Publicacions
Ma emá iques,
Vol
37
(1993),
271-284
.
GALOIS
H-OBJECTS
WITH
A
NORMAL
BASIS
IN
CLOSEDCATEGORIES
.
A
COHOMOLOGICAL
INTERPRETATION
A
bs ac
such
ha
J
.
N
.
ALONSO
ALVAREZ
AND
J
.
M
.
FERNÁNDEZ
VILABOA
In his
pape ,
o
a
cocommu a i e
Hop
algeb a
H
in
a
symme ic
closed
ca ego y
C
wi h
basic
objec
K,
we
ge
an
isomo phism
be ween
he
g oup
o
isomo phism
classes
o
Galois
H-objec s wi h
a
no mal
basis
and
he second
cohomology
g oup
H
2
(H,
K)
o
H
wi h
coe icien s
in
K
.
Using
his
esul ,
we
ob ain
a
di ec
sum
decomposi ion
o
he
B aue
g oup
o
H-module
Azumaya
monoids
wi h
inne
ac ion
:
BMinn(C,
H)
-B
(C)
®
12
(H,
K)
In
pa icula ,
i
C
is
he
symme ic
closed
ca ego y
o
K-mo-
dules
wi h
K
a
ield,
H
2
(H,
K)
is
he
second
cohomology
g oup
in oduced
by Sweedle
in
[21]
.
Mo eo e ,
i
H
is
a
ini ely
gen-
e a ed
p ojec i e,
commu a i e
and
cocommu a i e
Hop
algeb a
o e
a
commu a i e
ing
wi h
uni
K,
hen he abo e
decompo-
si ion
heo em
is
he
one ob ained
by
Bea ie
[5]
o
he
B aue
g oup
o
H-module
algeb as
.
P elimina y
A
monoidal
ca ego y
(C,
®,
K)
consis s
o
a
ca ego y
C
wi h
a
bi unc o
-
®
-
:
CxC
->
C and a
basic
objec
K,
and
wi h
na u al
isomo phisms
:
aABC
:A®(B®C)=(A(9
B)®C
l
A
:K®A-A
A
:A®K-A
(A
(9
aBCD)
o
aA(B(DC)D
0
(aABC
®
D)
=
aAB(C®D)
°
a(A®B)CD
(A(9
1B)oaAKB= A®B
272

J
.
N
.
ALONSO
ALVAREZ,
J
.
M
.
FERNÁNDEZ
VILABOA
I
he e
is
a
na u al
isomo phism
TB
:
A®B=B®A
such
ha
TAB
o
TB
=A
®
B,
TB®c
=
(B
(9
7-C)
o
(TB
®
C),
hen
C
is
called
a
symme ic monoidal
ca ego y
.
A
closed
ca ego y
is
a
symme ic
monoidal
ca ego y
in
which
each
unc o
-®A
:
C
,
C
has
a
speci ied
igh
adjoin
[A,
-]
:
C
->
C
([12],
[18])
.
Examples
:
1)
The
ca ego y o
se s
and
mappings
.
2)
The
ca ego y o
R-modules
o e
a
commu a i e
ing
R
.
3)
The
ca ego y
o
chain
complexes
o
R-modules
and
mo phisms
o
deg ee
0,
wi h
R
a
commu a i e
ing
.
4)
The
ca ego y
o
shea es
o
B-modules
o e
a
opological
space,
wi h
0
a
shea
o
commu a i e
ings
.
5)
The
ca ego y
o
cohe en
shea es
o
modules
o e
a
scheme
.
6)
The
ca ego y
o
all
R-g aded
modules
wi h
mo phisms
o
deg ee
0
(R
is
a commu a i e
g aded
ing)
.
7)
(R,
u)-Mod,
wi h
R
a
commu a i e
ing
and
o,
an
idempo en
ke nel
unc o
in
R-Mod
.
In
wha
ollows,
C
deno es
asymme ic
closed
ca ego y
wi h
equalize s,
co-equalize s
and
p ojec i e basic
objec
K
.
We
deno e
by
am
and
OM
he
uni
and
he
co-uni , espec i ely, o
he
C-adjun ion
M
®-
-1
[M,
-]
C
--->
C
which
exis s
o
each
objec
M
o
C
.
1
.

