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0 ¨pt 0 Z Z Z |ti| i ppq b ppq » ppq b cofib ´Σ S rtis ÝÑ S rtis¯ . âi 0 Here the tensor product is taken over the sphere spectrum, S rxs denotes the free E1-algebra i on a class x, and the classes ti live in degree 2p ´ 2. Modding out by p causes each of the above cofibers to split into two classes related by a Bockstein; modding out by p once more introduces the class τ0 and recovers Milnor’s computation of A˚ “ π˚pFp b Fpq, as an Fp-algebra. In the Cp-equivariant case our description involves a similar decomposition but is more complicated in two ways:

0 • Rather than extending the class ti to a map from S rtis using the multiplication on Z b Z, we will want to choose as generators a mixture of ordinary powers of ti and of norms, Nptiq, of ti.

• Rather than modding out by the relation ‘pti “ 0’ we will need to enforce the relation that ‘θti “ 0’, where θ is an equivariant lift of p to an element in nontrivial ROpCpq- degree. We will then also need to enforce the relation pNptiq“ 0. To make this precise, we will assume that the reader is comfortable with equivariant stable homotopy theory as used, for example, in [HHR16], and introduce the following conventions, in force throughout the paper:

• We will use ρCp to denote the regular representation of Cp.

2 • We will use λ to denote the representation of Cp on R “ C where the generator acts by e2πi{p.

λ´2 0 • We denote by θ : S Ñ S the map of Cp-spectra arising from the degree p cover Sλ Ñ S2. We’ll denote the cofiber of θ by Cθ. Note that the underlying nonequivariant spectrum of Cθ is the Moore space Mppq.

• If X is a spectrum, we will denote by NpXq the Hill-Hopkins-Ravenel norm of X, bp which is a Cp-equivariant refinement of the ordinary spectrum X . Z F • We denote by and p the Cp-equivariant Eilenberg-MacLane spectra associated to the constant Mackey functors at Z and Fp, respectively.

Cp • We use b to denote the symmetric monoidal structure on genuine Cp-spectra, Sp (often denoted by ^, the smash product).

1We learned this fact from John Rognes. One proof is to base change the equivalence 0 0 Z Z BP bS rv1,...s S » ppq to ppq and use that the Hurewicz image of the vi’s are pti, mod decomposables.

2 • The degree k map k Sλ Ñ Sλ is a p-local equivalence when pk,pq“ 1, so, when working p-locally, we will often make this identification implicitly. For example, we may write

p´1 ρCp 1` 2 λ Sppq “ Sppq .

• We use π‹X to denote the ROpCpq-graded homotopy groups of a Cp spectrum, so that, V ´W when ‹“ V ´ W is a virtual representation, πV ´W X “ π0MapSpCp pS ,Xq. Now we can give a somewhat ad-hoc description of the equivariant refinements of the building blocks in Z b Z.

Construction. Let x be a formal variable in an ROpCpq-grading |x|. Define a Cp-spectrum as follows: 2 1 |x| |x| pp´ q|x| |x|ρCp Tθpxq :“ Σ Cθ ‘ Σ Cθ ‘¨¨¨‘ Σ Cθ ‘ Σ Mppq, where Mppq is the mod p Moore spectrum. We denote the inclusion of the summand Σi|x|Cθ by i´1 i|x| x xˆ : Σ Cθ Ñ Tθpxq, the restriction ofx ˆ to the bottom cell by x, and the inclusion of the final summand by Nx. We denote by y |x|ρCp Nx : S Ñ Tθpxq the restriction of Nx to the bottom cell of the mod p Moore spectrum. Now suppose thaty R is a Cp-ring spectrum equipped with a norm NpRqÑ R. If we have a class x P π‹R such that θx “ 0, it follows that p ¨ Npxq“ 0 (see the proof of Lemma 4.4), so we may produce a map 0 0 S ‘pS rNxsb Tθpxqq Ñ R, which only depends on the choice of the nullhomotopy witnessing θx “ 0.

We can now state our main theorem.

