V.

Let

L

= A

u

V. Definition.

Let L

E

L.

Then

a) L is

topologically semisimple

if every closed subspace V of R, invariant under L, has an invariant complement.

b) L is

topologically nilpotent

if for all

k E

~ there exists a h

E

~ such that LhR c

mk.

c) L is called

topologically unipotent

if L - I is topologically nilpotent (here I denotes the identity in

A).

LEMMA

2.

L

E

L is topologicalZy semisimple (unipotent, niZpotent) if and only

if

L~,

the map induced on

R~!

is semisimple (unipotent, nilpotent) f01' aU

~ E ~ •

LEMMA 3 a. L E

L is topologically semisimple if and only if there exist

x

1, ••. ,xm

generating

m

(as ideal in R)

such that

LXi A . ~ ~ x. ·for A. E ~ ¢:, i=l, ••• ,m.

LEMMA 3 b. L

E

A(V)

is topoZogicaZly unipotent (nilpotent) i f and only i f

Ll

is unipotent (nilpotent).

These lemmata, as well the next theorem are proven for regular R in PRAAGMAN [4],

§

2. The proofs stay valid for arbitrary complete local rings. Let [A,B] = AB-BA as usual then:

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THEOREM l.a.

Let

F E

A,

then there exists a topologically semisimple

sF E A,

and a topologically unipotent

u F E A

such that

F

=

sFuF

and

[sF,uF ] O.

MOT'eoVer

sF

and

u F

are unique and satisfy

(sF) £,

THEOREM l.b.

Let

0

E

V. Then there exists a topologically semisimple

So

E

V

and a topologically nilpotent

no E

V

such that

0 =

so+no

and

[so,no] =

O •

..

~

V

So

d

no .

and

sat~s~y (sO) t~o~eo

er

an

are unt-que

~ J:

. £,

n

(O~) •

00 k

EXPONENTIALS AND LOGARITHMS.

Define for 0 E

V

exp 0 \' L. _D • Clearly k=O k!

exp 0 E

C,

and in fact exp D E A, which is proved in PRAAGMAN [4J, THEOREM 4 for regular R, but the proof does not depend on the regularity.

Denoting the set of topologically semisimple automorphisms (derivations) by SA ₍

Sv

_{) and the set of topologically unipotent automorphisms (nilpotent} deri-vations) by uA(nV) one proves without any problem:

LEMMA 4.a. exp

4.b. exp

nV

~ ~ u

A,

and

~'s ~

a t-Jec t.on.

b .. t'

4.c.

Let

D E

V,

then

s(exp D)

=

exp So, u (exp D)

00

n exp O.

For F E uA, define log F (I_F)k k • Clearly log FEe, and in fact

log F E nV (PRAAGMAN [4J, THEOREM 4), and log is the inverse of exp on

nV.

Clearly [D,LJ

=

0 implies [exp D,L] = 0, and i f F E u

A :

[F,L]

o

if and only if [log F,L]

=

o.

SCHAUDER BASES.

Let

xl

f ' " ,x

m generate m, then

{xu

I

u

E IN

~

} with the usual convention on multi-indices generate R as ¢-li~ear space.

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m {xu! E }

'l'here exists a subset S c :IN 0 such (1 S constitute a Schauder basis for R : if y E R then there exist unique y E t such that y =

I

y xU.

u

uEs a

In particular xS A.x., then

l l

I

e(a,S) Aaxa implies that e(a,S)

aEs

a

S

c(a,S) A x for all a and

S

and hence c(a,S)

~

0 implies AU AS. Similarly if D E

Sv

with D x. l

then c(o,S) ~ 0 implies

<

u,~

>

COI"Y'1UTATION.

Let F E

sA ,

and let x l ' ... ,x

m be a set of eigenvectors of F generating

m,

F x. = A.x .• Let LEe be defined by L

x~

=

L

l l l

uEs

Then [F,L] if and only if

~(o,B) ~

0 implies AU =

AB.

a

R,( 0 , B) x •

Similarly, let D E

Sv

be defined by D x.

=

~.x.,

then [D,L]

=

0 if and l l l

only if tea,S)

#

0 implies

<

a,~

>

=

<

S,~

>.

Both statements may be proven analogously to the statement on the e(o,S).

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§

2

THE TORSION-FREE DECOMPOSITION,

It will turn out that the decomposition in (topologically) semisimple and unipotent (nilpotent) parts does not provide a sufficient simplication of the problem. Therefore a further decomposition of the semisimple part will be introduced in this section. I shall start with a generalization of a theorem of LEWIS ([3J, section 5). The proof and its consequences will provide motivation for the introduction of characteristic subgroups and the

torsion-free decomposition.