An
objec
M
o
C
is
called
p o ini e
in
C
i
he
mo phism
[M

Om(K)
®M]
o
am(M
®M)
:
M
®M
,
[M,
M] =
E(M)
¡san
isomo phism,
whe e
M
=
[M,
K]
.

I ,
mo eo e ,
he
ac o iza ion
o
Qm(K)
:
M
®
1V1
--->
K
h ough
he
co-equalize
o
he
mo phisms
Qm(M)
®
M
and
M
®
([M,
(3m
(K) o
(Om(M)
®
M)]
o
am
(E(M)
®
M))
M
®
E(M)
®
M
->
M
®
M
is
an
isomo phism,
we
say
ha
M
is
a
p ogene a o
in
C
.
2
.

A
monoid
in
C
is
a
iple
A
=
(A,
7A,
PA)
whe e
A
is
an
objec
in
C and
MA
:
A®A
-> A,
7A
:
K
-~
A
a e
mo phisms
in
C
such
ha
MA
o(A
(9
7A)
=
A=
UA
o
(?1A
(9
A),
MA
0
(UA
®
A)
=
pA
-(A
(9
PA)
-
I
MA
o
Tá
=
PA,
hen
we
will
say
ha
A
is
a
commu a i e
monoid
.
Gi en
wo monoids
A
=
(A,
97A,
PA)
and
B
=
(B,
?7B,
[LB)
in
C,
:
A
->
B
is
a
monoid
mo phism
i
MB
o (
®
)
=
o
PA
and
o
7A
=
?7B
.
A
comonoid
(cocommu a i e),
D
=
(D, ED, 5
D
)
is
an
objec
D
in
C
oge he
wi h
wo
mo phisms
ED
:
D
-->
K,
5D
:
D
-->
D
®
D,
such
ha
(6
D
®
D)
o5
D
=
(D
®
6D)
o6
D
and
(ED
®
D)
o
6D
=
1
D=
(D
ED)
o
6D(TD
o
6D
=
6D)
.

I
D=
(D,
ED,
6D)
and
E
=
(E,
EE,SE)
a e
comonoids,
:
D
--->
E
is
a
comonoid
mo phism
i
(
®
)
o
S
D
=
5E
o
and
EE
0
=
ED
.
NORMAL
BASIS
IN
CLOSED
CATEGORIES
AND
COHOMOLOGY

