Pub
.
Ma
.
UAB
Vol
.
26
N4
3
Des
.
1982
POLYNOMIAL
AND
RELATEDALGEBRAS
AS
COHOMOLOGYRINGS
(REPORT
ON
RECENT
PROGRESS)
Le
p
be
a
p ime
and
La y
Smi h
§
1
.
Rings
o
In a ian s
as
Algeb aso e
he
S een od
Algeb a
P*
=
pIx1,
.
.
.,xn!
a
polynomial
algeb a
o e 2Z/p
.
A ques ion
o
basic
impo ance
is o
decidei P*
can
occu as
he2Z/p
cohomology
o
a
opological
space
.
O
cou sea
necessá y
condi ion
o
P*
o be
a
coho-
mology
ing
is
ha
i
be
an
uns able
algeb a
o e
he
S een od
algeb a
( o
p
í
2
he
gene a o s
x
j
all
ha e
e endeg ee,
so
he
Bocks ein
is
iden ically
ze o
and
onlyV*,
he
algeb a
o
S een od
educed
powe s
is
ele an )
.
By
assuming
hisex a
s uc u e
we
can
hen
y
o
ei he
cons uc
a
space
X
wi h
P*
-
H*(X
;
a/p),
o
y
o
use
highe
o de
cohomology
ope a ions,
and
o ,
ope a ions
in
ex ao dina y
cohomo-
logy o
p o e
no such
space
X
can
exis
.
Hindsigh
now
shows
ha
in
ac
ano he
app oach,
using ideas
om
Galois
heo y,
and in a ian
heo y,
p o ides
a
comple e
answe
o
he
ealiza ion
p oblem
o
non-
modula
polynomial
algeb as
Ole
say
ha
is
non-modula
i
This
hindsigh
sugges s
a
na u al
di ision
o
he
ealiza ionp oblem
;
namel
.
i s
cons uc
a
class
o
examples
o
uns able
polynomial
algeb as
o e
he
mod
p
S een od
algeb a,
and
hen
wo y
abou
which
o
.
hese
can
occu
as
cohomology
ings
.
One
elegan
way
o
cons uc ing
uns able
algeb as
o e
he
S een od
algeb a
is
p o ided
by
in a ian
heo y,
and
was
exploi ed
o good
ad an age
by
Cla k'and
Ewing
[6]
.
We
s a
wi h
G
p
P*,
P[x
1
,...
,xnl
deg
xi
jí
0
mod
2p
:
i
=
1,
.
. .
.,n
.)
a
ini e
g oup
G -+
GL(n
;
ZZ/p)

a
ai h ull
ep esen a ion
.
Le
V
:=
Vp e¿
7
7Z/p

be
he
ep esen a ion
space
n
o
p
and
o m
P
k
=
and
O
=
0
P(V)
=
P[V*]
he
g aded
polynomial
algeb a
on
he
dual
ec j
space
V*
o
V,
whe e
he
g ading
esul s
om
he
equi emen
:
deg
=
2
.
b
E
V
.
The ac ion
o
G
can
be
ex ended
o
P(V)
in
he
ob ious
way and
so
we
can
o m
he
ing
o
in a ian s
H*
:=
P(V)
G = {
E
P(V)Ig
=
dg
E
G}
.
(The
s udy
o
ings
o
in a ian s
wasin
ac an
im-
po an
local
indus y
in
U ingen
a ound
he
u n
o
he
cen u y,
so
i is
only
na u al
ha
I
should
spend
a
ce ain
app en iceship
in
his
a ea
.)
The
S een od
algeb a
ac s
on
P(V)
in
a
unique
way
compa able
wi h
he
Ca an
o mula
and
he
uns abili y
condi ion,
namely,
ia
he
condi ion

k
= 0
p
:
k
=
1
0
o he wise
:p
í
42
o
:
k
=
0
Sq
k
=

2
:
k=2

p=2
0

,

o he wise
(in
pa icula
O =Sq
1
=0)
Mo eo e
since
G
is
ac iog
by
linea
ans o ma ions
on
V
and
aising
o
he
p
h
powe
is
linea
in
cha ac e is ic
p,
i
ollows
ha
he
ac ion
o
G
commu es
wi h
he
ac ion
o
he
S een od
algeb a,
and
hence
H*
=
P(V)
G
inhe i s
omP(V)
he
s uc u e
o
an
uns able
algeb a
o e
he
S een od
algeb a
.
Example
-
i-