Theorem A. There are equivariant refinements

2 i´1 p ρCp ´λ Z Z ti : S Ñ ppq b ppq of the nonequivariant classes ti P π˚pZppq b Zppqq which satisfy the relation θti “ 0. For any choice of witness for these relations, the resulting map

Z 0 0 Z Z ppq b S ‘pS rNtisb Tθptiqq ÝÑ ppq b ppq âiě1 ` ˘ is an equivalence.

3 As an immediate corollary we have: Corollary. With notation as above, we have

0 0 F F 0 F F p b p » Λpτ qb p p b S ‘pS rNtisb Tθptiqq , âiě1 ` ˘ F F where τ0 is dual to the Bockstein, in degree 1 and Λpτ0q “ p ‘ Σ p. In particular, since F F F p b Cθ is indecomposable at odd primes, the spectrum p b p is not a direct sum of F ROpCpq-graded suspensions of p at odd primes.

σ´1 σ Remark. When p “ 2 we have an accidental splitting F2 b Cθ » Σ F2 ‘ Σ F2, where σ is the sign representation. F F F λ´1 Remark. One can show that p b Cθ b Cθ splits as p p b Cθq‘p p b Σ Cθq. It follows F F that p b p splits as a direct sum of cell complexes with at most 2 cells. Our result raises a few natural questions which would be interesting to investigate. Question 1. When specialized to p “ 2, how does our basis compare to the Hu-Kriz basis? Z Z F F Question 2. The geometric fixed points of ppq b ppq are given by p p b pqrb, bs, where b is the conjugate of b, a class in degree 2. It is possible to understand what happens to the generators ti and Nptiq upon taking geometric fixed points. One is left with trying to understand the remainingz class hit by tˆi on geometric fixed points. We don’t know what this should be. One guess that seems consistent with computations is that this class is given, up to conjugating the τi and modding out by pbq, by:

pi´1´pi´2 pi´1´1 τi´1 ` b τi´2 `¨¨¨` b τ0

It would be useful for computations to sort out what actually occurs. F Question 3. Is it possible to profitably study the p-based Adams spectral sequence using F F F this decomposition? Since p b p is not flat over p, one would be forced to start with the E1-term. But this is not an unprecedented situation (e.g. Mahowald had great success with the ko-based Adams spectral sequence). F F Question 4. Can one describe the multiplication on π‹ pb p in terms of our decomposition?

Relation to other work

As we mentioned before, we were very much motivated by the description of the C2-equivariant dual Steenrod algebra given by Hu-Kriz [HK01]. That said, our generators are slightly dif- ferent than the Hu-Kriz generators when we specialize to p “ 2. For example, the generator 2 t1 lives in degree 2ρC2 ´ λ “ 2, whereas the Hu-Kriz generator ξ1 lives in degree ρ “ 1 ` σ.

2 In this low degree, it seems likely that, modulo decomposables, we have uσξ1 “ t1 and that ξ1 is recovered 1`σ 2 from tˆ1 by restricting along F2 b S Ñ F2 b Σ Cθ.

4 Hill and Hopkins have also obtained a presentation of the C2n -dual Steenrod algebra, using quotients of BPR and its norms, which is similar in style to the one obtained here. At odd primes, Caruso [Car99] studied the Cp-equivariant Steenrod algebra, π‹mappFp, Fpq, essentially by comparing with the Borel equivariant Steenrod algebra and the geometric fixed point Steenrod algebra, and was able to compute the ranks of the integer-graded stems. There is also work of Oru¸c[Oru89] computing the dual Steenrod algebra for the Eilenberg- F MacLane spectra associated to Mackey fields (which does not include p). In the Borel equivariant setting, the dual Steenrod algebra is given by the action Hopf ˚ algebroid for the coaction of the classical dual Steenrod algebra on H pBCpq (see [Gre88]). There is also related work from the first and second authors. The first author produced F F a splitting of p b p in [San19] using the symmetric power filtration. This summands in that splitting were roughly given by the homology of classifying spaces, and were much larger than the summands produced here. The second author and Jeremy Hahn showed F λ λ`1 [HW20] that p can be obtained as a Thom spectrum on Ω S . The Thom isomorphism then reduces the study of the dual Steenrod algebra to the computation of the homology of ΩλSλ`1. Understanding the relationship between this picture and the one in this article is work in progress.

