Publicacions
Ma emá iques,
Vol
37
(1993),
19-44
.
Abs ac
P-LOCALIZATION
OF
SOME
CLASSES
OF
GROUPS
AUGUSTO
REYNOL
FILHO
The
aim
o
he
p esen
pape
is
o
s udy
he
heo y
o
P-
Localiza ion
o
a
g oup
in
a
ca ego y
C
such
ha
i
con ains
he
ca ego y
o
he
nilpo en
g oups
as
a
ull
sub-ca ego y
.
In
he
sec-
ond
sec ion
we
p esen
a
numbe
o esul s
on
P-localiza ion
o
a
g oup
G,
which
is
he
semi-di ec
p oduc
o
an
abelian
g oup
A
wi h
a
g oup
X,
in
he
ca ego y
G
o
all
g oups
.
I
ums
ou
ha
he
P-localized
(Gp)
is
comple ely
desc ibed
by
he
P-localized
Xp
o
X,
A
and
he
ac ion
w
o
X
on
A
.
In
he
hi d
sec ion,
we
p esen
he
cons uc ion
o
he
heo y
o
P-localiza ion
in
he
ca ego y
o
all
g oups
which
a e
ex ensions
o
nilpo en
g oups by
ini e
abelian
g oups
.
Ou
p oo
ollows
a he
closely
he
one
p e-
sen ed
in
[2,
chap e
II,
and
is
based
on
he
classical
in e p e a ion
o
he
second
cohomology g oup
o
a
g oup
.
In oduc ion
Since
Sulli an
i s
poin ed
ou
he
a ailabili y
and
applicabili y
o
localiza ion
me hods
in
homo opy
heo y,
he e
has
been
conside able
wo k
done on
u he
de elopmen s
and
e inemen s
o
he
me hod
and
on
he
s udy
o
new
a eas
o
applica ion
.
In
[2]
P
.
Hil on,
G
.
Mislin
and
J
.
Roi be g
cons uc ed
he heo y
o
P-localiza ion
o
nilpo en
g oups,
whe e
P
is
a
se
o
p imes
.
Some
ime
la e ,
P
.
Ribenboin
in
[3]
showed
ha
i
was
possible
o
localize
any
g oup
.
(The e
is
ano he
ap-
p oach
conce ning
P-localiza ion
in
g oup
heo y
de eloped
by
Bous ield
in
Topology
14
(1975)
133-150,
and
Mem
.
Ame
.
Ma h
.
Soc
.
1
0
(1977)
no
.
186,
bu ,
in
his
wo k,
we
jus
use
he
concep s
p esen ed
in
[2], [3]
and
[4])
.
The
cons uc ion
p esen ed
in
[3],
howe e ,
seems
o
be
qui e
abs ac
and
his
led
us
o
y
o
ob ain
a
mo e
explici
cons uc ion
o
he
P-localiza ion
o
a g oup
G
in
he
ca ego y
o
all
g oups
.
We
we e
success ul
when
G
is
a
semi-di ec
p oduc
o
a
ini e
abelian
g oup
A
by
2
0

A
.
REYNOL
FILHO
some
o he
g oup
X
.
In
addi ion,
we
managed
o
cons uc
heo y
o
P-
localiza ion
o
a g oup
in
he
ca ego y
C
o
g oups
which
a e
ex ensions
o
nilpo en
g oups
by
ini e
abelian
g oups
.
The
ques ion
conce ning
semi-di ec
p oduc
is
aken
up
in
Sec ion
2and
he
main
esul s
a e
2
.1,
2
.5
and
2
.10
which
could
be
s a ed
as
ollows
.
Le
P' be
he
complemen a y
o
P
in
he
se o
all
p imes
.
Theo em
(2
.1)
Le
N
-±->
G
-'»
X
be an
exac
sequence
o
g oups,
whe e
N
is
a
p-g oup
and
p
E P'
.
Then,
e
=
eooEP-localizes
G,
p o ided
ha
X

Xp
P-localizes
X
.
In
his
con ex ,
Theo em
2
.5
says
he
ollowing
:
Le
X

w
)
Au (A)
be
an
ac ion,
whe e
A
is
a
ini e
abelian
p-g oup
and
p E
P
.
Le
P
l
=
{q
E
P'
:
q
~~w(X)
~}
and
conside
Pi
he
mul iplica i e
se
gene a ed
by
P,
.
Se
H
he
sub-g oup
o
X
gene a ed
by
all
x
E
X
such
ha
he
o de
o
w(x)
belongs
o P,'
.
Le
wH
be
he
es ic ion o
w
o
H
and
F
=
l'F7,
whe e
is
he
smalles
posi i e in ege
such
ha
F'H
=
I"Hl
(He e
Fi
has he
o dina y
meaning
and
i s
de ini ion
may
be
ound
in
[2])
.
The e
is
an
ac ion
X

Au (A/ )
induced
by
w,
which
can
be
ac o ed
as
X

w
)

Au (A/ )
Le
G
=
Al
~,X
and
G'
=
A/
1
W
,Xp
Then, e
:
(a,,
x)
E
G
--->
(a+
F,
eo(X))
E
G'
P-localizes
G
.
Finally,
Theo em
2
.10
analyzes
he
si ua ion
in
which
A
is
a
ini e
abelian
g oup
.
Le
X
~
w
_->
Au (A) be an
ac ion,
whe e
A
is
a
ini e
abelian
g oup
.
Le
A
l
.
.
. .
,
A
be
he
p-p ima y
componen e
o
A
and
wi
:
X
--->
Au (Ai)
be
he
ac ions
induced
by
w
.
-
Le
G
=
A
]
c~X,Gi
=
A
2 ]

,X
and
ake
G
~
E
Xp
o
be
he
pull-
(EJp
back
o
he a ows
(Gi)
p

-*

Xp
.
(No ice
ha
(Gi)
p
a e
gi en
by
ei he
2
.1
o
2
.5
.
Then,
we
claim
ha
he
na u al
homomo phism
G
G
P-localizes
G
.
We
de o e
Sec ion
3
o
p esen
ou
esul e
conce ning
he
cons uc ion
o
he
heo y
o
P-localiza ion
o
a
g oup
in
he
ca ego y
C
o
g oups
P-LOCALIZATION
OF
SOME
CLASSES
OFGROUPS

