Publicacions
Ma emá iques,
Vol
36
(1992),
407-420
.
A
bs ac
2
.
3
.
COHOMOLOGY,
SYMMETRY,
AND
PERFECTION
EMILI
BIFET
*
We
explain
he
philosophy
behind he compu a ions
in
[BDP]
and
place
hem
in
a
wide
concep ual
se ing
.
We
also
ou line,
o
o ic
a ie ies,
he
esul ing
equi a ian
app oach
o
some
key
esul s in
ha
heo y
.
1
.
Symme y
In
many
si ua ions
ha
a ise
in
Algeb aic
Geome y
one
is
in e es ed
in
compu ing
he
mul iplica i e
s uc u e
o
he
cohomology
ing
H*
(X),
wi h
a ional
coe icien s
say,
o
some
algeb aic
a ie y
X
.
Examples
o
such
si ua ions
include
o ic
a ie ies,
comple e
quad ics,
comple e
symme ic
a ie ies
[DP1-2],
.
.
.
Some imes,
as in
he
examples
jus
men ioned,
he
a ie y
X
is
endowed
wi h
symme ies
ha
e lec
he
ac ion o
some
algeb aic
g oup
G
on
i
.
In
hese
cases
he e
is
a
ecipe,
inspi ed
by
he
wo k
o
M
.
F
.
A iyah
and
R
.
Bo
[AB1],
ha
o en
wo ks
:
Find
a
s ongly
G-pe ec
decomposi ion
o
X
(c
.
Sec ion
3 below
o
a
p ecise
s a emen
.)
In
he
examples
abo e,
his
is
simply
he
decomposi ion
in o o bi s
.
In
gene al he e
is
a
na u al
candida e
:
he
Kemp -Hesselink
s a i ica ion
o
X
[H], [K],
[N], a
na u al
ou g ow h
o
D
.
Mum o d's
Geome ic
In a ian
Theo y
.
Wi h
he
help
o his
decomposi ion
compu e
he
equi a ian
co-
homology
ing
HG(X)
.
(A
his
poin
i
may
also
be
na u al
o
apply
he
machine y
o
he
localiza ion
heo em
[AB2],
[Hs]
;
in
so
doing
one
usually
ob ains o he
in e es ing
desc ip ions
o
Hc(X)
.
)
Reco e
H*
(X)
om
HG
(X)
.
*This pape
is
dedica ed
o
he
memo y
o
my
iend
Pe e
Menal
.
I
chose
he
p esen
opic
o
his
occasion
because
i
was
he
subjec
o
ou
las
con e sa ion
.
I
miss
him
e y
much
.
40
8

