3.
~
-Rin~s.
We p r e s e n t
mainly
the theory
taken
consult
proven
from the p a p e r
this paper
here
[J~]
for a d d i t i o n a l
is an e x p o n e n t i a l
is an a l g e b r a i c
version
work
of Adams
3.1.
Definitions.
[~]
~ -rings.
The a l g e b r a i c
by A t i y a h
and Tall.
information.
isomorphism
of the p o w e r f u l
on fibre h o m o t o p y
Let R be a c o m m u t a t i v e
consists
of special
ring with
of a sequence
An
The reader
The main
for p - a d i c
identity.
A
should
to be
= J"(~)
in the
of v e c t o r bundles.
A -rin~
: R---) R, n ~ ~ ,
theorem
is
~ -rings w h i c h
t h e o r e m J' ( X )
equivalence
material
of maps
structure
such that
on R
for all
x,y E R
l°(x)
(3.1.1)
=
I
At(x) =
x
n(x+y
If t is an i n d e t e r m i n a t e
!3.1.2)
Then
3.1.1
shows
Ar(x)
n~ o
~n(x)tn"
~ n-r (y) .
~
that
~t
is a h o m o m o r p h i s m
powers
: R ---)
from the a d d i t i v e
I + R[[t]] + of formal
Exterior
n
r=o
we define
~ t (x) =
(3.1.3)
group
) = ~
power
of modules
have
I + R [[t]] +
group of R into the m u l t i p l i c a t i v e
series
formal
over
R with
properties
constant
like
term
3.1.1
I.
and we
28
shall
see l a t e r h o w e x t e r i o r p o w e r s
Grothendieck
A
ring.
~ -homomorphism
-operations.
We h a v e
Some f u r t h e r
axioms
behave well with
i-th e l e m e n t a r y
(3.1 .4)
~ -ring
the n o t i o n s
are n e e d e d
of
functions
Define polynomials
and
~ -operations
and c o m p o s i t i o n .
and let u i , v i be the
in X l , . . . , x p and y l , . . . , y q re-
with
integer
coefficients:
is the c o e f f i c i e n t
of t n in
(1+xiYjt) "
Pn,m(Ul,...,Umn)
(1+x
i I < ... < i
P n , m is of w e i g h t
is the c o e f f i c i e n t
• ...-x
11
m
T h e n Pn is a p o l y n o m i a l
of t n in
t).
Im
of w e i g h t
n in the u i and a l s o in the v i, and
n m in the u i. If we a s s u m e p ~ n, q ~ n in 3.1.4 and
p ~ mn in 3.1.5 t h e n n o n of the v a r i a b l e s
ui,v i involved
the r e s u l t i n g
of p,q.
polynomials
are i n d e p e n d e n t
~ - r i n g R is said to be s p e c i a l
following
a
A -subring.
t h a t the
y l , . . . , y q be i n d e t e r m i n a t e s
T[
A
on c e r t a i n
c o m m u t i n g w i t h the
I -ideal
to i n s u r e
P n (u I ' ' ' ' ' U n ; v I '''''Vn)
(3.1.5)
structures
on it is c a l l e d
to ring m u l t i p l i c a t i o n
symmetric
~i,j
structure
is a r i n g h o m o m o r p h i s m
respect
L e t x 1,...,xp,
spectively.
A -ring
groups.
ring R together with a
A
give
identities
hold
~t(1)
(3.1.6)
~m(
for x,y
if in a d d i t i o n
are
to 3.1.1
zero and
the
~ R
= I + t
~n(xy)
= Pn ( ~ I x .....
Anx;
~n(x))
= P m , n ( ~ I Xl..ol
A 1 y .... , A n y )
~ mnx)
29
One
called
can m o t i v a t e
3.1.6
n-dimensional
if
is c a l l e d
elements.
and
the x i , Y i are
i-th
the s e c o n d
product
third
identity
-ring
insure
considering
One
of
just
defines
function
that
many
theorems
A -ring
(I+ ~ a n t n ) o
Am(l+
Proposition
~
about
on
" A -structure"
...
ring
+
of f i n i t e
"'" + Yq
in a
+ Upt p
x,y.
and w e
see
If m o r e o v e r
one-dimensional
The
axioms
~ -rings
We
that
the
then
for a s p e c i a l
can be p r o v e d
formalize
the
this
by
remark.
of p o w e r
series.
by
= I +
is g i v e n
= I +
n. The
1+A[[t]] + by:
is g i v e n
(I+ ~ b n t n)
Zantn)
+
x..
1
is m u l t i p l i c a t i o n
"multiplication"
The
is a g a i n
elements.
structure
is
then
for such
for x =
one-dimensional
a
+ Xp and y = Yl
of the xj as above)
elements
A -ring
is a d i f f e r e n c e
= I + ult
is t r u e
is t r u e
"addition"
(3.1.7)
(l+xit)
of 3.1.6
3.1.6
...
x in a
of d e g r e e
element
one-dimensional
of o n e - d i m e n s i o n a l
identity
if e v e r y
= ~
elementary
An e l e m e n t
is a p o l y n o m i a l
If x = x I +
~t(x)
(u i the
~t(x)
finite-dimensional
dimensional
-ring
as follows.
P n (a I '" ..,an;b I ''" .,bn)tn.
by
X Pn,m(al .... 'amn)tn"
3.1.8.
1 + A[[t]] + is a
-ring
with
the
[~]
, p.
258.
structure
3.1.7.
Proof.
Compare
Using
only
if
Atiyah-Tal!
this
structure
~ t is a
one
sees
that
~ -homomorphism.
A is a s p e c i a l
Moreover
one
has
~ -ring
if and
the T h e o r e m
of
30
Grothendieck
that
I + A[[t]] + is a s p e c i a l
~ -ring
(Atiyah-Tall
to s h o w that c e r t a i n
~ -rings
are special.
Suppose
of o n e - d i m e n s i o n a l
loc.
cit.)
One can use
3.1.8
Proposition
3.1.9.
Let R be a
~ -ring.
in R are a g a i n
that p r o d u c t s
one-dimensional;
i__nnp a r t i c u l a r
L e t R I C R be the s u b r i n g g e n e r a t e d
R I is a
~ -subrin~ which
elements
I s h a l l be o n e - d i m e n s i o n a l .
by o n e - d i m e n s i o n a l
elements.
Then
is special.
Proof.
Every element
dimensional
of R I has
elements,
say x = x1+
is the i - t h e l e m e n t a r y
dimensional
the
lJ(Y).
before
=
elements.
~i
3.1.7
~ t(x)
facts
Hence
This
An(x-y)
At
A i(-x)
sums of one-
y = yl + ... +yq.
in the x. h e n c e a s u m of o n e 3
is an i n t e g r a l p o l y n o m i a l in
Ai(-y)
= ~i
Ai
~i(x)
function
hi(x)
An-i(-y)
~ R I. The r e m a r k s
I R I is a r i n g - h o m o m o r p h i s m
=
Then
A t(-x)
elements
and t h e n
and
A t A l(x)
and t h e s e
=
two
At ~i(x-y) = li At(x-y).
3.1.10.
~
(Atiyah-Tall
-ring
)R'
elements
such that the f a
is c a l l e d
That a
[4~]
R is s p e c i a l
finite-dimensional
f : R
Moreover
... +Xp,
x,y are
if x is a sum of o n e - d i m e n s i o n a l
~t
One can s h o w
that a
symmetric
show that
imply
Remark
the f o r m x-y w h e r e
) - and later we shall use this
if and o n l y
in R t h e r e
1
~ -ring
structure,
can be seen f r o m the f o l l o w i n g
if for any set a l , . . . , a n of
exists
a
~ -monomorphism
are sums of o n e - d i m e n s i o n a l
the s p l i t t i n g R r i n c i p l e
even
fact -
for s p e c i a l
if not s p e c i a l ,
Proposition
elements.
A -rings.
m a y be v e r y u s e f u l
due to G. Segal.
31
Proposition
3.1.11.
Let
~ -ring.
R be
a
Then
all
Z-torsion
elements
in R are n i l p o t e n t .
Proof.
Let
a be a p - t o r s i o n
element,
say pna
= O.
n
I =
At(o)
=
Then
n
~t(a)P
=
(1+at+...) p
n
n
--- 1+a p t p +...
mod
p A
n
and hence
ap
= p b for
a (pn+1)n
3.2.
=
some b e A. T h e r e f o r e
(p a b) n =
(pna)(an-lb)
= O
Examples.
a) T h e
integers
~t(1)
= 1 +
may
Z
be g i v e n
m n tn where
a
~ -ring
mI = I
The
structure
by d e f i n i n g
canonical
structure
on Z is
given by
A
(3.2.1)
This
canonical
following
combinatorial
Let
Aks
The
theory
elegant
be
way
functions,
b)
where
structure
the
At(m)
=
(1+t)
k(m)
=
(7)
Ak(_m)
=
(_1)k
by
interpretation:
subsets
A -rings
of h a n d l i n g
binomial
(I) = I + t
is s p e c i a l
set of all
of s p e c i a l
t
('m+k-lk )
3.1.9.
Let
It can be g i v e n
the
S be a set w i t h m e l e m e n t s .
be t h o u g h t
combinatorial
k. T h e n
IA k s ~ =
(~).
of as an e x t r e m e l y
identities
for
sets,
symmetric
etc.
L e t E , F be c o m p l e x
G-vector
G is a c o m p a c t
group.