Acknowledgements We thank Jeremy Hahn and Clover May for excellent discussions over the course of this project. We thank Mike Hill for generously sharing his time and his ideas, especially the idea to use the norm to build some of the generators of the dual Steenrod algebra. We apologize to those awaiting the publication of these results for the long delay.

1 Outline of the proof

To motivate our method of proof, let’s first revisit the classical story. We are interested in where the classes ti P π˚pZ b Zq come from, and why they are annihilated by p. Recall that the homology of CP 8 is a divided power algebra

8 H˚pCP q“ ΓZtβ1u

8 where β1 is dual to the first Chern class c1. Write βpiq :“ γpi pβ1q. Since CP “ KpZ, 2q, we have a map of spectra C 8 2Z P` Ñ Σ and hence a homology suspension map

8 σ : H˚pCP qÑ π˚´2pZ b Zq

8 which annihilates elements decomposable with respect to the product structure on H˚pCP q. 3 We can take ti :“ σpβpiqq. The relation pti “ 0 follows from the fact that pβpiq is, up to a

3Depending on ones preferences, this might be the conjugate of the generator you want; but we are only really concerned with these classes modulo decomposables.

5 p C 8 p-local unit, decomposable as βpi´1q in H˚p P q. In the equivariant case, we will proceed similarly.

Step 1. Compute the homology of KpZ,λq and use the homology suspension to define classes in π‹pZ b Zq. Step 2. Use information about the product structure on the homologies of KpZ,λq and KpZ, 2q to deduce relations for these classes, and hence produce the map described in Theorem A. Step 3. Verify that the map in Theorem A is an equivalence by proving that it is an underlying equivalence and an equivalence on geometric fixed points.

The first step is carried out in §2 and §3 by identifying KpZ,λq with an equivariant version of CP 8 and then specializing a computation due to Lewis [Lew88], which we review in our context. The second step is carried out in §4. The third and final step is carried out in §6 using a lemma proven in §5 that allows us to check that the map on geometric fixed points is an equivalence by just verifying that the source and target have the same dimensions in each degree.

1 2 Homology of BCpS

1 1 Recall that we have the Cp-space BCp S classifying equivariant principal S -bundles. The following lemmas give two useful ways of thinking about this space. P C 1 Lemma 2.1. The complex projective space p rzsq is a model for BCp S , where the generator 2πi{p of Cp acts on Crzs through ring maps by z ÞÑ e z. Here Crzs is the ordinary polynomial ring over C, and the projective space PpCrzsq“pCrzs ´ t0uq{pCˆq inherits an action in the evident way. 1 Z Lemma 2.2. The space BCp S is a model for Kp ,λq. Proof. The map PpCrzsq Ñ SP8pSλq to the infinite symmetric product, which sends a polynomial fpzq to its set of roots (with multiplicity), is an equivariant homeomorphism. The group-completion of the latter is a model for KpZ,λq by the equivariant Dold-Thom theorem. But SP8pSλq is already group- complete: the monoid of connected components of the fixed points is N{p “ Z{p. 1 Remark 2.3. The reader may object that the definition of BCp S makes no reference to λ, 1 k so how does BCp S know about this representation rather than λ for some k coprime to p? The answer is that, in fact, each of the Eilenberg-MacLane spaces KpZ,λkq coincide for such k: we have an equivalence of Z-modules

k ΣλZ » Σλ Z

6 whenever pk,pq “ 1. This follows from the computations in [FL04, Proposition 9.2], for example.

The filtration of Crzs by the subspaces Crzsďn of polynomials of degree at most n gives 1 a filtration of BCp S . Lemma 2.4. There is a canonical equivalence

1 Vk grkBCp S – S .

i´k where Vk “ 0 1 λ . À ďiďk´ Proof. This follows from a more general observation. If L is a one-dimensional complex representation, and V is an arbitrary complex representation, then the function assigning a linear map to its graph,

HomCpL, V q ÝÑ PpV ‘ Lq´ PpV q, is an equivariant homeomorphism. So it induces an equivalence on one-point compactifica- tions _ SL bV – PpV ‘ Lq{PpV q.