2
1
which
a e
ex ensions
o
nilpo en
g oups
by
ini e
abelian
g oups
.
The
ma e
could
be
desc ibed
as
ollows
:
Gi en
G
El
C
1,
he e
exis s
a
unique
ini e
abelian
sub-g oup
U
o
G
such
ha
G/U
is
nilpo en
and
I'2
=U
whe e
W
is
he
ac ion
a -
ached
o
he
ex ension
:
U
N
G
-»
G/U
.
Fu he mo e,
he e
ex-
is s
a
unique
g oup
U
and
an epimo phism
p
:
U
-»
U
and a
unique
bP
E
H
Z
((G1U)P
;
i7)
.
(bp
:
U
-
GP
-
(Gl
u)P)
a ached
o 1
yielding
commu a i i y
in
he
diag am
U G
Gw
I
PU

e

l
eo
jP
:

_

,--,

G
P

-~

(Gw)P
Unde
such
condi ions
(3
.12)
s a es
ha
G

>
G
P
is
a
unc o
and
e
is
a
na u al
ans o ma ion
o
unc o s
.
In
addi ion,
(3
.13)
also
s a es
ha
eP-localizes
G
.
In
Sec ion
1
we
in oduce
some
basic
esul s
needed
in
he
ollowing
sec ion
.
We
belie e,
ne e heless,
ha
Theo em
(1
.21)
is
impo an
on
i s
own
acco d
;
i
s a es
ha
i
X
-~
Au (A)
is
a
comu a i e
diag am whe e
X
~-
e-'>
X
P
P-localizes
X
and
A
is
a P-
local
ini e
abelian
g oup,
hen
eo*
:
H,1
P
(X
P
;
A)

H,
1
(X
;
A)
is
an
isomo phism
.
This
wo k
is
he
main
pa
o
he
au ho 's
Ph
.D
.
hesis
done
unde
he
guidance
o
P o esso
Pe e
John
Hil on
.
The
au ho
is
e y
much
indeb ed
o
P o esso
Dacibe g
Lima
Gonjal es,
a
whose
sugges ion
his
wo k
was
de eloped
.
1
.
P elimina ies
In
his
sec ion
we
in oduce
some
gene al
esul s
on
P-local g oups,
ac o ing
o
ac ions
and
some
p oposi ions
conce ning
g oup
cohomology
.
We
s a
by
ixing
he
no a ions
P'
=
{n
E
N*
:
p
1
n
=>
p E
P}
=
mul iplica i e
se
gene a ed
by
P
;
P'
is
he
complemen a y
o
P
in
he
se
o
all
p imes
.
We
also
ecall
ha
G
is
said
o
be
a
P-local
g oup

((dn
E
P'm)x
E
G
~-->
x'
E
G
is
bijec i e)
.
22

A
.
REYNOL
FILHO
Mo eo e ,
G

e
)
GP
P-localizes
G
E
l
C
1
in
he ca ego y
C
4=>(GP
E
IC
1,
GP
is
P-local
and
b'H
E
I
C
1,
H
P-local,
(V
E
HOm(G,
H»(3!
p
E
Hom(GP,
H))
which
yields
commu a i i y
in
he
diag am
G
--~,
H
el

l p

)
.
G
P
P oposi ion
1
.1
.
Le
E,,
...
,
E
,
K
be
P-local g oups,
whe e
P
is
a
se
o
p imes
.
Le
el
E
Hom(Ej,
K),
i
=
1,
. . . ,
.
In
hese
condi ions,
i
E
E
Hom(E,
K)
is
he pull-back
o
he
amily
(Ej)I<j< ,
hen
E
is
P-local
.
P oo
.
S aigh o wa d
.
P oposi ion
1
.2
.
Le
0
E
Hom(Y
F),
whe e
Y
is
a
P-local
g oup
and
F
is
a
ini e
g oup
.
Then,
(dy
E
Y)
we
ha e
o(O(y))
=n
E
Px
;
(o(O(y))
=
o de
o
O(y))
.
P oo
..
Le
yE
Y
and
suppose
ha
3q
E
P'
wi h
q
1
o(O(Y))
.
Then
we
may
conside
z
=
y
k
,
whe e
o(o(y))
=
q
.k
.
I
ollows
ha
o(o(z))
=
q
.
The
ac
ha
Y
is
P-local
and
q
E
P'
enables us
o
s a e
ha
d
>
0,
3z
E
Y
such
ha
z4
=
z
.
Thus,
O(z )g
=
O(z)
~
1
and
4'(z )g +'
-
O(z)
4
=
1,
SO
o(O(z ))
=
q +l,
y
>
0
.
In
pa icula
{O(z )
E
F
:
>
0}
is
in ini e
.
Howe e
his
is
impossible,
since
F
is
ini e
.
Co olla y
1
.3
.

Unde
he
condi ions
o
he
p e ious
p oposi ion,
we
ha e
1
O(Y)
¡
E
Px
(ie,
O(Y)
is
a
P- o sion
( ini e)
sub-g oup
o
F)
.
Rema k
1
.4
.
Acco ding
o
p oposi ion
(7
.1)
in
[4]
we
ha e
ha
a
ini e
g oup
F
is
P-local

F
is
a
P- o sion
g oup
.
The
ull
subca ego y
o
he
ca ego y
9
o
all
g oups
consis ing
o
all
nilpo en
g oups
is
deno ed
by
l
.
P oposi ion
1
.5
.
Le
X
--
w
->
Au (N)
be
an
ac ion,
whe e
X
and
N
a e
g oups
wi h
Au (N)
ini e
.
Then
3
!wp
such
ha
he
diag am
X
Au (N)
eo
i
XP
is
commu a i e

w(X)
is
a
P- o sion
sub-g oup
o
Au (N)
.
P oo
.
I
ollows
di ec ly
om
Co olla y
1
.3
and
Rema k
1
.4
.
P-LOCALIZATION
OF
SOME
CLASSES
OF
GROUPS