D#(n)
_=
P(Y)GL(V)
This
algeb a
was
o iginally
s udied
by
Dickson
who
showed
D
*
(n)
='
P[Y1,
. .
.,yn]
deg
yi

=
2(p
n
- p
n-i
)

1,
. .
.,n
La e
onwe
will
see
ha
D*(n),
which
we
e e
o
as
he
Dickson
algeb a,
plays
a c ucial
ole
in
he
classi ica ion
o
uns able
polynomial
algeb as
o e
he
S een od
algeb a
.
Fo nowle
me jus
men ion
ha
he
'
ac ion
o
he
S een od
algeb a
on D*(n)
is
comple ely
de e mined
by
he
o mulae
[17]
:
p
j
P
yk
wi h
an
analogous
o mula
o
p=2,
and
he
ac ha
he
Pp
J
gene a e
Example
2 (S een od-Wilke son)
:
The
polynomial
algeb a
in
ques ion
is A*
:=
P[x4,x2p+2]
whe e
j-k
=
n-1
j
=
n-1
)

p
>
2
o he wise
is
he
c ucial
o mula
.
In
[18]
S een od
e i ied
by
edious
calcula ion
ha
A*
admi s
an
uns able

*-
algeb a
s uc u e
.
(N
.
B
.
When
p =
3
A*=
H*(BSp(2)
;
7Z/3)
.)
In
[20]
Wilke sonobse ed
ha
whe e
2
is
he
subg oup
gene a ed
by (N
.
B
.

2l
p :=
p-adic
in ege s)
A
=
A*
-
P(V)
G
:
dim
ZZ/p
V
G
"
GL(n
;
2Z
p
)

-1
,
0
B
=
e+e
-1
,

1
p+1
e
+
e
-1
E
7L
p
)
and he
ac ion
o
G
on
V
is
ia
mod
p
educ ion
om
7L
p
.
(The
g oup
G
comes
om
he
Shepa d
and
Toddlis
[11]
.)
2
i
whe e
e
=
exp{-1
(N
.
B
.
one
needs
o check ha
Ou
p ima y
in e es
in
in oducing
his
cons uc ion
is
howe e
o
cons uc a
la geclass
o
uns able
polynomial
algeb as
o e
he
S een od
alge i a
.
Polynomial
ings
o
in a ian s,
howe e
in
cha ac e is ic
ze o=we e
long
known
o
a ise
om
he
canonical
ep esen a ion
o
he
We l
g oupo
a
compac
con_nec ed
Lie
g oup
on
he
uni e sal
co e ing
space
o
amaximal
o us
.
These
ep esen a ions
a egene a ed
by
eal
e lec ions
.
In
[11]
Shepa d
and
Todd
in oduced
a
complex
analog
o
e lec ions,
classi ied
all he
ini e
g oups
ha admi
complex
e lec ion ep esen a ions
and
showed
by
explici
calcula ion
ha
he
esul ing
ings
o
in a ian s
we e
polynomial
algeb as
(o e
T!)
Ewing
and
Cla k
exploi ed
he
wo ko
Shepa d
and
Todd
by
ca ying
he
Shepa d
and
Todd
classi ica ion
kicking
and
sc eaming
down
o
cha ac e is icp
.
To be mo e
'speci ic
one
in oduces
a
cha ac e is ic
ee
de ini ion
o
complex
e lec ions,
namely
"
De ini
ion
:
An
au omo phism
is
calleda
ps_eudó
e l
F
,
ec ion
i
1-p
has
ank
one
.
The
mo i a ion
o
his
is
clea
.
I
you
a e
going
o
ha e
a
e lec ion
ac oss
a
complex
hype plane,
. hen
he
o hogonalcomplemen
o
he
hype plane
is
a
complex
line
=
eal
2-plane,
so
we
can
alsomake
a
" unny
house
mi o "
by also
o a ing
he
imagein
he
o hogonalcomplemen
o
he
mi o
.
One
hen
p o es
[4]
[5]
.
Theo em
(Che alley-Bou baki)
:
Le
p
:
G->
GL(V)be
a ini e
dimensional
ep esen a ion
o
he
ini e
g oup
G
which
is
gene a ed
by
pseudo
e lec ions
.
I
IGI Y
0
mod
p,
whe e
p
is
he
cha ac e is ic
o
he
g ound
ield,
hen
whe e
:
i
hen
p
:
V
- V
p(V)
G
='