Lie
m ~ 0
of c a r d i a n l i t y
may
coefficients,
m
bundles
Then
over
exterior
the
powers
(compact)
G-space
A i of G - v e c t o r
X
32
bundles
satisfy
A°E
= I,
L e t KG(X)
(Segal
AIE
= E,
A n ( E ~ F) =
be the G r o t h e n d i e c k
[~%2] ). T h e n E z
f r o m the a d d i t i v e
over X into
therefore
group giving
A i [E] =
a
A -ring
Proposition
KG(X)
with
+
~
( A2E)t2+...
of i s o m o r p h i s m
A j (F)
classes
therefore
over X
is a h o m o m o r p h i s m
of G - v e c t o r
uniquely
bundles
to the
a map
) I+KG(X) [[t]] + : x ~--9 I+
I t : KG(X)
such t h a t
( AIE)t
I + KG(X) [[t]] + and e x t e n d s
Grothendieck
/%i(E)
r i n g of such G - v e c t o r b u n d l e s
) I +
semi-group
~ni=o
[ ~i(E)]
for E a G - v e c t o r
structure
bundle.
A1(x)t+...
These
A i yield
on KG(X).
3.2.2.
this
i -structure
is a s p e c i a l
~ -rin@.
Proof.
The proof depends
for g e n e r a l
on the so c a l l e d
G - is h i g h l y
Given vector bundles
Y and a G - m a p
is i n j e c t i v e
[~]
and f~E. s p l i t s
l
or K a r o u b i
For a d i s c u s s i o n
III,
[5]
splitting
exists
which
- especially
principle
a compact
into a sum of line b u n d l e s .
[I0~],
principle
of
; Karoubi
c) O t h e r v e r s i o n s
This
principle
says:
G-space
) X such t h a t the i n d u c e d m a p f W : KG(X) - - ) KG(Y)
U s i n g the s p l i t t i n g
ch.
non-trivial.
E I , . . . , E k o v e r X. T h e r e
f : Y
, 2.7.11
splitting
p.
3.2.2
A -operations
[~0~]
IV.
of t o p o l o g i c a l
See A t i y a h
193 for the case G = {I}
follows
essentially
in K - t h e o r y
f r o m 3.1.9.
see also A t i y a h
7.
K-theory
like real K - T h e o r y
or
[@]
,
33
Real-K-Theory
(Atiyah
[~]
), y i e l d special
d) A special case of b)
representations.
too.
is the r e p r e s e n t a t i o n ring R(G) of complex
Since r e p r e s e n t a t i o n s
cyclic subgroups and R(C)
~-rings
are d e t e c t e d by r e s t r i c t i o n to
for a cyclic group C is g e n e r a t e d by one-
d i m e n s i o n a l elements one can d i r e c t l y apply 3.1.9 to show that R(G)
is
special.
e) The Burnside ring acquires a
hi(s)
if we define
for a finite G-set S to be the i-th symmetric power of S. We
use the identity
A(G)
A -ring structure
~n(s+T)
as under b). This
See S i e b e n e i c h e r
=
~ i
h i (S)
An-i(T)
to extend this to
A -ring structure is in general not special.
[4W~] and the exercises
f) See A t i y a h - T a l l
to this section.
[4~] , I. 2 for the c o n s t r u c t i o n of a free
A-
ring on one generator.
3.3.
~ -operations.
We assume that R is a special
~ -ring.
Then R contains a subring iso-
m o r p h i c to Z for if I ~ R had finite a d d i t i v e order m, then
I = ~ t(o)
=
~t(m'1)
=
(1+t) m w o u l d give a c o n t r a d i c t i o n
coefficients of tm). A special
is given a
~-homomorphism
~ -ring R is called a u g m e n t e d i{ there
e : R
) Z. We call I = Ker e the
a u g m e n t a t i o n ideal;
it is a
u n i q u e l y x = e(x)
(x-e(x)) with e(x) E Z and x-e(x)
Define the
(3.3.1)
+
~ -ideal.
Any element x E R may be w r i t t e n
~ - o p e r a t i o n s on a special
;%t/(1_t) (x) =:
~t(x)
= I +
~t(x+y)
=
~t(x)
E I.
A -ring R:
~
n& I
Then
(3.3.2)
(compare
~t(y) .
~i(x)ti"
34
Moreover
one
has
n
(3.3.3)
(x) =
~ n(x+n-1) .
Proof.
Using
3.2.1
we
get
~t/(1-t)
(x) = I +
~
i& I
= I + [ j) I ( ~ i=I
j
= I +
We conclude
for
j > n,
most
from
i. e.
~
3.3.3
j>, I
that
,i+k-1 tk+i)
k~ o ~ k
)
A i(x) ( ~
~i(x) (j_i)j-1)t j
A J(x+j-1)t j
A J(x)
if x is n - d i m e n s i o n a l
= o for j > n i m p l i e s
then x-n
is of
~
~ J(x-n)=o
-dimension
at
n.
Suppose
R is an a u g m e n t e d
augmentation
is the
where
ideal
additive
ai E
group
I and
Proposition
I = ker
~ -ring with
e. W e d e f i n e
generated
~ ni ~
augmentation
the
e
~-filtration
nI
~
(al) "''"
by monomials
: R
) Z and
by:
Rn C R
n
. ~ r(ar)
n.
3.3.4.
(i)
R O = R, RI = I.
(ii)
Rm R n C
(iii)
R
is a
n - - --
Rm+ n .
~ -ideal
f o r n ~ I.
Proof.
(i) a n d
shows
(ii)
that
follow
directly
R n is an ideal.
To
from the definitions.
s h o w R n is a
(iii) : R = Z ~ , R I
A -ideal,
it is s u f f i c i e n t
35
to show
~r(
~ m(x )
~i(x+m-1)
because
e R m for x & I. F i r s t w e c o m p u t e
=
~l(x+m-i)
=
[
~i-S(m-i)
=
i
s=m
i
s=o
=
~S(x)
~,
i-s
~ i-S(m-s-1)
~r(~m(x))
=
s
~
Sometimes
define:
m because
we want
is an a u g m e n t e d
carries
the i n d u c e d
before,
In b e i n g
where
a i ~ I,
~
s+1. We use this
(x+m-1) .....
R with
the i d e a l g e n e r a t e d
each containing
= o.
the a u g m e n t a t i o n
is c a l l e d
-operations.
a special
ideal.
~ -ring
I as a u g m e n t a t i o n
We d e f i n e
the
by m o n o m i a l s
We
if t h e r e
ideal.
I then
~ - f i l t r a t i o n as
nI
n
~
(al)'... • ~ r(ar)
~ n i ~ n. We h a v e
(3.3.5)
11 = I, ImIn C
3.4. T h e A d a m s
operations.
Adams
introduced
which
are m u c h
in
[4 ]
Im+n,
certain
easier
to h a n d l e
L e t R be a s p e c i a l
A -ring.
%u
n
: R
in
~ rm(x+m-1))
Pr,m(Sl,...,Sm_1,o,...,o)
~ -ring
i
~ Rm
is a s u m of m o n o m i a l s
identity
special
(m-i)
= o for i ~ m ~
to w o r k o n l y w i t h
A ring I w i t h o u t
~
Ar(~m(x+m-1))
that Pr,m(Sl,...,Srm)
a t e r m s i for i ~
i-s
(x)
(m-i)
= Pr,m (
and o b s e r v e
for i ~ m
~ i ( I n) c In.
operations
derived
algebraically.
Define maps
)R,
n>p
I
f r o m the
A i
36
by
(3.4.1)
%42_t(x) = -t ~ t
~ t (x) =
~
( ~ t (x))/ ~ t (x)
n~ I ~n(x)tn"
n
A more e l e m e n t a r y way of d e f i n i n g the
is: Define the Newton
polynomial
Nn(S I '" . . , S n )
=
~
nj=1 xjn
where s. is the i-th e l e m e n t a r y symmetric function of the x..
l
3
~n(x)
(3.4.2)
= N n ( ~ 1 ( x ) .....
Then put
~n(x)).
We leave it as an exercise to show that the two d e f i n i t i o n s are
equivalent.
We w a n t to show that the
~n
are
A -ring homomorphisms.
means we have to verify certain identities b e t w e e n the
operations.
~ n_
This
and A j-
We use the v e r i f i c a t i o n p r i n c i p l e w h i c h says that it is
enough to verify the identities on e l e m e n t s w h i c h are sums of oned i m e n s i o n a l elements.
Atiyah-Tall
[~]
A formal proof of this p r i n c i p l e is given in
, I. 3.4, I. 4.5. Since in the applications the
-rings are f i n i t e - d i m e n s i o n a l
splitting p r i n c i p l e
and since we have to prove the
in order to show that something is a special
ring we do not prove the v e r i f i c a t i o n principle.
Proposition
3.4.3.
(i) If x is o n e - d i m e n s i o n a l then
(ii)
(iii)
~
n
~m
is a
~n
I -homomorphism.
=
~n
~m
= ~mn.
n
n
~2 x -- x .
~ -
37
(iv)
r
r
~ p (x) ~ x p mod p
(p prime).
Proof.
(i) follows
directly
(ii) Suppose
from 3.4.2.
x i, yj are one-dimensional.
because
R is special.
additive
From 3.4.1
homomorphism.
