A remark on N. Kuhn's unbounded strong realization
conjecture
Gérald Gaudens*
8 october 2003
Abstract
N. Kuhn has given several conjectures on the special features satisfied *
*by the singular coho-
mology of topological spaces with coefficients in a finite prime field, as *
*modules over the Steenrod
algebra [Ku]. The so-called realization conjecture was solved in special ca*
*ses in [Ku] and in com-
plete generality by L. Schwartz [Sc2]. The more general strong realization *
*conjecture has been
settled at the prime 2, as a consequence of the work of L. Schwartz [Sc3], *
*and the subsequent work
of F.-X. Dehon and the author [DG ]. In this note, we are interested in the*
* even more general
unbounded strong realization conjecture. We shall prove that it holds at th*
*e prime 2 for the class
of spaces whose cohomology has a trivial Bockstein action in high degrees.
1 Introduction
The singular cohomology of a topological space with coefficients in a finite pr*
*ime field is naturally
endowed with the structure of an unstable algebra over the Steenrod algebra. Th*
*at is, a graded ring
structure together with a compatible action of the Steenrod algebra [Sc1, p. 21*
*].
If an unstable module is isomorphic to the cohomology of some space, we say th*
*at this module is
topologically realizable.
N. Kuhn's conjectures [Ku ] tell that the realizable unstable modules have rat*
*her special algebraic
features. Namely, these conjectures tell us that the action of the Steenrod alg*
*ebra on the cohomology
of a topological space has to be `either very big or very small'.
The first of these conjectures [Ku , Realization conjecture, p. 321] was settl*
*ed by L. Schwartz. We
quote it as the following theorem.
Theorem 1.1 [Sc2, Théorème 0.1] Let X be a topological space. If the singular c*
*ohomology of X
with coefficients in a finite prime field is finitely generated as a module ove*
*r the Steenrod algebra, then
it is finite (as a graded vector space).
The more general strong realization conjecture [Ku , p.324] was only settled a*
*t the prime 2, as
consequence of the work of L. Schwartz [Sc3], and of F.-X. Dehon and the author*
* [DG ]. Let Ud be
the full subcategory of unstable modules annihilated by ~Td+1, the reduced Lann*
*es' functor iterated
(d + 1) times. The subcategory U0 is the subcategory of locally finite modules *
*[Sc1, Sc3].
Theorem 1.2 [Sc2, DG ] Let X be a topological space. If the singular cohomology*
* ~H*X of X with
coefficients in F2 is in Ud for some d, then ~H*X is in U0.
In this note, we turn our attention to the more general unbounded strong reali*
*zation conjecture.
Note. In the sequel, ~H*X always means the modulo 2 singular cohomology of th*
*e space X.
1.1 The unbounded strong realization conjecture
We denote by U the category of unstable modules over the modulo 2 Steenrod alge*
*bra. Every object
M of U is equipped with a natural decreasing filtration, the so-called nilpoten*
*t filtration [Ku , Sc3]:
_____________________M_=_M0___M1_ M2. .M.s Ms+1 . .0. .
*Laboratoire de mathématiques Jean Leray, Université de Nantes
1
We recall some properties of the nilpotent filtration. We say that an unstable*
* module is reduced if
the operator
Sq0 : M -! M, m 7-! Sq|m|m
is injective. If M has a compatible algebra structure, then M is reduced if an*
*d only if it has no
nilpotent elements. For each s, the module Ms=Ms+1is of the form sRsM where Rs*
*M is a reduced
module.
Another important property of the nilpotent filtration is that any unstable mo*
*dule is complete
with respect to its nilpotent filtration. This means that the natural map M -!*
* limsM=Ms is an
isomorphism. This can be seen from the fact that for each s, the module Ms is s*
*-connected.
An unstable module such that M = Ms is called s-nilpotent. A 1-nilpotent modul*
*e is simply called
nilpotent. An element of an unstable module is s-nilpotent provided it spans a *
*s-nilpotent submodule.