p[x
19
. .
.,x
n
]
deg
x
i
=

2d
i

i

=
1,
.
.
.,n,
IGI .=
-
d
1
.
.
.
dn
Thus in
he
non-modula
si ua ion
one
can
cons uc
lo s
o
example
o
uns able
polynomial
algeb as
o e
he
S een od
algeb a
.
By
u elizing
hei
mod
p
educ ion
o
Shepa d
and
Todd,
Cla k
and
Ewing
can
p o ide
he
ollowing
comple e
lis o
i educiable
examples
whe e
m
>
1
and
m
=
q
.
Numbe 1
1
Rank
n
I
O de
(n
+
1)!
Type
[4,6,
..
.,2(n
+
l)]
P imes
pl(n+1)!
2ak
n
q
,
m"_'n!
[2m,4m,
...
-
,
2(n
1)m,
2qn]
p;n!,p=1
mod
in
2b
2
2m
[4,2m]
m>2,p-=1modm
31
m
[2m)
p=_lmodm
4 2
24
[8,12]
p-1mod3
5 2
72
[12,24]
p-1mod3
6 2
48
[8,24]
p-1mod12
7 2
144
[24,24]
p-1mod12
8 2
96
[16,24]
p-1mod4
9 2
192
116,48]
p=_
1mod9
10
2
288
[24,48]
p=_1mod12
11
2
576
[48,481
p-1mod24
12
2
48
112,161
p-1,3mod8,
p
:P
3
13
2
96
[16,241
p=1mod8
14
2
144
[12,48]
p-1,19mod24
15
2
288
(24,481
p-1mod24
16
2
600
[40,60]
p-1mod5
17
2
1200
[40,120]
p=1mod20
18
2
1800
[60,120]
p-1mod15
19
.
2
3600
[120,120]
p-1mod60
2
360
[24,601
p
=_ 1,
4
mod
15
21
2
720
[24,1201
p-
1,
49
mod
60
22
2
240
124,40]
p=_1,9mod20
23
3
120
(4,12,201
p-1,4mod5
24
3
336
[8,12,281
p-
1,2,4mod7
25
648
[12,18,241
p=
--
1mod3
26
3
1296
[12,24,36]
p-=lmod3
27
3
2160
[12,24,60]
p-1,4mod15
28
4
1152
[4,12,16,241
p
2o 3
29
4
7680
(8,16,24,40)
p-lmod4,
po5
30
4
14,400 [4,24,40,601
p=1,4mod5
31
4
64
"
6!
[16,24,40,481
p_lmod4,
p-_5
32
4
2166!
[24,36,48,60]
p=_1mod3
33
5
72
.
6!
[8,12,20,24,36]
p-1mod3
34
6
1089!
[12,24,36,48,60,84)
p-lmod3,
p#7
35
6
72
.6!
[4,10,12,16,18,24]
p#2,3,
o 5
36
7 8
.9!
[4,12,16,20,24,28,361
p
,-2,3,5, o
7
37
8 19210!
14,16,24,28,36,40,48,601
p
¢
2,3,5, o
7
I
one
d ops
he
non-modula
es ic ion,
iz
.
DGI
¢
0,
hen
simple
examples,e
.g
.
E
P[Q1,
..
.,an]
=
p(V)
n
dim
V
=
n,

E
n
symme ic
g oup
show ha
he e
a e
ings
o in a ian s
ha
a e
poly-
nomial
.
In
ac in
he
s ic lymodula
case,
namely
when
G
is
a
p-g oup,
he e
is
a
cha ac e iza ion
o
he
g oups
and
ep esen a ions
[10]
o
which
P(V)
G
is
polynomial
.
Fo
example
onehas
long
known
:
E
xampl
e3
:

Le
UP(V)
=
be
he
subg oup
o
uppe
iangula
ma ices
.
Then
P(V)
U
P(
V
)

=

P[z
1
,-
,z
n
]
deg
.
.
=
2i(p
;
.n)
Fo
example
when
n
= 2
we
iad
E
GL(V)
P2
=
2
-
2-1 1
169
In
any
case
i
we
s a
wi h
:
hen
H*
:=
P(V)
G
is
a
polynomial
algeb a
o e
he
S een odalgeb a
o
be
ound
in
he
lis
compiled
by
Cla k
and
Ewing
and
mo eo e
he e
is
a
space
X
such ha
H*(X
;7Z/p)
=
H*
.
Le
me
summa ize
he
p eceeding
discussion
in
he
ollowing
esul
:
Theo em
2
(Cla k-Ewing)
:

