Moduli space of filtered lambda-ring structures
over a filtered ring
Donald Yau
March 8, 2002
Abstract
Motivated by recent works on the genus of classifying spaces of com-
pact Lie groups, here we study the set of filtered ~-ring structures over
a filtered ring from a purely algebraic point of view. From a global per-
spective, we first show that this set has a canonical topology compatible
with the filtration on the given filtered ring. For power series rings R[[*
*x]],
where R is between Z and Q, with the x-adic filtration, we mimic the
construction of the Lazard ring in formal group theory and show that the
set of filtered ~-ring structures over R[[x]] is canonically isomorphic to
the set of ring maps from some ü niversal" ring U to R. From a local
perspective, we demonstrate the existence of uncountably many mutually
non-isomorphic filtered ~-ring structures over some filtered rings, includ-
ing rings of dual numbers over binomial domains, (truncated) polynomial
and powers series rings over torsionfree Q-algebras.
Keywords: Filtered lambda-ring, filtered ring, truncated polynomial ring, power
series ring.
2000 Mathematics Subject Classification: 16W70, 13K05, 13F25.
E-mail: donald@math.mit.edu
Address: Department of Mathematics, MIT 2-588, 77 Massachusetts Avenue,
Cambridge, Massachusetts 02139, USA
________________________________________________________________________
1 Introduction
Let R be a ring, by which we always mean a commutative ring with unit. A ~-
ring (originally called special ~-ring) structure on R in the sense of Grothend*
*ieck
consists of functions ~i:R ! R (i 0) satisfying properties analogous to those
satisfied by exterior power operations. Examples of ~-rings include the complex
representation ring R(G) of a group G, the complex K-theory K(X) of a space
X, and the equivariant K-theory KG (X) of a G-space X. A filtered ring is a
ring R which comes equipped with a multiplicative decreasing filtration {In} of
ideals. A filtered ~-ring is a filtered ring R which is also a ~-ring for which*
* the
filtration ideals are all closed under the operations ~i for i > 0.
The main aim of this paper is to study the set of filtered ~-ring structures
over a filtered ring. Before discussing the results of this paper, let us first*
* provide
1
some motivations for this work, which comes from the author's recent study of
the genus of classifying spaces of compact connected Lie groups.
Recall that the genus of a nilpotent finite type space X consists of the ho-
motopy types of all nilpotent finite type spaces Y such that the p-completions
of X and Y are homotopy equivalent for all primes p and also their rationaliza-
tions are homotopy equivalent. The genus of a classifying space is a very rich
set. Indeed, Møller showed in [10] that the genus of the classifying space BG
of any non-abelian connected compact Lie group G is an uncountable set. The
case when G is SU(2) was first studied by Rector [12]. To understand these
uncountably many homotopically distinct spaces, one would like to find an al-
gebraic invariant which has the properties that (1) it can tell apart different
spaces in the genus of BG, and that (2) it is computable to some extent. An
algebraic invariant with the first property was found by Notbohm in [11], in
which he showed that K-theory ~-rings classify the genus of BG provided G is
a simply-connected compact Lie group. The question, therefore, became how
to compute these K-theory ~-rings and how to use them to answer topological
questions.
The author showed in [14] that K-theory filtered ring is an invariant of the
genus of a space X, provided X is simply-connected and has even and torsionfree
integral homology and a finitely generated power series K-theory filtered ring.
Here the filtration in K-theory comes from a skeletal filtration of the space.
More precisely, if X is such a space and Y is in its genus, then there exists a
filtered ring isomorphism from K(X) to K(Y ). This applies to, for example,
BSp(n) for any n 1, BSU(n) for any n 2, or any finite product of copies
of such spaces and of infinite complex projective space.
An immediate, purely algebraic consequence of these results, when applied to
the spaces BSp(n), is the following: For any positive integer n, the filtered p*
*ower
series ring Z[[x1, . .,.xn]] (the K-theory ring of BSp(n)) on n indeterminates,
where xi has filtration precisely 4i, admits uncountably many mutually non-
isomorphic filtered ~-ring structures.
These results just described also have significant topological implications.
In fact, by combining it with a theorem of Atiyah [1], the author [15] was
able to completely classify spaces in the genus of BSU(2) which admit non-
nullhomotopic maps from infinite complex projective space ("the maximal
torus") and to compute the maps explicitly in these cases. This extends the
classical result of McGibbon [9] and Rector [12] on maximal torus admissibility
of spaces in the genus of BSU(2).
Of course, different filtered ~-ring structures over an underlying filtered *
*ring
are already interesting from a purely algebraic point of view. But with the
above topological results in mind, it becomes clear that they should also be
studied because of their potential to provide information about spaces in the
same genus. With these motivations in mind, we now describe the main results
of this paper.
In the first two sections below we will take on a global perspective, in the
sense that we consider all the filtered ~-ring structures over a given filtered*
* ring
2
R simultaneously. We first address the following moduli problem:
Is there a canonical way to put a topology on the set of filtered ~-
ring structures over a given filtered ring R, which is in some sense
compatible with the filtration on R?
The answer to this question involves the universal ring of Witt vectors W(R),
sometimes called the big Witt vectors on R. The functor W on the category of
rings is actually (part of) a comonad. The following theorem, which gives an
affirmative answer to the above question, is an amalgamation of the results in
x2.
Theorem 1. The following statements hold.
o W restricts to a functor on the category of filtered rings.
o W is (part of) a comonad on the category of filtered rings.
Now regard W as a comonad on the category of filtered rings and let R be a
filtered ring. Then:
o W(R) has a canonical filtered ~-ring structure.
o R is complete (resp. Hausdorff) if and only if W(R) is so.
o There is a canonical bijection between the set of filtered ~-ring structur*
*es
on R and the set of W-coalgebra structures on R. In particular, the former
has a canonical topology as the subspace of the space of continuous maps
from R to W(R) consisting of the filtered ring homomorphisms which
make R into a W-coalgebra.
We will refer to this as the moduli space of filtered ~-ring structures on R.