2 3
Now
we
ake
A
Ñ
G
-'!»
X
an
exac
sequence
o
g oups,
whe e
A
is
abelian
.
Le
X
--
w
->
Au (A)
be
he
ac ion
de ined
by
p(w((x)
.a)
=
g.p(a)
.g-1
(whe e
E
(g)
=
x)
.
Fix a
collec ion o
p imes
P,
n
E
N
and
x
E
X,
and
de ine
:
en(x)
=
lA
+
w(x)
+
. .
.
+w(xn
-1
)
E
End(A)
.
Fo
his
endomo phism
we
ha e
:
Lemma
1.6
.
(,u(a)
.
g)
n
=
p(0(E(g))
.a)gn
;
b'g
E
G,
da
E
A,
dn
E
N
.
P ooE
I
is
easy
by
induc ion
on n
.
P oposi ion
1 .7
.
Le
P
be
a se
o
p imes
and
le
A
~
G
~
X
be
an
exac
sequence
o
g oups,
whe e
A
is
abelian
and
w
is
he
ac ion
a ached
o he
ex ension
.
Fix
he
condi ions
:
(i)
G
is
P-local
; (ii)
X
is
P-local
; (iii)
9
n (x)
E
Au (A),Vx
E
X,`dn
E
P"
.
Then,
i
wo
o
(i) ;
(ii)
;
(iii)
;
hold
so
does
he hi d
.
P oo
..
(ii)
+
(iii)
==>
(i)
.
Fix
n
E
P"
.
Le
g,
h
E
G
and
suppose
ha gn
=
hn
.
Then,
E(g)n
=
E(h)n
=>
E(g)
=
E(h)
since
X
is
P-local
.
So,
g
=
p(a)
.h
and
hn
=
gn
=
(p(a)
.g)n
=
p(0
.(E(h))
.a)hn
(Lemma
1
.6)
.
Hence
B
n
(E(h))
.a
=
0
.
So
a=0
andg=h
.
Likewise,
le
g
E
GAx
E
X
such ha
E(g)
=
xn
(XP-local)
.
The e-
o e,
e(g)
=
xn
=
E(hn)
.
The e o e
g
=
u(a)
.hn
.
Take
b
E
A
such
ha
a
=
Bn(e(h))
.b
.
Thus
g
=
F¿(B,(E(h))
.b)
.hn
=
(p(b) .h)n
due
o
1
.6
.
So
g
E
G

>
gn
E
G
is
bijec i e
.
(The
o he
implica ions a e
simila )
.
P oposi ion
1 .8
.
Le
P
be a
se
o
p imes
and
le
A
Ñ
G

X
be
an
exac
sequence
o
g oups,
whe e
A
is
ini e
abelian
.

Then,
Bn(x)
E
Au (A),
dx
E
X,
dn
E
P",
p o ided
ha ei he
G
is
P-local
o
A
and
X
a e
P-local
.
P oo
.
(I)
G
is
P-local
.
Fix n
E
P`
and
x
E
X
.
Suppose
ha
Bn
(x)
.a
=
0
.
Le
g
E
G
such
ha
E(g)
=
x
.
Then
(p(a)
.g)n
=
p(Bn(x)
.a)g n
=
gn
.
So
p(a)
.g
=
g
and
a
=
0
.
Thus
en(x)
E
Au (A)
since
A
is
ini e
.
(II)
A
and
X
a e
P-local
.
Fix n
E
P`
and x
E
X,
and
suppose
0,(x)
.a
=
0
.
Thus,
(w(xn)
-
lA)
.a
=
(w(x)
-
1A)00
n
(x)
.a
=
0
he e o e
w(x')
.a
=
a
.
On
he
o he

2
4

A
.
REYNOL
FILHO
hand,
o(w(x))
=
m
E
P'
(P op
.
1
.2)
.
The e o e
w(x)m
.a
=
a
.
As
gcd(m, n)
=
1,
i
ollows
ha
w(x)
.a
=
a,
whence
0
=
B n (x)
.a
=
n
.a
.
Thus a
=
0
since
A
is
P-local
.
So
8,,,(x)
E
Au (A)
.
a
Co olla y
1
.9
.
In
he
condi ions
o
he
p oposi ion
aboye
(1
.8),
G
is
P-local
<-==~
A
and
X
a e
P-local
.
E
-
"I
Nex
we
conside
a
spli
ex ension
N
>4
G

X,
whe e
N
is
a
i-
0
ni e
g oup
.
Le
X
---
w
->
Au (N)
be
he
ac ion
gi en
by
p(w(x)
.a)
=
u(x)
.p(a) .o,(x)-1
and
ake
Bn(x)
:
N
---
N
de ined
by
B
n
(x)
.a
=
1A(a)
.(w(x)
.a)
. . .
(w(x)n
-1
.
a),
x
EX
;
n
EN*
:
In
his
sligh ly
di e en
con ex
we
now
desc ibe
p ope ies
which
a e
qui e
simila
o
P op
.
1
.6,
1
.7,
1
.8
and
1
.9
.
Lemma
1
.10
.
(p(a)
.(x))n
=
P(
n
(x)
.a)
o
.
(X)n
;
dx
E
X
;
dn
E
N*
;
`da
E
A
.
P oo
.
See
P op
.
1
.6
.
P oposi ion
1
.11
.
Le
N
Ñ
G
<--<
X
be a
spli
sho
exac
sequence
o
g oups
and
le
w
be he
ac ion
de ined
by
he
spli ing
o
.

Fix
he
s a emen s
:
(i)
G
is
P-local
;
(ii)
X
is
P-local
;
(iii)
en(x)
is
a
bijec ion,
bx
E
X
;
`dn
E
P'x
.
Then,
i
wo
o
(i)
;
(ü)
;
(iii)
;
hold,
so does
he
hi d
.
P oo
.
See
P op
.
1
.7
.
E
P oposi ion
1
.12
.
Le
N
>-~
G
E--<
X
be
a
spli
sho
exac
sequence
0
o
g oups,
whe e
N
is
ini e
.
Then
Bn(x)
is
a
bijec ion,
dn
E
P",
dx
E
X
p o ided
ha
ei he
G
is
P-local
o
N
and
X
a e
P-local
.
P oo
.
I
G
is
P-local,
hen
he
p oo
ollows
as
(I)
P op
.
1
.8
.
So
le 's
suppose
ha
N
and
X
a e P-local
.
Le
w(X)
~-
i
>
Au (N)
and
G=
N
1
Zw(X)
.
w(X)
is
a
P- o sion
g oup
(Co
.
1
.3)
and
N
is
a
P- o sion
g oup
(Rema k
1
.4)
.
Thus
G
is
a
P- o sion
g oup
.
So
G
is
P-local
(Rema k
1
.4)
.
E
Then
aking he
sequence
N
~
G
<Z
w(X)
and
in oking he
i s
a
_
s a emen
o his
p oposi ion
we
conclude
ha
8,,,(T)
:
N