The next proposition now follows from [Lew88, Proposition 3.1]. 1 Z Proposition 2.5 (Lewis). The above filtration on BCp S splits after tensoring with , giving an equivalence Z 1 Z b BCp S` » te0, e1, ...u where i´k |ek|“ λ . 0ďàiďk´1 i´1 In particular, for i ě 1 we have |epi |“ 2p ρCp . We will also need some information about the multiplicative structure on homology. . Zˆ Lemma 2.6. Writing x “ y to mean that x “ αy for some α P ppq, we have

p . p . 1 e1 “ θep, and epi “ pepi` for i ě 1. 1 P C P C Proof. Using the model for BCp S given by p rzsq, we see that, in fact, p rzsq has the structure of a filtered monoid. It follows that the product in homology respects the filtration by the classes teiu. Thus, for i ě 0, we have:

p e i “ c e p ÿ i,j j jďpi`1

7 i`1 where the coefficients lie in π‹Z. When j ă p we see that the virtual representations |ci,j| have positive virtual dimension and their fixed points also have positive virtual dimension. The homotopy of Z vanishes in these degrees (see, e.g., [FL04, Theorem 8.1(iv)]), so we must have p 1 1 epi “ ci,pi` epi` where |c0,p|“ λ ´ 2 and |ci,pi`1 |“ 0 when i ě 1. In both cases, the restriction map on π‹Z is injective in this degree, so the result follows from the nonequivariant calculation.

3 Suspending classes

We begin with some generalities. If X is any Cp-spectrum, we have the counit

8 8 Σ` Ω X Ñ X which induces a map Z 8 8 Z σ : b Σ` Ω X Ñ b X, called the homology suspension. Just as in the classical case, σ annihilates decomposable Z 8 8 elements in π˚p b Σ` Ω Xq. Construction 3.1. For i ě 1, we define

2 i´1 p ρCp ´λ ti : S Ñ Z b Z

1 e i π2 i´1 Z S as the homology suspension of the element p P p ρCp p b BCp q. Here we use the identification 1 Z 8 λZ BCp S » Kp ,λq“ Ω Σ .

4 Two relations in homology

We begin with a brief review of norms, transfers, and restrictions. Remark 4.1 (Transfer and restriction). Given a nonequivariant equivalence pSV qe – Sn, we define e V n V e res : πV X Ñ πnX , px : S Ñ Xq ÞÑ pS –pS q Ñ Xq and

e n e V V n trV : πnX Ñ πV X, py : S Ñ X q ÞÑ pS Ñ Cp` b S – Cp` b S Ñ Cp` b X Ñ Xq. 0 For example, when V “ λ ´ 2 and X “ S , then trλ´2p1q“ θ. Changing the equivalence pSV qe – Sn has the effect of altering these classes by ˘1; in our case the representations in question have canonical orientations so this will not be a concern. Given a map X b Y Ñ Y we have a relation: trpx b respyqq “ trpxqb y.

8 Remark 4.2 (Norms). If a Cp-spectrum X has a map NpXqÑ X, then, given an underlying class x : Sn Ñ Xe, we may define a norm by the composite

Nx : NpSnq“ SnρCp Ñ NpXqÑ X.

e The underlying nonequivariant class is given by respNxq“ pgxq P πpnX . śgPCp Our goal in this section is to prove the following two lemmas.

Lemma 4.3. The classes 2 i´1 Z Z satisfy . ti P π p ρCp ´λp ppq b ppqq θti “ 0

Lemma 4.4. The classes 2 i 2 Z Z satisfy . Nptiq P πp p ´ qρCp p ppq b ppqq pNptiq“ 0 In fact, the second relation follows from the first.