273
3
.
Fo
a
monoid
A
=
(A,
T7A,
PA)
and a
comonoid
D
=
(D,
ED,SD)
in
C,
we
deno e
by
Reg(D,
A)
he
g oup
o in e ible
elemen s
in
C(D, A)
(mo phisms
in
C
om
D
o
A)
wi h
he
ope a ion
"con olu ion"
gi en
by
:
*
g
=
PA
o
(
®
g)
o
S
D
.
The
uni
elemen
is
ED
®
?1A
.
Obse e
ha
Reg(D,
A)
is
aü
abelian
g oup
when
D
is
cocommu a i e
and
A
is
commu a i e
.
4
.
De ini ion
.
Le
II
=
(C, qc,
pc)
be
a
monoidand
C
=
(C,
EC,
SC)
a
comonoid
in
C and
le
A
:
C
--->
C
be a
mo phism
.
Then
H
=
(C
=
(C,
EC,
6C),
II
=
(C,
77c,
7¿C),
C
,
A)
is
a
Hop
algeb a
in
C
wi h
espec
o
he
comonoid
C
i
Ec
and
Sc
a e
monoid
mo phisms
(equi alen ly,
7c
and
pc
a e
comonoid
mo phisms)
and
A
is
he
in e se
o
1c
:
C
--->
C
in
Reg(C,
C)
.
We
say
ha
H
is
a
ini e
Hop
algeb a
i
C
is
p o ini e in
C
.
5
.
De ini ion
.
(A,
WA)
=
(A,
?7A,
PA
;
WA)
is
a
le
H-module
monoid
i)
A
=
(A,
?7A,
PA)
is
a
monoid
in
C
.
ii)
(A,
c0A)
is
a
le
H-module
(WA
o
(C
®
WA)
=
WA
o(pc
®
A),
cO
A
o
( 7c
®
A)
=
A)
.
iii)
7A,
PA
a e
mo phisms
o
le
H-modules
(WA0(C077A)
=
7A0Ec
and
EPA
o(C
®
PA)
=
PA
0IPA®A, whe e
(PA®A
=
(WA
(9
WA)
o(C
,
A
0A)o(Sc®A®A))
.
We
say
ha
he ac ion
WA
o
H
in
A
is
inne
i
he e
exis s
a
mo phism
in
Reg(C,A)
such
ha
WA
=
PA
0(A
®
GIA
o
- A
))
0
(
®
-1
®
A) o
(Sc
®
A)
:
C
®
A
S
A, whe e
-1
is
he
con olu ion
in e se
o
.
6
.
De ini ion
.
I
H
is
a
cocommu a i e
Hop
algeb a
and
(A,
WA)
is
a
commu a i e
H-module
monoid,
hen,
we
say
ha
a
mo phism
u
in
Reg(C
®
C,
A)
is
a
2-cocycle
i
al
(Q)
*
03
( )
=
02
(o,)
*
04
(o
,
),
whe e
191
(U)
=
EPA
0
(00
Q),
192
(U)
=
Q
o
(PC
®
C),
a3(O )
=
o,
o
(C
®
pc) and
a4(Q)
=
Q®
EC
.
Two
2-cocycles
o,
and
y
a e
said
o
be
cohomologous,
w i en
-
-y,
i
he e
exis s
a
mo phism
E
Reg(C, A)
such
ha
o,
*
a
2
( )
=
al
( )
*
03
( )*y,
whe e
a
l
( )
=
WA0(C® ),
a2
( )
= opc
and
03
( )
=
0ec
.
T i ially,
"-"
is
an
equi alence
ela ion
.
The
se
o
equi alence
clases
shall
be
called
he
second
cohomology
g oup
o
he
cocommu a i e
Hop
algeb a
H
wi h
he
coe icien s
in
he
le
H-module
monoid
(A,
WA), and
will
be
deno ed
by
H
2
(H,
A)
.
I
o,
is
a
2-cocycle
in
Reg(C
®
C,
A),
hen
he
mo phism
=
u
*
a
l
(7 )
*
a
2
(7
-1
)
*a
3
(7 )
is
a
2-cocycle
in
Reg(C®C,
A) cohomologous
wi h
Q
such
ha
Qo
( 7c
®
C)
=
Ec
®
7A
=
Q
o
(C
®
7c),
whe e
i
=
o
-1
0
(C
®
7
7C)
is
a
mo phism
in
Reg(C,
A)
wi h
in e se
7
-1
=
u
o(C
®
lc)
.
Mo eo e ,
i
274

J
.
N
.
ALONSO
ALVAREZ,
J
.
M
.
FERNÁNDEZ
VILABOA
,y
is
cohomologous
wi h
u,
hen
he e
exis s
a
mo phism
í9
E
Reg
(C,
A)
such
ha
i9
o
)c
=
91A
and
Q
*
c92
(~)
=
al
(19)
*
o93(19)
*
5
.
Rema k
.
Le
C
he
ca ego y
o
K-modules
o e
a
ield
.
In his
case,
H
2
(H,
A)
is
he
second
cohomology
g oup
o
he
Sweedle 's
complex
q
{Reg(®C,
A)
;
Aq}q>0
Reg
(K,
A)
°o
.
Reg(C, A)
°1
.. . .
~'
Reg(®C,
A)
whe e
A
q:=
al
*
82
I
*