The functors ~Tcommute with the nilpotent filtration in the following sense [K*
*u , prop. 2.5, p. 331].
Let M be any unstable module. Let
M = M0 M1 M2. .M.s Ms+1 . .0. .
be the nilpotent filtration of M. Then the induced filtration of ~TM
~TM = ~TM0 ~TM1 ~TM2. .~.TMs ~TMs+1 . .0.
is the nilpotent filtration of ~TM, i.e. for all s,
~TMs = (~TM)s .
Furthermore, by exactness and commutation of ~Twith suspensions, we have a sequ*
*ence of equalities
and natural isomorphisms
sRs~TM = (~TM)s=(~TM)s+1= ~T(Ms)=~T(Ms+1) ~=~T(Ms=Ms+1) = ~T sRsM ~= s~TRsM*
* ,
that is, the functor ~Tand Rs commute for all s.
P `
Let n be an integer. Let n = i=12nibe the diadic expansion of n. We attach t*
*o n the integer
ff(n) = `.
Definition 1.3Let M be a reduced unstable module. We say that M is of weight le*
*ss than t if M is
trivial in all degrees ` such that ff(`) > t.
The weight w(M) of M is the integer (maybe infinite) such that M is of weight *
*less than w(M) but
not w(M) - 1.
A reduced module is of weight 0 if and only if it is concentrated in degree ze*
*ro. In this case, we say
that M is constant.
To understand the definition, we give the following examples.
Example 1.4 Let F(1) be the unstable submodule generated by the non zero degree*
* one class
in ~H*B(Z=2Z) = F2[u]. It is exactly the submodule of primitive elements of th*
*e Hopf algebra
H*B(Z=2Z) = F2[u]. A graded F2-basis for F(1) is given by the elements {u2i}i2*
*N. So F(1) is
zero in degrees ` such that ff(`) is strictly more that one. So the weight w(F(*
*1)) equals 1.
Example 1.5 It is easy to see that w(F(1) n) = n.
Example 1.6 The reduced cohomology ring ~H*B(Z=2Z) = F2[u] is of infinite weigh*
*t.
The next proposition shows the importance of the notion of weight.
Proposition 1.7[FS] A reduced unstable module M is in Un if and only if its wei*
*ght w(M) is less
or equal to n.
2
We can state the unbounded strong realization conjecture [Ku , p. 326] in a sl*
*ightly modified form.
unbounded strong realization conjecture. Let M be an unstable module such tha*
*t RsM is of
finite weight for each s. If M is topologically realizable, then the module RsM*
* is constant for all s.
The original conjecture of N. Kuhn is not stated in terms of weight, but in te*
*rms of functors [Ku ,
p. 325-326]. Let N il be the full subcategory of U of nilpotent unstable module*
*s. One can form the
quotient category U=N il. By [HLS ], it is known that U=N il is equivalent to *
*the full subcategory
F! of analytic functors of the category F of functors from finite dimensional F*
*2-vector spaces to all
F2-vector spaces (with natural tranformations as morphisms). In the category F,*
* one has a notion of
polynomial functor of degree n.
Let q : U -! F! denote the quotient functor U -! U=N il composed with the equi*
*valence of
categories U=N il ~=F!.
The point is that a reduced unstable module is of weight n if and only if q(M)*
* is polynomial of
degree n.
For the reader's convenience, we shall underline the proof of the fact that th*
*e strong realization
conjecture 1.2 is a consequence of the unbounded strong realization conjecture.*
* It relies on the
following lemma.
Lemma 1.8 An unstable module M is in Un if and only if RsM is in Un for all s.
Proof of lemma 1.8. Suppose M is in Un. As Un is a Serre subcategory [Sc3], t*
*he modules Ms and
Ms=Ms+1are also in Un for each s. So Ms=Ms+1= sRsM is in Un. But the functor ~*
*Tcommutes
with suspensions, so RsM is also in Un.
