ON THE BOREL COHOMOLOGY OF FREE LOOP SPACES
IVER OTTOSEN∗
Abstract
LetXbe a space and letK=H∗(X;Fp)wherepis an odd prime. We construct functors¯ and which approximate cohomology of the free loop space Xas follows: There are homomorphisms (K)¯ →H∗( X;Fp)and(K)→H∗(ET×T X;Fp). These are isomorphisms whenXis a product of Eilenberg-MacLane spaces of typeK(Fp, n)forn≥1.
1. Introduction
LetXbe a topological space andRa ring. The circle groupTacts on the free loop space Xby rotation of loops. The associated Borel cohomology groups are called string cohomology ofX[4]. We denote them as follows:
Hst∗(X;R)=H∗(ET×T X;R).
String cohomology as well as non equivariant cohomology of free loop spaces play a central role in geometry and topology. It is however often not possible to compute such cohomology groups.
WhenR = F2 =Z/2, M. Bökstedt and I found functors ofH∗(X)which approximateHst∗(X)and H∗( X)[2]. The purpose of this paper is to gen- eralize these functors to the caseR = Fp = Z/p wherepis any of the odd primes. Certain algebra generators in string cohomology are more difficult to construct in the odd primary case. Hence method and strategy differs from [2]
at various places.
The following application of the functors¯ andwill appear in the near future. There are two Bousfield cohomology spectral sequences. One conver- ging toH∗( X)and the other converging toHst∗(X). TheE2term of the first is isomorphic to the (non Abelian) derived functors of¯ and theE2term of the second is isomorphic to the derived functors of.
∗The author was supported by the European Union TMR network ERB FMRX CT-97-0107:
Algebraic K-theory, Linear Algebraic Groups and Related Structures.
Received January 15, 2002.
Notation. Fix an odd primep. We useFp-coefficients everywhere unless otherwise is specified.Adenotes the modpSteenrod algebra,Uthe category of unstableA-modules andK the category of unstable A-algebras. We let Alg denote the following category. An object in Alg is a non-negatively gradedFp-algebraAwith the property that ifa∈Aand|a| =0 thena=ap. The category of differential graded Fp-algebras is denoted DGA. For any A∈Algwe defineσ :A→Fpbyσ (x)= 1 for|x|odd andσ(x)= 0 for
|x|even. We also defineσˆ :A→Fp byσ(x)ˆ = 1−σ(x). The circle group is denotedT.
2. The approximation functor ¯
In this section we define a functor¯ : F →Alg which approximates the cohomology ringH∗( X)when applied toH∗X. HereFis a certain category which lies between K and Alg. The functor¯ lifts to an endofunctor on K which is nothing but an explicit description of Lannes’ division functor (−:H∗(T))K introduced in [5].
Definition2.1. LetF denote the following category. An object inF is an objectA∈Algwhich is equipped with anFp-linear mapλ:A→Awith the following properties:
• |λx| =p(|x| −1)+1 for allx∈A.
• λx=xwhen|x| =1 andλx =0 when|x|is even.
• λ(xy)=λ(x)yp+xpλ(y)for allx, y∈A.
FurthermoreAis equipped with anFp-linear mapβ:A→Aof degree 1 with the following properties:
• β◦β =0.
• β(xy)=β(x)y+(−1)|x|xβ(y)for allx, y∈A.
A morphismf : A → A in F is a morphism inAlg such thatf (λx) = λf (x)andf (βx)=βf (x).
Remark2.2. There are forgetful functorsK → F andF → Alg. For an objectKinK the mapλ: K →Kis defined byλx =P(|x|−1)/2xwhen
|x|is odd. The mapβis the Bockstein operation.
We let (v)denote the object H∗(T)in K. There is an associative and commutative coproductδ : (v) → (v)⊗ (v); v → 1⊗v+v⊗1. It comes from the product onTand has counitγ : (v)→Fpcoming from the unit 1→T.
Let⊥:K →K be the functor given byA→ (v)⊗A. The coproduct and counit above define natural transformationsδ:⊥ → ⊥2andγ :⊥ →Id
such that(⊥, δ , γ )is a comonad. A⊥-coalgebra is an objectKinK equipped with a morphismf :K→ ⊥(K)such that the following diagrams commute:
K−−−→ ⊥(K)f
❅
id ↓γ
K
K −−−−→ ⊥(K)f
↓f ↓δ
⊥(K)−−−−→ ⊥⊥(f ) 2(K).
Examples of⊥-coalgebras are cohomology ofT-spaces.