N
gi en
P-LOCALIZATION
OF
SOME
CLASSES
OF
GROUPS

25
by
0,,
(7
-
)

=

1N
.2(T)
.
. .
¡(7-n-
1)

=

1NT
. . .
T
n-1

is
a
bljec lon,

dT

E
w(X),
dn
E
P`
.
So,
b'x
E
X
V'n
E
P`
cae
ha e
ha
B,,(x)
is
a
bijec ion,
since
B
n
(x)
=
1N-w(x)
.
.
.
w(xn
-1
)
=
en(T),
whe e
T
=
w(x)
E
w(X)
.
Co olla y
1
.13
.
Unde
he
condi ion
o he
p e ious
p oposi ion
(1
.12),
G
is
P-local
~
N
and
X
a e
P-local
.
P oo
..
See
P op
.
1 .9
.
P oposi ion
1
.14
.
Le
N
-
G
~
X
be
an
exac
sequence o
g oups
.
Then,
G
and
X
P-local
==>
N
P-local
.
P op
.
I
ollows
di ec ly
om
he
de ini ions
.
a
F om
now
on
cae
es ablish
some
esul s
which
play
an
impo an
ole
in
Sec ion
3
.
Le
X
--
w
~+ Au (A) and
X
-
0
-~
Au (B)
be
ac ions,
whe e
A
and
B
a e
abelian
.
Le
also
a
E
Homz[
X
I
(A,
B)
(ie
a(w(x)
.a)
=
6(x)
.ca(a))
.
The
eade
in e es ed
in
mo e
de ails
abou
he
cons uc ions
in ol ed
in
he
p oposi ions
below
should
collec
ma e ial
in
[5,
chap e
II,
P opo-
si ion
4
.3
.],
o
ins an e
.
The
p oo s
o
he
nex
h ee
p oposi ions
ollow
easily
om
he
de i-
ni ions
acco ding
he
usual
echniques
.
P oposi ion
1
.15
.
Conside
he
diag am
whe e
A
and
B
a e
abelian
and
he
ocas
a e
exac
.
I he e
exis s
,3
E
Hom(G,
Q)
making
he
diag am
commu a i e,
hen
a
E
Homz[XI
(A,
B) and a
*
l
=
7*(
.
Con e sely,
i
a
E
Homz[X]
(A,
B)
and a
*
j
=
,
y*(,
hen
he e
does
exis
,l
E
Hom(G,
Q)
making
he
diag am
commu a i e
.
P oposi ion
1
.16
.
In
he
diag am
G
A
Ñ
G

X
«
1

T
,[ .L
N

1
-Y,
X
1
~
y
2
6

A
.
REYNOL
FILHO
he
ows
a e
exac ,
A
and
B
a e
abelian
and
T
and ~
yield
commu a i i y
.
Then,
he e
exis s
a
c oss
homomo phism
:
X
-->
B
such
ha
dg
E
G, O(g)
=
ms(g)
.T(g)
.
P oposi ion
1
.17
.
A
al
B
w
G
X
1T
1w
i
Q
Y
In
he
commu a i e
diag am
he
ows
a e
exac
and
A
and
B
abelian
.
Le
X

"'
i
B
be a
c oss
homomo phism
.
In
hese
condi ions
he
unc-
ion
G

0

Q
gi en
by
l(g)
=
mE(g)
.
T
(g),
`dg
E
G
is
a
g oup
homomo -
phism
.
Lemma
1
.18
.
Le
Q
be a
P- o sion
abelian
g oup
.
Then
H
<
,(Q)
is
a
P- o sion
abelian
g oup,
=
dq
>
0
.
(He e
Hq(Q)
means
he
homology
o
he
g oup
Q
wi h
in ege
coe icien s)
.
P oo
..
The
asse ion
is
eadily
checked,
since,
acco ding
he heo y
in
[2]
we
ha e
Hn(Q)p,
-
H

,(Qp,)
=
Hn((0))
=
(0)
;
n
>_ 1
(once
Q
is
P- o sion
abelian)
.
Lemma
1
.19
.
Le
N

G

Q
be
a
cen al
exac
sequence
o g oups
.
I
G
ac s
P-locally
on an
abelian
g oup
A, hen
Q
ac s
P-locally
on
H*
(N
;
A)
.
P oo
.
We
ecall
ha
i
G
ac s
on
A
by
means
o
w,
hen
he
ac ion
w
is
P-local
i
and
only
i
(dn
E
P`)
(Vx
E
G)B
n(x)
=
lA
+w(x)
+
+
w(xn
-1
)
E
Au (A)
.
Mo eo e ,
aking
z
E
G
such
ha
E(z)
=
x,
i
is
known
ha
he
induced
ac ion
o
Q
on
H'(N
;
A)
is
gi en
by
:
Q
Au (H
9
(N
;
A)),
whe e
Q(x)
=
w(z)
*
.( emembe
ha he
ex ension
N
,
G
-'»
Q
is
cen al)
.
Thus,
ixing
x
E
Q
and
pu ing
6n
(X)
=
1Hs(N
;A)
+
9
(x)
+

+
SZ(xn
-1
),
we
ge
:
'9n(X)
=
(1A)*+w(z)*+-
.
.+w(zn-1)*
=
[lA+w(z)+
.
.
.
+
w(zn
-1
)]
*
=
On(z)*
.
So
On(x)
is
an
isomo phism
.
Lemma
1
.20
.
Suppose
ha he
ac ion
X
--
w
-->
Aú (A)
is
P-local
and
X
is
a
P'- o sion
g oup
(A an
abelian
g oup)
.
Then,
w
is
i ial
.
P oo
.-
Se
x
E
X
.
By
hypo hesis,
3
n
E
P'x
such
ha
xn
=
1
.
SO,
0
=
w(xn)
-
lA
=
6
n
(x)o[w(x)
-
lA]
.
P-LOCALIZATION
OF
SOME
CLASSES
OF
GROUPS