Conversely, if RsM is in Un for all s, by exactness of ~Tit follows that M=Ms *
*(recall that the
nilpotent filtration is decreasing) is in Un for each s. In other words,
~Tn+1(M)=(~Tn+1(M))s = ~Tn+1(M)=~Tn+1(Ms) ~=~Tn+1(M=Ms) = 0
for each s. But ~Tn+1(M) is complete with respect to its nilpotent filtration, *
*hence
~Tn+1(M) = 0 .
It follows that M is in Un.
Now suppose we have an unstable module M which is realizable and is in Un, i.e*
*. such that ~Tn+1M =
0. By the preceding lemma, the module RsM is also in Un. But an unstable module*
* is of finite weight
n if and only if it is in Un.
So, the unbounded strong realization conjecture implies that RsM is constant f*
*or s 0. But a
reduced module is constant if and only if it is in U0. Hence, by the lemma, the*
* module M is in U0
and so the strong realization conjecture holds for M.
Another consequence of lemma 1.8 is to give another form of the unbounded stro*
*ng realization
conjecture:
unbounded strong realization conjecture. Let M be an unstable module such tha*
*t RsM is of
finite weight for each s. If M is topologically realizable, then M is locally f*
*inite.
1.2 The main result
Our main result is the following.
Theorem 1.9 Let X be a topological space such that ~H*X, the modulo 2 cohomolog*
*y of X has a
trivial action of the Bockstein operator in high degrees. If ~H*X is non const*
*ant, the first Rs~H*X
which is non zero is of infinite weight.
So we get in particular that the strong realization conjecture holds for the c*
*lass of spaces such that
the Bockstein acts trivially in high degrees.
3
Theorem 1.10 Let M be an unstable module such that RsM is of finite weight for *
*every s. Suppose
moreover that the Bockstein acts trivially on M in high degrees. If M is topolo*
*gically realizable then
M is locally finite.
One should compare theorem 1.10 with [Ku , theorem 0.1 and theorem 0.3] which *
*was our first
motivation to study the importance of Bocksteins in this situation.
To give background for our result, we recall the important example 0.11 of [Ku*
* , p. 326]. Let X
be the tthbar filtration of BCP1 . Then X is a nilpotent space such that for 1 *
* s < t, the module
Rs~H*X is of finite weight s. Our result shows that all cohomology classes whic*
*h reduce non trivially
in R1~H*X have a non zero Bockstein.
Assume the unbounded conjecture is true. If the cohomology ring ~H*X of a spac*
*e X is not locally
constant, then for some integer s, the reduced module Rs~H*X has to be of infin*
*ite weight. L. Schwartz
has provided precise conjectures [Sc3, conjecture 0.2, conjecture 0.3] about th*
*e value of s in special
cases. Our main theorem says that in the case of the vanishing of Bocksteins in*
* high degrees, the first
non constant Rihas to be of infinite weight. However, the example of N. Kuhn sh*
*ows that in general,
the value of s can be arbitrary high.
To prove theorem 1.9, we shall use, as in [DG ] the theory of profinite spaces*
* to be free of any
finiteness hypotheses. The Theorem 1.9 is a consequence of the more general
Theorem 1.11 Let X be a profinite space such that the modulo 2 cohomology of X *
*has no action of
the Bockstein operator in high degrees. If ~H*X is non constant, the first Rs~H*
**X which is non zero is
of infinite weight.
Theorem 1.11 implies Theorem 1.9 because the cohomology of a space is naturall*
*y isomorphic to
that of its profinite completion (which is a profinite space) as an unstable al*
*gebra [DG , p. 404, section
2.3 ].
2 Proof of the theorem 1.2
The proof of theorem 1.11 is by contradiction. Suppose there exists a profinite*
* space X such that
(i)the cohomology of X is not locally constant and for the lowest d such that R*
*d~H*X is non constant,
the module Rs~H*X is of finite weight,
(ii)the action of the Bockstein is trivial in high degrees in ~H*X.
We shall then find a contradiction. To this end, we use the same line of proof*
* that was used in
[Sc1, Sc2, DG ]. Let us recall how it goes. First, we let d be the minimal inte*
*ger such that Rs~H*X is
non constant.