The proof of this theorem consists of extending known results in the non-
filtered setting to the filtered one. In fact, it is a well-known fact that for*
* any ring
R, a ~-ring structure on R corresponds precisely to a W-coalgebra structure on
R, where W is regarded as a comonad on the category of rings. So what we will
do is to show that this correspondence restricts to the (non-full) subcategory *
*of
filtered rings and filtered ring homomorphisms. As in the non-filtered case, it*
* is
more convenient to work with a functor which is naturally isomorphic to W
via the so-called exponential isomorphism E.
Now one way to obtain filtered rings is to consider power series rings R[[x]]
with the x-adic filtration, where R is any ring and x is an indeterminate with
a fixed positive filtration. For example, the K-theory of BSU(2) is the filtered
power series ring Z[[x]] with x in filtration precisely 4. It is, therefore, na*
*tural
to ask the following representability question:
Is the functor which sends a ring R to the set of filtered ~-ring struc-
tures on R[[x]] (co)representable?
3
The reader will notice that this question is similar to one in formal group the*
*ory:
Is the functor which sends a ring R to the set of formal group laws over it
(co)representable? The affirmative answer to this formal group question is a
well-known result of Lazard, which says that this functor is co-represented by
what is now called the Lazard ring L. The isomorphism from Ring (L, R) (the
set of ring maps from L to R) to the set of formal group laws over R is given
as follows: It sends a ring map f :L ! R to the formal group law over R which
is the push-forward by f of the universal formal group law over L. We will
approach our representability problem in a similar way.
In order to exploit the analogy with formal group theory, we would like to
reduce a filtered ~-ring structure over R[[x]] to a sequence of power series _n*
*(x)
over R (the Adams operations) satisfying the properties:
1._1(x) = x,
2._m (_n(x)) = _n(_m (x)) = _mn (x),
3._p(x) xp (mod pR) for any prime p.
As we will see in x3 below, this is possible provided that we impose the heavy
restriction, Z R Q.
To state our next result, recall that a binomial domain is a Z-torsionfree
ring R for which the binomial symbol
` '
r = r(r_-_1)_._.(.r_-_n_+21)R Q
n n!
lies in R for any element r in R and positive integer n. For example, Z, Z(p)
(p-local integers), Zp (p-adic integers), and torsionfree Q-algebras (and hence
fields of characteristic 0) are binomial domains.
We then have the following theorem, which is a simplified version of the
main result (Theorem 3.3.1) in x3.
Theorem 2. There exists a binomial domain U such that for every ring R
between Z and Q, there is a canonical isomorphism
~=
R :Ring (U, R) -! {filtered~-ring structures overR[[x]]}.
The method of proof of this theorem has some similarity with the construc-
tion of the Lazard ring. Roughly speaking, one adjoins to Z variables corre-
sponding to the coefficients of the power series _p(x) and imposes appropriate
relations, though the actual construction is slightly more complicated than thi*
*s.
It should also be remarked that this ring U is non-trivial, since the power
series filtered ring Z[[x]] admits uncountably many mutually non-isomorphic
filtered ~-ring structures (see [14]).
The above two theorems are our main results about filtered ~-ring structures
over a filtered ring from a global perspective.
4
In x4 we take on a local perspective and consider the points in the moduli
spaces of filtered ~-ring structures over certain filtered rings. This consists*
* of
counting the number of points and, therefore, distinguishing between points in
the moduli spaces. We will consider the following filtered rings:
Ring of dual numbers B["] ("2 = 0), where B is a non-zero binomial domain,
Possibly truncated polynomial rings A[x] and A[x]=xN , where N 2 and
A is a non-zero, Z-torsionfree Q-algebra,
Power series ring A[[x]], where A is a non-zero, Z-torsionfree Q-algebra.
Each of these rings is equipped with the x-adic (or ä- dic) filtration.
Then we have the following theorem, which is extracted from x4 (see Theo-
rems 4.1.1 and 4.2.1).
Theorem 3. Each of these three filtered rings admits uncountably many mutu-
ally non-isomorphic filtered ~-ring structures.
A few remarks are in order. First, the conclusion in the theorem is stronger
than just saying that there are uncountably many points in the moduli spaces.
Second, the proofs of this existence result for the various rings are similar. *
*In
each case, we write down the Adams operations and use Wilkerson's Theorem
3.1.2 to conclude that they come from a ~-ring structure. Third, for the ring of
dual numbers over a non-zero binomial domain, we actually obtain a complete
classification of all of the isomorphism classes of filtered ~-ring structures *
*over
it.
This finishes the presentation of our main results. The rest of the paper is
organized as follows.
In x2 we begin by recalling the definition of a ~-ring and some properties
of the functor on the category of rings and prove Theorem 1 in this set-
ting. Then we transfer the result to the functor W by studying the exponential
isomorphism.
In x3 we first introduce the notion of a (filtered) _-ring, which one can
think of as a (filtered) ring with Adams operations. For a ring R between Z
and Q, a filtered ~-ring over R[[x]] is precisely a filtered _-ring over it. By
mimicking the construction of the Lazard ring, we then construct the ring U
which comes equipped with a ü niversal" _-ring structure over U[[x]] and prove
a more precise version of Theorem 2.
Finally, Theorem 3 is proved in x4. We will end this paper with the observa-
tion (Theorem 4.3.1) that, for any domain R (that is, a commutative ring with
unit without zero divisors), a filtered ~-ring structure on R[[x]] is determine*
*d by
any single Adams operation _p, p prime. This depends on a result (Proposition
4.3.2) about commuting power series which might be of independent interest.
5
Acknowledgment
I am indebted to Lars Hesselholt and Haynes Miller for sharing with me their
knowledge and ideas about ~-rings in particular and mathematics in general.
Theorem 1 and Theorem 2 are the results of questions of, respectively, Lars
Hesselholt and Haynes Miller. I also want to thank Kristine Baxter Bauer, Nitu
Kitchloo, Peter May, and Randy McCarthy for discussions about this work.
2 Moduli space of filtered ~-ring structures
We remind the reader that all rings considered here are commutative with unit,
and so by a ring we mean a commutative ring with unit.
The purpose of this section is to prove Theorem 1 in the Introduction. We
first recall the definition of a (filtered) ~-ring.
2.1 Filtered ~-ring
A filtered ring is a ring R equipped with a decreasing filtration
R = I0 I1 . . .
of ideals, called filtration ideals, such that InIm In+m for any n, m 0.