Proposition2.3.IfKis a⊥-coalgebra with structure mapf :K→ ⊥(K) thenKis a graded commutative DGA with degree−1differentialdgiven by
f (x)=1⊗x+v⊗dx, x∈K.
Furthermore, d(Pix) = Pidx for eachi ≥ 0 and d(βx) = −βd(x). In particulard(λx)=(dx)pandd(βλx)=0.
Proof. By the left of the above diagramsf may be expanded as stated.
By the right diagramd◦d =0. Sincef is a morphism inK we see thatdis Fp-linear, a derivation over the identity and that the stated relations hold.
Proposition2.4. Assume that the functor⊥:K →K has a left adjoint : K → K. Then there is a natural⊥-coalgebra structureη : → ⊥ on. For an objectB ∈K the mapηBis the image of the identity under the composite
HomK((B),(B)) HomK((B),⊥(B))
∼↓
= ∼=↑
HomK(B,⊥(B)) −−−−→δ∗ HomK(B,⊥2(B))
Proof. This is formally the same as the proof of [11] Proposition 3.4.
Definition 2.5. ForA ∈ F we define(A)¯ as the quotient of the free graded commutative and unitalA-algebra on generators
dx for x∈A
where|dx| = |x| −1, by the ideal generated by the elements d(x+y)−dx−dy,
(1)
d(xy)−d(x)y−(−1)|x|xd(y), (2)
d(λx)−(dx)p, (3)
d(βλx).
(4)
Note that(A)¯ is non-negatively graded sinced(xp)=0. We have defined a functor¯ :F →Alg.
Proposition 2.6. The functor ¯ : F → Alg lifts to a functor ¯ : K → K. Explicitely the A-action on (K)¯ is given byθ(x) = θx and θ(dx) = (−1)|θ|d(θx)forx ∈ K andθ ∈ A and the Cartan formula. The differentialdon(K)¯ is gradedA-linear.
Proof. LetdKdenote the gradedFp-vector space given by(dK)n=Kn+1. We writedxfor the element indKcorresponding toxinKhenced(x+y)= dx +dy. We define an A-module structure on dK byPidx = dPix and βdx = −dβx. Let S(dK) denote the free graded commutative algebra on theFp-vector spacedK. By the Cartan formulaS(dK) is anA-algebra and the symmetric productKS(dK)is anA-algebra. By definition(K)¯ = KS(dK)/IwhereI is the ideal generated by
1d(xy)−d(x)y−(−1)|x|xd(y), (5)
1(d(λx)−(dx)p), (6)
1d(βλx).
(7)
We verify thatA ·I ⊆I such that(K)¯ is anA-algebra. We have Pn(1d(xy)−dxy−(−1)|x|xdy)
=
i+j=n
(1d(Pi(x)Pj(y))−dPixPjy−(−1)|x|PixdPjy)
which is inI by (5) since the degree ofPi is even. Further β(1d(xy)−dxy−(−1)|x|xdy)
= −(1d(β(x)y)−dβxy−(−1)|βx|βxdy)
−(−1)|x|(1d(xβy)−dxβy−(−1)|x|xdβy) which is also inIby (5).
In anyA-algebra one has Pi(ap) = (Pi/pa)p wheni = 0 mod p and zero otherwise, since this is a consequence of the Cartan formula alone. So by Lemma 2.7 we have the following relation inS(dK)wheni=0 modp:
Pi(d(λx)−(dx)p)=d(Piλx)−(Pi/pdx)p=d(λPi/px)−(dPi/px)p. Fori =0 modpwe get zero. SoPi applied to an element of the form (6) lies inI. If we applyβto such an element we also land inIby (7). Finally Lemma 2.7 shows thatPi(1d(βλx))∈I and triviallyβ(1d(βλx))∈I.
We verify that(K)¯ ∈U. We must show thatPidx = 0 if 2i > |x| −1.
This holds if 2i > |x|sinceK ∈ U. If 2i = |x|we havePidx = dPix = d(xp) = 0. We must also show that βPidx = 0 when 2i+ 1 > |x| −1.
This holds if 2i + 1 > |x| since K ∈ U and if 2i + 1 = |x| we have βPidx = −dβPix = −dβλx = 0. Since the action on products are by the Cartan formula we have shown that(K)¯ ∈U.
Finally we check that(K)¯ ∈K. The Cartan formula holds by definition.
For|x|odd we haveP|dx|/2(dx)=dλx=(dx)pand the result follows.
Lemma 2.7. For any unstableA-algebra K and x ∈ K the following equations hold.