Le
p
:
G
-->
GL(n
;71/p)
be
gene a ed
by
pseudo
e lec ions
and
assume
(NMC)

p
+
IGI
Th
en
p
:
G
y
GL(n
;
7Z/p)
Algeb aic
Pa
:

The e
is
a
ep esen a ion
G
GL(n
;C)
o
G
as
a complex
pseudo
e lec ion
g oup,
such ha
he
polynomial
algeb as
P(O
e)
G

P(©
2Z/P)
G
n

n
ha e
he
same
ype
.

Topological
Pa
:

The e
exis a
a
space
X(V
;G)
such
ha
H*(X(V
;G))
=
P(V)
G
;
V
=O
7Z/p
he
ep esen-
n
a ion
spaceo
p
.
Rema ks
:
(1)

The
exis ence
o
a complex"li ing"
o
a
gi en
mod
p
ep esen a ion
can
be
explained
as
ollows
.
We
ha e
al eady
seen ha
(NMC)
allowsus
o
cons uc
a p-adicli ing
G<
GL(n
;7l
p
) .
Bu
G
being
a
ini e
g oupmeans
ha his
ep esen a ion
is
al eady
de ined
in
a
ini e
ex ension
o
Q
(simplyadjoin
enough oo s
o
uni y),
and
hence
o e
0
.
(2)

In
addi ion
Cla k
and
Ewing
de e mine
he
cha ac e
ields(N
.B
.
Since
hey
p o e
ha
he
Schu
index
is
always
1
i
doesn'
ma e
which
de ini ion
o
"cha ac e
ield"
one
is using)o
he
complex
hype plane
g oups
in
he
Shepa d-Todd
lis
.
Thus
s a ing
om
a
complex
hype plane
g oup
one
can
ead
o
o e
which
ini e
ields
7Z/p
i
admi a(pseudo
e lec ion)
ep esen a ions
.
Along
wi h
he
ques ion
o
ealizing
P(V)
G
as
a
cohomo-
logy
ing,
we
should
alsolooka
he
homo opyclassi-
ica ion
o mapa
be ween
suchspaces
.
The
cons uc ion
o e ed
by
Cla k
and
Elaing
deli e s
a
a he
explici
space
X(V
;G)
and
onecan
p o e
:
(see
[14])
(2)
he
S een odalgeb aac ion
on A* li s
o
an
uns able
ac ion
on P[xi,
.
.
.,xn],
and
(3)
P[y1,
. .
.,yn]
is closed
unde
his
li edac ion
.
Then
he e
exis s
a
opological
space
A
such
ha
H*(A
;
ZZ/p)
_
A*
The
cons uc ion
o
Cla k
and
Ewingp o idesmany
examples
o spaces
whose
a/p
cohomology
is
a
poly-
nomial
algeb a
.
Po
an
odd
p ime
p
ano he
e y
na u al
ques ion
o
s udy
is
ha o
ealizing
symme ic
algeb asi
.e
.,
ee
commu a i e
algeb as
as
cohomology
ings
.
A med
wi h
a
good
ealiza ion
heo em
o
symme ic
algeb as,
and
a
co esponding
classi ica ion
o
maps,
one
could
y
o
mimic
wi h
hese
spaces
as
building
blocks
he
upside
down
Pos niko
owe
(Sulli an's
minimal
model
cons uc ion)
o
ge
mo e
comple e
in o ma ion
abou
"Im{H*
:
Top
-
UnAl
The
minu e
one
s a s
o alk
abou
symme ic
algeb as,
heBocks einbeha iou
becomes
impo an
.
E en
in
he
simpeles
case
iz
and
P[x]
®
E[Y]

:

B
Y ~ 0
P[u]
0
E[ ]

8
:
u = 0
P on
.
3
:

Wi h
he no a ions
p eceeding
whe e
[X(V',G'),X(V11,G")]
=
Mo p ((V',GT),(V",G"))
cP
:
VI
-
Vil
MO p ((V',G'),(V",G"))
:_

(CP i)

1
:
Gl
--
,"
and
CP(g'
,
)=
$(g')CP( ')
V
g'

E
G',

'