A filtered ring homomorphism f :R ! S between two filtered rings is a ring
map which preserves the filtrations: f(InR) InSfor all n. The category of
filtered rings and filtered ring homomorphisms is regarded as a subcategory of
the category of rings. The filtration ideals are sometimes suppressed from the
notation.
A ~-ring is a ring R equipped with functions ~i:R ! R (i 0), called
~-operations, such that for any elements r and s in R, the following conditions
hold:
o ~0(r) = 1,
o ~1(r) = r,
o ~n(1) = 0 for all n > 1,
P n
o ~n(r + s) = i=0~i(r)~n-i(s),
o ~n(rs) = Pn(~1(r), . .,.~n(r); ~1(s), . .,.~n(s)),
o ~m (~n(r)) = Pm,n(~1(r), . .,.~mn (r)).
Here the Pn and Pm,n are certain universal polynomials with integer coefficients
(see Atiyah-Tall [2] or Knutson [6] for detail).
A filtered ~-ring is a filtered ring R which is also a ~-ring for which the
filtration ideals are all closed under the operations ~i (i > 0). A filtered ~-
ring map f :R ! S between two filtered ~-rings is a filtered ring map which
commutes with the ~-operations: f~i= ~if for all i.
6
2.2 Comonads and coalgebras
Since several results later on in this section use the notions of a comonad and*
* of
an algebra over a comonad, we will briefly recall the relevant definitions here.
The reader can consult MacLane's book [8] for more information on this topic.
A comonad on a category C is an endofunctor F :C ! C together with
natural transformations j :F ! Idand ~: F ! F 2, called counit and comulti-
plication, such that the following counital and coassociativity conditions hold:
F j O ~ = IdR = jF O ~ and F ~ O ~ = ~F O ~.
The natural transformations ~ and j are often omitted from the notation and
we speak of F as a comonad.
If F is a comonad on a category C, then an F -coalgebra structure on an
object X of C is a morphism , :X ! F X in C such that the following counital
and coassociativity conditions hold:
F , O , = ~ O , and j O , = IdX.
A map of F -coalgebras g :(X, ,X ) ! (Y, ,Y ) consists of a morphism g :X ! Y
in C such that ,Y O g = F g O ,X .
2.3 The functor
Here we recall only those properties of which we will need later on in this
section. A general reference for the functor (in the category of rings) is
Hazewinkel's book [4, Ch. 3].
If A is any ring, then (A), sometimes called the universal ~-ring associate*
*dP
to A, is the ring which as a set consists of the power series f(t) = 1 + i 1a*
*iti
in A[[t]]. AdditionPand multiplication are given as follows: With f as above and
g(t) = 1 + i 1biti, their sum and product, respectively, are defined by
" #
X X
f(t) + g(t)= 1 + arbs ti (where a0 = b0 = 1)
i 1 r+s=i
X
f(t) . g(t)= 1 + Pi(a1, . .,.ai; b1, . .,.bi)ti.
i 1
Here the Pi are the polynomials which appear in the definition of a ~-ring (see
x2.1). If ff: A ! B is a map of rings, then (ff): (A) ! (B) is defined by
X X
1 + aiti 7-! 1 + ff(ai)ti.
i 1 i 1
This makes into an endofunctor on the category of rings.
The functor is actually (part of) a comonad on the category of rings.
Consider the natural transformations of functors on the category of rings
j : ! Id and t: ! 2,
7
P P
which send f = 1 + i 1aiti2 (A) to, respectively, a1 and 1 + i 1~i(f)ti.
Here the ~i for i > 0 are defined by
X
(2.3.1) ~i(f) = 1 + Pj,i(a1, . .,.aij)tj 2 (A),
j 1
in which the Pj,iare the polynomials appearing in the definition of a ~-ring. T*
*he
functor together with these natural transformations, t and j, is a comonad
on the category of rings. Moreover, for any ring A, (A) is a ~-ring where the
~-operations are the ones in (2.3.1).
Now suppose that A is a ring with a certain ~-ring structure. Then one can
define the following map of rings
~t:A ! (A)
X
a 7-! ~i(a)ti
i 0
which makes A into a -coalgebra (that is, jO~t= IdAand (~t)O~t= tO~t).
There is a converse to this. Suppose that , :A ! (A) is the structure map
of a -coalgebra structure on A. Then there are operations ~i on A defined by
the equation X
~t(a) = ~i(a)ti = ,(a).
i 0
The ring A with these operations ~i (i 0) is then a ~-ring. Therefore, for
any ring A, there is a one-to-one correspondence between (resp. isomorphism
classes of) ~-ring structures on A and (resp. isomorphism classes of) -coalgeb*
*ra
structures on A. In the rest of this section, this correspondence will be exten*
*ded
to the filtered ring case.
2.4 Moduli space of filtered ~-ring structures via
In preparation for the main result of this section below, we will first show th*
*at
the functor and the comonad ( , t, j) restrict to the category of filtered
rings.
Lemma 2.4.1. restricts to a functor on the category of filtered rings, where
the filtration ideals on (R) for a filtered ring (R, {In}) are
8 9
< X =
(2.4.2) (In) = : 1 + aiti2 (R): ai2 In for everyi > 0.
i 1 ;
Proof.PTo check that this actuallyPmakes (R) into a filtered ring, let f(t) =
1 + i 1aiti and g(t) = 1 + i 1biti be in (In). Then
X X
f + g = 1 + citi with ci= arbs 2 In.
i 1 r+s=i
(Here we are using the convention a0 = b0 = 1.) So f + g lies in (In).
8
P
Now let h(t) = 1 + i 1diti be in (Im ). Then we have
X
f . h = 1 + Pi(a1, . .,.ai; d1, . .,.di)ti,
i 1
in which each Pi(a1, . .,.ai; d1, . .,.di) is an integral polynomial with each *
*sum-
mand involving at least one of a1, . .,.ai and at least one of d1, . .,.di. Thus
each coefficient Pi(a1, . .,.ai; d1, . .,.di) lies in In+m , and so f . h lies*
* in
(In+m ). This shows that (R) is a filtered ring with the (In) as filtration
ideals.
Finally, if ': R ! S is a filteredPring homomorphism,Pit is clear that the
map ('): (R) ! (S), sending 1 + aiti to 1 + '(ai)ti, preserves the
filtrations on (R) and (S).