CONCERNING A THEOREM OF LEWIS.

LE"I'IIS ([3J, section 5) proved that, in case of R is regular, every automorphism has a power which can be embedded in a complex analytic iteration group. I shall prove this, or rather an equiva-lent statement (see PRAAGMAN [5 ], THEOREH 2) for arbitrary R.

THEOREM 2.

Let

PEA,

then there exists an

h E :IN

such that

ph

has a logarithm.

Fr·oof.

I shall construct an E s

V

such that exp h D s

= (

s P) , and h

A.

x..

A,

the multiplica-~ multiplica-~

tive group generated by

{A

1, •••

,A

m} is a finitely generated subgroup of

i

r

_*

and hence

A

=

AO

~

A

l, where

AO

is a finite cyclic group and

A1

is free. Let

vo

generate

AO

and

v

_1/ " ' V

_r

A

l, and 0 ; LZ m + LZ r be defined by "Ct notation). and let 6 0 ; 0(0.) o(x)

Vo

0

v

LZ m + LZ / h LZ , h =

I

AO

I ,

(\lith a small abuse of

Let sD be defined by

<

<5 (a), p

>

X Ct , aES, where = log

v.

for some ~

s

choice of the dermination of log. D is topologically semisimple and satis-fies ;

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s ct 13

D(x .x ) SD (

I

c(y,ct+S)XY)

yEs

L

c(y,a+S)

<

o(y), P

>

xy yES

I

c(y,ct+S)

<

o(a) + 0(13), P

>

x\(since c(y,ct+S) " 0 implies ).,Y

yES

a+S

<

o{a) +

0(13),

p

>

x

Hence sD E

SV.

Further exp{h D)X, S

1 U

F

13 ₌

I

a Finally, i f x f(a,S)x ,

aES

0(0.) -F 0(13) and hence if

<

o(a),

Remarks.

hence o(a) +

0(13)

o

(y) )

h h S

: 0 A, x,

,

since

Vo

1, so exp(h D)=(

1 1

then f(a,S) = 0 i f Act _{" AS, hence if}

\l> -F

<

6(13),

\l

>.

So [ D, F] s u

=

o.

AU+S and

h

1. Since [sD,sF] = 0 and in virtue of [sF,logUF ] = 0 i t is clear that [D,F] O. This will be important in the next subsections.

2. As an immediate corollary one finds that F has a logarithm if 11\01 1.

The following example shows that this condition is not necessary Let R

=

C[[x]J, Fx

=

-x, then F == exp D, with Dx

=

nix.

*

A DECOt<lPOSITION.

Let F E A, then II(F) will be the subgroup of

t

generated by the eigenvalues of .Let tA be the subset of SA for which A(F) is finite,

f

Clnd

A

be tJw subset for whi.ch II (1") is free.

THEOREM 3.

Let

F E A,

then there

ex1,-S

. t t F E

A

,

E

f

A

and

u F E u

A

such

that

[ F, FJ '" t U [ F, FJ t f =: [fF/upJ =: 0, _F ₌₌ tFfFuF

and

if

_{L E}

C

then

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Proof·

Let

s D

be as

m

the proof of theorem 2. Then

f

F

exp

s

D

E fA, and

(fF

}-l ₌

_{tF EtA. The first part of the theorem now follows from remark}

₁

"above. Let LEe then [L,F]

= 0 i f

and only if [L,sF]

:::

[L,uF ]

₌ ₀

as

follows immediately from the well known corresponding property for spaces

of finite dimension.

,sF]

=

0

if and only if

~(a,S) ~

0

implies

A

a

=

AS

or

equivalenty aO(a)

=

00(S} and a(a} ::: a(S} which yields the second

of the

theorem.

Remapks.

3.

Note that [L,fF]

=

0

if and only if

I S

D]

=

O.

4.

The problem of finding a root or a logarithm of F has been reduced to

finding one of tF which commutes with fF and uF or equivalently with F.

t

5.

F and

are not uniquely determined by F : first of all \(F) is not,

and secondly the choice of

Vo

is significant. In the next subsection

I

will therefore introduce characteristic subgroups which do not depend on

the chosen decomposition in

a

torsion part

, a f

.ree part

f F

an

d

a unl-

.

u

potent part F.

CHARACTERISTIC SUBGROUPS.

