Publicacions
Ma emá iques,
Vol
34
(1990),
335-339
.
TRUNCATED
POLYNOMIAL
ALGEBRAS
OVER
THE
STEENROD
ALGEBRA
Abs ac
MOHAMED
ALI
I is
shown
ha
he
classi ica ion
o
polynomial
algeb as
o e
he
mod
p
S een od
algeb a
is
an
essen ially
di e en
p oblem
om
he
classi ica-
ion
o
polynomial
algeb as
unca ed
a
heigh
g ea e
han p
o e
he
S een od
algeb a
.
0
.
In oduc ion
.
Le
B
=
Zp[y2n

y2n2,
. . . ,
y2n,] be
a
polynomial
algeb a
o e
A(p),
he
mod
p
S een od
algeb a,
whe e
y
en
;
has
dimension
2ni
and
p
is
an
odd
p ime
.
I
each
ni
is
p ime
o
p,
he
esul s
o
[1]
and
[2]
imply
ha
he
s uc u e
o
B
is
well
unde s ood
;
in
pa icula
he
se
o
dimensions
{2n1,
2n
2 ,
..
.,2n,}
is
a
union
o
se s
gi en
in
he
Cla k-Ewing
lis
o
dimensions
in he
main
Theo em
o
[3]
.
Ea lie
a emp s
in
he
1960's
and
ea ly
1970's
o
classi y
he
se
o
dimensions
occu ing
in
B
o en
depended
only
on
he
A(p)-
algeb a
s uc u e
o
Bp+
1
=
Zp[y2n1,
y2n2,
. . . ,
yen,] p+1
,
he
polynomial
algeb a
unca ed
a
heigh
p
+
1
.
The
ques ion
o
de e mining
he
dimensions
o he
gene a o s
o
a
unca ed
polynomial
algeb a
A
=
Zp[x2n1,
X2n2,
.
.
.
.
X2n,]p+1
o e
A(p),
whe e
each
ni
is
p ime
o
p,
is
no
well
unde s ood
.
I
appea s
no
o
be
known
i
he
se
o possible
dimensions
in
he
wo
cases
coincide
as
is
ce ainly
he
case
when
=
1
.
The
pu pose
o
his
no e
is
o
se le
his
ques ion,
o
example,
Z11[xs,xlo]
12
suppo s an
,/4(11)-s uc u e,
bu
Z11[y6,y10]
does
no
.
The
ques ion
has
some
opological
signi icance
.
Fo
example,
i
a
p oduc
o
p-local
sphe es,
IIS(P)
,
1
<
i
<
,
suppo s an A(p)
s uc u e
in
he
sense
o
[5],
hen each
ni
E
{1,
2,

. .
p}
and
he e
exis s
a
unca ed
polynomial
algeb a
o e
A(p),
A
= Z
p
[x2n,
,
x2n
2,
. . .
,
x2nj
p+1
[4]
.

I
an
addi ion,
1
<
ni
<
p
o
each
i
and
he e
exis s
B
=Z
p
[y2n

y2n
2
,.. .,
y2n,],
hen
he
se
o
dimensions
{2n1,
2n
2
,.. .
,
2n,
.} is
gi en
by
he
Cla k-Ewing
lis
.

One
would
hen expec
ha ,
up
o
homo opy,
IlS
n
~
'
suppo s
he
s uc u e
o
a
opological
g oup
.
1
.
Cla k's
condi ion
.
Fi s
we
apply
a
well
known
heo em
o
A
.
Cla k
[2]
;
i is
clea
ha
he
p oo
holds
o
a
polynomial
algeb a
unca ed
a
heigh
g ea e
han
p
.
336

M
.
ALi
Theo em
1
.1
.
Le
A
be
a
polynomial
algeb a
unca ed
a
heigh
p
-1
-1
o e
¡he
mod
p
S een od
algeb a
.
I
2m
is
he
deg ee o
a
gene a o
o
A,
hen
ei he
m
-
0
mod
po
hese
exis s
a gene a o
o
dimension
2n
wi h
n
-
1
-
p
mod
I
we
es ic
a en ion
o
A=
Zp[s2n,x2,n]P+1
as
abo e,
ou ine
calcula ions
imply
ha , o
small
p imes,
he
pai s o in ege s {2n,
2m}
mus
lie
in
he
able
below
.
p
=
3

{4,4}
p
=
5

{4,4},{4,6},{4,8}
p
=
7

{4,4},
{4,6},
{4,8}
,
{4,12},
{6,6},
{6,12}, {8,12},
{12,12}
p
=
11

{4,4},{4,6},{4,8},{4,10},{4,12},{4,20},{6,10},
{8,20},{10,10},{10,20},{12,16},{20,20}
p
=
13