~2n( ~
Then xiY j is one-dimensional
one obtains
that
~
n
is an
Moreover
x i Z Yj) =
~un( }- xiY j) = ~ ~
n
(xiY j) = [ (xiYj) n
: ( ~ x n) ( [ Yjn) : ~2n( Z x i) ~ n( Z Yj).
~n(
A m( ~-xi )
: ~ n ( s m ( X I ..... Xr))
= Sm(X 1 ..... x n)
= z m ( [ x n) = Am( ~ n( ~- xi)) "
Now use the verification
(iii)
and
(iv) are likewise
As a consequence
n preserve
Proposition
the
immediate
we have
~n
from the verification
on a special
~ -ring.
principle.
Moreover
the
~ -filtration.
3.4.4.
Let I be a special
~ -rin 9. Assume
(i)
~2k(x)
- knx
(ii)
~k(x)
+ (-1)kk
(iii)
principle.
x E I n • Then the followin~
holds:
& In+ I
A k (x) 6 In+ I
~ k (x) + (-1)kkn-lx
~ In+ I.
Proof.
(i) We need only show that
~
k
(
m(a))
- k m ~ m(a) &
Im+ I for a ~ I,
38
because
i. e.
~k
is a
~ t(xi)
k(x i) =
~ -homomorphism.
= 1+xit,
then
) - km
= ~ k ( s m (x I ..... x r))
Sm((1+xl)k
x =
- I) - k m s m ( x l , . . . , X r
polynomial
of d e g r e e
x i and, by the v e r i f i c a t i o n
~k(x)-
~k-1(x)
implies
the result,
and x E I
(i) and
the r e s u l t
suitable
hence
~ m+1,
principle,
we o b t a i n
).
hence
(i) is t r u e
therefore
for
in g e n e r a l .
the w e l l - k n o w n
identity
A I (x)+...+(-I)k-1~) I (x) A k-1 ( x ) + ( - 1 ) k k A k ( x ) = o
because
~i(x)
6 I , li(x) 6 I for i ~ I,
n
n
n
(iii) F r o m
Thus
one,
- k m Sm(X I ..... x r)
(ii) F r o m the N e w t o n p o l y n o m i a l s
which
~ -dimension
one,
~m(x1+...+Xr)
- 1,...,(1+xr)k
is a s y m m e t r i c
~
1+x i has
~ -dimension
(1+xi)k - I and t h e r e f o r e
~k(~m(x1+...+Xr)
This
If X l , . . . , x r have
follows
universal
thus one gets
(ii) we o b t a i n k A k ( x )
if t h e r e
situations
the r e s u l t
are n a t u r a l w i t h
respect
3.5. A d a m s - o p e r a t i o n s
Let G be a f i n i t e
sentation
ring)
We a s s u m e
in R(G;F)
without
torsion,
One
E In+ I •
(One can p r o d u c e
e. g. free
A -rings;
s h o u l d n o t e that the a s s e r t i o n s
A -homomorphisms.)
on r e p r e s e n t a t i o n
g r o u p and R(G;F)
of f i n i t e l y
(-1)kkn(x)
is no k - t o r s i o n .
in g e n e r a l .
to
+
rings•
be the G r o t h e n d i e c k
(= r e p r e -
F[G]-modules
where
F is a field•
for s i m p l i c i t y
t h a t F has c h a r a c t e r i s t i c
zero.
Then elements
are d e t e r m i n e d
by t h e i r c h a r a c t e r .
the c o r r e s p o n d i n g
ring structure
operations.
character
on R(G;F).
generated
ring
ring.
We w a n t
Exterior
We i d e n t i f y
powers
to c o m p u t e
define
R(G;F)
with
a special
the a s s o c i a t e d
Adams-
A -
39
Proposition
3.5.1.
Let x e R(G;F).
Then
kx(g)
= x(g k) ,
g ~ G.
I__nnparticular
~k
= ~ k + ~GI
Proof.
Restrict to the cyclic group C generated by g. Pass to an algebraic
closure of F so that x I C = y-z where y and z are sums of one-dimensional representations.
The result then follows
account that for a o n e - d i m e n s i o n a l
xk(g)
from 3.4.3 taking
representation
into
x the relation
= x(g k) holds.
Now assume that F = Q [ ~ n ] where
unity.
Assume
to Z / n Z t
ponds to the field a u t o m o r p h i s m
Since characters
of F [ G ] - m o d u l e s
pk to such characters.
(i)
namely
take values
n-th root of
~GI. The Galois group
so that k mod n corres-
pk c h a r a c t e r i z e d
by p k ( ~ n )
= ~ k.
n
in Q [ ~ n~ we can apply
Let Q [~ n] be a splitting
famous theorem of Brauer it suffices
Proposition
is a primitive
that k is prime to the group order
GaI(Q [ ~] : Q) is isomorphic
see Serre
~n
field for G.
(By a
to take for n the exponent of G;
[ff~7] , P. 109). Then we show
3.5.2.
~ kx = pkx for x E R(G;Q [ ~ n]) and
(ii) If x is the character
irreducible
too
(k,IGl)
of an irreducibel
= I.
module then
k
x is
(a~ain k prime t_oo ~GI).
Proof.
(i) Let x be the character
of a matrix representation.
Restrict
to the
40
cyclic subgroup C g e n e r a t e d by g @ G. Then the m a t r i x for g is equivalent to a diagonal m a t r i x w i t h roots of unity u l , . . . , u r on the
diagonal.
Then
~ k(x) (g) =
Z u~ = pk( [ ui ) = pk(x(g)) "
(ii) Apply the Galois a u t o m o r p h i s m pk to a m a t r i x r e p r e s e n t a t i o n over
Q[{n]
Remark
3.5.3.
The Adams o p e r a t i o n are, of course,
nition.
Therefore
i n d e p e n d e n t of the field
of defi-
3.5.2 holds more generally.
3.7. The Bott c a n n i b a l i s t i c class e kLet R be a special
unity.
~ -ring and let
~ k be a p r i m i t i v e k-th root of
Let P(R) C R be the subset of f i n i t e - d i m e n s i o n a l elements
Then P(R) ~s an additive semi-group.
in R.
If x E P(R) we c o n s i d e r the pro-
duct
(3.7.1)
8k(X)
:= ~
u
~
-u
(x) 6 R ~
Z
Z [~
k
]
where the p r o d u c t is taken over all roots of tk-1 = o except I. We
identify R with its image in R ~
r[
~ r ~
Z [ ~ k] under the canonical map
I. Then 8k(U ) is c o n t a i n e d in R. ~ n
order to see this c o n s i d e r
the following d i a g r a m
R ~ Z Z Is I ..... S k _ 1 ]
- - 9
R ~ Z Z It I ..... tk_1]
I
;
$
a
1
~ a® z z [~k]
where t l , . . . , t k _ I are i n d e t e r m i n a t e s and Sl,...,Sk_ I are the e l e m e n t a r y
symmetric functions in the t.. The v e r t i c a l maps are induced by sub3
stituting for tl,...,tk_ I the roots of t k - I = o except I. Then
41
~.
3
~
(x) is s y m m e t r i c in the t. and since
]
-t.]
Z [s I ..... Sk_1] C
we see that
Z I t I ..... tk_1]
is an inclusion as a direct summand
~j
~ _ t . (x) ~ R ~ Z Z [s I ..... Sk_1] . But the map at the
J
b o t t o m is an injection too b e c a u s e Z
>Z [ ~ k] : n| ) n is a direct
injection~We
call it the Bott c a n n i b a l i s t i c class 8 k. The following
is immediate from the definition.
Proposition
3.7.2.
(i) If x is o n e - d i m e n s i o n a l then
ek(X).. = I + x +...
(ii) If x,y ~ P(R)
= k
k-1
then
8k(x+y)
Since ek(1)
+ x
= ek(X) Ok(Y)
@k is not in general a unit in R so that 8 k cannot
be e x t e n d e d to the additive subgroup g e n e r a t e d by f i n i t e - d i m e n s i o n a l
elements.
In the next section on p-adic
M -rings we find a remedy for
this defect.
3.8. p - a d i c
~ -rin@s.
Let p be a prime number.
Let Z
denote the p-adic integers.
One can
P
define Z
as the inverse limit ring inv lim z/pnz. If A is a finitely
P
g e n e r a t e d abelian group then A ~ z Z is c a n n o n i c a l l y isomorphic to the
P
p-adic c o m p l e t i o n of A
Ap
Tensoring with Z
:= inv lim A/p n A.
is an exact functor on the category of finitely
P
g e n e r a t e d abelian groups. (See A t i y a h - M a c Donald [~I] , Ch. 10 for
42
this and other back ground m a t e r i a l on completions.)
the p-adic topology:
A
carry
P
of zero is
Groups A
a fundamental system of n e i g h b o u r h o o d s
given by the subgroups pnA^. They are c o m p l e t e and H a u s d o r f f
P
topology.
If B is a special
augmented
~ -ring,
then, by definition,
there is a special
~ -ring R such that B = ker e w h e r e e is the augmentation.
Then we have the exact sequence
O-
~B ~
(because e : R---) Z splits)
Z ------)R (D Z
P
P
) Z
) O.
P
We w a n t to define the structure of a special
that B ~
in this
Z
P
is a
A -ideal.
We can extend the
~ -ring on R ~ Z
such
P
A i by c o n t i n u i t y if we
have shown
P r o p o s i t i o n 3.8.1.
The
~ i are continuous with r e s p e c t to the p-adic topology.
Proof.