. .
*
aq+2)4+1
and
o
each
E
Reg(®C,
A),
aI( )
=WA-(C® )
az( )=
o(C®i-20C®u
c
(9
C®q
-
i+l0
C)
Oq+2
( )
=
®
EC
A
~
Reg(
q
®
1
C,
A)
~~
. . .
([
21
1)
.
7
.
De ini ion
.
(B,PB)
=
(B,''IB,MB
;
PB)
is
a
igh
H-comodule
monoid
i
:
i)
B=
(B, nB,
MB)
is
a
monoid
in
C
ii)
(B,
PB)
is
a
igh
H-comodule
((PB®C)OPB
=
(B(9SC)OPB
;
(B®
-c)
o
PB
=
B)
.
iii)
PB
:
B
-->
B
®C
is
a
monoid
mo phism
om
(B,
?7B,
N-B)
o
he
p oduc
monoid
BH
=
(B
®
C,
?7B
®
?1c,
(AB
®
pc) o
(B
®TB
®
C))
( ha
is,
PB
o
?1B
=
71B
®
?
1c
and
PB
o
P,
B
=
(MB
(9
¡ c)
o
(B
®
TB
C)
o
(PB
(9
PB))
F om
now
en
we
assume
ha
H
is
a
ini e
cocommu a i e and com-
mu a i e
Hop
algeb a
.
8
.
De ini ion
.
A
igh
H-comodulemonoid
(B,
PB)
is
said
o
be
a
Galois
H-objec
i
and
only
i
:
i)
The
mo phism
-YB
:=
(MB
(9
C)
o
(B
(9
PB)
:
B
(9
B
->
B
®C
is
an
isomo phism
.
ii)
B
is
a
p ogene a o
in
C
.
Fo
example,
in
he
case
o
(R,
u)-Mod,
a
commu a i e
H-comodule
monoid
is
a
couple
(B,
PB),
whe e
B
is
acommu a i e
(R,
u)-algeb a
and
PB
:
B
-,
B
L
H
:=
Q,
(B
(9
R
H)
is
a
mo phism
o
algebas
and
i
de ines
a igh
H-comodule
s uc u e
o e
B
.
whe e
NORMAL
BASIS
IN
CLOSED
CATEGORIES
AND
COHOMOLOGY

275
(B,
PB)
is
a
Galois
H-objec
i
and
only
i
B
is
a (R,
u)-p ogene a o
and
he
mapping
pB
:
B#H
->
Hom(B,
B)
a ising
om
he
le
B#H-
0
module
s uc u e
on
B
is
an
isomo phism
([15,
(1 .3 .17)])
.
I
a
Galois
H-objec
is
isomo phic
o
H
as
an
H-comodule
hen
we
say
ha
i
has
a no mal
basis
.
I
B
1
and
B2
a e
Galois
H-objec s,
:
B
I
->
B2
is
a
mo phism
o
Galois
H-objec s
i i is
a
mo phism
o
H-comodules
(PB,,
o
=
(
®C)
o
PB
l
)
and
o
monoids
.
I
(A,
PA)
and
(B,
PB)
a e
H-comodule
monoids, hen
A
o
B,
de ined
by
he
ollowing equalize
diag am
al
A
B
AoB

->
A®B
i
:i
A®B®C
02
AB
aAB
=
(A
(9
TB)
o
(PA
(9
B),
and
aAB
=A®
PB
is
an
H-comodulemonoid
o
be
deno ed
by
(A
o
B,pAB)
.
I
mo eo e
(A,
PA)
and
(B,
PB)
a e
Galois
H-objec s,
hen
(A
o
B,
pAB)
is
also
a
Galois
H-objec ,
whe e
pAB
is
he
ac o iza ion
o
he
mo phism
aAB
o
iAB
(o
02
AB
o
iAB)
h ough
he
equalize
iAB
®
C
.
The
se o
isomo phism
classes
o Galois
H-objec s
(wi h
a
no mal
basis),
wi h
he
ope a ion
induced
by
he
one
gi en
abo e,
is
an
abelian
g oup
o
be
deno ed
by
Galc(H)(Nc(H))
.
The
uni
elemen
is
he
class
o
(II,
SC)
and
he
opposi e
o [(B,
PB)]
is
[(B
°P
,
(B
(9
A) o
PB)]
whe e
B'P
=
(B,
7B,
MB
o
7-B
)_
Rema k
.
In
he
case
o a
ini ely
gene a ed
p ojec i e,
commu-
a i e
and cocommu a i e
Hop
algeb a
H
o e
a
commu a i e
ing
R,
Galc(H)
is
he
g oup
o
Galois
H-objec s
in
he
sense
o
S
.
Chase
and
M
.
Sweedle
in
[9]
.
9
.
P oposi ion
.
[(B,PB)]
E
NC(H),
hen,
he e
is
a
2-cocycle
o,
in
Reg(C
®
C,
K)
sa is ying
a
o
( 7C
®
C)
=
EC
=
Q
o(C
®
?1C)
.
P oo
.
Le (B,
PB)
a
Galois
H-objec wi h
a
no mal
basis
.
Thenwe
ha e
an
isomo phism
-
YB
:
B®B
->
B®
C
and
an
H-comodule
isomo phism
:
C
---~
B
.
The e o e
he
mo phism
o
H-comodules
=
(EC
®
B)
o
(
-1
®
)
o
(?7B
(9
C)
:
C
->
B
is
in
Reg(C,
B)
wi h
in e so
.Í
-1
=MBo(B®EC®B)o(B®
-1
® )o(
-
yB
l
(S
?
7C)o(?7B®C)