Proof of Lemma 4.4 assuming Lemma 4.3. Since p “ trp1q, the class pNptiq is the transfer p i i p of the class resptiq into degree p2p ´ 2qρCp . Notice that p2p ´ 2qρCp ´ |ti | “ λ ´ 2 (after identifying the λk suspensions with λ for pk,pq“ 1), and the transfer of 1 into this degree is θ, so we have p pNptiq“ θti “ 0.

p . Proof of Lemma 4.3. By Lemma 2.6, we have e1 “ θep so that θt1 “ σpθepq “ 0, since σ annihilates decomposables. For the remaining classes, consider the commutative diagram

/ λ KpZ,λq` Σ Z

rθs θ   / 2 KpZ, 2q` Σ Z

8 where rθs“ Ω pθq. Thus, to show that θti “ 0 for i ě 2, it is enough to show that rθs˚epi is Z Z decomposable in π‹p ppq b Kp , 2q`q for i ě 2. Write Z Z Z ppq b Kp , 2q` “ ppqtγipβ1qu 8 where the elements γipβ1q are the standard module generators of H˚pCP ; Zq, and write βpiq “ γpi β1. To show that rθs˚pepi q is decomposable for i ě 2, it is enough to establish the following two claims:

. pi´1θ i (a) rθs˚pep q “ pipp´1q´1 βpiq, and uλ p . (b) βpi´1q “ pβpiq.

9 8 Claim (b) is just the classical computation of the product in homology for H˚pCP , Zq. For claim (a), let ιλ denote the fundamental class in cohomology for KpZ,λq and ι2 the same for ˚ KpZ, 2q. Then we have rθs pι2q“ θιλ by design, and hence

˚ j j j rθs pι2q“ θ ιλ.

The map on homology is now determined by the relation

j j j Z xrθs˚epi , ι2y“ θ xepi , ιλy P π‹ ppq.

Since θj is a transferred class, the value above is also a transfer, and hence determined by its restriction to an underlying class. But resprθsq“rps and we clearly have rps˚prespepi qq “ 1 i pi´ θ p βpiq, which agrees with the restriction of pipp´1q´1 βpiq. This completes the proof. uλ 5 Digression: Detecting equivalences nonequivariantly

The goal of this section is to establish a criterion for detecting equivalences of Z-modules. We recall that ΦCp Z » Fprbs where the class b in degree 2 arises from taking the geometric fixed points of the Thom class λ 2 uλ : S Ñ Σ Z. Proposition 5.1. Let f : M Ñ N be a map of Z-modules which are bounded below. Assume the following conditions are satisfied:

(i) f is an underlying equivalence.

ΦCp ΦCp (ii) πjM and πjN are finite dimensional of the same rank, for all j.

ΦCp ΦCp (iii) π˚M and π˚N are graded-free Fprbs-modules. Then f is an equivalence.

We will deduce this proposition from the following one, which relates geometric and Tate fixed points.

Proposition 5.2. Let M be a Z-module which is both bounded above and below. Then the natural map M ΦCp rb´1sÑ M tCp is an equivalence.

Proof of Proposition 5.1 assuming Proposition 5.2. By assumption (i), it is enough to check ΦCp ΦCp that f is an equivalence; by assumption (ii), it is enough to check that π˚pf q is an ΦCp ´1 injection; and by assumption (iii) it is enough to check that π˚pf qrb s is an injection.

10 Again by (i), the map f tCp is an equivalence. So, from the diagram

M ΦCp rb´1s / N ΦCp rb´1s

  tCp / tCp M » N we see that it is enough to check that the vertical maps are injective on homotopy. More generally, we show that whenever X is a bounded below Z-module, the map

ΦCp ´1 tCp π˚X rb sÑ π˚X is injective. Indeed, by Proposition 5.2 and the fact that the Tate construction commutes with limits of Postnikov towers (see, e.g., [NS18, I.2.6]), we have

ΦCp ´1 » tCp tCp lim pτďnXq rb s Ñ limpτďnXq » X . n ` ˘ n Therefore, we need only check that

ΦCp ´1 ΦCp ´1 π˚X rb sÑ π˚ lim pτďnXq rb s n ` ˘

ΦCp ΦCp is injective. Since the maps X ÑpτďnXq have increasingly connective fibers, we can ΦCp ´1 replace the left hand side by plimn π˚pτďnXq qrb s and reduce to showing that