E
V'
This
classi ica ion
o
maps
comes
in
handy
when
one
ys
.
o
use
he
spaces
X(X
;G)
as
"building
blocks"
o
cons uc
spaces
ealizing
o he
in e es ing
uns able
algeb as
o e
he
S een od
algeb a
as
cohomology
ings
.
He e
[12]
o
example
is a
sample esul
in
his
di ec ion
.
(This
esul has
also
beenob ained
independen ly
by
Howa dHille
.)
P op
.
4
:
Le
A*
=
P[x1,
. .
.,xn]/(Y11
. .
.,yn)
be
a
g aded
comple e
in e sec ion, ha
is
an
uns able
algeb a
o e
he
S een od
algeb a
.
Assume
ha
:
n

n
(1)

(_T
deg
x
i
)(7F
deg
Yi)
í
G(p)
;
i=1
i=1
he wo
examplesbeha e
e y
di e en ly,
as
Aguadé
has
shown
.
As
a
sampleo
his
esul s
one
has
[3]
.
P o
-
p
.
5
:
(J
.
Aguadé)

Suppose
p
en
odd
p ime,
and
is
an
uns able
algeb a
o e
he
S een od
algeb a
.
Le
2d
=
:
deg
x
hen
dip-1
and
all
suchS*
occu
as
cohomology
ings
.
Ske ch
o
P oo
:

To
see
ha dlp-1
we
w i e
(3y
=
x
.
The e
is
he
Adam
ela ion
so
applying
his
o
y
gi es
whe e
P
d
y = 0
by
uns abili y
.
Bu
hissays
P
1
ac s
non i ially
on S*
do
dIp-1
.
To
cons uc
he
examples
whe e
Py
= x
Aguadé
p oceeds
as
ollows
.
Le
CE
7l/p
x
=
2Z/p-1
be
a
gene a o ,
and
S*
=
E[y]
0
P[x]
P
1
.
=d -1
=
(
d
-
1)

P
d
+
Pd ,
p
1
p
P
d -1
y
=

(d
-
1)P
Pd y

+

Pd Py
=
0 +
p
d
(x
)
=
xp
se
§

:=
ep_1/d
.
Then
e
induces
an
ac ion
o 7Z/d
on
K%Z/p
;1)
= B
7Z/p
.
Le
Y

:=
K(E/p
;1)
/g
be
he
o bi
space
.
One
hen
has
(whe e
deg
u =
1)
H*
(Y
;
2Z/p)
=
H*(BTl/p)
1
=

(E[u]
®
P[Pu])1
-

E(u(p
u)
d-1
)

0

P(([3u) d
)
as
equi ed
.
Recallinghow he
cons uc ion
o
Cla k
and
Ewing
is
he
many a iable
gene aliza ion
o
he
one
a iable
cons uc ion
o
Holzsage
[9]
and
Sulli an
[19]
one
is
emp ed
o
y
o
p oceed
analogously
s a ing
wi hAguadé's
cons uc ion
o
p o e
:
P op
.
6
:
Suppose
S*
-

E(y
1
,. . . .
yn
)
0
P(Py1,
.
.
.,pyn)
is an
uns able
algeb a
o en
he
S een od
algeb a
whe e
n
(NMC)

71
-
deg
Pyi
i
0(p)
.
i=1
Then he e
exis s
a
space
Y
such ha
H*(Y
;7Z/p)
2
S*
.

The
ideao
he
cons uc ion
would
be
o
s a
wi h
p

:
G
y
GL(n
;
2Z/p)
.
Le
V
:=
®
a/p
be
he
ep esen a ion
n
space
o
p
.
The
ac ion
o
G
on
V
inducesa
ee
ac ion
on
BV
=
K(V
;1)
so
we
can
o m
he
o bi space
Y
:=
BV/G
B(G
x
p
V)
.
As
in
P oposi ion
(1),
i
p
+
IGI

one
ob ains
H*(Y
;
a/p)
_
H*(BV
;
a/p)
G
N
[E(V)
0
P(PV)
]G
Howe e
i is
almos
ne e
he
case
ha
[E(V)
®
P(PV
N
E(Y)
0
P(PY)
whe e
P(HY)
N
P(8V)
G
,
(see
o
example,
[4
;
Ex])
e en
in
he
nices
uses
.
Co
example,
pick
an
enoi
-
ous
p ime
p
.
Le
E
l
ac
on
V
:=
®a/p
ia he
adjoin
n
ep esen a ion
.
Then
one
sees
he
ob ious
map
is-simplyno
e en
monic
.
To
1,....
n
o
V
and
ecall
Bu
cp

:
E(a1,
. .
.,an)
0
P(Pa1,"
.,wn)
-
.