{4,4},{4,6},{4,8},{4,12},{4,14},
{4,24},{6,6},{6,10},
{6,12},{6,24},{8,8},{8,12},{8,16},{8,24},{12,12},
{12,18},{12,24},{16,24},{24,24}
Compa ing
his
lis
wi h
he
lis
o
possible
dimensions
o
he gene a o s
o
a
polynomial
algeb a
B
=
Zp[y2n,
y
2
,
n
]
o e
he
S een od
algeb a
[3],
we
see
ha
he
only
pai s
which
do
no
appea
in
he
la e
a e
{6,10}
and
{8,
20}
when
p
=
11
and
{6,10},
{8,16}
and
{12,18}
when
p
=
13
.
We
mus
he e o e
conside
he
possibili y
o
de ining
he
ac ion
o
he
S een od
algeb a
on
A=
Z
p
[x
2n
,
x
2
,
n
]P+
1
in
hese
i e
cases
.
2
.
S een od
heo em
.
In his las
published
pape ,
S een od
conside ed he
ques ion
o
de ining
he
cyclic
educed
powe s
on
g aded
polynomial
algeb a
.
We
belie e ha his
esul s
ha e
ne e
been
applied
.
We
summa ize
hem
in
ou
con ex
.
Le
A
be
a
g aded
algeb a o e
he
(uns able)
mod
p
S een od
algeb a con-
cen a ed
in
e en
dimensions
.
Recall
ha
he
cyclic
educed
powe s
a e
homo-
mo phisms
pq
:
A
2n
-s
A2n+2q(p
-
1)
sa is ying
:
(1)
p
°
=
Iden i y,
(2)
pqx
=
xP
i
2q
=
dimx,
(3)
pqx
=
0
i
2q
>
dim
x,
(
4
)
p9(xy)
=El-0
pixp4-'y,
(5)I a<pb,
TRUNCATED
POLYNOMIAL
ALGEBRAS

33
7
1 l

a+

(p
-
1)(b
-
)
-
1

a+b-
0
a-p

)p

p
Now
le
A
=
Z
p
[x2n,
,
x2np
... .
,x2n,
]
a+l
.

Suppose
ha
o
each
i,
he e
a e
de ined
homomo phisms
Pq
by
se ing
P
g
x2ni
=
q,i,
0
<
q
<
2ni,
whe e
,i
E
A
has
dimension
2ni
+
2q(p
-1),
Pni
x2n
;
=
XZni
and
P
g
x2n
i
=
0
o
q
>
ni
and
ex ending
Pq
o e
A
by
linea i y
and
he
Ca an
o mula
(4)
.
Theo em
6
.2
and
Lemma
4
.1
o
[7]
imply
ha
hese
Pq
de ine
an
ac ion
o
he
S een od
algeb a
p o ided
ha
R(a,
b)x2
ni
=
0
o
all i
whe e
(a,
b)
=
(p
,
b),
>_
0
.
We
now
es ic
a en ion
o
he
i e
examples
A=
Zp[x2n,x2m]
n+1
lis ed
abo e
whe e
n
<
m
.
Suppose
ha
P
I
x2n
=
91,n,P
l
x2m
=
h
1 ,
m
.
We
de ine
Yx2n
induc i ely
o
1
<
i
<
n
by
equi ing
R(1,
i
-
1)x2n
=
0,
o
equi a-
len ly,
P
I
x2n
=
(
i)
-1
P
1
P
i-l
x2n,
P
n
x2n
=x2n
wi h
simila
de ini ions
o
x
2
,
n
.
Dimensional
easons
imply
ha
p-
=
0 in
all
cases
as
2mp
-
2n
<
2p(p
--
1)
and
he e o e
R(p
,
b)
=
0
o
>
0
.
Thus
ó
show
ha
he
de ini ión
o
he
P'
de ines
an
ac ion
o he
S een od
algeb a
i
is
su icien
o
check
ha
wi h
he
chosen
91,n
and
hl,,n,
R(I,n
-
1)x2n
=
0and R(1,
m
-
1)x
2
,,
=
0
.
Theo em
2
.1
.
(a)
p
=
11,
ZI1
[x
s
,
x
1o
]
12
and
Zll
[x8,
X211
12
suppo
an
ac ion
o
he
S een-
od
algeb a
which
is
unique
up
o
algeb a
isomo phism
o e
he
S een od
algeb a
.
(b)
p
=
13,
Z13[xg,x10]14,
Z13[x12,
x18]
14
and
Z13[x8,x1B]
14
do no suppo
an
ac ion
o
¡he
S een od
algeb a
.
We
conside
he
examples
in
u n
.
(a)
(1)
p
=
11,
{6,10}
.
We
i s
p o e
uniqueness
.
Assume
ha
ZI
I
[x
s
,
xlo]
12
suppo s
an
A(p)-ac ion
.
Fo
dimensional
easons
p
l
x
s
=
ax
s
xi
o1
P
xio
=
%x
lo
+
-
yx i
.
Rou ine
compu a ions
show
ha
p
3
xó
=
(3!)
-1
[2ay
2
x6
l
+
a(a
+
20)(a
+
4P)xsxi
o
+
5a
(a
+
2p)xsxi
o
]
=
xsl
The e o e
ay
2
=
3
and
a+20
=
0
.
We
can
choose
y
o
ha e
any
non-ze o
alue
.
So we
se
y
=
5
.
Then
a
=
1
and
,Q
=
5
.
We
deduce
ha
i
A
suppo s an
ac ion
o
he
S een od
algeb a,
he
ac ion
is
unique
up
o
algeb a
homomo phism
o e
he
S een od
algeb a
and
we
can
se
P
I
xs
=
xsx
2
o,
p
1
XIO
=
5(x3
1