ΦCp ´1 ΦCp ´1 plim π˚pτďnXq qrb sÑ lim π˚ pτďnXq rb s n n ` ˘ is injective. Finally, this reduces to showing that the kernel of

ΦCp ΦCp ´1 lim π˚pτďnXq Ñ lim π˚ pτďnXq rb s n n ` ˘ consists of elements annihilated by a power of b. This is clear because, for each j, the system ΦCp tπjpτďnXq un is eventually constant. Proof of Proposition 5.2. Let E denote the full subcategory of Z-modules M for which

M ΦCp rb´1sÑ M tCp

is an equivalence. Then E is stable, closed under retracts, and closed under suspending by representation spheres. ΦCp ´1 tCp ΦCp The map M rb s Ñ M is one of Z “ Fprbs-modules, and hence one of Fp- modules, so it must be a retract of

pM{pqΦCp rb´1s“ M ΦCp rb´1s{p Ñ M tCp {p “pM{pqtCp .

Thus M{p P E if and only if M P E. So, by replacing M with M{p and considering the ♥ Postnikov tower, we are reduced to proving the proposition in the case where M P ModZ is F a Mackey functor which is a module over p.

11 e In particular, M is an FprCps-module. Let γ denote the generator of Cp so that FprCps“ p j e e Fprγs{p1 ´ γq . Let FjM Ď M be the sub-Mackey functor generated by p1 ´ γq M Ď M . This is a finite filtration with associated graded pieces given by Mackey functors with trivial underlying action. So, since E is a thick subcategory, we are reduced to the case when M is F a discrete p-module with trivial underlying action. For the next reduction we recall some notation. If N is any Mackey functor, denote by

NCp the Mackey functor N b Cp` and, if A is an abelian group, denote by Atr the Mackey functor whose transfer map is the identity on A and whose restriction map is multiplication by p. We also recall that the transfer extends to a map of Mackey functors tr : NCp Ñ N. Now consider the two exact sequences 0 Ñ imptrqÑ M Ñ M{imptrqÑ 0 e 0 Ñ kerptrqÑ M tr Ñ imptrqÑ 0 If N is any Mackey functor with N e “ 0, then N P E since then N “ N ΦCp is bounded above and hence N ΦCp rb´1s “ 0. Thus, from the exact sequences above, we are reduced to the case where M is of the form V tr for an Fp-vector space V (with trivial action). Now F 2´λF 2´λ recall that p pqtr “ Σ p and hence V tr “ Σ V . So we are reduced to showing that the constant Mackey functor V lies in E, where V is an Fp-vector space with trivial action. This certainly holds for V “ Fp, and in general we have Φ ΦCp F Cp V » p bFp V, since geometric fixed points commutes with colimits, and

tCp FtCp V » p bFp V by direct calculation. (Notice this holds even when V is infinite-dimensional). This completes the proof.

6 Proof of the main theorem

We are now ready to prove the main theorem. Recall that we have constructed classes

2 i´1 Z Z ti P π p ρCp ´λp ppq b ppqq,

and shown that θti “ 0 and pNptiq“ 0. With notation as in the introduction, let X “ S0 ‘pS0rNt sb T pt qq i ` i θ i ˘ and 0 0 X “ S ‘pS rNtisb Tθptiqq âiě1 ` ˘

Then, choosing nullhomotopies which witness θti “ 0, we get a map: Z 0 0 Z Z f : ppq b S ‘pS rNtisb Tθptiqq ÝÑ ppq b ppq âiě1 ` ˘ The main theorem is then the statement:

12 Theorem 6.1. The map f is an equivalence.

Proof. Combine Proposition 5.1 with the two lemmas below. Lemma 6.2. The map f e is an underlying equivalence.