P(V)
G
P(a1,
..
.,an)
0 E
E(V)
see
his
choosea
basas
1
+
. . .
+
ñ
=
2
+
...
+
ñ
=
so
p(a19
. .
.Pa
n
)
E
ke
cp
.
Thus
a
p oo
o
P op_
6
mus
p óceed
alóng
o he
lines
.
The
p oo
in
[13]
uns'mo e
o less
as
ollows
:
Begin
as
be o e
wí h
G
<
GL(V)
.
Le
*k
:
X(V
;G)
-
X(V
;G)
be
he
map
induced
by
he
mo phism
( ecall
P op
.
3)
X
k

:

(V,G)

_

(V,G)I
ak( )
=
k ,
X
kg
=g
.
Fo m
he
ibe
squa e
(A :=
diagonal
map)
Y
kX
(1,
*
k)
X,
X
x
X
X
:_
(V,G)
de ining
Yk
.
Then
o
k =
p+1 one
inds
whe e
H*
(Y
p+
11
a/P)
=
E(Y)
0
P(PY)
P(PY)
=POV)G
.
So
o
all he
examples
o
ings
o in a ian s,
and
ela ed
ings,
which
we
ha eshown
o
occu
as
cohomo-
logy ings
ha esa is ied
he
non
modula i y
condi ion
.
We
ha e
howe e ,
a
leas
as
algeb a
o e
he
S een od
algeb a,
he
o he
`
ex eme
case
o
P(V)
Up
(V)
,
e c,
namely
P(V)
G
whe e
G
is
a p-g oup
.
The
ollowing
esul se les
he ealiza ion
ques ion
o
hese
modula
ings
o
in a ian s
in
he
nega i e
[15]
.
P o
p
.

Le
p
be an
odd
p ime
and

G
<
GL(n
;
7L/p)
a
p-g oup
such ha
is
a
polynomial
algeb a
.
Then
R*
canno
a ise
as
he
7l/p
cohomology
o
a
space
.
Thus
a
polynomial
algeb a
al
whe e
deg
x
i
=
2p
,
i
=
1,
. .
.,n,
and
a
leas
one
a
l
posi i e,
ha
occu s
as
a
ing
o
in a ian s
can
ne e
be
he
cohomology
algeb a
o
a
space
.
The e o e
we
canno
sepa a e
he
ealiza ion
ques ion
in o
a
non-
modula
heo y,
bu
mus mo e
om
a
non-modula
heo y
o
a
"mixed"
heo y
.
A
p o o ype
example
he e
is
he
Dickson
algeb a
18
4
R*
:=
P(V)
G
;
V
=
O
Tl/p
n
P*

.=
P(x
1
, .
..
.
x
n
)
D*
(n)
:=
p(V)GL(V),j
P[Y1,
. .
.,yn]
deg
y
i=
2p
n
-
2p
n-i
;

i
=
1,
. .
.,n
.
One
eason
o
singling
ou
D*(n)
o
special
s udy
is
he
ollowing
esul
p o ed
join ly
wi h
Bob
Swi ze ,
in en
a emp
o
cla i y
he
wo k
o
Adams
andWilke son
o
be
discussed
in
he
nex
sec ion
.
P op
.
8
(join
wi h
R
.M
.
Swi ze )
:

Le
H*
E
UnId/V
*
(
:=
Uns able
In eg al
Domein
o e
he
S een od
algeb a
.)
ANASC
ha
H*
i
P(V)
G
o
some
G
<
GL(V),
whe e
deg
=2
d
E
V,
is ha H* be
a ini e
algeb aic
ex ension
in
UnId/,R*
o
D*(n)
.
Finally
he
ealiza ion
ques ion
o
D*(n)is
se led
by [17]
P o
p
.

(join
wi h
R
.M
.
Swi ze )
:

ANASC
ha
D*(n)
occu
as
a
cohomology
algeb a
is
:
o
n=2
and
p<3
.
N
.B
.