Piλx=
λ(Ppix), i =0 modp 0, otherwise (8)
Piβλx=
βλ(Ppix), i =0 modp (βPi−1p x)p, i =1 modp
0, otherwise
(9)
Proof. We just prove (8) since the proof of (9) is similar. When|x|is even both sides in the equation are zero. Assume that|x|is odd. By the instability condition Piλx = 0 when 2i > p(|x| −1)+1. When i is divisible by p this inequality implies 2i ≥ p(|x| −1)+por 2pi ≥ |x|and since|x|is odd
2i
p >|x|. SoPi/px=0 and the equation holds in this case. If 2i=p(|x| −1) thenPiλx=λ2x=λ(Pi/px).
Finally assume that 2i < p(|x| −1). Then we can apply the Adem relation:
PiP|x|−12 x =
[pi]
t=0
(−1)i+t
(p−1)|x|−1
2 −t −1 i−pt
Pi+|x|−12 −tPtx.
The instability condition shows thatPi+|x|−12 −tPtx =0 unlessi≤pt. But the binomial coefficient is zero wheni < pt. So we get zero wheni =0 modp and the term corresponding tot =i/pwheni =0 modp.
Proposition2.8. The functor¯ :K →K is left adjoint to⊥:K →K; B→H∗(T)⊗B. Thus there is an equivalence of functors¯ ∼=(−:H∗(T))K. The differentiald : (A)¯ → ¯(A), associated to the natural⊥-coalgebra structure, is given byd(x)=dxforx ∈A.
Proof. We can define natural maps as follows wherex∈A: F : HomK((A), B)¯ ←→HomK(A,⊥(B)):G
F (f )(x)=1⊗f (x)+v⊗f (dx),
G(g)(x)=γ ◦g(x), G(g)(dx)=(α⊗1)◦g(x)
whereα : (v)→Fpis the additive map of degree−1 given byv →1 and 1→0. It is easy to verify thatF◦G=idandG◦F =id. The description of dfollows by using these explicit adjunction formulas in the composite defining ηin Proposition 2.4.
Proposition2.9. For any spaceXthere is a morphism inK (and inDGA) e:(H¯ ∗X)→H∗( X); e(x)=ev0∗(x); e(dx)=dev0∗(x) whereev0 : X →X;ω → ω(1). This morphism is natural inX and it is an isomorphism ifX=K(Fp, n)withn≥0. IfH∗Xis of finite type andY is any space then there is a commutative diagram
(H¯ ∗X)⊗ ¯(H∗Y )−−−−→ ¯∼= (H∗X⊗H∗Y )
e⊗e↓ e↓
H∗( X)⊗H∗( Y )−−−−→∼= H∗( (X×Y )) where the lower horizontal map is the Künneth isomorphism.
Proof. The proof of Proposition 3.9 in [11] goes through with the obvious changes. Thus the isomorphism statement is a consequence of [5] 1.11.
3. The approximation functor
In this section we describe the functor:F →Algwhich gives an approx- imation toH∗(ET×T X)when applied toH∗X. We also define a natural transformationQ:→ ¯which corresponds to the mapH∗(ET×T X)→ H∗( X)induced by the quotient. We do however not go into the topological interpretations here.
Definition3.1. ForA∈Fwe define(A)as the free graded commutative Fp-algebra on generatorsφ(x),q(x),δ(x)forx∈Aanduof degrees
|φ(x)| =p|x| −σ (x)(p−1),
|δ(x)| = |x| −1,
|q(x)| =p|x| −1−σ(x)(p−3),
|u| =2
modulo the ideal generated by
(10) φ(x+y)−φ(x)−φ(y)+σ(x)
p−2
i=0
(−1)iδ(x)iδ(y)p−2−iδ(xy), (11) δ (x+y)−δ(x)−δ(y),
(12) q(x+y)−q(x)−q(y)+ ˆσ (x)
p−1
i=1
(−1)i1
iδ(xiyp−i), (13) (−1)σ(a)σ(c)ˆ δ(a)δ(bc)+(−1)σ (b)ˆσ (a)δ(b)δ(ca)
+(−1)σ(c)ˆσ(b)δ(c)δ(ab), (14) φ(ab)−(−up−1)σ (a)σ (b)φ(a)φ(b),
(15) q(ab)−(−up−1)σ (a)σ (b)(uσ (b)q(a)φ(b)+(−u)σ(a)φ(a)q(b)), (16) q(x)p−up−1q(λx)−φ(βλx),
(17) δ(a)φ(b)−δ(abp)−δ(aλb)+δ(ab)δ(b)p−1, (18) δ(a)q(b)−δ(abp−1)δ(b)−δ(aβλb),
(19) δ(x)u, (20) q(βλx), (21) δ (xp)
wherea, b, c, x, y∈Kand|x| = |y|.