We have shown that sends a filtered ring (map) to a filtered ring (map).
Naturality is pretty clear. So restricts to a functor on the category of filt*
*ered
rings. This finishes the proof of the lemma. |___|
From now on whenever we speak of (R) as a filtered ring, we mean the
filtration in (2.4.2).
Theorem 2.4.3. The natural transformations t and j preserve filtrations
when applied to filtered rings. Therefore, ( , t, j) restricts to a comonad on
the category of filtered rings.
Proof. In view of Lemma 2.4.1 and the fact that ( , t, j) is already a comonad
on the category of rings, it suffices to prove the first assertion, which is pr*
*etty
clear for j. To see that t is a filtered ring map when applied to filtered rin*
*gs
(R, {In}), one only has to note that the Pj,iare integral polynomials without
constant terms, and so ~i(f) (i > 0) lies in (In) whenever f 2 (R) does. |__*
*_|
This proof also yields the following result, which is a filtered analog of t*
*he
fact that, for any ring R, (R) has a canonical ~-ring structure.
Corollary 2.4.4. Let (R, {In}) be a filtered ring. Then the filtered ring (R)
with the operations ~i as in (2.3.1)is a filtered ~-ring.
Proof. Indeed, what remains to be proved is that ~i( (In)) (In) for any
i > 0, n > 0, which was already observed in the proof of Theorem 2.4.3. |__*
*_|
The next result shows that the filtered ring (R) has topological properties
similar to those of R; namely, one of them is complete (resp. Hausdorff) if and
only if the other is so. Recall that a filtered ring (R, {In}) is said to be co*
*mplete
(resp. Hausdorff) if the natural map
p: R ! lim-R=In
9
of filtered rings is surjective (resp. injective). Note that this map p is inje*
*ctive
if and only if the intersection \n 1In is 0. This has relevance, for example, in
topology: If X is a space in the genus of a torsionfree classifying space BG of
a simply-connected compact Lie group G, then its integral K-theory K(X) is
complete Hausdorff (see [14]).
Proposition 2.4.5. Let (R, {In}) be a filtered ring. Then (R, {In}) is complete
(resp. Hausdorff) if and only if ( (R), { (In)}) is complete (resp. Hausdorff).
Proof. From the definition of (In), it is easy to see that the condition
"
In = (0) R
n 1
is equivalent to the condition
"
(In) = (0) (R).
n 1
That is, R is Hausdorff if and only if (R) is Hausdorff.
For the assertion about completeness, consider the commutative diagram of
filtered rings and filtered ring maps:
(2.4.6) (R) __fl_//lim (R)= (In)
MM -
MMMM fi|
ffMMM&&MMfflffl||
lim- (R=In)
The map fl is induced by the projection maps (R) ! (R)= (In) and ff and
fi by R ! R=In. It is clear that for each n, the map (R) ! (R=In) is
surjective with kernel (In); thus fi is an isomorphism. Moreover, it is evident
that p: R ! lim-R=In is surjective if and only if ff is so. The fact that fi is*
* an
isomorphism now implies that p is surjective if and only if fl is so. That is, *
*R is
complete if and only if (R) is so.
This finishes the proof of the proposition. |___|
Now we are ready for the main result of this section, which is that the
correspondence between ~-ring structures and -coalgebra structures over a
ring has a filtered analog.
Theorem 2.4.7. Let (R, {In}) be a filtered ring. Then the set of filtered ~-ring
structures on R is canonically isomorphic to the set of -coalgebra structures
on R, where is regarded as a comonad on the category of filtered rings.
Proof. First suppose that R is given a filtered ~-ring structure. Then for an
element r in In, ~i(r) belongs to In for every i 1, so that
X
~t(r) = 1 + ~i(r)ti
i 1
10
lies in (In). Therefore, the map ~t:R ! (R) is a filtered ring homomor-
phism. Since this map (between rings) already makes R into a -coalgebra
when is regarded as a comonad on the category of rings, it also gives rise to*
* a
-coalgebra structure on the filtered ring R when is regarded as a comonad
on the category of filtered rings.
Conversely, suppose that , :R ! (R) is the structure map of a -coalgebra
structure on the filtered ring R. Then it yields a ~-ring structure on R by
declaring that , = ~t. To see that it is actually a filtered ~-ring, we must sh*
*ow
that ~i(In) In for any i > 0, n > 0. Now if r is in In, then
X
,(r) = ~t(r) = 1 + ~i(r)ti
i 1
lies in (In). Thus, ~i(r) belongs to In for every i 1. This shows that the
~-ring structure on R given by , is a filtered ~-ring.
This finishes the proof of the theorem. |___|
By slightly modifying the above proof, one can also classify the isomorphism
classes of filtered ~-ring structures over a filtered ring.
Corollary 2.4.8. The set of isomorphism classes of filtered ~-ring structures on
a filtered ring R is canonically isomorphic to the set of isomorphism classes of
-coalgebra structures on R, where is regarded as a comonad on the category
of filtered rings.
If R is a filtered ring, then the filtration generates a topology on R, so in
this case both R and (R) are topological spaces. The set of continuous maps
from R to (R) is then a topological space with the compact-open topology.
Now Theorem 2.4.7 yields the following result, which is what the title of this
paper refers to.
Corollary 2.4.9. Regard as a comonad on the category of filtered rings. Then
the set of filtered ~-ring structures on a filtered ring R has a canonical topo*
*logy
as the subspace of the space of continuous maps from R to (R) consisting of
the filtered ring homomorphisms which make R into a -coalgebra.
For a filtered ring R, the set of filtered ~-ring structures on R with the
topology in Corollary 2.4.9 is called the moduli space of filtered ~-ring struc*
*tures
on R.
2.5 Moduli space of filtered ~-ring structures via W
We now show that the results in the previous section can be restated with
the functor W in place of . More information about the functor W (in the
category of rings) can be found in Hazewinkel's book [4, Ch. 3].
11
For a ring A, the universal ring of Witt vectors on A, denotedQW(A), is the
ring defined as follows. As a set W(A) is the countable product 1i=1A. There
are functions
wn :W(A) ! A (n 1)
defined by, for an element a = (ai) in W(A), the formula
X n_
wn(a) = dadd.
d|n
If a = (ai) and b = (bi) are elements of W(A), then their sum and product in
the ring W(A) are the unique elements c = a+W b and d = a.W b, respectively,
such that for any n 1 the following equations hold:
wn(c) = wn(a) + wn(b)
wn(d) = wn(a)wn(b).