Proof. First observe that, by our construction in the proof of Lemma 4.4, the map Nptiq p´1 restricts to the map ti tˆi, since the nullhomotopy witnessing pNptiq “ 0 was chosenz to p restrict to the nullhomotopy chosen for pti that came from the already chosen nullhomotopy of pti. The upshot is that the map

0 0 S ‘ S rNtisb TθptiqÑ Z b Z

restricts on underlying spectra to the map

0 S rtis{pptiqÑ Z b Z

obtained just from the relation pti “ 0 and extended via the multiplicative structure. In particular, on mod p homology f e induces a ring map

Fprtisb ΛpxiqÑ Fprξisb Λpτiq.

e We know that ti maps to ξi and that βxi “ ti, so that βpf˚pxiqq “ ξi. Modulo decomposables, τi is the only element whose Bockstein is ξi. So xi must map to τi, mod decomposables. It follows that f e is a mod p equivalence, and hence an equivalence.

ΦCp ΦCp Lemma 6.3. pZ b Xq and pZ b Zq are free Fprbs-modules, finite-dimensional in each degree, and isomorphic as graded vector spaces over Fp.

Proof. If Y is any Cp-spectrum, then

Z ΦCp F ΦCp F F ΦCp p ppq b Y q “ prbsb Y » prbsbFp p p b Y q

is a free Fprbs-module. Applying this in the cases Y “ X and Y “ Z, we see that each is a free Fprbs, evidently finite-dimensional in each degree. So it suffices to prove that

ΦCp Fp b X – Fp bpFprbsq

as graded vector spaces. Notice that we can write, as graded vector spaces,

F ΦCp F p p´1 p b Xi – prdpi´1q,ξisbFp Λpσi´1, τiq{pdpi´1q,dpi´1qτi,dpi´1qσi´1, σi´1τiq,

i´1 i´1 where |σi´1|“ 2p ´ 1 and |dpi´1q|“ 2p . Indeed, tˆi, on geometric fixed points, gives rise

to two classes; one we are calling dpi´1q and the other we are calling σi´1. Similarly, Nptiq, on geometric fixed points, gives rise to two classes: one we are calling ξi and the otherzτi, in their usual degrees. The relations are the ones needed to ensure that the monomials not arising from geometric fixed points of elements in Xi are omitted.

13 It follows that we have an isomorphism of graded vector spaces

1 F ΦCp F F p p´ pbX – prξn : n ě 1sbFp prdpiq : i ě 0sbFp Λpσj, τk : j ě 0,k ě 1q{pdpiq,dpi´1qτi,dpiq σi, σi´1τiq. We are trying to show that this is isomorphic, as a graded vector space to F F F F p b prbs– prξn : n ě 1sbFp Λpτi : i ě 0qbFp prbs.

We may regard each vector space as a module over Fprξn : n ě 0s in the evident way, and hence reduce to showing that the two vector spaces F V “ Λpτi : i ě 0qbFp prbs and F p p´1 W “ prdpiq : i ě 0sbFp Λpσj, τk : j ě 0,k ě 1q{pdpiq,dpi´1qτi,dpiq σi, σi´1τiq i i are isomorphic. (Here recall that |σi| “ |τi|“ 2p ´ 1, |b|“ 2, and |dpiq|“ 2p ). Let I range over sequences pa0, a1, ...q with 0 ď ai ď p ´ 2, J range over sequences 1 pε0,ε1, ...q with εi P t0, 1u, K range over sequences pκ0, κ1, ...q with κi P t0, 1u, and let K 1 1 1 range over sequences pκ0, κ1, ...q with κi P t0, 1u. We impose the following requirements on these sequences:

• Each sequence has finite support.

1 1 • If κi “ 1, then κi “ 1. (So K is otained from K by changing some subset of 1s to 0s). • J ¨ K “ I ¨ K “p0, 0, ...q. That is: I and K have disjoint support and J and K have disjoint support.

Then V has a basis of monomials

i i aip κipp´1qp M “p b qτ p b qτ 1 I,J,K ź J ź K iě0 iě0 and W has a basis of monomials

1 1 pκi´κiqpp´ q N “ d σ p d qτ 1 1 I,J,K I J ź piq K r s iě0

1 1 1 where K r1s“p0, κ0, κ1, ...q. These have the same number of basis elements in each dimen- sion, so V – W .

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