Remark3.2. We have some immediate consequences of these relations:
By (10), (11) and (20) we haveφ(0)=q(0)=δ(0)=0. By (14) and (15) we haveq(an)=nφ(a)n−1q(a)such thatq(ap)=0. By (21) we haveδ(1)=0 so by (21) and (17) we findδ(λb) = δ(b)p. By (18) andδ(1) = 0 we have δ(βλb)=0. By (14), (15) and (17) the algebra(A)is unital with unitφ(1).
Sinceδ(xp)= q(xp) = 0 we see that(A)is non-negatively graded. We have defined a functor:F →Alg.
Lemma3.3. LetK∈F andx, y∈K with|x| = |y| =n. The following relations hold in(K)¯ :
(22)
p−1
i=1
(−1)i+11
id(xiyp−i)
=(x+y)p−1d(x+y)−xp−1dx−yp−1dy, neven
(23)
p−2
j=0
(−1)j+1(dx)j(dy)p−2−jd(xy)
=(d(x+y))p−1(x+y)−(dx)p−1x−(dy)p−1y, nodd.
Proof. We verify (22) and omit the proof of (23) which is similar. Sinced is a derivation we have
p−1
i=1
(−1)i+11
id(xiyp−i)=
p−1
i=1
(−1)i+1(xi−1yp−1dx−xiyp−i−1dy).
By splitting the sum in two at the minus sign and substitutingj =i−1 in the first of the resulting sums we see that the above equals the following:
p−2
j=0
(−1)jxjyp−j−1dx+
p−1
i=1
(−1)ixiyp−i−1dy
=
p−1
t=0
(−1)txtyp−t−1(dx+dy)−xp−1dx−yp−1dy.
For 0≤t ≤p−1 we have thatt! is invertible inFpand also p−1
t
t!=(p−1)(p−2) . . . (p−t)=(−1)tt! modp.
Thus we havep−1
t = (−1)t. Substituting this in the above and using the binomial formula the result follows.
Proposition 3.4. For A ∈ F there is a natural morphism inAlg as follows:
Q:(A)→ ¯(A); φ(x)→xp+λx−x(dx)p−1,
δ(x)→dx, q(x)→xp−1dx+βλx, u→0. Furthermore,Im(Q)⊆ker(d :(A)¯ → ¯(A)).
Proof. We check that the formulas for Qmap the relations (10)-(21) to zero. Formula (23) and the additivity ofx → xpshows that (10) is mapped to zero. It is trivial that (11) is mapped to zero. By (22) and the additivity of x→βλxit follows that (12) is mapped to zero.
Taking the derivative of products and permuting factors we find the follow- ing equations:
d(a)d(bc)=d(a)d(b)c+(−1)σ(b)d(a)bd(c), d(b)d(ca)=(−1)σ (a)(σ (b)+ ˆˆ σ (c))ad(b)d(c)
+(−1)σ(c)+ ˆσ (a)(σ(b)+σ(c))ˆ d(a)d(b)c, d(c)d(ab)=(−1)σ(c)(ˆ σ(a)+σ (b))ˆ d(a)bd(c)
+(−1)σ(a)+ ˆσ (c)(σ (a)+ ˆσ(b))ad(c)d(b).
After some reductions (13) follows from these.
One easily checks that (14) and (15) are mapped to zero in each of the cases σ (a) = σ(b)= 0,σ(a)= σ(b) = 1 andσ (a) = ˆσ(b)= 1. It also follows by small direct computations that (16)–(21) are mapped to zero.
4. The morphismQand cohomology of (A)¯
In this section we define an additive transformationτ :¯ →which corres- ponds to theT-transfer fromH∗( X)toH∗(ET×T X). The mapQgives a morphism from(A)/(u)to the cycles in(A)¯ . Via this a map8similar to the Cartier map [3] is defined. It turns out that(A)/(u)∼= ker(d)when8is an isomorphism. Parts of the material presented here correspond to section 8 in [2]. We letAdenote an object inF.