276

J
.
N
.
ALONSO
ALVAREZ,
J
.
M
.
FERNÁNDEZ
VILABOA
and
sa is ying
o
IC
=
?7B
.
Indeed
:
* -1=MBo
(eco
B0
ECO
B)o( -1®B® -1® )o
°(?IB0-YB1®7Ic)o( 0C)-SC=
=MBo(Ec®B®ec0B)o( -1®B® -1® )o
o(B®[-YB1°'YB]®C)°(1]B®?]B0B®97c)o =
=
(ec
®B®
ec)
o
( -1
®
®
C)
°
([PB
°
nB]
®
C)
=
EC
0
77B
-1* =lLB°(B®[Eco -1]0
B)o(-yBl0 )0(?7B®6c)=
=MBo(B®[sco -1]® )o(B0PB)°-YB1°(17B®C)-
=MB°(B®Ec® )o(B06c)o(B® -1)°-YB10(77B®
C)=
=(B®ec)°-YB°7B10(?7B®C)=
=
EC
®
?IB
because
is
an
H-comodule
isomo phism and
he
equali ies
:
(ec
®
B)
o
( -1
®
) o
(?IB
®
?)c)
=
(Ec
0
B)
o
( -1
0
) 0
PB
°
71B
=
?7B
(yBloC)o(B06c)=(B®PB)0-YB1
T i ially,
is
a
mo phism
o
H-comodules
and
o
?1c
=
77B-
The
mo phism
Q
=
(P,B
o (
®
))
*
( -1
®
P,C)
:
C®
C
->
B
ac o s
h ough
he
equalize
K'IBB
~B®C
B077C
because
H
is
a
Hop
algeb a,
(B,
PB) an
H-comodule
monoid,
a
mo -
phism
o
H-comodules
and
he
equali y
PB
o
-1
=
(
-1
®
A)
o
- C
o
6C
([14,
(2
.3)])
.
Mo eo e ,
he
ac o iza ion,
á
,
o
u
is
in
Reg(C®C,
K)
wi h
in e se
he
ac o iza ion
o
he
mo phism
u
1
=
(
°P-C)
*
(PB
°TB
°
(
-1®
-1)
)
C
0C
-
B
h ough
he
equalize
?7B
.
NORMAL
BASIS
IN
CLOSED
CATEGORIES
AND
COHOMOLOGY