lo
+
xs)
.
The e o e
le
91,3
=
xsxio,
hi,s
=
5(xó
+
.
xs)
and
de ine
P
g
xs,
PgXIO
as de-
sc ibed
abo e
.

The
calcula ion
abo e
implies
ha
R(1,
2)X6
=
0
and
so
we
need
only check
ha
R(1,4)xl0
=
0
.
Rou ine
calcula ion
e i ies
ha
his
is
ue
.
(a) (2)
p
=
11, {8,
20}
.
Again
assume
ha
Z11
[x8,
x12]
12
suppo s
an
A(p)-
ac ion
.
Fo
dimensional
easons,
p1 x8
=
"8x20,
p
1
x20
=
0x20
+yx8
.
Rou ine
compu a ions
show
ha
p
4x8
=
(4!)
-1
[a(a
+
0)(a
+
2
0)(a
+
30
)x8
x2o
+
a7(a
-
p)(a
+
2p)x6x6o
`

+
ay
2(a
+
30)x8
1
]
=
x81
.
The e o e
by
sui able
choice
o x2o,
we
can assume
ha
a
=
1
.
I
ollows
ha
p=
5,
7
=
4
.
Replacing
x8
by
x8
i
necessa y,
we
can assume
ha
y=
4,
and
he e o e
i
A
suppo s
an
ac ion
o
he
S een od
algeb a,
he
ac ion
is
unique
up
o
algeb a
homomo phism
o e
he S een od
algeb a
and
we
can
se
p1
x8
=
x8xlo,
p1
x2o
=
5x2
0
+
4x8
.
The e o e
le
91,4
=
XSx20,
hijo
=
5x30
+
4x8 and
de ine
0
9
x6,
pgx2o
as
desc ibed
ábo e
.
.
Clea ly
R(1,
3)x8
=
0
and
so
we
need
jus o
check
ha
R(1, 9)X20
=
0
.
Rou ine
calcula ion
e i ies
ha
his
is
ue
.
(b)
(3)
p
=
13,
{6,10}
.
Again
suppose
ha
Z13[x6,x1o]
14
suppo s
an
A(p)
ac ion
.
Fo
dimensional
easons
p1
x
6
=
ax
55
+
px
3
o,
p
1
x10
=
yxsxlo
.
Rou ine
calcula ions
show
ha
03x6
=
(3!)
-1
[6a
3
x6
3
+
p{6a
2
+
(4a
+
3y)(5a
+
3y)}x610
+
0
2
(5a
+
3y)x6
1