If ': A ! B is a ring map, then W(f): W(A) ! W(B) is defined by sending
(ai) to ('(ai)). Then W is an endofunctor and the wn :W ! Id are natural
transformations of functors on the category of rings.
The functor W is related to via the following map, called the exponential
isomorphism:
~=
E(A): W(A) -! (A)
Y
(a1, a2, . .).7! (1 + aiti)
i 1
This is a ring isomorphism for any ring A. Actually E :W ! is a natural
transformation of functors on the category of rings. This natural transforma-
tion implies that W is also a comonad on the category of rings, in which the
counit is jE and the comultiplication is (E2)-1 tE. Here E2: W 2! 2 is the
composition
W 2 E-!~= W -E-!~= 2.
The natural transformation E, in fact, gives an isomorphism
~= 2 -1
( , t, j) -! (W, (E ) tE, jE)
of comonads on the category of rings.
The following result will imply immediately that the results in the previous
section, in particular Theorem 2.4.7 and its corollaries, are still valid with
replaced by W everywhere. For a filtered ring (R, {In}), denote by W(In) the
subset of W(R) consisting of elements (ri) with each ri2 In.
Theorem 2.5.1. Let (R, {In}) be a filtered ring and let r = (ri) be an element
of W(R). Then r lies in W(In) if and only if its image under E(R) lies in
(In). Therefore, W(R) with the ideals W(In) is a filtered ring and W and
(W, (E2)-1 tE, jE) restrict to, respectively, a functor and a comonad on the
category of filtered rings. Moreover, E is an isomorphism of comonads on the
category of filtered rings and E(R) is a filtered ring isomorphism.
12
Proof. Note that the image of r under E(R) is
Y
E(R)(r) = (1 + riti) = 1 + r1t + r2t2 + Q3t3 + Q4t4 + . ...
i 1
Here for n 3 the coefficient Qn has the form:
iX j
(2.5.2) Qn = ri1. .r.ik+ rn,
where the sum inside the parentheses is taken over all partitions, i1+. .+.ik =*
* n,
of n with 1 i1 < i2 < . .<.ik < n. This shows that if r lies in W(In), then
its image under E(R) lies in (In).
Conversely, suppose that E(R)(r) lies in (In). From the definition of the
exponential isomorphism E, it is easy to see that the elements r1 and r2 belong
to In. Suppose by induction that rj lies in In for every j < m. Then it follows
from (2.5.2)that Qm - rm lies in In, which implies that rm lies in In since Qm
does.
This finishes the proof of the first assertion. The other assertions follow
from this, Lemma 2.4.1, and the two paragraphs immediately preceding this
theorem. |___|
3 Filtered ~-rings over power series rings
The purpose of this section is to study (co)representability of the functor whi*
*ch
sends a ring R to the set of filtered ~-ring structures over R[[x]]. The main r*
*esult
is Theorem 3.3.1, which implies Theorem 2 in the Introduction. Throughout
this section, the indeterminate x is given a fixed positive filtration d, and R*
*[[x]]
is given the x-adic filtration. First we recall a result of Wilkerson about Ada*
*ms
operations determining a unique ~-ring structure, which we will need later on
in this section and also in the next section.
3.1 Wilkerson's Theorem
Recall that a ~-ring R has Adams operations _n :R ! R (n 1) defined by
the Newton formula
(3.1.1)_n(r)-~1(r)_n-1(r)+. .+.(-1)n-1~n-1(r)_1(r)+(-1)nn~n(r) = 0.
They satisfy the following properties:
o all the _n are ~-ring maps,
o _1 = Id,
o _m _n = _mn = _n_m ,
o _p(r) rp (mod pR) for each prime p and element r in R.
13
If R is a filtered ~-ring, then the corresponding Adams operations are filtered
~-ring maps. There is a partial converse to this. Suppose that (R, {In}) is a
filtered ring with a ~-ring structure in which the Adams operations are filtered
ring maps. If the filtration ideals Im are all closed under dividing by intege*
*rs
(that is, nr 2 Im implies r 2 Im for n, m > 0, r 2 R), then the Newton formula
3.1.1 implies that they are also closed under the operations ~i for i > 0. For
example, the power series ring R[[x]] with the x-adic filtration has the proper*
*ty
that each filtration ideal is closed under dividing by integers.
Adams operations are easier to work with because they are (filtered) ring
maps. So it is tempting to ask if a ring R equipped with endomorphisms _n
(n 1) satisfying the last three properties above actually has a ~-ring struct*
*ure
with the _n as Adams operations. The answer is yes, as the following theorem
of Wilkerson [13, Prop. 1.2] shows, provided that R is torsionfree as a Z-modul*
*e.
Theorem 3.1.2 (Wilkerson). Suppose that R is a torsionfree ring equipped
with ring homomorphisms _n :R ! R for n 1 satisfying the properties:
1._1 = Idand _m _n = _mn ,
2._p(r) rp (mod pR) for each prime p and element r in R.
Then there is a unique ~-ring structure over R with the given _n as Adams
operations.
Since in this section our attention is on Adams operations rather than ~-
operations, the following notion will be useful.
3.2 Filtered _-ring
A _-ring is a ring R equipped with ring homomorphisms _p: R ! R (_pRif we
wish to specify R), p primes, such that
o _p(r) rp (mod pR) for each element r in R and prime p,
o _p_q = _q_p for any primes p and q.
A _-ring map f :R ! S between two _-rings is a ring homomorphism f :R ! S
which commutes with the structure maps: f_pR= _pSf for each prime p.
A filtered _-ring is a _-ring R which is also a filtered ring for which the
structure maps _p are filtered ring homomorphisms. A map between two filtered
_-rings is a filtered ring homomorphism which is also a _-ring map.
Any filtered ~-ring gives rise to a filtered _-ring by using its Adams opera-
tions _p. Conversely, Wilkerson's Theorem 3.1.2 implies that a filtered _-ring
structure on a torsionfree filtered ring R in which the filtration ideals are c*
*losed
under dividing by integers, determines a unique filtered ~-ring structure on R.