E
.
BIFET
This
ecipe
is
jus
one
mo e
ins ance
o
he
old
philosophy
o
using
any
symme ies
ha
may
be
p esen
in
he
p oblem
in
o de
o
simpli y
i
.
We
shall
wo k,
o
simplici y,
wi h
equi a ian
cohomology
de ined
in
e ms
o
homo opy
quo ien s
Le
.
he
Bo el
cons uc ion
(see
Sec ion
2
below
.)
Bu
he
knowledgeable
eade
could
subs i u e
h ougliou
HG(X)
by
he
smoo h
~-adic
cohomology
o
he
algeb aic s ack
de e -
mined
by
he
G- a ie y
X
.
He
would
hus gain
he
ad an age
o
ha ing
a
comple ely
algeb aic
heo y
wi h,
as a
bonus,
an
in e es ing
a i hme ic
wis
[B2]
.
In
he
examples men ioned
ea lie ,
he ecipe
wo ks and
gi es
e y
explici
esul s
.
We
shall
conside
below
in
some
de ail
he
case
o o ic
a ie ies,
bu he
eade
is
ad ised
o
look
a
[BDP]
o
a
ho ough
ea men
o
h ee
examples
om
he
p esen
poin
o
iew
.
In
ac
one
o
he
aims
o
his
pape
is
o
be e
explain
he
philosophy
behind
hose
compu a ions
and
o
place
hem
in
a
wide
concep ual
se ing
.
Ano he aim
o
he
pape
is
o
ou line
in
he
las
sec ion
an
"equi a i-
an "
app oach
o
some
key
esul s
in
he
heo y
o
o ic
a ie ies
.
This
app oach
cla i ies,
I
belie e,
he
na u e
o
h ee
esul s
.
The
ex
o
he
i s
h ee
sec ions ollows
closely a alk
deli e ed
a
he
Uni e si y
o
Copenhagen
in
July
1
.989
on
he
occasion
o
he
Zeu hen
Symposium
.
I
would
like
o
hank S
.
Kleiman
and
A
.
Tho up
o
o ganizing
ha
con e ence
and
c ea ing
a
e y
iendly
a hmosphe e
.
2
.
Cohomology
Suppose
ha
a
opological
g oup
G
ac s
p ope ly
on
a
space
X
.
The
(equi a ian )
cohomology
o
he
G-space
X
should
be
he
cohomology
o
he
quo ien
X/G
.
Un o una ely,
unless
G
is
ac ing
eely
on
X,
he
usual
opological
quo ien
does
no
p o ide
a
use ul
heo y
.
In
gene al,
i is
necessa y
o ind
he
igh
no ion
o
quo ien
.
The
bes
no ion
is
p obably
ob ained
by
aking
he
quo ien
in
he
2-ca ego y
o
oposes
[SGA4]
.
Howe e ,
o
simplici y,
we
shall
wo k
he e
wi h
a
homo opy
quo ien
XG
.
(
In
a
pu ely
algeb aic
con ex
he
ole
o
XG
would
be
played
by
he
algeb aic
s ack
de e mined
by
he
G- a ie y
X
;
he
equi a ian
cohomology would
jus
be
he
smoo h
~-adic
cohomology
o
his
s ack
.
)
Be o e
we
can
desc ibe
his
no ion
o
quo ien ,
howe e ,
i
is
necessa y
o
look
closely a
he
case
whe e
X
is
a
poin
Le
.
he
heo y
o
cha ac e is ic
classes
.
Recall
ha
a classi ying
space
o
p incipal
G-bundles
is
by
de ini ion
a
space
BG
oge he
wi h
a
uni e sal p incipal
G-bundle
EG
o e
i
.
By
uni e sal
we
mean
ha
isomo phism
classes
o
p incipal
G- ib e
bundles
COHOMOLOGY,
SYMMETRY,
AND
PERFECTION

409
o e
a
nice
space
X
co espond
na u ally
o
homo opy
classes
o
maps
om
X
o
BG
.
The
co espondence
is
gi en
by
pulling-back
he
uni e sal
bundle
EG
.
The
cohomology
ing
H*
(BG)
is
by
de ini ion
he
ing
o
cha ac e is ic
classes
o
G
.
Example
1
.
G=
C'
.
This
is
simply
he
heo y
o
line
bundles,
and
he
classi ying
space
is
he
in ini e
p ojec i e
space
PC
.
I s
cohomology
ing
H*(BG)
=
Z[cl]
is
a
polynomial
ing
in
one
a iable
o
deg ee
wo
.
Example
2
.
G=T
= C
lx
. . .
x
C"
(an
algeb aic
o us
.)
In
his
case
we
ha e
:
BT-
BC
x
x
. . .
x
BC
x
and
i s
cohomology
is
a
polynomial
algeb a
in
se e al
a iables
(as
many
as ac o s
.)
In
ac ,
i
X(T)
deno es
he
g oup
o
algeb aic
cha ac e s
o
T,
hen
H*(BT)
-
Sym*X(T)
.
In
pa icula
H
2
(BT)
-
X
(T)
.
Example
3
.
G
=
GL

(C)
.
The
classi ying
space
is
he
in ini e
di-
mensional
G assmannian
and
H*
(BG)
=
Z[cl,
.
.
. ,
c,,]
whe e
he
a iables
e2,
1 <_ i <_
n,
ha e
deg ee
2i
and
co espond
o
he
Che n
classes
.
In
gene al,
i
T
is
a
maximal
o us
in
G,
we
ha e
(
wi h
a ional
coe icien s
)
H*
(BG)
=H*(BT
)
W
whe e
W
=
NG(T)/T
is
he
Weyl
g oup
o
(G,
T)
and
he
igh
hand
side
deno es
he
sub ing
o
in a ian s
.
We
a e
now
eady
o
desc ibe
he
homo opy
quo ien
men ioned
ea -
lie
.
This
is
gi en
by
he
Bo el
cons uc ion
XG
ob ained
a e
exchang-
ing he
ib e
G
o
he
uni e sal
bundle
EG
wi h
X
Le
.
XG=EGxGX=(EGxX)/G
whe e
G
ac s
by
g
-
(e,
x)
=
(eg
-1
,
gx)
.
De ini ion
.
The
equi a ian
cohomology
HG(X)
is
by
de ini ion
he
cohomology
o
he
Bo el
cons uc ion
XG
.
No e
ha
he e
is
a
ib a ion
(2
.1)