Definition4.1. LetIδ(A)⊆(A)denote the idealIδ(A)=(δ(x)|x∈A). Proposition4.2.There is anFp-linear map of degree−1as follows τ :(A)¯ →(A); a0da1. . . dan→δ(a0)δ(a1) . . . δ (an), a0→δ(a0) whereai ∈Afor eachi. It has the following properties:
τ(Q(α)β)=(−1)|α|ατ(β) for α ∈(A), β ∈ ¯(A), Q◦τ=d, τ◦Q=0. Note thatτ◦d=0andIm(τ)=Iδ(A).
Proof. We must show that τ is well defined. The relations arising from (1), (3) and (4) are respected since we have the same relations in(K)withd replaced byδ. We must verify that the following relation is respected:
a0da1. . . dai−1d(aiai+1)dai+2. . . dan
=(−1)(k+ ˆσ (ai))σ (ai+1)a0ai+1da1. . . daidai+2. . . dan
+(−1)(k+1)σ (ai)a0aida1. . . dai−1dai+1. . . dan
wherek= |da1. . . dai−1|. It suffices to check that
xd(yz)=(−1)σ (y)σ (z)ˆ xzd(y)+(−1)σ(y)xyd(z) is respected. This follows by (13) after some work with the signs.
By definition we have Q◦τ = 0. By direct computations one sees that τ(Q(α)β) = (−1)|α|ατ(β)whenαequalsφ(x),q(x)orδ(x)andβ equals a0da1. . . danora0. The general case follows from this. In particularτ◦Q=0 sinceτ(1)=0.
Definition4.3. Let L(A) = (A)/(u)and(A) = L(A)/Iδ(A). Ex- plicitely,(A)is the free graded commutativeFp-algebra on generatorsφ(x), q(x)forx∈Aof degrees|φ(x)| =p|x| −σ(x)(p−1),|q(x)| =p|x| −1− σ (p−3)modulo the relations thatφandqare additive and
φ(ab)=(1−σ (a)σ (b))φ(a)φ(b), (24)
q(ab)= ˆσ (b)q(a)φ(b)+ ˆσ (a)φ(a)q(b), (25)
φ(βλx)=q(x)p, (26)
q(βλx)=0. (27)
SinceQ(Iδ(A))⊆d(A)¯ we may define anFp-algebra map8by the follow- ing diagram whereP denotes the canonical projection:
L(A)−−−−→P (A)
Q↓ 8↓
(A)¯ −−−−→ ¯(A)/d(A)¯
Sinced◦Q=0 we have in fact defined a morphism8:(A) →H∗((A))¯ . Remark4.4. Sinceτ◦d =0 we can defineτas a map on(A)/d¯ (A)¯ . We have a commutative diagram as follows:
(A)−−−−→ ¯8 (A)/d(A)¯ −−−−→τ L(A)−−−−→P (A)
↑ Q↓ 8↓
(A)¯ −−−−→ ¯d (A) −−−−→ ¯(A)/d(A)¯ where the compositeτ ◦8vanishes and ker(P )=Im(τ).
Theorem4.5. Assume that the map8 : (A) → H∗((A))¯ is an iso- morphism. Then so isQ:L(A)→ker(d :(A)¯ → ¯(A)).
Proof. The diagram is formally the same as the one above Theorem 8.5 of [2]. So the same diagram chase gives the result.
There is a filtration(A)⊇u(A)⊇u2(A)⊇. . .with associated graded objectGr∗(A)given byGri(A)=ui(A)/ui+1(A). Consider the follow- ing composite of surjective maps:
(A)−−−−→ui· ui(A)−−−−→Gri(A), i≥1
The idealIδ(A)+u(A)⊆(A)is send to zero so we get a surjectiveFp-linear mapui·:(A)→Gri(A).
Proposition4.6. For eachi≥1there is a uniqueFp-linear map8i such that the following diagram commutes:
(A) −−−−−→ui· Gri(A)
8↓ 8i↓
H∗((A))¯ −−−−−→ui⊗− ui⊗H∗((A))¯ If8:(A)→H∗((A))¯ is an isomorphism then
Gr∗(A)∼=ker(d)⊕(u⊗(A)) ⊕(u2⊗(A)) ⊕ · · ·. Proof. The following elements generate theFp-vector spaceGri(A): (28) uiφ(x1) . . . φ(xn)q(xn+1) . . . q(xn+m)+ui+1(A)
wheren, m≥0 andxj ∈Afor allj. (Ifnormequals zero we have an empty product which equals 1 by definition.) We can describe the relations among these generators. Firstly they are additive in each variablexj. Secondly there is a relation corresponding to each of the relations (24)–(27) for example uiφ(x1) . . . φ(xtxt) . . . φ(xn)q(xn+1) . . . q(xn+m)
=(1−σ (xt)σ(xt))uiφ(x1) . . . φ(xt)φ(xt) . . . φ(xn)q(xn+1) . . . q(xn+m) moduloui+1(A). If the map8iexists such that the diagram commutes it must send (28) to
ui⊗8(φ(x1) . . . φ(xn)q(xn+1) . . . q(xn+m)).