Fo
n
=
1
he
example
a edue
o
Holszage and
Sulli an
.
Fo
n
=2
hey
a e
all
classical,
iz
.,
(CP(W),
BSU(3)
when
p=2
The
case
P(V)
GL(2
;ZZ/3)
has
ecen ly
been
ealizad
by
A
.
Zab odsky
[23]
.
We
collec
some ac s
abou
V
[2
;
§
51
.
(1)

I

1
,...
.
n
is
a
7l/p
basis
o
V
hen
1
, . .
.,
n
a e
algeb aically
independen
.
(2)

The
elemen s
o
V
a e
uns able,
so
(3)

he
ac ion
o
IP
*

on

P*

commu es
wi h
he
ac ion
o
GL(n
;
IF
p
),
so
P*
:=
P[ 1,
. .
.p n]
E
UnId/
>
*
D*(n)

:=
P*
GL(n
;I
p
)
<
P*
<
E*
a e
inclusions
in
UnId/19*
(4)

e e y
x
E
P*
is
in eg al
o e H*
.
Le
A* be
he
algeb aob ained
omH* by
adjoining
1,
.
. .
,
n
.
Theh
A*
E
UnId/iq
*
and
we ha e
he
inclusions
H*
<
A*
>
P*
>
D*(n)
.
Suppose
ha we
knew
ha A*
>
D*(n)
we e
an
algeb aic
ex ension
.
Then
A*
>
P*
is
algeb aic
.
Bu
by
he
algeb aic
closu e
heo em
P*
is
algeb aically
closed
and
hence
A*
=
P*
whence
H*
<
A*
=
P*
is
a
sepa ablealgeb aic
ex ension
.
Conside
he
Galois
g oup
G
:=
Gal(E*>
F(H*))
.

Clea ly
G<
GL(n
;IFp
)
because
he
elemen s
o
G
de ine
linea
ans o ma ions
o V
and
an
au omo phism
o E*
ixing
F(H*)
is
uniquely
de e mined
by
i s
ac ion
on
V
.
Now
we
claim
H* P*
G
=IF
P[ 1#
. .
., niG
To
see
his no e
he
inclusion
H*
<
P*
G
is
clea
.
On
he
o he
hand
because
E*
>
F(H*)
is a
Galois
ex ension
e e y
x
E
P*
G
lies
in
F(H*)
.
Fu he mo e
x is
in eg ál
o e
H* by
(4)
abo e
.
H* is
howe e
a
polynomial
algeb a,
hence
in eg ally
closed,
and
hus
x
E
H*,
i
.e
.
P*
G
<
H*
so
F
ollows
.
Hence
ou
p oblem educes
o
showing
A*

>

D*
(n)
is
an
algeb aicex ension
.
In
ac
we
show
ha
i
is an
in eg al
ex ension
by an
a gumen
li ed
om
Adams
and
Wilke son
[2]
.
Fi s
o
all
ecal1
D*(n)
_
IFp[y,,
.
.
.lyn]
.
Le
(y1,
.
.
.,yn)
deno e
he
ideal
o A*
gene a ed
by
y1'
. .
.,y
n
.

I we
can
show ha
A*/(y1,
...
.
yn
)
is ini e
dimensional
as a
ec o
space
o e IF
p
,
hen
in
ac an
easy
a gumen
ia
induc ion
o e
he
g ading,
shows
ha
e e y
elemen
o A*
is
in eg al o e
D*(n)
.
To
see
ha
A*/(y1,
. . . .
yn)
is
ini e
dimensional
o e IF
p
we
p oceed
as
ollows
.
Le
zE
A
2d
wi h
d
q(
0
mod
p
.
Then
[2
;
2
.3]
he e
is an
elemen
b
E
-
9
*
such
ha
Now
he
de i a ion
is
P
en
(bz)

=
zp
n
ynPeo
+
yn-1P
l+
. .
.+
y1Pen-1
+
pen
and
hence anishes
on E*
and he e o e
ce ainly
on
.
A*
<
E*
.
Thuswe ha e
z
pn
=
P
en
(bz)
=
-ynPen(bz)-

y1Pen-l
(bz)
E
(y1,
.
.
.,yn)
By
cons uc ion
A*
is
a
ini ely
gene a edE
p
algeb a,
wi h
gene a o s
a
l
,.
.
. y
a
m

whose
deg ees
a e
ela i ely
p ime
o
p
.
By
he
p eceedingcalcula ion
he
na u al
map
is
su jec i e
.
Bu
IFp[al,
. .
.,am]
y
(ap
.,
. . .
.am)
IFp[al,
...
.
a
m
]/(ap