X
-+
XG
-->
BG
.
410

E
.
BIFET
The
spec al
sequence
o
his
ib a ion
is
he
key
e
he
heed
s ep
in
he
ecipe
.
This
is
based
on
wo k
o
P
.
Deligne
[De],
V
.
A
.
Ginzbu g
[G],
F
.
Ki wan
[K],
. . .
He e
ollow
some
o he
p ope ies
o
equi a ian
cohomology
wi h
a-
ional
coe icien s
[Hs],
[AB2]
:
a)
I
G
ac s
eely
en
X,
hen
HG(X)
=
H*(XIG)
.
b)
I
T
is
a
maximal
o us
in
G
and
W
=
NG(T)/T
is
he
Weyl
g oup,
hen
HG(X)
=
HT(X)w
whe e
he
igh
hand
side
is
he
sub ing
o
W-in a ian s
.
c) I
X
has
a
single
o bi ,
hen
HG(X)
-
H*
(B
H)
whe e
H
is
he
s abilize
o
any
poin
.
d)
I
K
is
a
maximal
compac
subg oup
o
G,
hen
Hc(X)
=
Hix(X)
.
One
o
he
easons
equi a ian
cohomology
is
easie o
compu e
han
o dina y
cohomology
is
ha
i
has
many
mo e
"poin s"
.
Le
me
y
o
explain
his
.
Mos
succes ul
calcula ions
o
cohomology
achie e
hei
objec i e
by
exp essing
he
cohomology
o
he
space
unde
conside a ion
(e .g
.
p ojec i e
space
P')
in
e ms
o
ha
o
spaces
o
which
i is
al eady
known
(e .g
.
cells
.)
Ul ima ely,
howe e ,
hey
educe
he
compu a ion
e
ha
o
he
cohomology
o a
poin
.
I
one
hinks o o dina y
cohomology
as
being
he
case
G=
1
o
he
equi a ian
one,
hen
i is
clea
ha
he
poin s
coincide
wi h
he
o bi s
.
Thus
in
he
equi a ian
heo y
e e y
o bi
gi es
ise
o a
"poin ",
and
he e
a e
as
many
poin s
as
he e
a e
conjugacy
classes
o
subg oups
in
G
.
The
equi a ian
cohomology
o
such
a
poin
H
is
p ecisely
he
ing
o
H-cha ac e is ic
classes
Le
.
he
cohomology
o
he
classi ying
space
o
H
.
I
ollows
ha
in
he
equi a ian
heo y
he e
is
much
mo e
eedom
o
mo emen
.
Ano he
impo an
ea u e
o
equi a ian
cohomology
is
ha
he e
is
a
heo y
o
equi a ian
Che n
classes
.
A
G-linea iza ion
o
a
ec o
bundle
F
o e
X
is
an
ac ion
u
:
G
x
F
->
F
which
is
linea
en
he
ib es
and
u ns
he
p ojec ion
7
:
F
--->
X
in o
a
G-equi a ian
map
Le
.
7
(g
-
x)
=
g
-
7 (x)
o
e e y
g
E
G
and
e e y
xE
F
.
No e
ha
he
homo opy
quo ien
FG
p o ides
us
wi h
a
ec o
bundle
o e
XG
.
The
equi a ian
Che n
classes o (F,
u)
a e
by
de ini ion
he
Che n
classes
o
FG
.
This
akes
a mos
simple
o m
o
a
line
bundle
o e
an
o bi
.
In
his
case
he
equi a ian
Che n
class
c(L, u)
is
de e mined
by
he
(3
.2)
COHOMOLOGY,
SYMMETRY,AND
PERFECTION