Given i and N chose k
o such that
k
(~
J ) is d i v i s i b l e by pN for k ~ k °
and I $ j $ i. Then
Aj(pkx)
= p j ( A 1 ( p k ) .....
~ j(pk) ;
A 1 ( x ) .....
Aj(x))
is c o n t a i n e d in pN R if k ~ k ° and I ~ j ~ i b e c a u s e Pi is of weight J
in the first j variables.
~i(y)
for k >/ k .
o
_
~i(x)
=
~
If x-y = pk z then
i
j=1
l i-j (y)
~ j (pkz) ~ pN R
43
T h e p r o o f of t h i s P r o p o s i t i o n
shows
that
if a ~ Z
is the l i m i t of
P
a sequence
(an), a n E Z then
lim
A l(anX)
=
A l ( l i m anX)
=
A l(ax)
and h e n c e
(3.8.2)
After
is f i n i t e l y
is
A t(x) a
a ~ Zp
~t(ax)
=
~t(x) a
x E R
~k(ax)
= a ~k(x).
~ -topology
Proposition
as an a b e l i a n
a p-adic
A = B ~Zp
group;
of some
moreover
o__nnB is f i n e r t h a n the p - a d i c
some e x a m p l e s
of p - a d i c
~ - r i n ~ A to be a
~ -ring B which
we r e q u i r e
that
topology.
~ -rings.
3.8.3.
Let X be a f i n i t e
o__nn~(X)
r e m a r k s we d e f i n e
the c o m p l e t i o n
generated
We n o w d e s c r i b e
the
=
these preliminary
-ring which
the
t(ax)
connected
is c o n t a i n e d
~ -topology
CW-complex.
in the n - t h
is d i s c r e t e
T h e n the n - t h
~ -filtration
skeleton-filtration.
a n d ~(X) ~
Z
In p a r t i c u l a r
is a p - a d i c
~ -rin@.
Proof.
L e t X n be the n - s k e l e t o n
Sn~(X)
is d e f i n e d
to be the k e r n e l
: ~(X)-----~(xn-I).
m e n t x = [E] n
~
i ~ ~ (y) = ~ (i_y)
i m p l i e s the result.
I(G).
~
E is an
(E-n+l)
be the representation
complex numbers.
kernel
=
n
L e t R(G)
)Z
skeleton
of the r e s t r i c t i o n
A n y e l e m e n t of K(X n-l)
(n-~) w h e r e
n
L e t R(G)
on X. T h e n the n - t h
map
is r e p r e s e n t e d
(n-])-dimensional
= O.
filtration
The r e l a t i o n
bundle~
by an eleHence
S n S m & Sn+ m t h e n
r i n g of the f i n i t e g r o u p G o v e r the
: x ~ - 9 d i m x be the a u g m e n t a t i o n
T h e n w e can c o n s i d e r
three
topologies
on R(G):
with
44
(i)
Proposition
Let
G be
(ii)
The
(iii)
The
I__nn p a r t i c u l a r
Then
I(G) ~
~
topology.
-topology,
the n e x t
Proposition
Let
~ -tin@ which
a
finite
topologies
Zp is a p - a d i c
3.8.5.
with
I(G)-adic
the
Proposition
I be
topology.
defined
by the
~
-filtration.
3.8.4.
a p-group.
We use
The p - a d i c
(i),
for the p r o o f
Then
the
and
(iii)
coincide.
~ -rin@.
is @ e n e r a t e d
~ -dimension.
(ii),
of
3.8.4.
by ~ f i n i t e
I-adic
number
topology
of
elements
coincides
with
the
-topology.
Proof.
By d e f i n i t i o n
maximal
applied
m+1
~ -filtration
~ -dimension
m+1
(-x) ~
shows
Ikm+1 ~
Proof
of
Put
of the
to the m o n o m i a l s
-
~
m+1
(x) m o d
12
finite
I n c I n. Let m be the
set of g e n e r a t o r s
in the g e n e r a t o r s
we
obtain
Im+ I ~
we have
1 2. By
lie
I. T h e n
in 1 2. S i n c e
induction
one
I k.
By
3.8.5
the t o p o l o g i e s
(ii)
and
(iii)
coincide.
Let
Then
(x-e(x)) m -- xm-e(x) m
because
must
for
3.8.4.
I = I(G).
m = ~G~.
of a g i v e n
we have
m is a p - p o w e r .
~mx
By 3.5.1
~ x m m o d p R(G).
we have
Putting
m o d p R(G)
~mx
these
= e(x)
facts
a n d by
together
3.4.3
we
(iv)
obtain
45
(x-e(x)) m ~ e(x)
- e(x) m ~ O
mod p R(G)
This shows Im = p I, hence the I-adic topology
topology)
is finer than the p-adic topology.
(see A t i y a h
[&]
than the I-adic.
(and therefore the
M -
One can show that mI < 12
), so that the p-adic topology is also finer
(This last fact also follows from l o c a l i z a t i o n theorems
to be p r o v e d later in this lecture.)
As a slight g e n e r a l i z a t i o n of 3.8.4 we m e n t i o n
Proposition
3.8.6.
Let G be a p - g r o u p and X a c o n n e c t e d finite G-CW-complex.
~G(X) ~
Proof
Zp i__ss~ p-adic
~ -rin~.
(~G(X) = kernel of xl
Then
> d i m x)
(sketch).
From the fact that X is a finite G - C W - c o m p l e x one shows by induction
over the number of cells that KG(X)
group.
By 3.8.5 the
topology.
is a finitely g e n e r a t e d abelian
~ - t o p o l o g y coincides w i t h the ~ G ( X ) - a d i c
Let X ° be the e q u i v a r i a n t z e r o - s k e l e t o n of X. The kernel N
of r : KG(X)----)KG(X°)
is n i l p o t e n t
~
R(Gx),
(compare Segal
5.1). M o r e o v e r KG(X°)
M
of X °. Put I = ~G(X).
By A t i y a h - M a c Donald
[~%2]
, Proposition
the product taken over the orbits
[~]
, T h e o r e m 10.11, the
p-adic t o p o l o g y on rI is induced from the p-adic topology on KG(X°).
Hence from 3.8.4 we see that for some t, rI t c
I t c pI + N. But if N k = O then I tk c
prI, or equivalently~
(pI+N) k c pI. This shows that the
I-adic topology is finer than the p - a d i c topology.
Now we continue w i t h the general d i s c u s s i o n of p-adic
A = B ~Zp.
its closure.
If B n is the n-th
~ -ideal of B we let A(n)
From 3.8.1 we o b t a i n that the A(n)
are
M -rings
= Bn ~
~ -ideals.
Zp be
By
46
d e f i n i t i o n of a p - a d i c
A(n), n ~
~-ring
the t o p o l o g y defined by the system
I, is finer than the p-adic topology;
in p a r t i c u l a r this
t o p o l o g y is also H a u s d o r f f and one has
(3.8.7)
A(n)
A ~ inv lim A/A(n).
contains the n-th
the p-adic topology.
(3.8.8)
We observe
A(n)/A(n+1)
because
~
Z
P
need not be closed in
~ -ideal A n of A but A n
~
(Bn/Bn+ I) ~
Zp
is exact on finitely g e n e r a t e d abelian groups.
From 3.4.4
and 3.8.8 we obtain
Proposition
3.8.9.
A(n)/A(n+1)
is a p-adic
For a
e A(n)/A(n+1)
~ -ring.
The p r o d u c t of two elements
is zero.
we have
~k(a )
(-i) k-1 k n-1
=
a
~uk(a) = kna.
we shall show that
k
~
a certain constant c(k,n)
k
(x) =
(3.8.10)
~k
(x+k-1)
acts on A(n)/A(n+1)
as m u l t i p l i c a t i o n with
i n d e p e n d e n t of the ring A. From
one computes
c(k,n)
=
k
~
i=I
(_i)i-I
.n-1
.k-l)
i
(k-i
In order to analyse these numbers we put
~t(x)
= I + fn(t)x
47
where
fn(t ) =
is a c e r t a i n
series with
formal power
~
~j=1
(-I)J-I
series
in Z[[t]].
If we d i f f e r e n t i a t e
fn(t)
is a c t u a l l y
= t(1-t)
a polynomial
fn(t ) =
In p a r t i c u l a r
~
m
The operation
We d e s c r i b e
finer.
cation
group,
the
-- o on A ( n ) / A ( n + 1 )
for m > n.
topology
a r e A(r),
in the f i l t r a t i o n
topology
= 1+a+b+ab
with neighbourhood
(1+pnA+A(n)
is a group.
basis
In ~
specified.
in the p - a d i c
topology
(A(n) I n ~ I) w h i c h
is
I + a, a 6 A, w i t h m u l t i p l i It is a c o m p a c t ,
of I g i v e n by
I).
~ - r i n g s A. A
if not o t h e r w i s e
converges
the set I + A of s y m b o l s
(1+a) (1+b)
equivalently
c(j,n) tj .
of the B o t t m a p 8k for p - a d i c
~ r ~ I ar' w i t h
Therefore
n
~ k"
a variant
it c o n v e r g e s
f'(t)n
of d e g r e e
[ nj=1
s h a l l a l w a y s be the p - a d i c
A series
since
to t we o b t a i n
formula
so that fn(t)
topology
isa geometric
= t .
formally with respect
fn+1(t)
3.9.