=
x63
The e o e
a
3
=
1,
p{6a
2
+
(4a
+
3y)(5a
+
3y)}
and
p2 (5a
+
3y)
=
0
in
Z13
.
This
is
possible
only
i
p
=
0,
bu
his
implies
ha
0
9
x10
E
(x
6
)
which
is
no
possible
as
p
5
xlo
=
X13
.
Thus
Z
13 [x
6
,
x10]
14
will
no
suppo
an
A(13)
ac ion
.
10
(b) (4)
p
=
13,
{8,16}
.
Again
suppose
ha
Z13
[X8,
x16]
14
suppo s an
A(p)
ac ion
.
Fo dimensional
easons
p1 x8
=
ax8
-1-
pxi6+yxáxls,
0'x16
=
6x8X16+
/Zx8x16
+
0x8
.
Rou ine
calcula ions
show
ha
p4
x8
=
(4!)
[{7a
4
+
12a0B
2
+
8ayic0
+
6a
2
8y
+
11y202
+
608
2
/
+
y8p
2
+
2BZyb}x83+
{9a
2
yN,
+
3app
+
6ayp
2
+
PBp2
+
yp
3
+
80
,
yó
+
3a3y+
4cey
2
0
+
9a
2
pB
+
8aO-y6
+
py0
2
+
12pO
2
6}x8
1
x16+
{5x
3
0
+
110
2
0
2
+
60Byp
+
11apO-y
+
4a
2
y
2
+
6ay
2
p
+
8a
2
-yb
+
6«060+
a
2
pp
+
8ayóp
+
30Bóp
+
7yóp
2
+8&
3
+
9B-y
2
6
+
80y0
+
2By
3
}x8
16
+
{a
2
py
+
11poy
2
+
6ay
3
+
7y
3
[1
+
11a-Y26
+
12PB6y
+
5,Y
2
6FU
+
120-yp2+
6a
2p6
+
100ó
2
p
+
12apbp
+
12-í 62,U
+
6p c
2
ó
+
11ap
2
B
+
3p2B c}xáxi6+
{90,
2
0
2
+
6p
2
0y
+
5apy
2
+
3py
2
~
+
apy6
+
110
2
06
+
ap
2
F¿
+
9py6,¿+
90
2
0
2
+
lly
4
+7^í
'3
6
+
3y262
+
6_Y63
+3
apb
2
+
9ppb2}x8xi6+
{5ap
2
y+0
3
B+4yp
2
p+0y
3
+4py
2
ó+12pyó
2
+10«0
2
6+0
2
Pb+1106
3
}xáxi6+
{llapa
+
120
3
0
+
302y2
+
602y6
+
90262
}x8x66]
=x63
TRUNCATED
POLYNOMIAL
ALGEBRAS

33
9
One
can
show
ha
his
implies
ha
0
=
0,
and
so
p
g
xls
E
(x8)
which
is
alse
as
p
8
xls
=xis
.
Thus
Z13[x8,x16]
14
will
no
suppo
an A(13)-ac ion
.
(b)
(5)
p
=
13,
{12,18}
.
Simila ly
as in
he
las
examples,
we
can
see ha ,
o
dimensional
easons,
p
1
x
1
2
=
axil
+Oxi81
il
l
xl8
=
yxi2x18 and
hen
p
s
x12
=
(6!)
-1
[{
8
a
i
xiz
+
(
3a
'
y
+
9a
l
y
2
+4a
2
y
3
+
4ay
4
+
6,y5
}x
10
x12
i8
+ i
2
{5a
4
+7a
2
y+5a
2
y
2
+
3
ce
-
y 3
}xi2x
i8
+,~
3
{7a
3
+9a
2
y-~
lla-y
2
+y
3
}xi2x
6 8
+
q4
{7a
2
+
12a-y+ 6y
2
}X12X881
=
X13
12
.
This
is
ue
only
i
,Q
=
0,
bu
his
implies
ha
p
e
x
1
8
E
(x12)
which
is
impossible
as
p9
x18
=
X18
and
so
Z13[x12,x18]
14
will
no
suppo
an
.4(13)-ac ion
.
Re e ences
1
.

J
.F
.
ADAMS
AND
C.W
.
WILBERSON,
Fini e
H-spaces
and
algeb as o e
he
S een od
algeb a,
Ann
.
Ma h
.
111
(1980),
95-143
.
2
.

A
.
CLARK,
On
I 3
o
ini e
dimensional
H-spaces,
Ann
.
Ma h
.
78
(1963),
193-195
.
3
.

A
.
CLARK
AND
J
.
EwING,The
ealiza ion
o
polynomial
algeb as
as co-
homology
ings,
Paci ic
J
.
Ma h
.
50
(1974),
425-434
.
4
.

J
.R
.
HUBBUCK
AND
M
.
MIMURA,
Ce ain
p- egula
H-spaces,
A ch
.
Ma h
.
49
(1987),
79-82
.
5
.

N
.
IWASE,
H-spaces
wi h
gene a ing
subspaces,
P oc
.
A
.
Roy
.
Soc
.
Edin
.
( o
appea )
.
6
.

J
.
STASHEFF,
Homo opy
associa i i y o
H-spaces,
I
and
II,
T ans
.
Ame
.
Ma h
.
Soc
.
10
8 (1963),
275-292
and
293-312
.
7
.

N
.E
.
STEENROD,"Polynomial
algeb as
o e
he
algeb a
o
cohomology
op-
e a ions,"
H-spaces,
Neucha el
(Suisse)
1970,
Lec u e no es
in
Ma h
.
196,
1971,
pp
.
85-99
.
Depa men
o
Ma hema ical
Sciences
Abe deen
Uni e si y
Abe deen
AB9 2TY
SCOTLAND
Rebu
el
18 de
Desemb e
de
1989