411
iso opy
ac ion
(cha ac e )
o
he
s abilize
on
he
ib e
o
L
o e
he
poin
.
Ac ually
hese no ions
ind hei
mos
na u al
o mula ion
when
exp essed
in
e ms
o
algeb aic
s acks
.
Fo
example a
G-linea ized
Gx-
module
is
simply
a
module
o
he s uc u e
shea
o
he
algeb aic
s ack
de e mined
by
he
G- a ie y
X
.
3
.
Pe ec ion
Le
X
be a
smoo h
complex
algeb aic
a ie y,
and
le
he
algeb aic
g oup
G
ac
on
X
.
Suppose
S
C
X
is
a
closed
G-in a ian
smoo h
sub a ie y
and
le
U
=
X
-S
be
he
complemen a y
open
se
.
Unde
hese
condi ions,
he e
is
a
long
exac
sequence
( he
equi a ian
Thom-
Gysin
sequence,
see
[AB1]
o
example
)
. . .

HG
2codimS(S)
is
HG(X)
-
HG(U)

. . .
Mo eo e
he
composi e
o
he
maps
hen,
o
example,
we
ha e
HG
2codimS(S)
HG(X)
1
es ic ion
Hc(S)
is
mul iplica ion
by
he
Eule
class
e(NSIx)
(=
op
Che n
class in
his
con ex )
o
he
no mal
bundle
N
s
1
x
.
I
his
long
exac
sequence
spli s
in o
sho
exac
sequences
0
->
HG
2codimS(S)
_,
Hc(X)
_
HG(U)
->
0
bG(X)
=
bG(U)
+
bG
2codimS(S)
and
one can
deduce
he
equi a ian
Be i
numbe s
o
X
om
hose
o
S
and
U
.
In
[AB1]
A iyah
and
Bo
made
he ollowing
undamen al
obse a-
ion
:
I
e(NS
I
x
)
is
no
a
ze o-di iso
in
he
ing
H*
(S),
hen
he
mo phisms
s
a e injec i e
and
he
long
exac
sequences
(3
.1)
spli
in o
sho
exac
sequences
(3
.2)
.
This
mo i a es
:

412

E
.
BIFET
De ini ion
.
We
say
ha
a
decomposi ion
X=
Si
U
. .
.USN
is
s ongly
G-pe ec
i
:
1)
Each
S
i
is
bo h
smoo h
and
G-in a ian
.
2)
Fo
each
k,
Xk
=S1
U
.
.
.
USk
is
an open
subse
o
X
.
3)
Fo
each
pai
(Sk,
Xk),
k
>
1,
he
Eule
class
e(NS,_IX,)
is
a
non-ze o
di iso
.
In
[AB1]
a
decomposi ion
is
de ined
o be
G-pe ec
i
he
long
ex-
ac
sequence
de e mined
by
each
pai
(Sk,Xk)
spli s
in o
sho
exac
sequences
.
In
his
case
one
has
an
iden i y
o
equi a ian
Poinca é
se ies
I
is
clea
ha s ongly
G-pe ec
implies
G-pe ec
.
An
immedia e
consequence
o
he
de ini ion
is
P oposi ion
.
I
{S2}1<á<N
is
a
s ongly
G-pe ec
decomposi ion o
X,
hen
o
e e y
pa ial
union
Xk
he
mo phism
induced
by he
es ic-
ions
o
he s a a
(3
.3)

HG(Xk)
-
H
Hc(SZ)
1<¡<k
is
injec i e
.
PC(x)
_

2
.codimSi
.
PG(S2)
1<i<N
P oo
..
Fo
k
=
1,
i is
ob ious
.
Suppose
i
holds
o
k
-
1
;
i
su lces
o
show
ha
he
mo phsm
a e
injec i e
.
Conside
he
diag am
Hi-2codimSk
(sk)
G
HG(Xk)
-
HG(Xk-1)
x
H
*
(Sk)
HG(Xk-1
U
sk)