For n = I this
sum
f1(t)
recursion
3n - 1 (1_tt
_ )
topological
(1+pnA In ~ 0), or
48
Let k be a natural number prime to p. Consider
is a primitive
adic numbers.
k-th root of unity in an algebraic
The product
Zp [ ~ k~ where
closure of the p-
~ (l-u) over all roots u of t k - I = O
except
I is equal to k, hence a unit in Z . Therefore
P
Zp [ ~ k ] and hence u/(u-1) ~ gp [ ~ k] . The series
~u/(u_1) (a) = I +
converges
u/(u-1)
+
~k(a)
the product
=
~
1-u is a unit in
~2(a) (u/u-l)2 + ...
P
~ u / ( u _ 1 ) (a) in this m u l t i p l i c a t i v e
(3.9.1)
where
~1(a)
in the p-adic topology on I + A ~ Z
an element
~ k
~u/(u_1) (a)
Z
[~ k] hence defines
P
group. We define
E I + A~D Zp[~k~
is taken over all roots of t k - I = O except
I. The
Zp-algebra
Zp [~ k] is free as Z -module with Z -I as a direct summand;
P
P
therefore A = A ~ Z
Zp C A ~ Z Zp [5 k ] as a subring. (As to the
P
P
freeness of the module: Let L ~ Qp [ t] be an irreducible polynomial
with L( ~ k ) = O. Then L divides
is factorial we can choose
GauB-Lemma.
polynomial
We claim:
that a coefficient
according
tk - I = O
(compare
Proposition
3.9.2.
for L a monic polynomial
0k.
since Zp
It ] , by the
P
Then Zp [~ k ] ~ Zp [t]/L and the right-hand side is clearly
a free module.)
k(a)
the cyclotomic
~ k(a) E
in Z
I + A. This follows
of a monomial
in the
to definition
3.9.1
~
i
from the fact
(a) in the expansion
is symmetric
of
in the roots of
3.7).
The ma~
9k
from the additive
: A ---~
I + A
compact group A into the m u l t i p l i c a t i v e
I + A is a continuous
homomorphism.
compact group
It commutes with the Adams operations
49
and m a p s A(n)
into
I + A(n).
Proof.
k is a h o m o m o r p h i s m : d i r e c t l y from 3.3.2 and 3.9.1. S i n c e
n
n
n
n
pN
~
~ k ( p a) = ( ~ k ( a ) ) p
and (l+a) p ~ I +
+ A(N) if ( ) ~
for
I ~ i ~ N we
commutes
with
kA(n)
Remark
C
see
~ i it c o m m u t e s
the
manner:
(m,a) (n,b)
=
(I+A)/(I+B)
and
A(n)
is a
~
j
~ -ideal
can
adjoin
group
Then
Z ~ A define
I + A =
if I + B and
an i d e n t i t y
in the
a multiplication
{(1,a) la ~ A ~ C
I + A are g r o u p s
Z × A.
If
then
~ I + A/B.
~ -rings.
A is s a i d
(3.10.1)
to be o r i e n t e d
~ t(a)
terminology
ideal
we
additive
(mn,mb+na+ab).
Oriented
~ -ring
identity
On the
B C A is an ideal
This
~ k" S i n c e
with
Since
3.9.3.
standard
A
continuous.
N
I + A(n).
If A is a r i n g w i t h o u t
3.10.
~ k is p - a d i c a l l y
that
0 mod p
of the
Proposition
has
special
the
=
if
~1_t(a)
following
augmented
,
a ~ A.
reason:
Suppose
finite-dimensional
A is the
augmentation
A -ring
R. T h e n
3.10.2.
A is o r i e n t e d
of d i m e n s i o n
if and
only
n say,
if for e v e r y
~r(x)
=
finite-dimensional
A n-r(x)
for all
element
x,
r.
Proof.
If 3.10.1
r(x)
=
is s a t i s f i e d
~n-r(x)
for a I and
implies
a 2 then
~ t (x) = t n
for a l - a 2 too.
i/t(x)
The
a n d this
equation
yields
50
~t(x-n)
=
=
1_t (x-n)
=
~ t/(1_t) (x-n)
tn
=
=
A t/(1_t) (x) (1-t)n
(l-t)/t (x)
~ (1_t)/t(x-n)
=
(1-t)/t
(x) (I+(I-t)/t) -n
t n
(1-t)/t (x)
Note
that
n m u s t be
because
A n(x)
~r(x)
=
We call
the
augmentation
= I, so that
~ n-r(x)
from
R an o r i e n t e d
x-n
of an n - d i m e n s i o n a l
~ A.
The
same
element
calculation
x
gives
3.10.1.
A -ring
r
if
(x)
A n-r(x)
=
whenever
x
is n - d i m e n s i o n a l .
Example
Let
3.10.3.
KOG(X)
compact
be the G r o t h e n d i e c k
G-space
vector
bundle
is the
G-vector
orientable
X where
then
ArE
KSOG(X)
is an o r i e n t e d
oriented
=
orientable
X x ~---)X
~
An-rE.
~ E - F
~ -ring
G-vector
Lie
group.
if the
n-th
with
trivial
bundles
over
the
An n-dimensional
G-
exterior
G-action
power
A nE
on ~ .
If E is
ideal
is an
Hence
E KOG(X) I E,F
and the
associated
orientable }
augmentation
~ -ring.
If x is a o n e - d i m e n s i o n a l
~1(x)
of real
G is a c o m p a c t
E is c a l l e d
bundle
ring
=
containing
~°(x)
element
= I. T h e r e f o r e
essentially
only
one
in the o r i e n t e d
should
even-dimensional
think
of
elements.
~ -ring
such
then
a ring
as
51
We now consider a refinement
oriented
of the operations
~ -ring R (a p-adic oriented
@k(resp.
~ -ring A).
Let x ~ R be an element of dimension
2m. Let k be an odd integer.
Let J a set of k-th roots of unity u # I which contains
u,u
km
-I
exactly one element.
~u~
j(1-u)-
2m
(Since k ~
is an algebraic
~ k ) for an
from each pair
I(2) we have u # u
integer because
-I
.) The product
~u~1(1-u)
= k.
Therefore
(3.10.4)
where
km ~ u E
~ k is a primitive
oriented
~ - u (x) (1-u)2m ~ R [ ~ k ]
k-th root of unity.
The fact that R is
implies
(3.10.5)
Therefore
J
A _u(X) (l-u) -2m =
3.10.4
is independent
A _i/u(X) (I-I/u) -2m
of the choice of J. We call this
element
or
ek
Proposition
(x)
3.10.6.
(i)
If x and y are even-dimensional
(ii)
The square __
of
8°r(x)
k
then
e r(x+y)
Or(x ) or
= @k
ek (Y)"
is @k(X) "
--
or
(iii) 8 k (x) E R.
Proof.
(i) follows directly
from the definitions,
see that 8~r(x)
over
Q.
from the analogous property
using
is formally
3.10.5.
of
A t"
(ii) follows
(iii) Using 3.10.5 again one can
invariant under the Galois group of Q( ~ k )
52
If A is an oriented p-adic
~ -ring one defines
the square root of
k by
or
~ k (x) =
(3.10.7)
Using
~t
Proposition
(i)
=
u/u-1
(x)
holds
3.10.8•
=
~ °r(x)
k
The square of
~ k°r(x) E
(iii)
~
u E J
~ 1-t one shows that the following
~ ~ r(x+y)
(ii)
7~
9 or(
k Y ).
or
~ k (x) i s
9k(X).
I + A.
We now compute O °r(z)
k
for a two-dimensional
A _u(Z)
= I - uz + u 2. If we formally write
A_u(Z)
= (1-ux) (1-uy)
(3.10 9)
If we m u l t i p l y
z = x+y with xy = I then
and therefore
~ _u(Z ) (I_u)-2
•
element z. We have
1-ux
=
these expressions
Y
1-u
according
-1
1-u x
-I
1-u
to the definition
of 8or,
k tz),
we obtain
(3.10.10)
ekr(z)
= ky (k-I)/2
7Fu(1-ux)
L(1-u)-1
= y (k-1)/2(1+x+...+xk-1)
= (x(k-1)/2 + x(k-3)/2
This
last expression may also be w r i t t e n
(3.10.11)
xk/2
xi/2
-
x
-
x
-k/2
-I/2
+ ... + y(k-1)/2).
53
where we use this at this point merely as a suggestive
having x I/2 defined.
Actually
8k°r(z)
formula without
is an integral polynomial
in z:
The polynomial
Pk (t) =
is contained
P5(t)
~u@
J (t-(u+u-1))
in Z [t] and has degree
(k-I)/2,
= -I+I+t 2. One has for a 2-dimensional
(3.10.12)
A proof
e k°r(z)
follows
e. g. P3(t)
= 1+t,
z
= Pk(Z)
from the identity
t k - 1 P k ( t 2 + t -2) = (1+t+...+t2k-1)/(1+t)
which can be seen by observing
of degree
that both sides are monic polynomials
2k-2 having the 2k-th roots of unity = +I as roots.
From 3.10.10 one obtains
e k°r(z)
(3.10.13)
3.11. The action of
We consider p-adic
= I +
for a 2-dimensional
~Iz+
9 k on scalar
~(k-I)/2 z .
K -rin~s.