We necessarily have that d 1 [Ku , prop. 0.8 and cor. 0.9]
We can suppose that ~H*X is d-nilpotent (see the discussion of [DG , beginning*
* of section 7.2]) and
as connected as we want.
It follows from the hypotheses that Rd~H*X is of finite weight f > 0. We shall*
* then perform Kuhn's
reduction [DG , section 7.1]. That is to say, by using Lannes' theory in the fr*
*amework of profinite
spaces, we can suppose that f = 1.
Using this fact, we shall then construct a family (ffi,d)i ~ of special classe*
*s in ~H*X satisfying a
certain set of conditions (Hd).
Then we follow this classes for 0 s d in the iterated loop spaces sX usin*
*g the map induced in
cohomology by the evaluation map Z -! Z. We shall show that the induced class*
*es (ffi,s)i ~in
sX satisfy a similar set of conditions (Hs).
We show that for s = 1, the set of conditions (H1) implies that the cup square*
* of ffi,1is trivial for
i ~, for some integer ~.
The final step is to show that we also have that the cup square of ffi,0is tri*
*vial for i ~. A
carefull reading of [DG ] should convince the reader that the last step of our *
*proof is in some sense
4
already contained in [DG ]. However, [DG ] was not written in a way to make evi*
*dent this assertion.
Furthermore, we feel that we have been a little sketchy at some important point*
*s [DG , p. 425]. So we
offer here to give some of the details lacking in [DG ].
The contradiction follows from the fact that the set of condition (H0) says in*
* particular that the
cup square of ffi,0is non trivial for i ~, thus giving a contradiction.
2.1 Kuhn's reduction with trivial Bocksteins
Let Y be a profinite space. Let RY be the Bousfield-Kan functorial fibrant repl*
*acement of Y ([Mo ],
see also [DG , section 2.4]). We denote by Y the homotopy cofiber (in the homo*
*topical algebra of
profinite spaces) of the natural map
Y -! Map(B(Z=2Z), RY ) .
Let f 1 be the weight of Rd~H*X. We consider the space f-1X.
Lemma 2.1 The space f-1X satisfies
(i)the unstable module Rd~H* f-1X is of weight 1,
(ii)the action of the Bockstein is trivial in high degrees in ~H* f-1X.
Proof. It follows from [DG , section 5] that
~T~H*X ~=~H* X
as unstable modules.
As the nilpotent filtration commutes with ~T, it follows that for all s and t
~TsRt~H*X ~=Rt~Ts~H*X .
On the other hand, we know that M is of weight k if and only if
~Tk+1M = 0 andT~kM 6= 0 .
We only need to prove that the action of the Bockstein is also trivial in ~Tf-*
*1~H*X ~=~H* f-1X.
But this is a consequence of proposition A.2.
2.2 Technicalities
We need the following lemma.
Lemma 2.2 For 1 ` d, the module R`~H*X` is of weight one.
Proof of lemma 2.2. If d = 1 the lemma is clearly true from the hypotheses, o*
*therwise we prove
lemma 2.2 by induction on :
Lemma 2.3 Let Y be a profinite space such that ~H*Y is h-nilpotent, h 2. The*
*n Rh-1~H* Y and
RhY have the same weight.
Proof of lemma 2.3. We use the Eilenberg-Moore spectral sequence which calcul*
*ates ~H* Y from
~H*Y . Its Es,*2-term is a subquotient of ~H*Y s, which is sh-nilpotent. Becaus*
*e the subcategory of
t-nilpotent modules is a Serre subcategory, it happens that E-s,*1is also sh-ni*
*lpotent.
Let Fs~H* Y be the Eilenberg-Moore filtration, whose associated graded is the *
*abutment of the
Eilenberg-Moore spectral sequence. We have
Es,*1= s(Fs=Fs-1)~H* Y ,
as unstable modules, hence (Fs=Fs-1)~H* Y is (hs - s)-nilpotent.
Because the Eilenberg-Moore filtration is convergent, we have that ~H* Y=Fs~H**
* Y is at least (hs -
s)-nilpotent.