HG(Xk-1)
HG(Sk)
-
P oo
.
Since
HT(P)
.
Bu ,
i
COHOMOLOGY,
SYMMETRY,
AND
PERFECTION

41 3
Now,
i
71(a)
=
0,
hen
he e
is
a
b
E
HG
2codiMSk
(Sk)
such
ha
a
=
~(b)
.
Bu ,
om
0
=
(a)
=
(«b))
=
b
U
e(NS,
1x
k
)
i
ollows
ha
b
=
0
and
he e o e
a
=
«
b)
=
0
.
a
Thus,
in p inciple,
i
ose
knows
he
cohomology
ings
o
he s a a
and
one
con ols
he
injec ion
abo e,
i is
possible
o
desc ibe
he
cohomology
ing
o
X
.
This
is
he
eason
we
singled
ou
his
no ion
o
special
conside a ion
.
A iyah
and
Bo
also
gi e
an
in ini esimal
c i e ios
o
e(NSIx)
o
be a
non-ze o
di iso
(see
[AB1
P oposi ion
13
.4
.])
I
is
p o ed
in
[K]
using
his c i e ios
ha
he
Kemp -Hesselink
s a i ica ion
[H]
o
a
G- a ie y
is
s ongly
G-pe ec
in
he
abo e
sense
.
Le
us
enuncia e
his
las
c i e ios
in
he
case
o
an
o bi
:
P oposi ion
.
Le
C
be
a
G-o bi
in he
smoo h
algeb aic
a ie y
X
.
Choose a
poin
P
E
C
and
iden i y
C
wi h
G/Gp,
whe e
Gp
is
he
s abilize
o
P
.
I he
iso opie
ac ion
o
a
maximal
o us
T
in
Gp
on
he
no mal
space
NOIx(P)
has
no non-ze o
ixed poin s,
hen
e(NoIx)
is
a
non-ze o
di iso
.
Hc(0)
=
Hcp
(P)
-
HT(P)
is
an
embedding,
i
su ices
o
see
ha
e
=
e(N(DIx)
is
non-ze o
in
NoIx(P)
=

®

Cxi
X¡EX(T)
is
he
weigh
decomposi ion,
hen
e=l1xi
:7É
o
.
Since
all
xz
a e
non-ze o
.
The
conside a ions
abo e
mo i a e
:
De ini ion
.
We
say
ha
X
is
a
pe ec
embedding
( egula
embedding
in
[BDP],
bu
his
was a
bad
choice
o
which
I
plead
guil y)
p o ided
a)
Each
o bi
closu e
C
is
smoo h
and
i
is
he
ans e sal
in e sec-
ion
o
he
codimension
one
o bi
closu es
ha
con ain
i
.
b)
Fo
e e y
P
E
0,
he
s abilize
Gp
has
a
dense
o bi
in
he
no mal
space
NOIX(P)
.
To
any
pe ec
embedding
X
we
associa e
a
simplicial
complex
Cx
=
(V,
S)
as
ollows
:
41
4

E
.
BIFET
1
.
V
=
{
1
O
is
an
o bi o
codimension
one}
.
2
.
I'
C
V
is
a
simplex
i ,
and
only
i ,
E
(No e
ha
?
is
a
simplex
.)
O
7
~
?
.
I
is
easy
o
show
ha
he
simplexes
a e
in
one- o-one
co espondence
wi h
he
o bi s
.
I
is
clea
ha
he
decomposi ion
o
X
in o o bi s
is
in his
case
s ongly
G-pe ec
.
The
algeb aic
a ie ies
men ioned
abo e,
namely
o ic
a ie ies
and
comple e
symme ic
a ie ies
(in
pa icula
comple e
quad ics),
p o ide
examples
o
pe ec
embeddings
.
In
[BDP]
an
explici
desc ip ion,
based
on
hese
ideas,
is
gi en
o
he
equi a ian
cohomol-
ogy
ing
o
any
pe ec
embedding
.
I
should
be
possible
o
ex end
hese
esul s
o
he
case o
a
well
beha ed
G- a ie y
and
he
Kemp -Hesselink
s a i ica ion
.
4
.
An
example
:
o ic a ie ies
We
shall
illus a e
he
gene ali ies o
he
p eceding
sec ions
wi h
he
conc e e case
o
o ic
a ie ies
.
Le
T
be
an
algeb aic
o us
.
A
o ic
a ie y
is
a no mal
algeb aic
T- a ie y
X
con aining
T
as
a
dense
o bi
( his
includes
he
equi emen
ha
he
s abilize
a
he
poin s
o
his
o bi
be
i ial
.)
Re e en es
[D],
[F],
[O]
p o ide
e y
good
exposi ions
o
he
basic
heo y o
hese
a ie ies
.
A
simple
example
is
he
a ine
plane
wi h
he ac ion
o
T=
C"
x
C
X
gi en
by
( 1, 2)(xl>x2)
=
( 1X1, 2x2)
This
ac ion
has
ou
o bi s,
namely
he
o igin,
he
wo
punc u ed
axes
and
he
o us
T
i sel
.
This
can
be
gene alized
o
he ac ion
-
(x1,
x2)
=
( x1
x
l,
x2
x2)
de e mined
by any
in eg al
basis
{X1,
X2}
o
he
cha ac e
g oup
X
(T)
-
Z
2
.
This
ac ion
induces
an
ac ion
on
he
ing
o
egula
unc ions
on
he
a ine
plane,
he
ing o
polynomials
in
wo
a iables,
gi en
by
(
)
(x)
=
(
-1
x)
.
The
weigh
unc ions (Le
.
he
unc ions
such
ha
=
X
o
some
cha ac e
X
called
he
weigh
o )
a e
p ecisely
he
monomials
a-Xi
1
X2
2
and
hei
weigh
is
n1(
-
X1)+
n2
(
-
X2)
.
These
weigh s
span
a
COHOMOLOGY,
SYMMETRY,
AND
PERFECTION