Let F

E

A,

and let

6

0

and

6 be as defined in the

o

proof of THEOREM 2. Let

L

(F)

ker 6

m 0

0

c z:; •

Then

L

(F) does not depend on

the choice of

va; and neither on the choice of

x

1, •••

,x

m

Let L E

C

then L(L) is the subgroup of

z:;

generated by the set

{n-r~ E ilL. m c(a·;I3)

cf

0

or

£«(X,r{) I O},

where

o!:i u";ual LXr5

I

aEs

LEMMA 5.

Let

F

E

tA

and

L

E

C. Then

[F,L]

L

(L) c

La

(F) •

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Proof.

[F,L]

°

if and only if

~(a,S)

#

°

implies A

a

=

AS or equivalantly

a-SEker 00

EO(F);

henee if and only if ElL)

C EO(F).

(Note that eta,S)

~

°

also implies AU

=

A

P

!).

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§

3

ROOTS.

In this section I shall derive a criterion for the existence of an

n-th root

of an automorphism F, that is an automorphism G

This criterion will be formulated in terms of the characteristic subgroups of

F.

Therefore

I

start with an investigation into the structure of these groups.

THE STRUCTURE OF THE CHARACTERISTIC SUBGROUPS,

Since 00 (Zl m) is a finite group there exists an eO

E

Zlm and a subgroup

V

C Zlm such that Zlm

Zl eO til V and

r:

0 (F) h Zl eO til V. Let p be any number. I f p divides h then EO(FP ) (hlp) Zle

O

til

V,

and if p and h are relatively

then EO (FP ) 1:0 (F) •

And the other way around, if G

P

==

F,

and

P

divides h then necessarily (G) phZle

O til V, while if p does not divide h one might have 1:°(G)

On the other hand if F and G are automorphisms, then

a FG x

L

9 (

i3 ,

Ct) Fx

i3

==

GEs

So 1:(FG) == E(F) + 1:{G), and in g(S,a)

L

yEs y fey,S) x . F.

o

1: (F).

If F

~

Gn, then E(F} == E(G) C EO(G), which is a sublattice of [,0(1") of finite

index, the exact index on the prime factorizations of hand n. To avoid these complications in the formulation of the theorem define }; (F)

~

{a E Zl m i n a E I:(F)}. Then the condition E{F) c 1:0 (G) which is

n

necessary for F == Gn, translates to

r:

(F) C EO (F) •

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LEMMA 6.

Let

F

and

G

E A,

then

Z(F) C (G)

implies that

L (F') C Z (G ).

o

n

fI, z::; eO $

v,

then £ (g.c.d(hrn» == h. Let a E E (F) satisfy (l E h k eO ffi V for some k E Z::;. Let

f3

E En (F), then

np E hke

O $ V, hence hkE n z::; I so n

I

(g.c.d(h,n» k and hence

B E

£ z::; eO $ V. So L (F) C LO (G)

AUTOMORPHISMS HAVING ROOTS.

It turns out that the condition L (F) C IO(F) not n

only is necessary for the existence of an n-th root of F, but also sufficient

THEOREM

4. Let

F

C

A.

F

an

root

if

t

(F') c: L

o

(F).

n

So assume L (F) C ZO{F). Choose a torsion-free decomposition F n

t f u

F. F. Ft 2nik/h as constructed in THEOREM 3. Let

Va

be a generator of Ao(F),

Va

== e let

n

_P

m(p) be

n = the decomposition of n in factors, and let pEp

h

n

plh

pm(p) , and choose q E z::; such that q

n

plh

m(p)

p 1 mod i . This

is possible since

£

and hn/£ are

Then h 7.l prime. Define tG

E C

by 1, and

6

0 (V) tF

_{x .}

a

ffi

V

for some eO and

V. o.

Let

a

E Z(F) then

a

E mh eO $ V. If g.c.d(mh,n} j 1 then mh/j

E

h z::; hence

m E j Z::;. Bu t g. c • d ( j sh , n ) j implies that g.c.d1h,n) divides j, hence hj

E

fI, z::; 1 and hence a

E

£

::z

eO ffi V.

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t t

This implies in the first place that G E

A,

and secondly that ,F] ~

O.

In view of remark 4 this proves the theorem.

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§

4

LOGARITHMS.

A criterion for the existence of a logarithm may be formulated by a

similar condition as for the existence of a root. As in the preceding section, I start with deriving a necessary condition, and prove afterwards that this is also sufficient.

t h

LOGARITH~1S

OF

ROOTS

OF UNITY.

Let F

E

A,

and F = I ('"

I

AO (F)

I

=

h) 1. then

any D such that exp D

=

F much have its eigenvalues in (2ni/h)~. Let D be such a derivation and let TO : tzm +

~

be defined by Dxa

=

(2'1fi"C(a)/h)xa• If exp D

=

F, then 6

0(0',)

=

TO(a} mod h, so "CO is onto and hence ker TO has rank m-1. Since [L,D]

=

0 if and only if l:(L} c ker TO' one can easily prove that to have a logari thm

~

(F) C V where

r.o

(F) h LZ eO Ell V for some V.