5
From the short exact sequence
F-1~H* Y -! ~H* Y -! ~H* Y=F-1~H* Y
and from [DG , Corollaire A.3], we have that Rh-1~H* Y and Rh-1F-1~H* Y are of *
*the same weight.
We know that
Rh-1F-1~H* Y ~=Rh F-1~H* Y=F0~H* Y = RhE-1,*1
and so we need to compare RhE-1,*1and Rh~H*Y .
But E-1,*1is isomorphic to the quotient of ~H*Y by B, the union of the images *
*of the differentials.
The image of the differential dr is easily seen to be at least ((r + 1)(h - 1) *
*+ 2)-nilpotent (see [DG ]).
Hence, the union of the image of the differentials is at least (2h - 1)-nilpote*
*nt. We have a short exact
sequence:
B -! ~H*Y -! E-1,*1.
A new application of [DG , Corollaire A.3] gives that RhE-1,*1and Rh~H*Y are o*
*f the same weight
and the lemma follows.
Lemma 2.4 The module R0F-1~H*X0 is of weight 1. The module R0F-2~H*X0 is of we*
*ight 2.
Proof of lemma 2.4. We have an isomorphism
R0F-1~H*X0 ~=R1 (F-1=F0)~H*X0 ~=R1E-1,*1.
The module E-1,*1is a quotient of ~H*X1 by an at least 2-nilpotent submodule B.
So we have an exact sequence
B -! ~H*X1 -! E-1,*1.
By the lemma 2.2, the module R1~H*X1 is of wght 1 which proves the first asser*
*tion.
The module R0(F-2=F-1)~H*X0 is isomorphic to R2 2(F-2=F-1)~H*X0 = R2E-2,*1. Th*
*e module
E-2,*1is a subquotient of (~H*X1) 2. So we have modules B C (~H*X1) 2 such *
*that C=B = E-2,*1.
The module B is the union of all the images of the differentials and C is the s*
*ubmodule of infinite
cycles. One estimates that B is at least 3-nilpotent. Hence by [DG , corollai*
*re A.2] implies that
R2E-2,*1is isomorphic to R2C. On the other hand the functor R2 preserves monomo*
*rphisms [DG ,
proposition A.1] and so R2E-2,*1is isomorphic to some submodule of R2((~H*X1) 2*
*. We note at last
that
R2(~H*X1) 2 = i+j=2Ri(~H*X1) Rj(~H*X1) = R1(~H*X1) R1(~H*X1)
As R1(~H*X1) is of weight one, the module R2(~H*X1) 2 is of weight 2, and so a*
*re R2E-2,*1and
R0F-1~H*X0.
From the short exact sequence
F-1~H*X0 -! F-2 -! F-2=F-1~H*X0 2E-2,*1
[DG , Corollaire A.3], and the preceding remarks, we find that R0F-2~H*X0 is of*
* weight 2.
2.3 Construction of classes
This lemma is a special case of [DG , proposition 7.2].
Lemma 2.5 Let M be a reduced module of weight 1. Let j be the unity of the ad*
*junction M !
~TM H~*B(Z=2Z). Then j factorizes by the submodule ~TM F(1). Moreover, the ke*
*rnel and cokernel
of
j : M ! ~TM F(1)
are locally finite.
6
We apply this lemma to M = Rd~H*X, which we can suppose to be of weight,1 by p*
*roposition 2.1.
Then it follows that there is a cyclic submodule of the form F(1) 2 in M, gener*
*ated by some ~ff,of
degree 2,. We can suppose , as big as we want. So we pick up some ~ ,.
We lift up s~ff~to a class ff~,dof degree 2~+ d through the epimorphism (~H*X*
*)s -! sRs(~H*X),
and we define recursively, for i ~ i
ffi+1,d= Sq2ffi,d .
We get some classes (ffi,d)i ~satisfying the following set of conditions:
8 i *
>> the Bockstein acts triviallyfonf ,
>: i i,d
fori ~, we haveSq2ffi,d= ffi+1,d.