2
7
Then,
w(x)
=
lA
( o
0
n
,(x)
E
Au (A))
.
The
nex
heo em
is
s a ed
in
he
ca ego y
97
.
Theo em
1
.21
.
Conside
he
commu a i e
diag am
X
~
Au (A)
whe e
X
is
a
nilpo en
g oup,
A
is
a
P-local
ini e
abelian
g oup
and
w,
wp
a e
ac ions
.
91
Then
we
ha e
Hj
p
(Xp
;
A)

HW
(X
;
A)
.
e
o
P oo
..
(Induc ion
on
c
=
nil
X
.)
I
X
is
abelian,
we
ake
he
ollowing sho
exac
sequences
:
0
->
Ke (eo)
____>
X
eó,
eo(X)
->
0
. . .
(1)
0
->
eo(X)
-
e,
->
Xp
--->
Coke (eo)
-+
0
.
.
.
(2)
(1)
yields a
spec al
sequence
(Lyndon-Hochschild-Se e)
whe e
E2
,
s
=
H'(eo(X
)
;
H'(Ke (eo)
;
A))
.
No icing ha
Ke (eo)
ac s
i ially
on
A
we
a e
allowed
o
say
ha
he
ollowing
sequence
is
exac
0 -->
Ex (H
S
_1(Ke (eo))
;A)
>
Hs(Ke (eo
;A))
--->
Hom(H,(Ke (eo)
;A)
-~
0
.
Since
Hom(P'- o sion,
P-local)
=
(0)
=
Ex (P'- o sion,
P-local)
and
(ds
>0)H
s
(Ke (eo))
is
P'- o sion
(Lemma
1
.18)
we
conclude
ha
E2's
=
(0),
ds
>
0,
whence
he
spec al
sequence
collapses
.
Thus,
H
,
(eo
(X)
;
A)
=
E2'o
=
E ,
-
H,,,
(X
;
A)
.
The e o e,
we
ha e
go
ha
eó
is
an
isomo phism
.
Likewise,

(2)

yields

ano he

spec al

sequence

whe e

E
2
>
s

=
H (Coke (eo)
;
H'(eo(X)
;
A))
.

He e

Coke (eo)

ac s

i ially

on
H
-
'(eo(X
)
;
A)
.
This
may
be seen
om
Lemmas
1
.19
and
1
.20
acco ding
o
he
ol-
lowing
a gumen
:
due
o
P oposi ion
1 .5
wp(Xp)
is
a
( ini e)
P- o sion
g oup
.
So,
Y
=
A
1
iwp(Xp)
(whe e
wp(Xp)
y
Au (A))
is
a
ini e
P-g oup
.
So
Y
is
P-local,
and
hen
i
ollows
ha
wp
ac s
P-locally
on
A
(use
he
same
a gumen
ha
he
one
in 1
.12)
.
Now,
by
Lemma
1
.19,
we
ha e
ha
Coke (e
o
)
ac s
P-locally
on
Hs(eo(X)
;A)
.
As
Coke (eo)
is
P'- o sion,
ou s a emen
now
ollows
om
Lemma
1 .20
.
34

A
.
REYNOL
FILHO
Finally,
we
analyse
he
si ua ion
in
which
A
is
(only)
a
ini e
abelian
g oup
.
Le
X
"->
Au (A)
be an
ac ion,
whe e
A
is
a
ini e
abelian
g oup
.
I
¡
A
l=
pá'
. . .
p

,
hen
A
i
=
pi-p ima y
componen
and
Au (A)
-
l
Au (A
i
)
.
i=I
Thus,
he e
is
(uniquely
de e mined)
wi
:
X
--->
Au (A
i
)
;
i
=
1,
. .
.
.
Le
G=A

W
X
;
G
i
=A
i
X
;
G
j>
X
;
G
i
~>
X
as
usual
.
I is
well-known
ha
E
is
he
pull-back
o
(Ei)1<i< -
Le
G

Gi be
he
p ojéc ion
and
G
~
Xp
be
he
pull-back
o
he
(EjP
a ows
(Gi)
p
--)
Xp
;
i
=
1,

.
.
.
Since
(E_i)p
o
(Qj)p
=
1X
P
,
he e
does
exis
(only one)
Q
E
Hom(Xp,
G)
such
ha
~ i
o
Q
=
(ui)
p, Vi
(he e
Wi
is
he
usual
p o-
jec ion)
.
Likewise,
i is
plain
ha
3
!
E
Hom(G,
G)
such ha
~ io
=
ei~i(ei
Gi
-)
(Gj)p
P-localizes
Gi)
.
I
ollows
ha
é
=
eOE
and
=
Qeo
.
We
ecall
ha
G
is
P-local
by
p op
.
1
.1
.
Mo eo e ,
3

E
Hom(Gp,G)
such ha
7 ioo
=
(7 i)p,
Since
(Ei)po(7 i)p
=
Ep,,di
(In
pa icula
7 i
is
an
isomo phism)
.
By
unique
ness
we
ha e
go
=
Oe
;
i~o
=
ep
;
OQp
=
and
E

=
1X
P
as
well
.
(So
G=CIXP)
.
Finally,
le
C
=
ke

-
®
ke (Ei)p
;
i=1
Ti
:C
,
G,N
_
=ke Ep
;p'
:N-~Gp,e, ,0,de ine
e
:A->N,
A
->
C
and
i
:
N
--->
C
by
es ic ion
.
Le
B=
ke
0
.
Soon
we
a e
going
o
show
ha
3
!
e'
E
Hom(C,
N)
such
ha
e'7
=
é
.
We
a e
able,
a
las ,
o
cons uc
he
ollowing
commu a i e
diag am
:

whe e
P-LOCALIZATION
OF
SOME
CLASSES
OF
GROUPS

35
B
K=Ke
Diag am
2
.6
Lemma
2
.7
.
is
an epimo phism
.
P oo
.
This
ollows
om
he
ac
ha
E
K
=
®
K¡
(K¡
=
ke
i-1
In o de
o
jus i y
all
he
indica ions
in
he
diag am,
we
s ill
need
wo
lemmas
.
ke (Ejp
N
(Gilp
~-»
Xp
aken
in
he
conjunc ion
wi h
he
cases
p e iously
analysed
.
Lemma
2
.8
.
é
IK
=
0
.
P oo
.
Since
36

A
.
REYNOL
FILHO
we
jus
ha e
o
show
ha
é
1x
;=
0,
Vi
.
This
ollows
om
he
diag am
( i
is
a
spli ing
a ached
o
S i)
K
i
~-a