27
7
The
mo phism
Q
:
C
®
C
--->
K
is
a
2-cocycle
.
Indeed
:
77B
0
(191(5~ )
*
0307))
=
=( * -1)o(a ®Pc)o(c®TC0C)o(6c®6c)o
o(c®ñ~ 0PC)o(cocoTC
0c)o(c®6c®6c)=
=PBo(B® -1)°PBo o(& ®PC)o(c®TC®C)o(6c®6c)o
o(c(9
®Pc)o(cocoTC
0c)o(c®6c®6C)=
=P-Bo(B® -1)0PB0PB-(
® )o(c®Q 0PC)0
o(coC®TC0c)0(c®6c(9
6c)=
=P-Bo(B® -1)0PBOPB0(PB0C)o(
® (9
)=
=PBo(B® -1)°PBo o(15~ ®PC)o(c®TC®c)o(6c®6c)o
o(ú ®PC®C)o(C®TC®C®C)o(6c®6c®C)=
=( * -1)0(ú ®Pc)o(c®TC0C)0(6C06c)o
o(a 0PC0c)o(c®TC
0C0c)o(6c®6c(9
c)=
_
?7B
o
(a2(5~ )
*
a4(5~ ))
because
is
a
mo phism
o
H-comodules,
H
a
cocommu a i e
Hop
algeb a
and
he
equali y
o(a (9
Pc)o(c®TC
(9
C)o(6c(9
6C)=PB-( ® )
and
hen,
since
77B
is
a
monomo phism,
ú
is
a
2-cocycle
.
T i ially,
Q
o
(77C
®
C)
=
EC
=
o
(C
®
77C)
.
Rema k
.
I
[(B1,
PB,
)]
=
[(B2,
PB2)]
E
Nc(H)
hen
he e
is
an
iso-
mo phism
o
H-comodule
monoids h
:
B1
-
B2
.
Clea ly,
Q
,
=
Qho
,
.
(No ice
ha
h
o 1
E
Reg(C,
B2)
wi h
in e se
h o
i
1)
.
Mo eo e ,
a l
-
Z57 z
.
Indeed
:
The
mo phism
e
=
(ho l)* a
1
:
C
-->
B2
ac o s
h ough
he
equalize
?7B2
:
PB2Oe=(PBZ®PC)o»o l)®T
2(9
ín)o(6c® 21®C)o
o(C0(TCo6c))o6c=(B2® )C)0e
and
hen, he e
exis s
a
mo phism
é
:
C
->
K
such
ha
7B2
o
é
=
e
.
Clea ly,
é o
i1c
=
K
.
Mo eo e ,
é
is
in
Reg(C,
K)
wi h
in e se é-1,
he
ac o iza ion
o
e-1
=
2
*
(h
o
i
1)
h ough
he
equalize
17B2
.
278

J
.
N
.
ALONSO
ALVAREZ,
J
.
M
.
FERNÁNDEZ
VILABOA
We
also
ha e
ha
:
l
B
2
°
(
j
j
*
a2
(e))
_
-
Y"B2
°
( 1B2
®
7B2)
°
(-j b0
*
a2(e))
_
=
MB
2
°
(MB2
®
MB2)
°
[(h
°
i)
®
(h
° l)
®
((h °
l
1 )
*
(h
°
l))®
®
2
' ]
-
(
C
OCOSC)o(CoCopc)°(COT80C)°(5c0SC)=
_
MB2
°
(iíBZ
®
B2)
°
(PB2
®
(1IB2
°TBZ)
282)°
°((ho l)O(h° l)0( 2
1
* 2)0 2
1
O 20 2
1
)°
°(cOc0C0b
c
oli
c
)o(CococoTcoc)°
*(COCob
c
ob
c
)o(COTCOC)o(5C05
C
)=
-
Y"B2
°
(MBZ
®
B2)
°
((nB2
0
é)
®
(77B2
O
~)
®
( 7B2
°
Q 2))°
o(coTC0C)°(6C0bC)=
=
/BZ
°
(al
(e)
*
1
93
(e)
*
Z
7
2
)
and
hen, since
7B
2 is
a
monomo phism,
-5
-
-
5 2
10
.