14
The following preliminary result gives necessary and sufficient conditions in
order that R[[x]] be a filtered _-ring.
Lemma 3.2.1. Let R be a ring between Z and Q. Then a filtered _-ring (=
filtered ~-ring) structure over R[[x]] determines and is determined by power
series _p(x) in R[[x]], one for each prime p, such that for any primes p and q
the following conditions hold:
o _p(0) = 0,
o _p(x) xp (mod pR),
o _p(_q(x)) = _q(_p(x)).
Proof. The structure maps of a filtered _-ring structure on R[[x]] clearly gives
rise to such a sequence of power series.
Conversely, suppose given a sequence of power series _p(x) satisfying the
above properties. Since any ring endomorphism on R[[x]] must restrict to the
identity map on R, the power series _p(x) can be extended uniquely to filtered
ring endomorphisms _p on R[[x]]. It is clear that these filtered ring maps sati*
*sfy
_p_q = _q_p
for any primes p and q. Also, the hypothesis about _p(x) implies that
_p(ff) ffp (mod pR[[x]])
for any power series ff over R, since this is true when ff is in R and _p is the
identity map on R. Therefore, the power series _p(x) (p prime) determine a
unique filtered _-ring structure on R[[x]]. |___|
3.3 The representability question
Before we state the main result of this section, note that if f :R ! S is a
ring homomorphism and if R[[x]] is equipped with a filtered _-ring structure
in which each of the structure maps _pRrestricts to the identity on R, then it
can be pushed forward to a filtered _-ring structure {_pS} over S[[x]] as follo*
*ws.
(Note that we are using the subscripts R and S for filtered _-ring structures
over R[[x]] and S[[x]], respectively.) Denote by f*(_pR(x)) the power series ov*
*er
S obtained by applying f to each coefficient of _pR(x), and take _pSto be the
unique filtered ring endomorphism on S[[x]] which restricts to the identity map
on S and satisfies
_pS(x) = f*(_pR(x)).
In this case we also write f*(_pR)for the map _pS.
For two rings R and S, the set of ring homomorphisms from R to S is denoted
by Ring (R, S).
We are now ready for the main result of this section, which implies Theorem
2 in the Introduction.
15
Theorem 3.3.1. There exist a binomial domain U and a filtered _-ring struc-
ture {_puniv}over U[[x]] such that the following statements hold.
1.Each _punivrestricts to the identity map on U.
2.For every ring R between Z and Q, the natural map
R :Ring (U, R) -! {filtered~-ring structures overR[[x]]}
f 7-! {f*(_puniv)}
is an isomorphism.
Proof. The ring U is defined to be the quotient
Z {v(p, i,q1, ...,}qn)=J
U = ___________________(Z-torsions).
In other words, take the polynomial ring Z {v(p, i,q1, ...,}qn), divide out by
the ideal J (to be defined below), and then divide out by the ideal generated
by the Z-torsions in the resulting quotient Z {v(p, i,q1, ...,}qn)=J. Here the
v(p, i,q1, ...,aqn)re independent indeterminates, where n 0, i 1, and p, q1,
. . ., qn any primes. The ideal J is generated by the elements w(p, q,al)nd
V (p, i, q1, . .,.qn+1) defined as follows. We first define
(
pv(p, i) ifi 6= p
u(p, i)=
1 + pv(p, p)ifi = p.
The elements w(p, q,al)re then defined by the equation
(3.3.2)
" #i " #i
X X X X X
u(p, i) u(q, j)xj - u(q, i) u(p, j)xj = w(p, q,xl)l.
i 1 j 1 i 1 j 1 l 1
The elements V (p, i, q1, . .,.qn+1) are defined to be
V (p, i, q1, . .,.qn+1)v=qn+1(p, i,q1,-...,vqn)(p, i,q1, ..., qn)
(3.3.3)
- qn+1v(p, i,q1, ...,.qn, qn+1)
Here p, q, q1, . .,.qn+1 are primes, n 0, and l, i 1.
It follows immediately from the definition of the elements V (p, i, q1, . .,*
*.qn+1)
that each element u in U satisfies the congruence condition
(3.3.4) up u (mod pU).
Since U is also torsionfree, Wilkerson's Theorem 3.1.2 implies that it is a bin*
*o-
mial domain.
Now consider the following power series over U[[x]]:
X X
_puniv(x) = u(p,i)xi = (1 + pv(p, p))xp + pv(p, i)xi.
i 1 i 1, i6=p
16
These power series _puniv(x) extend uniquely to filtered ring endomorphisms on
U[[x]] if one insists that each _punivrestricts to the identity on U. To see th*
*at
{_puniv}is a filtered _-ring structure on U[[x]], first note that (3.3.2)implies
that
_puniv(_quniv(x))= _quniv(_puniv(x))
for any primes p and q. So _puniv_qunivand _quniv_punivcoincide as filtered ring
endomorphisms. Moreover, since one has that
_puniv(x) xp (mod pU[[x]]),
the condition
_puniv(ff) ffp (mod pU[[x]]) for everyff 2 U[[x]]
is equivalent to (3.3.4), which we already know is true. Therefore, {_puniv} is*
* a
filtered _-ring structure on U[[x]] in which each structure map restricts to the
identity map on U. This proves the first assertion.
Now we consider the second assertion. To prove the surjectivity of R , let
R be a ring between Z and Q with a given filtered _-ring structure {_p} on
R[[x]] which is determined by the power series (see Lemma 3.2.1)
X
_p(x) = r(p, i)xi.
i 1
We want a ring map f :U ! R which carries the filtered _-ring structure
{_puniv}on U[[x]] to the given one on R[[x]]. We first specify the values of f *
*on
the generators v(p, i)to be
(
r(p, i)=p ifi 6= p
f(v(p, i)) =
r(p, p)- 1=p ifi = p.
This makes sense because
(
0 (mod pR) ifi 6= p
r(p, i)
1 (mod pR) ifi = p.
The values of f on the other generators of U are then inductively determined
by the formula:
qn+1
f v(p, i, q1, ...,-qn)f v(p, i, q1, ..., qn)
(3.3.5) f v(p, i, q1, ...,=qn+1)________________________________q.
n+1
This shows that the map R is surjective.