But this formula gives a well defined map by the above identification of the relations among the generators.
The mapui⊗ −is an isomorphism so if8is also an isomorphism we see thatui·is injective. By definitionui·is always surjective so the result follows.
Definition4.7. LetnFpdenote the category of non-negatively gradedFp- vector spaces. Define the free functorSF :nFp→F to be the left adjoint of the forgetful functorF →nFp.
Remark4.8. We haveSF(V⊕W)=SF(V )⊗SF(W). Furthermore there is an explicit description as follows
SF(V )=SAlg
V ⊕βV∗≥1⊕
i≥1,ν∈{0,1}
βνλi
βVeven,∗≥2⊕Vodd,∗≥2
whereSAlgdenotes the left adjoint of the forgetful functorAlg→nFp. Theorem4.9. The map8:(A)→H∗((A))¯ is an isomorphism when Ais a free object inF.
Proof. By the results in the appendix section 10 it suffices to show that 8is an isomorphism whenA = Fn = SF(Vn),n≥ 0 whereVn is the free Fp-vector space on one single generatorxnof degreen.
We haveF0 = Fp[x0]/(x0p−x0)and (F¯ 0) = F0 with zero differential such thatH∗((F¯ 0)) = F0. On the other hand(F 0) ∼= F0with generator φ(x0). So8is an isomorphism since8(φ(x0))=x0p=x0.
Further, F1 = (x1)⊗Fp[βx1] withλx1 = x1. Since(dx1)p = dx1we can use the idempotents from Remark 4.11 below to get a splitting
(F¯ 1)=
i∈Fp
ei(F¯ 1).
For eachiwe havedei =0 and(dx1)ei =iei. Alsodβx1=dβλx1=0. Thus d(x1@(βx1)rei)=@i(βx1)rei. It follows thatH∗(ei(F¯ 1))= 0 fori =0 and H∗(e0(F¯ 1))= F1such thatH∗((F¯ 1))= F1. Since8(φ(x1))= x1e0and 8(q(x1)) =βx1we see that8is surjective. The relationsφ(βx1)= q(x1)p and q(βx1) = 0 shows that φ(x1) and q(x1) generate (K) so 8 is also injective.
Assume thatnis even andn ≥ 2. In the following we write [−] for the functor which takes a set to the vector space it generates. We have
Fn =SAlg[xn, βxn, λiβxn, βλiβxn |i≥1]
and we find that(F¯ n)=Fn⊗SAlg[dxn, dβxn]. We change basis such that
the differential becomes easier to describe:
(F¯ n)=SAlg[xn, dxn]⊗SAlg[βxn, dβxn]
⊗SAlg[λiβxn−(dλi−1βxn)p−1λi−1βxn, βλiβxn|i ≥1]. By the Künneth formula we find thatH∗((F¯ n))equals
SAlg[xnp, xnp−1dxn]⊗SAlg[λiβxn−(dλi−1βxn)p−1λi−1βxn, βλiβxn|i ≥1]. The algebra (F n) is generated by the classes φ(xn), φ(λiβxn), q(xn) andq(λiβxn)wherei ≥ 0. We see that8maps these generators to the free generators for the cohomology of(F¯ n). Hence8is an isomorphism. The case wherenis odd andn≥3 is similar.
Lemma4.10. There is an isomorphism of rings as follows α:Fp[x]/(xp−x)→(Fp)p; x→(0,1,2, . . . , p−1)
whereFp[x]is the polynomial ring in one variablexof degree zero and(Fp)p is thep-fold Cartesian product ofFpby itself.
Proof. Use the factorizationxp −x =
n∈Fp(x −n)and the Chinese remainder theorem.
Remark4.11. Leten = α−1(0, . . . ,0,1,0, . . . ,0)with the 1 on thenth place forn∈Fp. Clearlyenem =0 forn=m,e2n = enand
en =1. Also xen=nen. Finding eigenvectors forxf (x)=nf (x)and normalizing one gets the following:
e0=1−xp−1, em= −
p−1
i=1
x m
i
, m=0.