Now, define for 0 ` d,
X`= d-`X
so that Xd = X and X0 = dX.
The evaluation morphism Y - ! Y induces for any profinite space Y a map evY *
*: ~H*Y - !
~H* Y ~= ~H* Y . For 0 i d - 1, we define recursively classes (ffi,`)i ~in*
* ~H*X` by
ffi,`= evX`+1(ffi,`)
We prove by downward induction that
Proposition 2.6The classes (ffi,`)i ~satisfy, for 0 ` d and i ~:
8 i *
>> the Bockstein acts triviallyfonf ,
>: i i,`
fori ~, we haveSq2ffi,`= ffi+1,`.
Proof of proposition 2.6. The assertion on the degree of (ffi,`)i ~follows fr*
*om the definitions. The
second point is a consequence of the following lemma (see [DG , proposition A.4*
*]).
Lemma 2.7 Let Y be a profinite space such that ~H*Y is `-nilpotent for ` 1. *
*Then ~H*Y is (` - 1)-
nilpotent and the evaluation morphism induces a monomorphism
Rd~H*Y ,! Rd ~H* Y ~=Rd-1 Y
The third and fourth points are consequences of the Steenrod algebra linearity*
* of the evaluation
morphism. Namely, it follows from
(Sq1 ffi,`-1) = Sq1 ffi,`-1= Sq1evX` (ffi,`) = evX` (Sq1 ffi,`) = 0
that the Bockstein acts trivially on ffi,`, and
i 2i 2i 2i
(Sq2 ffi,`-1) = Sq ffi,`= Sq evX` (ffi,`) = evX` (Sq ffi,`) = evX` (ffi*
*+1,`) = ffi+1,`-1
i
shows how Sq2 acts on ffi,`.
2.4 The cup-square of ffi,1is trivial
For ` = 1, the classes ffi,1have degree 2i+ 1, and the unstable algebra structu*
*re gives for i ~,
i+1 1 2i 1
ffi,1[ ffi,1= Sq2 ffi,1= SqSq ffi,1= Sq ffi+1,1= 0 .
So to sum up the situation, we have a profinite space X1 = d-1X and classes (*
*ffi,1)is~uch that
for i ~,
7
(i)the class ffi,1is of degree 2i+ 1 in ~H*X1 ,
(ii)the class ffi,1reduces non trivially in R1(~H*X1) ,
(iii)the Bockstein acts trivially on ffi,1,
i
(iv)we have Sq2ffi,1= ffi+1,1,
(v)the cup square ffi,1[ ffi,1is trivial.
Suppose that we are able to prove that the same set of conditions holds for (f*
*fi,0)i,~0then, we have
the following contradiction
i
0 = ffi,0[ ffi,0= Sq2 ffi,0= ffi+1,06= 0 .
So we need to prove that ffi,0[ ffi,0= 0. To this end, we proceed exactly as i*
*n [DG , Sc3]. But
before, we need some technical results.
Note. We could have define directly the classes ffi,1. However, it would not h*
*ave shorten much the
proof. So we proceed as in [Sc2, Sc3, DG ]
2.5 The cup square ffi,0is trivial
We use the Eilenberg-Moore spectral sequence which relates ~H*X1 to ~H*X0 = ~H**
* X1.
We know that for i ~, the cup square ffi,1[ ffi,1is trivial. So ffi,1[ ffi,*
*1defines an element
of E-1,*2. For degree reasons, the higher differentials coming from E-1,*2are t*
*rivial and so, the cup
square ffi,1[ ffi,1induces a permanent cycle, which never bounds for nilpotence*
* reasons (see [DG ,
section 7.4]). Let wi,`an element of ~H*X0 detected by this permanent cycle.
i
We want to compare Sq2 wi,0to ffi+1,0[ ffi,0. In the very same way than the 1-*
*cycle ffi,1 ffi,1
induces an infinite cycle, the 1-cycle ffi,1 ffi+1,1+ ffi+1,1 ffi,1induces al*
*so a permanent non trivial
cycle. On the other hand, the shuffle product on the E2-term of the Eilenberg-M*
*oore spectral sequence
converges to the cup product on the E1 -term, so this cycle detects the cup pro*
*duct ffi+1,0[ ffi,0. By
Cartan's formula, i
Sq2(ffi,1 ffi,1) = ffi,1 ffi+1,1+ ffi+1,1 ffi,1
is a permanent cycle, andithe compatibility of the Eilenberg-Mooreispectral seq*
*uence with Steenrod
operations shows that Sq2(ffi,1 ffi,1) converges to Sq2 wi,0.