,
..
.,

am)
is
isably
ini e
dimensional
.
194
A*/(yl,
.
.
.,yn)
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.
J
.F
.
Adams,On
he
non-exis en e
o elemen so
Hop
In a ian
One,
Ann
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o
Ma h
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72(1960),20-104
2
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J
.F
.
Adams
and
C
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Wilke son,
Fini e
H-Spaces
and
Algeb as
o e
hé
S een od
Algeb a,
Ann
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o
Ma h
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J
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Aguadé,
Cohomology
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Two
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Z
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177(1981),
289-296
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He mann
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C
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Che alley,
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o
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gené a ed
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Re lec ions,
Am
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J
.
o
Ma h
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77(1955)
778-782
6
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A
.
Cla k
and
J
.
Ewing,
The
Realiza ion
o
Polynomial
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as
Cohomology
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J
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o
Ma h
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425-434
7
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G
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Cooke,
Cons uc ing
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wi h
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ia
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Loop
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o
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101(1979),515-542
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G
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Cooke
and
L
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Mod
p Decomposi ions
o
co-H
Spaces
and
Applica ions,
Ma h
.
Z
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157(1977),
155-177_
9
.
R
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Holzsage ,
H
Spaceso
Ca ego y
<
2,
Topology9
(1970),
211-216
10
.
H
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Nakajima,
Modula Rep esen a ions
o
p-G oups
wi h
Regula
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o
In a ian s,
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Jap
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56
(1980),
469-473
11
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G.C
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Shepha d
and
J
.A
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Todd,
Fini e
Uni a y
Re lec ion
G oups,
Canadian
J
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o
Ma h
.
6(1954)
274-304
12
.
L
.
Smi h,
A
No e
on
he
Realiza ion
o
G aded
Comple e
In e sec ion
Algeb as
by
he
Cohomology
o
a
Space,
Qua
.
J
.
o
Ma h
.
( o
appea )
13
.
L
.
Smi h, On
he
Realiza ion
and
Classi ica ion
o
Symme ic
Algeb as
as
Cohomology
Rings,
IHES
P ep in
1981
14
.
L
.
Smi h,
On
he
Homo opy
Classi ica ion
o Maps
be ween
ce ain
spaces wi h
Polynomial
Cohomology,
Gb ingenP ep in
1981
C

T

C-4+l,

nn

+L,o
ATnn

Rcal
i
7n+
;
nn

n
M
di
11
a
Ri
naR

n
In a ian s
as
Cohomology
Algeb as
PAMS
( o
appea )
16
.
L
.
Smi h
and
R
.M
.
Swi ze ,
Polynomial
Algeb as
o e
he
S een od
Algeb a,
Va ia ions
on
a
Theme
o
Adams
and
Wilke son,
G6 ingen
P ep in
1981
17
.
L
.
Smi h
and
R
.M
.
Swi ze , On
he
Realiza ion
and Non
Realiza ion
o
Dickson
Algeb as
as
Cohomology
Rings,
IHES
P ep in
1981
18
.
N.E
.
S een od,
Polynomial
Algeb as
o e
he
algeb a
o
CohomologyOpe a ions,H-Spaces
(Neucha el)
Sp inge
LNM
196(1971),
85-99
19
.
D
.
Sulli an,
Gene ics
o
Homo opy
Theo yand
he
Adams
Conjec u es,
Ann
.
o
Ma h
.
100(1974),
1-78
20
.
C
.
Wilke son,Some
Polynomial
Algeb as
o e
heS een od
Algeb ak1%,
BAMS
79(1973),
1274-1276
21
.
C
.
Wilke son,
Classi ying
Spaces,
S een od
Ope a ions
and
Algeb aic
Closu e,
Topology
16(1977),
227-237
22
.
C
.
Wilke son,
In eg alClosu e
o
Uns able
S een od
Ope a ions,
J
.
Pu e
and
Applied
Algeb a
13(1978),
49-55
23
.
A
.
Zab odsky,
On
he
Réaliza ion
o
In a ian Subg oups
o n,(X),
Je usalem
P ep in
I
wan
o
hank
Bob
Swi ze
o
his
con inuous
suppo ,
encou agemen ,
and
collabo a ion,
wi hou
which
his, and
many
o he
p ojec s,would
ha ebeen
impossable
.
Ma hema isches
Ins i u
de
Uni e si á
Bunsens ape
3/5
D-3400
Gb ingen
Bundes epublikDeu schland