5. Steenrod diagonal elements
In this section we use the functorR+of [6] to define a functorR :K →K. We needR for a description ofgiven in the next section. LetK denote an unstableA-algebra and considerFp[u] with|u| = 2 an object inK by the isomorphismFp[u]∼=H∗(BT).
Definition5.1. Forx ∈K and@ = 0,1 we defineSt@(x)∈ Fp[u]⊗K
by St@(x)=u−@σ (x)ˆ
i≥0
(−up−1)[|x|/2]−i⊗β@Pix.
Note that the terms where the total exponent ofuis negative hasβ@Pix =0.
LetR(K)⊆Fp[u]⊗Kbe the sub-Fp-algebra generated byu⊗1 andSt@(x) for allx ∈Kand@ =0,1.
Theorem 5.2. For each θ ∈ A one has θR(K) ⊆ R(K). Thus R is a functorR :K →K. The explicit formulas are as follows wheren=[|x|/2]
and@=0,1:
PiSt@(x)=
t
(p−1)(n−t)+@σ(x) i−pt
u(p−1)(i−pt)St@(Ptx)
−@(−1)σ (x)
t
(p−1)(n−t)−1+σ (x) i−pt−1
·u(p−1)(i−pt)−1+(2−p)σ (x)St0(βPtx), βSt@(x)=(1−@)uσ(x)ˆ St1(x).
Proof. The formula for the Bockstein operation follows directly by the definition ofSt@(x). We use results from [6] to prove the other formula. By [13] we have thatFp[u, u−1] is anA-algebra withβ =0 and
Piuj = j
i
uj+i(p−1); i, j ∈Z; i ≥0.
Here the following extended definition of binomial coefficients is used where r ∈Randk ∈Z.
r k
=
r(r−1) . . . (r−k+1)
k! , k >0
1, k=0
0, k <0
LetA= (a)⊗Fp[b, b−1] with|a| =2p−3,|b| =2p−2 be theA-algebra introduced in [6] (2.6). That isβa=band
Pi(bj)=(−1)i
(p−1)j i
bi+j,
Pi(abj−1)=(−1)i
(p−1)j−1 i
abi+j−1.
Note that we have changed the names of the generators. In [6] they were named uandvinstead ofaandb. We define an additive transfer map as follows:
τ :A→Fp[u, u−1]; bj →0; abj−1→(−up−1)ju−1. Note that|τ| = −1. A direct verification shows thatτ isA-linear.
A functorR+from the category of gradedA-modules to itself is constructed in [6]. In the case of an unstableA-algebraKit comes with anA-linear map f :R+K→BA⊗Kdefined by [6] (3.1), (3.2). The composite
R+K−−−−→f σ A⊗K−−−−−→Bτ⊗1 BFp[u, u−1]⊗K is given by
sbk⊗x → −s
j
(−up−1)k−ju−1⊗βPjx,
sabk−1⊗x →s
j
(−up−1)k−ju−1⊗Pjx.
Especiallysbn⊗x → −suσ (x)St1(x)andsabn−1⊗x →su−1St0(x)where n=[|x|/2]. The formulas [6] (3.4), (3.5) for theA-action onR+M gives the following formulas for theA-action onuσ(x)St1(x)andu−1St0(x):
Pi(uσ(x)St1(x))=
t
(p−1)(n−t) i−pt
u(p−1)(i−pt)−σ(x)St1(Ptx)
−
t
(−1)σ(x)
(p−1)(n−t)−1 i−pt−1
·u(p−1)(i−pt−σ(x))−1St0(βPtx), Pi(u−1St0(x))=
t
(p−1)(n−t)−1 i−pt
u(p−1)(i−pt)−1St0(Ptx).
This proves the result directly forσ (x) = 0 and@ = 1. By the Cartan for- mula applied to uu−1St@(x) we have that PiSt@(x) = uPi(u−1St@(x)) + upPi−1(u−1St@(x)). By combining this with the formulas above we get the result in the other cases.
6. A pullback description of the functor
In this section we describe(K)as a pullback in the case whereK is a free object inK. We start by a result on cohomology of Eilenberg-MacLane spaces.
Recall that a sequence of integersI =(@1, s1, @2, s2, . . . , @k, sk, @k+1)with si ≥0 and@i ∈ {0,1}is called admissible ifsi ≥psi+1+@i+1andsk ≥1 or if k=0 whenI =(@). The degree ofIis defined as|I| =
@j+
2sj(p−1) and the excess is defined recursively bye((@, s), J )= 2s+@− |J|. We use the following notationPI =β@1Ps1β@2Ps2. . . β@kPskβ@k+1.