Ai

Ñ

Gi

~i'

G
el

1
el

1
e
i
1 , 1

(
)P
ke (Ei)P
,--,
(Gi)P

GP
So
we
ha e
e'
E
Hom(C,
N)
wi h
e'
=
é
.
Lemma
2
.9
.
(i)7(W(x)
.a)
=W(eo(x))
. (a)_

;dx
E
X
;b'a
E
A
(ii)é(w(x)
.a)
=
wp(eo(x))
.e(a)
}
P oo
.
Bo h
s a emen s
a e
eadily
checked
om
he
de ini ions
.
Theo em
2
.10
.
In
he
condi ions
abone,
G
--
-~>
G
P-localizes
G
.
P oo
:
Le

Gp
de ined
by
0(p(c)
.~j
;(z))
=
p'e'(c)
.ap(z)
;
cE
c
l
zEXp
.
0
E
Hom(G,
Gp)
by
p op
.
2
.2,
so
ha
i
is
plain
ha
3
.
P-localiza ion
on
he
ca ego y
C
Th oughou
his
sec ion
we
cons uc
he
heo y
o
P-localiza ion
o
a
g oup
in
he
ca ego y
C
o
g oups
which
a e
ex ensions
o
nilpo en
g oups
by
ini e
abelian
g oups
.
(al hough
we
s ill
use he
same
no a ion
G
~
Gp
o
P-localiza ion
in
he ca ego y C)
.
P oposi ion
3
.1
.
Le
A
~
G
~
X
be
an
exac sequence
o
g oups,
whe e
A
is
abelian
ini e
and
X
is
nilpo en
.
Le
X
_
w
->
Au (A)
be
he
ac ion
a ached
o
he
ex ension
and
suppose
I'W
=
A
.
Le
also
,0

E

Hom(G,K)
and
B

Ñ

K

-'»

Y
be
an exac
se
quence,
whe e
B
is
ini e
abelian
and
Y
nilpo en
.

Then,
he e
exis
a
E
Hom(A,
B)
and
,y
E
Hom(X,
Y)
which
yield
commu a i i y
in
he
diag am
A
al
B
X
1y
Y
P-LOCALIZATION
OF
SOME
CLASSES
OFGROUPS

37
P oo
.
Le
H
=
op(A)
<
Y
.
I
ollows
om
I`
=
A
ha
H
C
[Y,
H]
.
So
H
C
[Y,
H]
C
I
,2
Y
and
hen,
by
induc ion,
H
C
I'
k y,
bk
>_
2
;
whence
H
=
{1}
since
Y
is
nilpo en
.
This comple es
he
p oo
.
P oposi ion
3
.2
.
dG
E
l
C
1,
3
!
U
=
U(G)
<
G,
U
ini e
abelian
wi h
G/U
nilpo en
such
ha
F
ue
,
=
U,
p o ided
ha
w
is
he ac ion
a ached
o
he
ex ension
U
>-~
G
-»
G/U
.
P oo -
Le
A
>'-"->
G
»
X
be an
ex ension
whe e
A
is
ini e
abelian
and
X
is
nilpo en
.
Le
SZ
:
X
->
Au (A)
be
he
ac ion
a ached
o
his
ex ension,
and
se
F
=
F',
whe e
is
he
smalles
posi i e
in ege
such
ha
F =
I'
1
.
Le
U=
la(F)
<G
.
So
U
is
ini e
abelian
.
Fu he mo e,
A/F
>->
G/U
-»
X
is
exac
and
X
ac s nilpo en ly
on
A/F,
so
ha
G/U
is
nilpo en
.
I 's
also
plain
ha
F
u
e
,
=
U,
i
w(gU)u
=
gug
-1
.
Finally
we
poin
ou
ha
he
uniqueness
ollows
in
a
s aigh o wa d
way
om
p oposi ion
3
.1
.
Now
le
p
be
a
p ime
and
C
p
be
he
ull
sub-ca ego y
o
C
o
all
g oups,
which
a e
ex ensions
o
X
by
A,
whe e
A
is
a
ini e
abelian
p-g oup
.
Co olla y
3
.3
.
G
E
1
C
p
1
=>
U=
U(G)
is
a
ini e
abelian
p-g oup
.
P oo
.
In
ac ,
U
=
u(F)
and
F
is
a
sub-g oup
o
A
.
a
Co olla y
3
.4
.
G
ESC
I ;
G
is
nilpo en
~
U=
U(G)
=
{1}
.
P oo
.
(==)
G
E¡
l
1==>
w
:
G/U
--->
Au (A)
is
nilpo en
==>
U
=
F2
_
.
.
.
=
FW
1
=
{1}
.(c
=
nil
w)
.
(4--=)
I
is
ob ious
.
Co olla y
3
.5
.
G
E
l
p

11
p,
q
p imes,
p
qL
q,
such
ha
G
E
1
C
p
1
n
¡C
.1
.
P oo -
I
ollows
om
co
.
3
.3
and
co
.
3
.4
.
We
now
de ine
Gp
El
C
1
p o ided
G
El
C
1 .
Fix
~
:
U
>'->
G
-'»
G/U
whe e
U
=
U(G)
is
de ined
by p op
.
3
.2
;
w(gU)u
=
gug
-1
and
G/U
(G/U)p
P-localizes
G/U
in
77
.
We
conside
3
cases
:
Le
p
be
a p ime and
suppose
i s ly
G
E
l
C
p
1 .
I)
p
E P'
.
Se e
=
eo
o
e,
G
»
G/U
Pa'
(G/U)p
.
38

A
.
REYNOL
FILHO
Then we
ha e
:
HW(G/U
;
U)
-
-'-*
-H"-,(G/U
;
U/I')
So
we
mus
conside
(0)
-
(GIU)P
=
(G/U)P
We
should
poin
ou
ha
7
*
j
=
e*I
P
=
0
.
II)
pEP
.
Le
P
l
=
{q
E
P'
:
q
1 1
w(G/U)1
1},H
=<x
E
G/U
:
o(w(x))Pj
>,F
=
F(H) and
w
:
G/U
---->
Au (U/F)
as
de ined
jus
a e
heo em
1
.21
.
Co olla y
1
.25
allows
us o claim
ha
3
!
ac ion
wP
making commu a i e
he
diag am
Gl
u
-i