41
5
cope
a
in
X(T)
®R
.
Con e sely
we
can
eco e
he
a ie y,
including he
T-ac ion,
by
aking
he
Spec um
o
he
monoid
algeb a
C[a
nX(T)]
o ,
wha
is
essen ially
he
same,
he
algeb a
homomo phisms
om
C[a
n
X(T)]
o
C
.
In
gene al,
a ine
o ic
a ie ies
can
be cons uc ed
as
ollows
.
We
deno e
Y(T)
=
Hom(C",
T)
he
g oup
o
one-pa ame e
subg oups
o
he algeb aic
o us
T
.
(Recall
ha
he e
is
a
pai ing
X(T)
x
Y(T)
-~
Z
gi en
by
aking
(X,
w) o
be
he
unique
in ege
such
ha
X(M( ))
=
(X,w)
o
all
E
C
X
.)
Fi s
we
conside
a
cone
a
=
{Al il
+
. . .
+

1
A2
E
R,
Ai
>
0 o
e e y
i}
in
Y(T)R=
Y(T)
®Z
R
whe e
{mi,
.
.
.
,m

,,}
a e
ini ely
many
one-
pa ame e subg oups
(ac ually
we
also
ask
ha
he
cone
o
.
ha e
a
e ex
Le
.
a
n
(-a)
=
0
.)
Nex
we
in oduce
i s
dual
in
X(T)R=
X(T)
OZR
gi en
by
a
=
{X E
X(T)R
1
(X,
)
>
0 o
e e y
mEa}
.
Then
we
exp ess
he
monoid
a
n
X(T)
in
e ms
o a
ini e
numbe
o
gene a o s
(4
.1)
o, nX(T)=N-XI+
. .
.+N-XN
( ha
his
can always
be done
is
a
consequence
o
Go dan's
lemma
[D],
[F],
[O]
.)
Finally
we
cons uc
he
a ine
model
X
Q
by
aking,
as
in
he
case
o
he
a ine
plane,
he
Spec um
o
he
monoid
algeb a
o
equi alen ly
he
(scheme
heo e ic) closu e
o
he
image
o
he
map
T
-+
CN
sending
o
( Xl
,
. . . ,
XN
)
.
A
o ic
a ie y
is
ob ained
by
glueing
oge he
he
a ine
models abo e
along
T-in a ian
open
subse s
.
Think
o
example
o
he p ojec i e
plane ob ained
by
glueing he e
copies
o
he
a ine
plane
along
he
open
o bi
and
iden i ying
he
punc u ed
axes
in pai s
.
O
cou se,
he e
we
a e
glueing
no
only
he
spaces
bu
also
he
T-ac ions
so
he
ac ions
on
ou
he e
a ine
planes
ha e
o
be
compa ible
.
Fo una ely
he
cones
in o-
duced
ea lie
allow
his
compa ibili y
o be
exp essed
in
simple
e ms
.
De ini ion
.
Le
T
be an
algeb aic
o us
.
A
collec ion
E
o
cones
as
abo e
in
Y(T)R
is
said o
be
a
an
whene e
i
sa is ies
he
ollowing
p ope ies
a)
E e y
ace o
a
cone
a
in
E
also
belongs
o
E
.
b)
The
in e sec ion
o
any wo
cones
in
E
is
a ace
o
bo h
o
hem
.