~ -rings A with trivial multiplication,
A(n)/A(n+1)
in P r o p o s i t i o n
3.8.9,
k n and
multiplication
by
~k
~2 z + ... +
z the identiy
on which
~k
like
is m u l t i p l i c a t i o n
by
(-1)k-lk n-1. Then we have seen in 3.8.
that
~t(x)
where
fn(t)
= I + fn(t)x
in an integral polynomial
defined by the recursion
formula
54
fl (t) = t,
Therefore
~k
fn+1 (t) = t(1-t)f'n(t) .
is g i v e n by
9k(X)
=
7~ u
We h a v e to c o m p u t e
the r a t i o n a l
Z
the s u m b e i n g
(1+Xfn(u_--Ul)) = I + x
U
number
I u fn(u_~Ul)
(Galois theory)
fn(u_~Ul
-- ) =: bn(k)
t
t a k e n o v e r the k - t h roots
of u n i t y u # I. Put hn(t)
=
= fn(t_-~tl) .
Proposition
We h a v e
3.11.1.
the f o l l o w i n ~
identity between
formal power
series
in x and t
over Q
n
log(1
+ ~
(1-eX))
x
=
n~, I hn(t)
~
"
(The m e a n i n g
log(1+y)
t
1-t
of the left h a n d side is: Use the p o w e r s e r i e s
2
3
= Y - ~ 2 + ~--3 - ... and r e p l a c e y w i t h the p o w e r s e r i e s
(1-e x) w h i c h
has no c o n s t a n t
term.)
Proof.
We put
n
K(t,x)
where
with
the gn(t)
respect
:= i o g ( 1 +
are c e r t a i n
(1-eX))
power
to t and x and o b t a i n
series
=
~ n2, I gn (t)
x
in t. We d i f f e r e n t i a t e
K(t,x)
55
dK
dt
e
X
teX_l
I
dK
te x
I--r_.
dx
teX_1
hence
dK
dt
t
dK
dx
t
1-t
n
We
apply
this
coefficients,
differential
thus
equation
to
gn(t)
these
If we
are p r e c i s e l y
replace
an i d e n t i t y
the b
n
the
formal
= tgn_ I (t)
with
power
formulas
a k-th
series
root
for the h
of u n i t y
in x o v e r
n
~- n), I b n(k)
= log "~u-1
ekX_ 1
kx
x
xn--[ =
Z
u#1
1-ue x
I
1-u
- log ~
log
1-ue
1-u
(1+eX+...+e
(k-1) x)
eX_1
log
n
=
if we
use
the
~
n>. I (kn-1)
expansion
log
an
x
eX-1
x
_~
n~1
xn
an ~
n
"
.
u ~ I we
Q(i~k).
follows
= log
and c o m p a r e
t
1-t
recursion
t in 3.11.1
between
(k) as
x
n~,1 gn (t) ~
obtaining
gl (t)
and
~
We
obtain
compute
56
The
a
are
n
defined
easily
expressed
yields
tiate
the
t
t
-I
1
=
defining
+
Bernoulli
Bm
tm
~
numbers
B
m
which
are
I
- ~, B2m+1
series
x
I nan
of
the
n-1
n!
-
-
Bn
n
a
= O
with
n
I
I - -x +
"
for
m~
respect
~/_.. n ~ . o
I.
to
x
Bn
If
x we
we
differen-
obtain
n-1
n.I
then
Collecting
these
Proposition
k
now
we
n
~
I
a I = --2
I,
obtain
3.11.2.
: A(n)/A(n+1)
We
for
computations
--~
I + A(n)/A(n+1)
x ~---)I
come
rational
to
oriented
functions
(3.11.3)
The
Z
BI =
an
the
of
m ~ I
immediately
"7
z. n ~
and
terms
by
e
This
in
previous
calculations
m
(kn-1)
B
~
n
~ -rings.
hn(t)
h
+
is
one
proves
(t -I)
=
(-I) m
fm(t)
=
(-I) m
yield
the
x
map
.
From
by
hm(t)
fm(t)
the
recursion
induction
formula
for
57
Proposition
Let A be
or
k
3.11.4.
an o r i e n t e d
p-adic
Remark
coefficients
c has
A(2n-1)
= A(2n)
The
to e x t e n d
a unit.
connection
completion.
only
can
R be
k+2.
~ k+1
This
8 k and
finds
+ c
gives
by i n d u c t i o n
9 k"
finite-dimensional
such
elements
be accomplished
the
an a u g m e n t e d
formal
special
B = ker
generated
elements
one must
have
by passing
x.
In o r d e r
t h a t @k(X)
~ -topology
e(x)
~ t(x-e(x))
I -ring with
we
as an a b e l a i n
to t h e p - a d i c
augmantation
e
: R---) Z
assume:
group
b y x I = I, x 2 , . . , x m
..,m.
on B is f i n e r
= dim x whenever
is a p o l y n o m i a l
~-
dim
is
setting.
e. M o r e o v e r
= d i m x r for r = I,
We t h e n h a v e
moreover
(_l)k(k+1)
least
for
of
sometimes
ideal
_7 ~ r ( a ) (l-t) r o n e
finite-dimensional.
e( x r)
The
at
defined
(i) R is f i n i t e l y
(iii)
~ k +
between
We d e s c r i b e
and augmentation
(ii)
.
I.
it to n e g a t i v e s
This
are
(_1)k
for n ~
=
_
M -filtration
The map e k was
which
B2n
~
x
(k2n-1)
7 zr(a)tr
in
k =
Let
I +
is t h e m a p
3.11.5.
Equating
3.12.
Then
: A ( 2 n ) / A ( 2 n + 1 ) ----~ I + A ( 2 n ) / A ( 2 n + 1 )
x~---~
where
~ -rin~.
(x-e(x))
than
the p-adic
topology.
x is f i n i t e - d i m e n s i o n a l
in t of d e g r e e
~
d i m x.
and
~ d i m x, h e n c e
58
Proposition
3.8.5
-topology.
shows
The
that
the B-adic
ring A = B~
Z
topology
coincides
is a p - a d i c
~ -ring,
with
by
the
(iii)
above.
P
Proposition
3.12.1.
Let
) R ~
i : R
Zp b e
finite-dimensional
the
canonical
x 6 R the
element
map
and
i 8k(X)
(k,p)
= I. T h e n
is a u n i t
in R ~
for
Zp.
Proof.
If d i m x = n t h e n
a n d r -I
exists
I+A C B ~ Z
eSkx
in Zp.
= 8kex
: @ k n = k n.
Therefore
Put
r = k n,
r - l i e k x = l+a,
is a m u l t i p l i c a t i v e
subgroup.
If
then
(r,p)
a 6 B ~
Z . But
P
(1+a) (1+b) = ] t h e n
P
r
-I
(1+b)
We may
e'
: R ~
is the
inverse
now extend
Zp---~
of
iSkx.
~k to a h o m o m o r p h i s m
Zp is i n d u c e d
by e
R --~ Zp ~
: R----gZ then,
R.
If
for x , y
dimensional
e'@k(x-y ) = kex-ey
T h e r e f o r e 8k
induces
ek
Proposition
The
a homomorphism
: B ---~ I + A,
A = B~
Z
.
P
3.12.2.
followin~
dia@ram
is commutative:
B
i/k\,\ Sk
A
9 I+A
~k
(k,p)
= I
finite-
= I
59
Proof.
Let m = dim x. Then
~t(x-m)
-~ t/(t-1)
(x-m)
A _t(m)
and the d e f i n i t i o n of e k and
ek(X)
and hence 8k(x-m)
=
is a p o l y n o m i a l of degree ( m. Using
9k
=
9k
i(x-m).
~ -ring.
A_t(x)
~ k we obtain
3 13. D e c o m p o s i t i o n of p-adic
Let A be a p-adic
=
i(x-m)
This
ek(m)
suffices for the proof.
~ -rin~s
A fundamental
system of n e i g h b o u r h o o d s
zero for the p-adic topology may be taken as
natural numbers ~
are c o n s i d e r e d as a (dense)
(pnA + A(n) I n ~
of
I). The
subset of the p-adic
numbers.
Proposition
3.13.1.
The map
IN x A
----) A
:
(k,a)
~-----9
~k(a)
i__ssu n i f o r m l y continuous.
Proof.
Let M = 2N and suppose p
M
divides s. If X l , . . . , x r have
~'-dimension
one then
k+s
( Z x i) -
~,k(z xi)
N
= p
SI +
SN
= ~ (1+xi)k((1+xi)S-1)
60
where
S. is a s y m m e t r i c
3
Hence
given
N ~
function
I we h a v e
shown
of w e i g h t
that
there
~ j in the x
exists
l
for
j = I,N.
M ~ O such
that
pMls
implies
~k+S(x
for all x w h i c h
verification
Hence
) _
are
~2k(x)
for
special
is u n i f o r m l y
is a h o m o m o r p h i s m
We can n o w
pN A + A(N)
a sum of e l e m e n t s
principle
our m a p
6
in the
extend
of
~ -rings
continuous
second
the m a p
~ -dimension
this
in the
variable
one.
By the
for
all x.
variable.
Since
holds
first
it is u n i f o r m l y
k
(k,a) ~--~ ~
it
continuous.
(a) by c o n t i n u i t y
to a m a p
k
x A ---)A, d e n o t e d w i t h the same symbol. T h e r e f o r e
~
: A--)A
is
P
d e f i n e d for all k ~ Z
as a c o n t i n u o u s h o m o m o r p h i s m . M o r e o v e r we still
P
have
~k
~ 1 =
~kl.