Again i t turns out that this condition is also sufficient. Again some notation {a E

~

m

I

3 n E IN such that na E l:(F)} is the smallest direct

summand of

~

m containing l:(F).

AUT~ORPHISMS

HAVING A LOGARITHM.

THEOREM 5.

Let

F E

A. Then F has a logarithm

if

and only

if

l: (F) C l:°{F}.

00

Proof.

Let D be a logarithm of F, then [sD, == 0. Let the subgroup of

¢

generated by the eigenvalues of sD "'1

n

2niQ

Then [ ,F]

m

o

I let 6

0 '" (2ni/h) ~, and "CO : ~ -+ 2Z as above.

o

implies L(F} C ker TO' and since ker TO is a direct summand of tzm, (F) cker TO.

ker TO C ker 6

0 = E?F}.

On the other hand since 6

0 factors over TO' clearly So E (F) C l:°(F) which proves the only if part.

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On the other hand, if 1:"" (F)

c: LO

(F), then there exist eOEZZ; m and V

c:

ZZ; m

m 0

such that ZZ;

==

eO ZZ;

+ V , E

(F)

==

h eO ZZ;

(j) V

and E(F)

c:

v.

2nik/h

tn

e

generate AO(F), and define

t a

O. Then (exp n)x

==

0, and hence the theorem is proved.

a x (T

o

(a)2'lfik/h)

a

x , where

a

x , and 1:(F)

c:

ker TO implies

Remark

6.

Note that this criterion does not refer to a particular choice of

the logarithms of A(F), as does the 'smooth additional monomial' condition

(see

REICH

[6J).

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§

5

ROOTS, LOGAR ITHIVlS AND ITERATIONS.

In this section I establish the connection with iterations of auto-morphisms. But first a simple consequence of the preceding sections will be proved.

ROOTS AND LOGARITHMS.

Since ~oo(F) = U

nEJN

readily follows from THEOREM 4 and 5.

~ (F) the following theorem now n

THEOREM 6.

Let

F

E

A,

then

F

has a logarithm if and only if

F

haD an n-th

root

for aU n

E IN •

ITERATIONS.

A complex (real, rational, fractional) iteration of F

E A

is a 1

group homomorphism from

¢

(JR,

f2, -

Z;:;) to

A

such that 1 -+ F. n

If F

=

exp D, t -+ exp t D is an iteration of F, which is even analytic in t. Analytic iterations have been studied by LEWIS [3J, REICH and SCHWAIGER [9J, by BUCHER [1J in connections with continuous iterations, by REICH and

KRAUTER [7,8J in relation to fractional iterations. A good survey is REICH [6J. In PRAAGMAN [4,5J I studied the connection between iterations and logarithms.

As a direct consequence of THEOREM 6 a large number of statements on iterations follows. For if F has a logarithm D, then exp t D is an iteration, complex and analytic, of F. On the other hand if F

t is a rational iteration of F then F

1/n is an n-th root of F. Hence the question posed by REICH and SCHWAIGER ([9J. page 617) is answered, as well as the question whether the existence of a rational iteration implies the existence of an analytic one

(REICH [6J, §6, PRAAGMAN [5J).

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REFERENCES.

[lJ

BUCHER, W.

]

HUMPHREYS, J.E.

[3J

LEWIS, D.

] PRAAGMAN, C. [5J PRAAGMAN, C.

[6] REICH, L.

]

REICH, L. and

KRAUTER, A.R.

[8J

REICH, L. and

KRAUTER, A.R.

]

REICH, L. and

SCHWAIGER, J.

Kontinuerliche Iterationen formal-biholomorpher

Abbildungen. Ber. Mathem.-Stat. Sektion im

Forschungszentrum Graz, 97 (1978).

Linear algebraic groups. Springer GTM 21, New York (1975).

On formal pOwer series transformations.

Duke Math. J.5

(1939) 794-805.

Iterations and logarithms of formal automorphisms.

Aequ. Math. 29 (1985) (to appear).

Iterations and logarithms of automorphisms of complete

local rings.

To appear in the Proceedings 5th Int.

Symp. on Iteration theory, Lochau 1984.

Iteration problems in pOwer series rings.

Proceedings

4th Int. Symp. on Iteration theory, Toulouse 1982.

Coli. Int. CNRS 332 Paris (1982).

Roots and analytic iteration of formaUy biholomol

1