The Cartan formula gives
Sq2i(ffi,0[ ffi+1,0)=Pp+q=2iSqpffi,0[ Sqqffi+1,0
so i i P i
Sq2(ffi,0[ ffi+1,0)=(Sq2ffi,0) [ ffi+1,0+ p<2iSqpffi,0[ Sq2 -pffi+*
*1,0
= ffi+1,0[ ffi+1,0+ Pp<2iSqpffi,0[ Sq2i-pffi+1,0.
Let z be defined by
X p 2i-p
z = Sq2iSq2i!i,0- ffi+1,0[ ffi+1,0= p<2iSqffi,0[ Sq ffi+1,0.
If 0 < p < 2i, then Sqpffi,0is in degree ` such that ff(`) > 1. the element ff*
*i,0is in F-1~H*X0 by
definition, thus so is Sqpffi,0. But R0F-1~H*X0 is of weight one and this impli*
*es that Sqpffi,0reduces
to zero in R0F-1~H*X0. In other words, Sqpffi,0is nilpotent.
i-p
If p = 0, then Sq2 ffi+1,0is of weight greater than 2. The same argument show*
*s that if p = 0, the
i-p
element Sq2 ffi+1,0is nilpotent.
i-p *
* p
So for p < 2i, either Sqpffi,0or Sq2 ffi+1,0is nilpotent and so is the cup pr*
*oduct Sq ffi,0[
i-p
Sq2 ffi+1,0.
8
i
We conclude that for p < 2i, the element Sqpffi,0[ Sq2 -pffi+1,0reduces to zer*
*o in R0F-2~H*X0. So
i 2i
Sq2Sq wi,0and ffi,0[ ffi,0project to equal elements of R0F-2~H*X0.
i 2i *
* 2i 2i
On the other hand, the decomposition of Sq2Sq [Sc3, lemme 5.7, p. 554] implie*
*s that Sq Sq wi,0
projects to an elementiofiweight greater than three in R0F-2~H*X0. But R0F-2~H**
*X0 is of weight 2
by lemma 2.4, so Sq2Sq2wi,0reduces to zero in R0F-2~H*X0. Thus ffi+1,0[ ffi+1,0*
*reduces to zero in
R0F-2~H*X0 for i ~.
In other words, the element ffi,0[ ffi,0is nilpotent for i ~. Thus for some t
Sqt0(ffi,0[ ffi,0) = Sqt0ffi,0[ Sqt0ffi,0= 0
but i+t
Sqt0ffi,0[ Sqt0ffi,0= ffi+t,0[ ffi+t,0= Sq2 ffi+t,0= ffi+t+1,06= 0*
* .
This is a contradiction.
A Trivial Bockstein actions and Lannes' functor
The material of this section is well-known. It is already used in [Ku , proposi*
*tion 1.3, p. 328] and first
proved by M. Winstead [W ]. We thank gratefully J. Lannes who explained us the *
*following proof.
Let M be an unstable module. The notation M n stands for the submodule of M of*
* elements of
degree greater than n. We say that the action of the Bockstein is trivial in de*
*gree greater than n if
Sq1M n = 0.
Proposition A.1Let M be an unstable module. The action of the Bockstein in M is*
* trivial in degree
greater than n if and only if the action of the Bockstein in TM is trivial in d*
*egrees greater than n.