If ~ d e n o t e s the c o m p a c t t o p o l o g i c a l g r o u p of
Z
p-adic
units
then A becomes
By H e n s e l ' s
cyclic
group
splits
into
Lemma
Z
of o r d e r
a topological
c o n t a i n s the roots of x p-I
P
p-1 g e n e r a t e d by d, say. T h e
eigenspaces
of
A =
~
Ai:
is so b e c a u s e
A may
- I = O. T h i s
additive
is a
group
A
d
(3.13.2)
(This
~ -module.
be
p-2.
l=O
A
1
A i
considered
dx:dix
as
Z
[C] module,
where
C is
P
the
cyclic
algebra
group
generated
by T and T a c t i n g
~
d;
[ C] splits c o m p l e t e l y b e c a u s e Z
contains
P
P
d
of unity). S i n c e
~
is a ring h o m o m o r p h i s m we h a v e
(3.13.3)
Z
as
A i Aj C Ai+ j
and the
the
group
(p-1)-th
roots
61
so that A becomes
a Z/(p-1)-graded
duction m o d p
Z ~ ---9 Z/pZ.
p
commutes
~d.
with
Then
ring.
Let U be the kernel
U acts on each group A
of the re-
because
l
u~ U
Put
(3.13.4)
A
(n)
= A. n
i mod
(p-l).
1
A(n).
1
Then
Proposition
Ai(n)
3.13.5.
= Ai(n+1)
Proof.
if
It follows
multiplication
n~
3.14.
i mod
The
says that
d
~
acts
hand,
on Ai(n)/Ai(n+1)
as
by defirJtion of A.,
1
if the q u o t i e n t
isomorphism
to the m a i n
result
is n o n - z e r o
it acts
we must
~ k"
in the theory
~ k is an i s o m o r p h i s m
J' (X) = J" (X) of Adams
bundles
that
(p-l).
is the a l g e b r a i c
computation
3.8.9
by d i. Hence
exponential
We now come
This
from
by d n. On the other
as m u l t i p l i c a t i o n
have
n ~
if k g e n e r a t e s
reformulation
[Z]
, which
of the g r o u p J(X)
of p-adic
~ -rings w h i c h
the p-adic
of A t i y a h - T a l l
[~]
fibre
homotopy
in the
classes
p ~ 2. An integer
~ -ring.
The group
k is a t o p o l o g i c a l
Zp is t o p o l o g i c a l l y
generator
if and only
cyclic
3.14.1.
Let A be a p - a d i c
0 m o d p-1.
~ -ring
(p#2).
Let k g e n e r a t e
Assume
the p-adic
that A(n)
units.
= A(n+1)
Then
if
if k g e n e r a t e s
(Z/p2) ~.
n ~
of v e c t o r
over X.
Let A be a p-adic
Theorem
(p~2).
of the t h e o r e m
is one essential step
of stable
units
for
62
~k
: A ---~
I + A
is an i s o m o r p h i s m .
Proof.
We have
exact
A = inv
rows
lim A / A ( n ) ,
(see 3.9.2
and
3.8.7.
We h a v e
a commutative
~k
[.
--3
A/A(n) ------>0
'k
I~'k
I
$
4,
Therefore
case
1 + A(n)/A(n+l)----~
it s u f f i c e s
~ k is the m a p
of the p a r t i c u l a r
assumption
computed
yon
only
Theorem
in Z
a~
have
the n u m b e r s
Staudt
a unit
we
to p r o v e
ring,
with
3.9.3)
0.---> A(n) /A(n+1) - - - ~ A / A ( n + I )
0-----)
diagram
1 + A/A(n+I)---)1
the
theorem
1+d(k,n)a
hence
where
d(k,n)
the
in 3.11.2
d(k,n)
case
and
(Borewicz-Safarevic
if k is a p - a d i c
for A ( n ) / A ( n + 1 ) .
is an i s o m o r p h i s m
to c o n s i d e r
~
if d(k,n)
n ~
O mod
, p.
.
In that
is a unit.
p-1.
that
(p-l),
By
We have
f r o m the
410)
and n ~ O
0
Zp is i n d e p e n d e n t
it f o l l o w s
[30]
generator
+ A/A(n)---,~
Clausen-
d(k,n)
Actually
is
it
P
has
been
3.11
and
observed
the
by A t i y a h - T a l l
Clausen-von
Staudt
needs
to p r o d u c e
a p-adic
n ~
(p-l)
~k
O
and
example
in a m o m e n t
Example
3.14.2.
Let
R(Z/p;Q)
two
irreducible
~]
, p.
theorem
~ -ring
such
is an i s o m o r p h i s m .
and
thereby
be the G r o t h e n d i e c k
modules:
The
trivial
that
the
results
is not n e c e s s a r y .
that
We
completing
ring
283
A(n)/A(n+1)
shall
One
such
I, and
an
of T h e o r e m
of Q [ Z/p ] - m o d u l e s .
module
only
~ O for
describe
the p r o o f
V
of
which
There
3.14.1.
are
splits
as
63
W + W 2 + ... + W p-I
ideal
I is the free g r o u p
By 3.5 the A d a m s
k
= 0 if p/k.
an i s o m o r p h i s m
and e v a l u a t i n g
(l-t) p -
and
on a s i n g l e g e n e r a t o r
Evaluation
of c h a r a c t e r s
: x|
at g m a p s
=
~i((1-t)
~
acts as k n and as identity.
Hence
~ r(_p)
-topology
for n ~ 0
in I n is
coincide.
(p-l).
only
0
Since k g e n e r a t e s
(p-l)
=
~
(-I) p-I
where
and the p - a d i c
u p ~ u ( ( 1 - ~-/~)
=
k p-I
=
I + kP-1-1
case).
~ k,
(k,p)=1,
p the l o w e s t
(v-l) (p-l)< n $ v(p-1).
topology
~ k on I n / I n + 1 ~
i -u P
the p - a d i c
=
by pn
in this
Zp
for I n / I n + I is the i m a g e of pr.
9 k ( - p ) -I =
(1-u)P
(cyclic
(p-l) b e c a u s e
( ~ p-1(_p))V
=
u
0
~ p-1(_p)
We n o w c o m p u t e
A generator
9k(p)
(short c a l c u l a t i o n )
= 0 for r ~ p and p i z r ( - P )
if n ~
Since
I n / I n + I = Z/p for n ~
(k,p)=1,
g of Z/p g i v e s
p acts on I n / I n + I as m u l t i p l i c a t i o n
I n / I n + I is n o n - z e r o
p o w e r of p a t t a i n a b l e
= id if
+ wit),
the r i g h t h a n d p o l y n o m i a l
(-t) p. T h e r e f o r e
Since
k
) -p. We have
~ t(wi-1)
P-li=1
~
at a g e n e r a t o r
~ p = 0 we see that I n / I n + I is a p - g r o u p
Moreover
x = I + W +...+ wP-I-p.
are g i v e n as follows:
~
for I ~ r ~ p-1.
H e n c e the a u g m e n t a t i o n
operations
I ---) pZ
~t(x ) =
into
o v e r the c o m p l e x n u m b e r s .
and the
In/In+1
Hence
{1_uu)p)-1
_
• p
P
u n i t s m = p- I (kp- I_i ) is an i n t e g e r
prime
to p. We o b t a i n
r-1
~ k ( p r) =
so that
~k(p) p
r-1
=
(1+mp) p
~ k is on I n / I n + I the m a p
I n / I n + I = Z/p this
is an i s o m o r p h i s m .
~
I + m p r m o d pr+1
~k(a)
= 1+ma ~ I + I n / I n + I. S i n c e
64
Remark
3.14.3.
We know
from
3.11.
the map
a~--~ I +
that
(kn-1)
for n = r(p-1)
~ k in t h e e x a m p l e a b o v e is
B
n a a n d t h a t p B n is p - i n t e g r a l . W e o b t a i n
B
m
pB n ~
=_
describe
is o n e
certain
of A i n t o e i g e n s p a c e s
? k induces
instances
~ -ring.
_ m ( P B n ) m o d p.
Staudt
where
Hence
congruences.
the h y p o t h e s i s
of T h e o r e m
operations
described
a splitting
(i = O , I , . . . , p - 2 ) .
Then
a map
3.13.5
Proposition
In 3.13 w e h a v e
A i of A d a m s
k
we
can
apply
: Ao-----9
the Theorem
1 + A°
to it:
3.14.4.
L e t A be a p - a d i c
units.
of t h e v o n
~
is f u l f i l l e d .
Let A be any p-adic
and by
B
((1+mp) r_1 ) - ~n H m r p n
n
-I m o d p. T h i s
We n o w
3.14.1
B
- -n
n
~ (kn-1)
g -rin~,
p ~ 2. L e t k b e
a ~enerator
of t h e p - a d i c
Then
~k
: Ao
) I + A°
is a n i s o m o r p h i s m .
Proposition
Let A be
3.14.5.
a p-adic
A(n)/A(n+1)
~ -rin~.
= 0 for n ~ 0
Assume
that
hu k = id for
(k,p)
= I. T h e n
(p-l).
Proof.
For x ~ A(n)/A(n+1)
we have
x =
~
kx = k n x
a n d kn-1
~ Z~ for n~ O(p-1).
P
65
Let A be a p - a d i c
~ -ring.
AC =
(3.14.6)
{ a I ~ka
A U = A/N,
(I+A)
r
=
~ k commutes
N =
= a, all k ]I
{a
- ~ka
{1+a I ~ k a
(I+A) p =
Since
Put
(I+A)/M,
with
= a, all k ]
M =
the A d a m s
)r
{ (1+a)/
~k(1+a)
operations
: A ------'*
( 9k
I a E A, all k } .
we have
I a 6 A, all k ] .
i n d u c e ~ maps
(I+A) p
(3.14.7)
(gk) p : Ap . . . .