Au (U/ )
/
AI
P
(Glu)P
Taking
he
na u al
p ojec ion
U
--

U/F,
we
ha e
ha
3
!
e*I
P
=
7
*
l
whe e
Hwp((G1U)P
;
U/F)
Once
mo e
i
is
shown
by p op
.
1
.15
ha he e
is
a
commu a i e
diag am
U
>
P
+
G

Glu
le
leo
IP

U/
,--,
G
P
--

(Glu)P
¡el
U
lCPw
o
.
p
bP
such
ha
A
his
poin
i
is
impo an
o poin
ou
ha
we
ha e
de ined
G
E
Up
1
C
1-+
G
P
E
l
C
1
and
his de ini ion
is
"good"
since
G
E
l
C
p
1
n
1
C
q
¡==¿-
G
E
17
(co
.
3
.5)
and
hen
U=
{1}
(co
.
3
.4)
In
pa icula ,
his
cons uc ion
ex ends
he
one
made
in
[2]
.
Example
3
.6
.
Le
w
:
7L
-->
Au (Z/3
®
7G/5)
gi en
by
w(1)
.a
=
2a
and
w(1)
.b
=
2b
.
Le
G=
(Z/3
®
7G/5)
'j
~,7Z
.
Then F2
=
Z/3
®
7G/5
=
A,
whence
G
11
17
1
.
Howe e
G
E
l
C
1
and
since
U=
c(A),
i
ollows
ha
G
1
Up
1
C
p
1 .
This
example shows
ha
P-LOCALIZATION
OF
SOME
CLASSES
OF
GROUPS

39
III)
G
OCI
Up
ICpI-
Now
U
is
no
longe
a
P-g oup
.
Ne e hless,
Also,
and
es
=
(wl,
.. . ,
wl)
.
We
ha e
1
E
H
2
(Gl
u
;
U)
SP
E
H,,p(G/u)P
;
U)
whe e
is
de ined
by
(I)
o
(I1)
.
Also,
whe e
Ui
is
he
pi-p ima y
componen
o
U
.
G/U
--->
Au (U)
=

Au (Ui)
i=1
(7 i
.
,. .
,7 *
)
i-1
whe e
U
-H Ui
is
he
usual
p ojec ion
.
No ice
ha
I'2
=U
-4
I'
2
w

wi
=
U¡
;
Vi
=
1,
. . . ,
.
Le
i
=
S i
*
l
and
conside
he
commu a i e
diag am
(1 1*
,.
.
.
,7 -
)
(7
1
*
, .
.
.
,7
*
)
Si
Ui
%+ Gi
-»
G/U
Pi

1
Pi

Po
(
i)P
:

Ui

>li-x>

(Gi)p

(
P

(G/u)P
(Si)i
E
®í-1
H~2
i
(Gl
U
;
Ui)
®x =1eo-1)(® z=1
Pi*)
(Si)P
E
®i=1H(w
;)n((G/U)P
;
Ui
and 3
!
Sp
such
ha

,7F
**)1p
=
((ji)p)i
p o ided
ha

i
is
he
usual
p ojec ion
and
p
=
®ipi
.

40

A
.
REYNOL
FILHO
Diag am
3
.7
By
de ini ion,
we
ha e
ha
IP
is
he
pull-back
o
he
a ows
since
As
o
G
-j
Gp
de ined
by
(I)
;(II)
;(III)
we
ha e
he
nex
wo
p opo-
si ions
.
P oposi ion
3
.8
.
e
is
P-su jec i e
.
P ooi
Ob ious
.
Suppose
we
ha e
P oo
.
I
ollows
di ec ly
om
he
de ini ions
.
Co olla y
3
.9
.
G
--
e
>
Gp
---
9
-->
K,
wi h
K
P-local
.
Then
e
=
ge ==>
=
g
.
P oposi ion
3
.10
.
G
P-local
==
:>
e
is
an
isomo phism
.
P oo
.
We
ha e
3
cases
o
analyse
.
The
only
one
which
is
no
ob ious
is
(II)
.
U

G
-» G/ l
i
1

1
.e

leo
,--
UI
,
GP
-~
(Glu)P
G
P-local
=>G/U
P-local
(co
.
1
.9)
.
So
w(G/U)
is
a
P- o sion
sub-g oup
o
Au (U),
since
Au (U)
is
ini e
and
G/U
is
P-local
(co
.
1
.3)
.
The e o e
H
=
{1}
and
F
=
{l}
.
The e o e
n
=
lu
.
So
e
is
an
isomo phism
.
P-LOCALIZATION
OF
SOME
CLASSESOF
GROUPS

41
Nex
we
conside
a commu a i e diag am
U(G)
=

U

G

G/U
la
la
17
U(K)
=

V

K

K/
Le
us
de ine
á
E
Hom(U,
V)
induced
by
(apu
=p a
;
U
P
»U)
.
We
ake
U=
®U(p)
and
V
=
®V(p)
;
p-p ima y
decomposi ions
.
P oposi ion
1
.26
assu es ha
a(FU(p))
C
FV(p),
since
a(U(p))
C
V(p)
.
Ac ually,
we
conside
~(P)

U(p)

'--'

G(p)

-
Gl
U
1
alu(P)
S(p)

V
(P)

K
(p)

-

K
and
hen
use
p op
.
1
.26
o
~(p)
_
7 (p)
*

and «p)
=
7 (p)
*
(
.
We
de ine,
by
es ic ion,
«p)
:
U(p)
-
V(p),
whence
we
ha e
PU(P)
l
U(p)
=

U(p)/ u(P)
and
inally
á
=
®pd(p)
.
A
his
poin
we
s a e
a
undamen al
p oposi ion
.
U(p)

a~P)

V
(P)
l
PV(P)
V
(p)/FV(P)
=
V
(P)
P oposi ion
3
.11
.
á
is
an
homomo phism
o
modules
.
P ooL
Le
us
conside
he
comu a i e
diag ams
.
G
U
Au U)
eo
1

/
wa
(G/U)P
K
->
Au (9)
eo
1

/sap
(Kl )P
whe e
wP
and
gP
a e
he
ac ions
gi en
by
he
ex ensions
U
Ñ
G
P
-»
(G/U)P
1IX
17P
V
Ñ
KP
-»
(K )
P
4
2