To show that R is injective, suppose that f and g are two ring maps from
U to R such that
f*(_puniv(x)) = g*(_puniv(x))
for each prime p. We must show that f = g. The hypothesis implies that
f(u(p, i)) = g(u(p, i))
for each prime p and i 1. Since R is torsionfree, this in turn implies that
f(v(p, i)) = g(v(p, i)).
17
The formula (3.3.5)now shows that f and g must agree on all the generators
v(p,i,q1,...,qn)of U, whence f = g.
This finishes the proof of the theorem. |___|
4 Examples
In this final section we study points in the moduli spaces of filtered ~-ring
structures over certain filtered rings. In each case we will exhibit uncountably
many mutually non-isomorphic filtered ~-ring structures, which in particular
implies that the moduli space has uncountably many points. Unlike in [14] where
the existence of uncountably many mutually non-isomorphic ~-ring structures
over the power series ring Z[[x1, . .,.xn]] was obtained by a combination of
topological results, our method here is purely algebraic. We will write down
directly Adams operations and then use Wilkerson's Theorem 3.1.2 to lift them
to a ~-ring structure.
We begin with
4.1 Rings of dual numbers
Recall that a binomial domain is a torsionfree ring B for which the binomial
symbol ` '
b = b(b_-_1)_._.(.b_-_n_+21)B Q
n n!
lies in B for any element b in B and positive integer n. Examples of binomial
domains include the integers, the p-local integers, the p-adic integers, and Z-
torsionfree Q-algebras (and hence fields of characteristic 0).
Let " be a square-zero variable: "2 = 0. The ring B["], known as the ring of
dual numbers over B, is given the ä- dic filtration; that is, I0 = B["], I1 = (*
*"),
and In = (0) for any n 2. Note that B["], as a B-module, is free of rank 2,
and in particular it can be a finitely generated abelian group.
We now show that, despite the "small" size of B["] as a B-module, its moduli
space of filtered ~-ring structures has many points.
Theorem 4.1.1. The filtered ring B["] of dual numbers over any non-zero
binomial domain B admits uncountably many mutually non-isomorphic filtered
~-ring structures.
Proof. Let R denote the filtered ring B["]. We will first show that any sequence
(ap) of elements in B indexed by the set of primes with each ap p-divisible
in B gives rise to a filtered ~-ring structure on R. So suppose that (ap) is
such a sequence. Define an associated sequence of filtered-ring endomorphisms,
_n :R ! R (n 1), as follows. Set _1 = Idand if n has prime decomposition
n = pe11. .p.ekk, then for any elements ff, fi 2 B, we set
_n(ff + fi") = ff + fiae1p1.a.e.kpk".
18
It is clear that the _n are filtered-ring endomorphisms such that
_n_m = _m _n = _mn .
Also, for any prime p we have that
_p(ff + fi")= ff + fiap"
ff (mod pR)
ffp + fip"p (mod pR)
(ff + fi")p (mod pR)
It then follows easily from Wilkerson's Theorem 3.1.2 that there is a unique
filtered ~-ring structure on R with these _n as Adams operations.
To finish the proof, we will now show that filtered ~-ring structures on R
arising from different sequences are not isomorphic. So suppose that two filter*
*ed
~-ring structures S1 and S2 over R corresponding to sequences (ap) and (bp),
respectively, are isomorphic as filtered ~-rings. We claim that ap = bp for each
p. Indeed, if ': S1 ! S2 is a filtered ~-ring isomorphism, then ' must send "
to u" for some unit u 2 B*. Therefore, since
'_p = _p',
we have that
uap" = ubp",
whence ap = bp.
Now since B is a non-zero, characteristic 0 ring, there are uncountably many
distinct sequences (ap) in it with each ap p-divisible. Thus, together with what
we just proved above, we conclude that there are uncountably many distinct
isomorphism classes of filtered ~-ring structures on R.
This finishes the proof of the theorem. |___|
Note that the above argument actually gives a complete classification of all*
* of
the isomorphism classes of filtered ~-ring structures when B is any ring between
Z and Q. To see this, notice that if _p is the Adams operations of a filtered
~-ring structure over B["], then
_p(m=n) = m=n
for any element m=n 2 B. Now since we have that
_p(") = ap" "p = 0 (mod pB)
for some element ap in B, so ap must be p-divisible in B. This means that for
any elements ff and fi in B, one has that
_p(ff + fi") = _p(ff) + _p(fi") = ff + fiap".
Now if n has prime decomposition, n = pe11. .p.ekk, then the fact that _r_s =
_s_r = _rs for any r and s implies that
_n(ff + fi") = (_p1)e1. .(._pk)ek(ff + fi") = ff + fiae1p1.a.e.kpk".
19
Thus, the given filtered ~-ring structure over B["] is among those constructed
in the proof of Theorem 4.1.1. Let us state this as a separate corollary.
Corollary 4.1.2. Let B be any ring between Z and Q. Then a filtered ~-ring
structure on the filtered ring B["] of dual numbers must have, for any prime p,
Adams operation of the form
(4.1.3) _p(a + b") = a + bap"
for some p-divisible element ap in B.
Conversely, let (ap) be a sequence of elements in B indexed by the set of
primes such that ap is p-divisible in B. Then there is a unique isomorphism cla*
*ss
of filtered ~-ring structure over B["] with Adams operations satisfying (4.1.3).
4.2 Possibly truncated polynomial rings and power series
rings
One might wonder whether or not Theorem 4.1.1 still holds if the ring of dual
numbers is replaced by a general truncated polynomial ring A[x]=xN (N 2), a
polynomial ring A[x], or even a power series ring A[[x]]. As we will see shortl*
*y,
there is indeed a corresponding result in each of these cases, but we have to
restrict the ground rings to non-zero, Z-torsionfree Q-algebras. As usual, we
filter the rings A[x]=xN , A[x], and A[[x]] by the x-adic filtration.
Theorem 4.2.1. Let A be any non-zero, Z-torsionfree Q-algebra and let R be
one of the following filtered rings:
o the truncated polynomial ring A[x]=xN (N 2),
o the polynomial ring A[x], or
o the power series ring A[[x]].
Then R admits uncountably many mutually non-isomorphic filtered ~-ring struc-
tures.