Theorem
)(I+A)~
3.14.8.
I_~f p # 2 and k is a v e n e r a t o r
r
(~ k)
of the p - a d i c
and
units
then the m a p s
3.14.7
( ~ k) r
are isomorphisms.
Proof.
One
first
adic
shows:
If 0 --~ X --9 Z --) Y ---) O is an exact
~ -rings and the T h e o r e m
Z, The f o l l o w i n g
diagram
with
is true
exact
sequence
for X and Y, then
rows
(ker- coker
it is true
sequences)
commutative
0
>X
P
----) Z
I
$
r
i
$
Im
--
r
)Y
i
$
P____)
r
X
ir
$
----~ Z F'----~ Y r - - 9 0
i
$
of p-
&
o---~ (1+x) --~ (1+z) --9 (1+Y) --} (1+x)r-~ (I +Z)r,--> (1+Y)r~o.
is
for
68
One
applies
the
fi~e
lemma.
(To e s t a b l i s h
the ker-
coker
sequence
note
that
I- ~ k
o----~
is e x a c t
x
} x
if k is a g e n e r a t o r
of the p - a d i c
for A ( n ) / A ( n + 1 ) :
For n ~ O ( p - 1 )
for n ~
~k
first
O(p-1)
part
of the p r o o f
inv
that
We now
discuss
cyclic,
The
but
operation
Q/
-I
~
Proposition
the T h e o r e m
equality
"invlim"
are needed.
is a l r e a d y
analogous
group
{~
of
I I
is d e f i n e d
= O,
is t r u e
is true
(A(n)/A(n+1))p
an i s o m o r p h i s m
(I+A)/(I+A(n))
on c o m p a c t
results
2-adic
is;
The Theorem
by
3.14.1.
for all A / A ( n ) .
= O;
By the
From
(inv lim A / A ( n ) ) P
for
is e x a c t
9 xr----~o
units).
A(n)/A(n+1)
lira (A/A(n) [~) =
and an a n a l o g o u s
(Note
itself
) x
e.g.
the T h e o r e m
groups.)
for p = 2 w h e r e
units
for A follows.
oriented
P = Z 2 is not
3 is a g e n e r a t o r .
for p - a d i c
~ -rings,
(topologically)
Since
see
~ -rings
-I E Zp the
3.13.
3.14.9.
I f A is an o r i e n t e d
p-adic
~ -ring
then
-I
=
id.
Proof.
If x has
~ -dimension
I =
so that
~/
-I
AO(2+2x)
I then
=
1+x has
~2(2+2x)
I
(x) = 1+---x - I = x.
sum of o n e - d i m e n s i o n a l
elements.
Hence
Now
~-dimension
=
AI(I+x) 2 =
the P r o p o s i t i o n
apply
I. T h e r e f o r e
(l+x) 2
is true
a "verification
for a
principle".
67
Theorem
3.14.10.
Let A be
of
an o r i e n t e d
~ / { ~
I~
p-adic
(p a n y p r i m e ) .
Let k be a ~enerator
. Then
or
~k
induces
~ -rin~
: A
}
I + A
isomorphisms
or ~
(~k)
or
(gk)C
and
or
~k
is an i s o m o r p h i s m .
If p = 2 then
Proof.
L e t p = 2. W e h a v e
By 3 . 1 1 . 5
this
to show that A(n)/A(n+1)
group
is z e r o
case
because
if n = 2rd,
the Clausen-von
If o n e w a n t s
Staudt
to a v o i d
~ k°r in a s p e c i a l
d' ( k , n ) ~ Z ~.
P
3.15.
Thom-isomorphism
Let G be a compact
compact
class
~kt(E)
t(E).
as
I, t h e n k n = I + 2 r+2 c, c odd,
B
I I " H e n c e (kn-1) __nn
2n = c
d 2B n and b y
2B2m-
in 3 . 1 4 . 2 .
X.
If M(E)
E
2. T h e r e f o r e
Staudt
theorem
as
one
can
of
compute
= d(k,n) ~ Z ~P
3.14.8.
Thom
G-vector
space
bundle
of E w e h a v e
is a f r e e K G (X) - m o d u l e
with
have
type
a uniquely
Z 2.
or
Ok , Ok .
is the
we must
d' (k,n) ~
2d' (k,n)
in the p r o o f
> X a complex
and ~G(M(E))
with
-I m o d
For p # 2
can proceed
group,
Therefore
= Q%(E)t(E)
=
and the maps
Lie
G-space
2. So let n = 2m. T h e n
B
(kn-1) ~ n ~ Z 2 b y 3.1 I . 4 . In
the Clausen-von
So o n e
t(E)~ ~G(M(E))
generator
of Z / { +
theorem
case
hence
the
d odd and r ~
k is a g e n e r a t o r
isomorphically.
if n -_-- I m o d
or
k (a) = I + d' ( k , n ) a a n d d' (k,n)
this
is m a p p e d
a relation
determined
of t h e
element
over
the
Thom
a single
~k(E)~KG(X).
68
Proposition
The
equality
Proof.
Both
addition
@k(E)
O k and ~k
suffices
: ~G(ME)
For
more
vector
this
bundle
is a free
For k odd
@k°r(E)
one
Qk°r(E)
principle
bundles
section.
it
E. Let
Then
s~t(E) = I-E
= s~(~k(E)t(E))
e.
from
= ~ k ( E ) (l-E).
g. at X a c o m p l e x
projective
is a n a l o g o u s
ingredients.
8n w h i c h
has
(Atiyah
then
E, h a v i n g
E
t(E) ~
[10]
) says
~°r(E)
@k
We d e f i n e
If k is odd
Let
but
slightly
> X be a real
a Spin(8n)-structure.
a Thom-class
on t(E).
because
situation
we
also
~OG(M{E))
that
With
and
again
the
~OG(M(E))
by the e q u a t i o n
have
defined
a Spin-structure,
in
3.10
is o r i e n t a b l e .
3.15.2.
and
E a G-vector
bundle
with
In p a r t i c u l a r
Spin(8n)-structure
~r(E)
__is i n . p e n d e n t
the e q u a l i t y
of the
Spin-
for odd k.
Proof.
Using
10.3,
Theorem
3.16.
Comments.
section
certain
the
defines
periodicity
= "°r(E)@k holds.
structure
(look
and h o m o m o r p h i c
splitting
line
zero
: s ~ ~kt(E)
We d e s c r i b e
KOG (X) - m o d u l e
element
for
by the
or
@ k the
and
of d i m e n s i o n
Bott
Prooosition
This
bundles
°r(E)t(E)
(E) : "Qk
~kt
induced
~k(l-E)
topological
equality
= ] + E + . . . + E k-I
Spin-structure
generalized
the
maps
3.7.2.
complicated.
G-vector
the
@k(E)
N o w use
real
be
for b u n d l e
By the
to p r o o f
I-E k :
implies
space).
holds.
are n a t u r a l
) KG(X)
and t h e r e f o r e
This
: ~k(E)
to m u l t i p l i c a t i o n .
therefore
s#
3.15.1.
basic
3.10.10
B on p.
is b a s e d
results
a proof
81 and
is c o n t a i n e d
Theorem
on A t i y a h - T a l l
of A d a m s
[I]
in Bott
C" on p.
[9~]
,
, Proposition
paper
axiomatizes
89.
. That
[2]
[~I]
. The
reader
should
89
also
study
vectors,
the ~ l a t i o n - s h i p
and Hopf-algebras
to i n v e s t i g a t e
properties
the
for
related
9k
to
suitable
u E U.
Given
the
)R
there
x
that
gives
tensor
there
ring
a special
3. S h o w
We
be
of the n u m b e r
also mention
in A t i y a h - S e g a l
an i s o m o r p h i s m
product
way
of
that
exists
1
exists
the
interesting
theoretical
exponential
[13]
on t h e w h o l e
of s p e c i a l
such that
a free
X-rings
the m a p s
~ -ring
~-rings
R and
such
a special
; this
ring
A,B
A---) A ~
G-set
in g e n e r a l
not
induce
let
special.
k-ring
f(u)
~-ring
following
is
(under a
is a s p e c i a l
B, B --9 A ~ B
universal
property:
homomorphism
: x.
and x 6 ~
(splitting
A i(s)
U on one g e n e r a t o r
is a u n i q u e
S m R such
on S i n d u c e s
a
~ -ring
x6 R there
~ -ring
E S are one-dimensional
If S is a f i n i t e
by the
that
if R is s p e c i a l
~ A i(s)
special
is c h a r a c t e r i z e d
IM I : i. The G - a c t i o n
S J
obtained
in a c a n o n i c a l
This
: U
4.
numbers.
It w o u l d
Witt-
A-homomorphisms.
2. S h o w
the
significance
).
groups,
Exercises.
I -ring
f
but
[~[]
formal
hypothesis).
I. S h o w t h a t
are
topological
~-rings
I -rings,
(Hazewinkel
of the B e r n o u l l i
isomorphism
3.17.
between
that
then
x = Xl+...+x n where
principle).
be t h e
set of
a G-action
structure
n-dimensional
on
subsets
Ai(s).
on A(G) . T h i s
M c S with
Show
that
structure
the
is