A
.
REYNOL
FILHO
We
oug h
o
ge
ha
á(wp(z)Z)
=
Qp(yp(z))
.á(a),
bz
E
(G/U)p
and
baEU
.
Fix
z E
(G/U)p
and
á
E
U
.
G/U
nilpo en

n
E
P`
such ha
zn
=
eo(x)
.
Then,
á(wp(zn)
.á)
=
U(wp(eo(x))
.-j)
=
á(w(x)
.á)
=
a(w(x)
.a)
(by
de ini ion)
=
p a(w(x
)
.a)
=
p (9(y(x))
.a(a))
(a
is
a
homomo phism
o
modules)
_
Q(y(x))
.a(a)
=
gp(eo(y(x)))
.5(a)
=
QPyp(zn)
.á(á)
.
.
.
(*)
.
On
he
o he
hand
o(wp(z))
=
m
E
Px,
(p op
.
1
.2),
since
(GJU)p
is
P-local
and
Au (U)
is
ini e
.
So,
gp(yp(zn)m)
.á(j)
=
U(WP(z
n
)
m
.Q)
_
Z!(á),

E
U
.
The e o e,
gp(yP(z
m
)
n
)I
IX(U)
-
1
IX(U)
.
S ill,
aking
in o
accoun
ha
in
he
exac
sequence
(p
:
V
>-->
Kp
~>
(K/V)
p
;
Kp
and
(K/V)p
a e
P-local
we
can
s a e
ha _
Bn(yp(zm))
_
1V
+
gp(i'P(z
m
))
+
.
. .
+
gp(yp(zm)n-1)
E
Au (V),dn
E
P'x
and
,
YP(z
m
)
E
(K1V)p
.
Thus
Va
- E
U
we
ha e
:
0
=
á(¿í)
-
gp(yp(zm)n)
.Cj(Q,)
_
[les
-
pp(i
y
p(z
m
))
n
ja(a)
=
en(Í
y
p(z
m
»
0
[
1
V-
Qp(yp(zm))]
.a(Z¡)
.
The e o e,
5
(
-
j)
-
QP(yp(z
m
))
.a(a)
=
0,
whence
Pp(yp(z
m
»
Iá((1)
=
1
-a(5)
.
Finally
3
,
s

E

9G
such
ha
m
+
sn

=

1

(gcd(m,
n)

=

1)
.
The e o e
á(wp(z)
.
á)
=
á(WP(z
n
)s
oWP(z
m
)
.
j)
=
á(WP(zn)s
.á)
=
gp(_YP(z
n
)
3
)
.
á(á)
=
gp(i'P(z
n
)s)
.
qP(7P(z
m
)
)
.5
(C
1
)
=S2p(yp(z))
.a(-j)
.
Theo em
3
.12
.
G,
K
E
I
C
1 ;
3
!
/3p
E
Hom(Gp,
Kp)
yelding
commu-
a i i y
in
he
diag am
I
:
U

»

G

~>

G/U
Sp

>i~>
Gp
>
(G1U)P
V

>,
IK

~>

K/
V
Cp
'IYp
Sp
:
V

>~>

Kp

-"*
P

(K/V)p
P oo '
The
uniqueness
ollows
om
he
co olla y
3
.9
:,
Fo he
exis ence
we
obse e
ha
eóyP~p
=
.y*e*(p
=
y
*
p
"
~
(de i-
ni ion
o
(p)
=
p
"
y
*
~
=
p
"
a*(1) (p op
.
1
.16)
=
á
*
pu*
=
ce
*
e*Ip
(de
.
o
~p)
=
e*U*jp
.
I
ollows
ha
y*~p
=
c**
jp
due
o
he
ac
ha
H
Z
((G/U)P
;
V)
--°
>
P-LOCALIZATION
OF
SOME
CLASSES
OF
GROUPS

43
H
2
(G/U
;V)
(Th
.
1
.21)
.
So by
p oposi ion
1
.16,
3
E
Hom(Gp,
Kp)
yielding
commu a i i y
in
he
" on ace"
o
he
diag am
.
Thus
Te
and
eO
make
commu a i e
he
diag am
¿

e
U
GGl
u
p oa
1

e
U
e/3

1
eo
V
,--,
KP
-
(K1 )P
Use
o
he
p oposi ion
1 .17
shows
ha
0
:
G/U

>
V
a c oss
ho-
momo phism
such ha
e l(g)
=
0e(g)
.Te(g),dgE
G
.
Howe e ,
HI((G/U)p
;V)

el
.
HI(G/U
;
V)
(Th
.
1
.21)
.
So
9
=
0'
eO
+
S ,
whe e
w
(x)
=
-
x
. ,
E
V
.
Se ing
S
:
(G/U)p

>
V,
6 (z)
=
-z
. ,
i
ollows
ha
S
o
eO
=
S
and
he e o e
0
=
Opoeo, whe e Op
=
BP
+
6
.
Now
eO(g)
=
9Peoe(g)
.Te(g)
=
-POpEpe(g)
.Te(g),
dg
E
G
.
Thus
de ining
,QP
:
Gp
->
Kp
by
~3p(z)
=
0pep(z)
.- (z),dz
E
Gp,
i
ollows
om p op
.
1 .18
ha
~3p
E
Hom(Gp,Kp)
and
~3pe
=
eO
.
Besides,
p,QP
=
-
ypEP
and
Op7!
=
U
.
Rema k
.
The
heo em abo e shows
us
ha
G
---~
Gp
is
a
unc o
and
e
is
a
na u al
ans o ma ion
o
unc o s
.
Theo em
3
.13
.
G
e
>
Gp
P-localizes
G
in
C
.
P oo
.
Le
G,
K
E!¡
C
1,
wi h
K
P-local,
and
0
E
Hom(G,
K)
.
Owing
o
p oposi ion
3
.1,
he e
exis s
a
commu a i e diag am
U(G)
=

U

G

Gl
la 10 17
U(K)
=

V

>~>

K

~>

Kl
Now
using
h
.
3 .12
we
conclude
ha
3
!
~
3p
E
Hom(Gp,
Kp)
such
ha
Ope
=
ei
.
So
i
is
enough
o ake

=
e
-1
o
OP
Gp
-
Kp
Ap
(p op
.
3
.10)
The
uniqueness
ollows
om
co
.
3
.9
.