P oposi ion
.
I
o,
is
a
2-cocycle
in
Reg(C
o
C,
K)
such
ha
o,
°
(71c
®
C)
=
ec
=
o
(C
®
7c),
hen
(C,
=
(C,
i7c,
wc
o
=
(o-
®
pc)
°
(C
®
Tc
®
C)
°
(S
c
®
bc))
;
Sc)
is
a
Galois
H-objec
wi h
a no mal
basis
.
P oo
.
T i ially,
(C
Q
,
Sc)
is
an
H-comodule
monoid
.
The
mo phism
,
yco
=
(loco
o
C)
o
(C
®
bC)
:
C
®
C
->
C®C
is
an
isomo phism
wi h
in e se
ycó
=
(l-zcoc)°(Cou-1OA®C)°(COPCOCOSC)°(scOaOCOC)o
o(Cob
c
oC)o(Co6C)
Indeed
:
lyc
o
°
^
Yco
=
=(wcoC)°(PC(S
o

-1
OaoC)0(C0C0PC0C0bc)o
o(CocoPC(S
aocoC)o(COTCocobcoC)°
°(u0b
c
0b
c
06
C
)o(COTC0C0C)°(6
C
OS
C
OC)o(CO6
C
)=
=(PC0C)o(PC0o
-1
OA0C)0(CoC0PCo5COC)o
°(COTC0[mc°(COA)°6
C
lo6
C
)0(COCOS
C
OC)o
"
(o-(S
s
c
0S
C
)0(COTc0C)0(6
C
0b
c
)=
NORMAL
BASIS
IN
CLOSED
CATEGORIES
AND
COHOMOLOGY

27
9
=(PC0C)o(PC0a
-1
®a®
c)o(C®TC®[TCo5
C
]®C)o
o(ao6C®Co6C)o(c®TC®6C)o(6
C
05
C
)=
=
(Pc®C)
o
(C®[PC
o
(COA)
o 6
C
]®C)o
o(u0C0C®o,
-1
®C)o(C®C0C®TC®C®C)o
o(c0C0bc05c0C)0(c07506
C
)o(6
C
05
C
)=
=(0'®c®u
-
1
®C)o(COCO
[TCosc](96
C
)o
0(c®TC®c)o(sc(9
sc)=
=([0'*0,-1]oCoc)o(C®Tc
(9
c)o(6
C
(9
5C)=
=C®c
¡YCo
o
i'Co
=
=
(01®PC®C)
o
(PC(9TC®C®C)
o
(C®C®PC®5C®C)o
o
(C07C®c®sc)o(scou-1®sc®C)o(C®MC®C®a®C)o
o(c®C®a®C®5
C
)o(5
C
®5
C
oC)o(co5
C
)=
(U®PC®C)
o
(C®TC®C®C)
o
(PC®PC®[TC
oa
C
]®C)o
o
(C
(9Tc
®C
®
S
c
)
o
(SC
®
Q
-1
®
[(A
®
. )
o
TC
oS
c
]
®
C)o
"
(co ic®C®C®
C)o(5
C0A®C(9
6
C
)o
"
(cobcoc)o(Cobc)=
=(0'®C(9
C)o(c®TC(SC)0(PC®Nic(9
6C)o
o(c®TC®[PCo(A(9
c)o6
C
]0
c)
0
(6C®0,-1®a(9
6C)o
o
(C®4
C
®C®6
C
)
o
(sC®a(gCOC)
o
(C®6
C
®C)
o
(C®6C)
_
=(o ®C0C)o(lic®Tc0C)o(C®Tco6C)o
o([TCo6
C]®01-
1®a®
C) 0
(C0
PC
®C®C(9
C)o
o(c(9
C®a®C®C®
C)0(6
C
0b
c
(9
C0C)o
o(c®[TCo6
C
]0
C)o(C05C)=
=(c®o,®C)o(c®C0U
-
'05
C
)0(c®[b
C
0P
C
]®CoC)o
o(c®C®a®C®
C)o(b
C
(9
sc0C)0(c0bc)=
=(c®o ®o
-
1(9
C)o(c®C(9
TC®CoC)o
o(c®s
c
®s
c
oC)o(c®N-C(9
6C)0(6C®a®C)0(c06C)=
=(c®[0'*u-1](9
C)o(C(SPC05C)o(5C®a®c)o(c06C)=
=C®c
and
hus
(C
o
,
S
c
)
is
a
Galois
H-objec wi h
a
no mal
basis
.
Rema k
.
I
o,
and
-y
a e
wo
2-cocycles
such
in
P oposi ion 10
and
cohomologous, hen
(é®C)
o5C
:
(C
o
,
SC)
->
(C,,,
S
c
)
is
an
isomo phism