Proof. The proof of this theorem proceeds in exactly the same way as in the
proof of Theorem 4.1.1, so we will not give all the details. The only notable
difference is that the congruence relation
_p(f(x)) f(x)p (mod pR)
for any f(x) in R is now trivially true, since p is a unit in A and hence in R.
This is the reason why we need A to be a Q-algebra and not merely a binomial
domain. |___|
It should be remarked that in Theorem 4.2.1 for a polynomial ring, the
assumption about Q-algebra cannot be omitted. Indeed, as shown by Clauwens
[3], the polynomial ring Z[x] admits essentially only two ~-ring structures.
20
4.3 A Hasse principle for power series ~-rings
In this final section we make an observation about distinguishing between non-
isomorphic filtered ~-ring structures over a filtered power series ring.
In Theorem 4.2.1 when R is the filtered power series ring A[[x]], the various
filtered ~-ring structures over it which we constructed are easily distinguisha*
*ble
by considering the leading coefficients in their Adams operations applied to
x. However, there are other situations in which one would like to distinguish
between different filtered ~-ring structures over a filtered power series ring *
*where
the corresponding Adams operations have the same leading coefficients. A case
in point is the genus of the classifying space BSU(2). As the author shows
in [14], spaces in its genus (which there are uncountably many of them up
to homotopy, see [10]) all have the same K-theory filtered ring, Z[[x]] with x
in filtration precisely 4, up to isomorphism. Therefore, for each one of these
K-theory filtered ~-rings, the Adams operation _p, p any prime, satisfies the
congruence condition
_p(x) p2x (mod x2).
But as Notbohm shows in [11], these ~-rings are mutually non-isomorphic. This
raises the question:
How can filtered ~-ring structures over a filtered power series ring be
distinguished when their corresponding Adams operations have the
same leading coefficients?
Here we give an answer to this question when the ground ring is a domain
(that is, commutative ring with units without zero divisors). This applies, for
example, to the situation above when the filtered ring is Z[[x]].
Theorem 4.3.1. Let R be a domain and let S1 and S2 be two filtered ~-ring
structures over the filtered power series ring R[[x]] equipped with the x-adic *
*fil-
tration, where x is in some fixed positive filtration. Denote the Adams operati*
*ons
of S1 and S2 by _n1and _n2, respectively. Assume that for each prime p there
exists an element ffp 2 R, which is neither 0 nor a root of unity, such that
_p1(x) ffpx _p2(x) (mod x2).
If ': S1 ! S2 is a filtered ring homomorphism (resp. isomorphism), then it is
a filtered ~-ring map (resp. isomorphism) if and only if there exists a prime p
such that '_p1= _p2'.
This Theorem is an immediate consequence of the following observation
about commuting power series, which is a slight generalization of a result due
to Lubin [7, Prop. 1.1].
Proposition 4.3.2. Let f(x) and g(x) be power series over a field k. Suppose
that f(x) ffx g(x) (mod x2) for some element ff 2 k which is neither 0 nor
a root of unity. Then for every element c 2 k there exists a unique power series
h(x) over k satisfying the following conditions: (1) h(0) = 0, (2) h0(0) = c, a*
*nd
(3) h(g(x)) = f(h(x)) as power series.
21
Accepting this proposition for the moment, let us prove the theorem.
Proof of Theorem 4.3.1.Since the ö nly if" part is obvious, we now consider
the "if" part.
Note that for any prime q the power series _q1('(x)) and '(_q2(x)) have
the same linear coefficient, both have 0 constant term, and both satisfy the
last condition in Proposition 4.3.2 with g = _p1and f = _p2. Therefore, by
uniqueness we conclude that _q1('(x)) and '(_q2(x)) coincide as power series,
and so '_q1is equal to _q2' as filtered ring endomorphisms on R[[x]].
This proves the theorem. |___|
Since the proof of Proposition 4.3.2 is an elementary degree-by-degree argu-
ment, which is almost exactly the same as that of Lubin's [7, Prop. 1.1], we wi*
*ll
not give the details. The only important point in the argument is that one can
divide by (ffj-ff) in k for any integer j 2, which is possible by the hypothe*
*sis
on ff.
References
[1]M. F. Atiyah, Power operations in K-theory, Quart. J. Math. Oxford 17
(1966) 163-193.
[2]M. F. Atiyah and D. O. Tall, Group representations, ~-rings and the J-
homomorphism, Topology 8 (1969) 253-297.
[3]F. J. B. J. Clauwens, Commuting polynomials and ~-ring structures on
Z[x], J. Pure Appl. Algebra 95 (1994) 261-269.
[4]M. Hazewinkel, Formal groups and applications, Pure and Applied Math-
ematics 78, Academic Press, New York-London, 1978.
[5]K. Ishiguro, J. Møller, and D. Notbohm, Rational self-equivalences of
spaces in the genus of a product of quaternionic projective spaces, J. Math.
Soc. Japan 51 (1999) 45-61.
[6]D. Knutson, ~-rings and the representation theory of the symmetric group.
Lecture Notes in Math. 308, Springer-Verlag, Berlin-New York, 1973.
[7]J. Lubin, Nonarchimedean dynamical systems, Comp. Math. 94 (1994) 321-
346.
[8]S. MacLane, Categories for the working mathematician, 2nd ed., GTM vol.
5, Springer-Verlag, New York, 1998.
[9]C. A. McGibbon, Which group structures on S3 have a maximal torus?,
Lecture Notes in Math. 657 (1978) 353-360.
[10]J. M. Møller, The normalizer of the Weyl group, Math. Ann. 294 (1992)
59-80.
22
[11]D. Notbohm, Maps between classifying spaces and applications, J. Pure
Appl. Algebra 89 (1993) 273-294.
[12]D. L. Rector, Loop structures on the homotopy type of S3, Lecture Notes
in Math. 249 (1971) 99-105.
[13]C. Wilkerson, Lambda-rings, binomial domains, and vector bundles over
CP(1), Comm. Algebra 10 (1982) 311-328.
[14]D. Yau, On adic genus, Postnikov conjugates, and lambda-rings. Preprint,
2001. arXiv:math.AT/0105194
[15]__-, Maps to spaces in the genus of infinite quaternionic projective space,
Progress in Math. (Skye - Scotland, 2001), Birkhäuser Verlag, to appear.
Available at (http://www-math.mit.edu/~donald/papers.html).
23