Lemma6.1.The cohomology ring of the Eilenberg-MacLane spaceK(Fp,n) can be written in the following form whenn≥2:
H∗(K(Fp, n))=SF[PIιn|Iis admissible, e(I)≤n−2, @1=0].
Furthermore,H∗(K(Fp,1))=SF[ι1]andH∗(K(Fp,0))=SF[ι0].
Proof. The casesn= 0,1 are trivial. Assume thatn≥ 2 and define the set A(n)= {I |I is admisseble, e(I)≤n−1,|I| +nis odd}.
Remark that ifI ∈A(n)then(0, (|I| +n−1)/2, I)∈A(n). To see this write I ∈A(n)asI =(@, s, I). Thene(I)=2s+@− |I| ≤n−1 or equivalently 2sp+2@ − |I| ≤ n−1 such that the sequence (0, (|I| +n−1)/2, I) is admissible. Its excess isn−1 and its degree plusnis odd sincep−1 is even.
By Cartan’s computation (a special case of [9], Theorem 10.3) we have that H∗BnFpis the free graded commutative algebra on the set
B= {PJιn |J is admissible, e(J ) < nor(e(J )=nand@1=1)}.
Assume thatPIιn belongs to the set in the statement of the lemma. Then PIιn andβPIιn belongs toB. By the remark we see that if|I| +nis even thenβ@λiβPIιn∈Band if|I| +nis odd thenβ@λiPIιn∈Bfor@ =0,1 and i≥1.
Conversely, assume thatPJιn ∈B. Ife(J )≤n−2 ore(J )= n−1 and
@1=1 it is clearly one of the generators described in the lemma. It suffices to handle the casee(J ) = n−1, @1 = 0 since the casee(J )= n, @1 = 1 then follows. WriteJ as J = (0, s, J)where e(J ) = 2s− |J| = n−1. Then 2s =n+|J|−1 such thatPJιn=λPJιnande(J )≤e(J). We can continue this process until the next@equals one or the excess drops belown−1.
Proposition 6.2. For any object K in K there is natural morphism of Fp-algebrasA:(K)→Fp[u]⊗Kdefined by
φ(x)→St0(x), q(x)→St1(x), δ(x)→0, u→u⊗1. The image of this morphism isIm(A)=R(K).
Proof. We check that (10)–(21) are mapped to zero by the formulas defin- ingA. Sinceδ(x)is mapped to zero this is trivial for all elements except (14), (15), (16) and (20).
By the Cartan formula and|ab|
2
= |a|
2
+|b|
2
+σ(a)σ(b)one verifies that
St0(ab)=(−up−1)σ(a)σ (b)St0(a)St0(b),
St1(ab)=(−up−1)σ(a)σ (b)(uσ(b)St1(a)St0(b)+(−u)σ(a)St0(a)St1(b)) such that (14) and (15) are mapped to zero. Lemma 2.7 implies that (16) and (20) are mapped to zero.
Proposition6.3. IfKis a free object inK thenker(A)=Iδ(K). Proof. Assume thatK=SK(V )for a non negatively graded vector space V. We must show thatA¯ :(K)/Iδ(K)→Fp[u]⊗Kis injective.
The algebra(K)/Iδ(K)has generatorsφ(x),q(x)forx ∈Kandu. The relations are thatφandqare additive and that (14), (15), (16) and (20) equals zero. Let {vs | s ∈ S} denote a basis for V. By Lemma 6.1 we find that K=SF(W)whereW is the graded vector space with basis
B = {PIvs |Iadmissible, e(I)≤ |vs| −2, @1=0, s∈S}.
We see that the following elements are algebra generators for (K)/Iδ(K) wherea∈B0,b∈B1,v∈Bodd,∗≥3,w∈Beven,∗≥2andi≥0:
u, φ(a), φ(b), q(βb), φ(βv), φ(λiv), q(βv), q(λiv), φ(w), φ(λiβw), q(w), q(λiβw).
We claim that these generators are mapped to algebraically independent elements inFp[u]⊗K. By the formulas defining A we see that it suffices to check this claim in the case whereV is one dimensional. So assume that K=SK[ιn] where|ιn| =n.
For anynwe haveu→u⊗1. Forn=0 we haveφ(ι0)→1⊗ι0and for n= 1 we haveφ(ι1) → 1⊗ι1,q(ι1) → 1⊗βι1 so in these two cases the claim holds.
Assume thatn≥2. The algebra generators are mapped as follows modulo