Because ~TM is a submodule of TM we have :
Corollary A.2Let M be an unstable module. If the action of the Bockstein in M i*
*s trivial in degree
greater than n, then the action of the Bockstein in ~TM is also trivial in degr*
*ees greater than n.
Before proving the proposition A.1, we recall (see [LZ] or [Sc1, p. 27]) the d*
*efinition of the double
M of an unstable module M. The module M is the unique unstable module M such*
* that:
(i)the module M is zero in odd degrees,
(ii)for any `, M2`is M`,
(iii)the natural map : M -! M which maps m to m = Sq0m is linear with respe*
*ct to the
Steenrod algebra.
In other words:
Sq2` m = Sq`m .
It is evident from the definition that the action of the Bockstein is trivial *
*on M. Conversely, we
have
Lemma A.3 Let M be an unstable module such that the action of the Bockstein is*
* trivial in each
degree. We denote by Moddand Meventhe odd and even degree parts of M as graded *
*vector spaces.
Then M splits as a module over the Steenrod algebra as
M = Modd Meven .
proof of lemma A.3. This lemma is the consequence of the following facts:
(i)the Steenrod algebra is generated as an algebra by the squares Sqi,
(ii)we have for any odd square the Adem relation
Sq2n+1= Sq1Sq2n .
9
When the action of the Bockstein is trivial, it follows that Moddand Mevenare *
*unstable submodules
and that the vector space decomposition M = Modd Mevenis in fact a Steenrod al*
*gebra module
deccomposition.
Lemma A.4 Let M be a module such that M is zero in odd degrees. Then M is of t*
*he form M1
for a unique unstable module M1. Let M be an unstable module such that M is zer*
*o in even degrees.
Then M is of the form M = M2 for a unique module M2.
proof of lemma A.3. Let us prove the first assertion. It follows from the defi*
*nitions that M1 has to
be defined by M`1= M2`. Furthermore, we also have no choice for the Steenrod al*
*gebra structure
on M1. It remains only to show that this actually defines an action of the Stee*
*nrod algebra, which
amounts to the definition of .
To prove the second assertion, we remark that for any module M concentrated in*
* odd degrees,
the operator Sq0is trivial. But The triviality of this operator is the exactly *
*the obstruction for the
algebraically desuspending an unstable module. So M is of the form M = M0for a*
* unique M0. Now
M0 is concentrated in even degree and by the first part, we have that M0 = M2 *
*for a unique M2.
So, we have:
M = M0= M2 .
We return to the proof of proposition A.1.
Proof of proposition A.1. Let M be an unstable module having trivial action o*
*f the Bockstein in
degrees greater than n.
We have a short exact sequence of unstable modules
M n -! M -! M=M n .
By exactness of the T functor, we get an exact sequence:
TM n -! TM -! TM=M n . (1)
Lannes' T functor admits a natural splitting as
T ~=~T Id
hence the exact sequence 1 splits into two short exact sequences
M n -! M -! M=M n and~TM n -! ~TM -! ~TM=~TM n = ~T(M=M n)
Now M=M n is a bounded module, so ~T(M=M n ) = 0. On the other hand, M n has t*
*rivial action
of Bocksteins and so, by lemma A.3,
M n = (M n )even (M n )odd.
Now, lemma A.4 ensures that
M n = M1 M2
and so
~TM n = ~T( M1 M2) = ( ~TM1 ~TM2)
because the functor ~Tcommutes to suspensions and to .
It follows that ~TM n has trivial action of Bocksteins in each degrees. Finall*
*y, TM = M ~TM has
trivial action of Bocksteins in degrees greater than n.
The converse is a consequence of the aforementioned splitting of the T functor.
Aknowledgements. The author thanks gratefully V. Franjou, J. Lannes, L. Piriou*
*, and L. Schwartz
for their interest and helpful comments. This work was partly done in Munster, *
*while the author was
guest of the SFB 478 `Geometrische '.
10
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Gérald Gaudens
Laboratoire de mathématiques Jean Leray
2, Rue de la Houssinière
44000 Nantes, France
e-mail. Gerald.Gaudens@math.univ-nantes.fr
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