Detecting β elements in iterated algebraic K-theory of finite fields
aa r X i v : . [ m a t h . K T ] O c t DETECTING THE β -FAMILY IN ITERATED ALGEBRAIC K-THEORYOF FINITE FIELDS GABE ANGELINI-KNOLL
Abstract.
The Lichtenbaum-Quillen conjecture (LQC) relates special values of zeta func-tions to algebraic K-theory groups. The Ausoni-Rognes red-shift conjectures generalize theLQC to higher chromatic heights in a precise sense. In this paper, we propose an alternategeneralization of the LQC to higher chromatic heights and prove a highly nontrivial casethis conjecture. In particular, if the n -th Greek letter family is detected by a commutativering spectrum R , then we conjecture that the n + 1 -st Greek letter family will be detectedby the algebraic K-theory of R . We prove this in the case n = 1 for R = K ( F q ) p where p ≥ and q is prime power generator of the units in Z /p Z . In particular, we prove thatthe commutative ring spectrum K ( K ( F q ) p ) detects the β -family. The method of proof alsoimplies that the β -family is detected in iterated algebraic K-theory of the integers. Conse-quently, one may relate iterated algebraic K-theory groups of the integers to modular formssatisfying certain congruences. Contents
1. Introduction 11.1. Conventions 41.2. Acknowledgements 52. Overview of the toolkit 52.1. The THH-May spectral sequence for K ( F q ) p β -family in iterated algebraic K-theory of finite fields 93.1. Detecting v and β BP ∧ V (1) -THH-May spectral sequence 123.3. Detecting the β family in homotopy fixed points of topological Hochschildhomology 163.4. Detecting the β -family in iterated algebraic K-theory 24References 271. Introduction
Following Waldhausen [36], the famous Lichtenbaum-Quillen conjecture states that themap(1) K n ( A ; Z /ℓ Z ) → K ét n ( A ; Z /ℓ Z ) from algebraic K-theory to étale algebraic K-theory is an isomorphism for n sufficiently largewhere A is a nice regular ring with ℓ invertible in A and ℓ is an odd prime [21, 28]. Since alge-braic K-theory satisfies Nisnevich descent and étale algebraic K-theory satisfies étale descent,the question can be translated into the question of whether the map from motivic cohomology to étale cohomology is an isomorphism in a range. In this way, the conjecture was resolvedby M. Rost and V. Voevodsky as a consequence of their proof of the Bloch-Kato conjecture .Thomason showed in [33] that K ét n ( A ; Z /ℓ Z ) ∼ = β − K n ( A ; Z /ℓ Z ) under the same conditionson A where β is the Bott element in K ( A ; Z /ℓ Z ) . From the perspective of homotopy theory,we may therefore view the map (1) as the map on π n induced by the map of spectra S/ℓ ∧ K ( A ) → v − S/ℓ ∧ K ( A ) where S/ℓ is the cofiber of multiplication by ℓ . Here the map v : Σ p − S/ℓ → S/ℓ is a v -selfmap which has the property that no iterate of it with itself is null homotopic. This allows usto define v − S/ℓ as the homotopy colimit of the diagram
S/ℓ v −→ Σ − p +2 S/ℓ v −→ Σ − p +4 S/ℓ v −→ . . . of spectra. The effect of inverting the Bott element is the same as the effect of inverting v by work of Snaith [31] as interpreted by Waldhausen [36, Sec. 4].The original motivation of the Lichtenbaum-Quillen conjecture was to relate algebraic K-theory groups to special values of zeta functions. For A the ring of integers in a totally realnumber field F and ℓ an odd prime, Wiles proved that quotients of étale cohomology groups of A [1 /ℓ ] recover special values of the Dedekind zeta function ζ F [38]. The Lichtenbaum-Quillenconjecture then gives a correspondence between algebraic K-theory groups and special valuesof Dedekind zeta functions. Notably these special values correspond to the v -periodic partof S/ℓ ∗ K ( A ) because they are detected in v − S/ℓ ∗ K ( A ) .As another specific example, consider the algebraic K-theory of finite fields F q where q is aprime power that topologically generates the ring Z × ℓ and ℓ is an odd prime (or equivalently q generates the units in Z /ℓ Z ). D. Quillen [27] computed K n ( F q ) for all n and after localizingat ℓ , there is an isomorphism K s − ( F q ; Z ( ℓ ) ) ∼ = Z /ℓ ν ℓ ( k )+1 Z where s = ( ℓ − k and ν ℓ ( k ) is the ℓ -adic valuation of k . The order of the group K s − ( F q ; Z ( ℓ ) ) corresponds exactly to the ℓ -adic valuation of the denominator of B s / s where B s is the s -thBernoulli number. Recall that Bernoulli numbers are the coefficients in the Taylor series xe x − X s ≥ B s x s s ! and the special values of the Riemann zeta function satisfy ζ ( − s ) = ( − s B s / ( s +1) for s ≥ .This example is intimately tied to stable homotopy theory as well. J.F. Adams showed thatthe image of the J-homomorphism from the homotopy groups of the stable orthogonal group tothe stable homotopy groups of spheres is highly nontrivial and the classical Bott periodicityin the homotopy groups of the stable orthogonal group corresponds to periodicity in thestable homotopy groups of spheres [1]. In fact, the ℓ -local image of the J-homomorphismexactly corresponds to the image of the map π ∗ ( S ( ℓ ) ) → K ℓ − k − ( F q ; Z ( ℓ ) ) when ℓ is anodd prime. The image of J therefore bridges the fields of homotopy theory and numbertheory. The spectrum H F q detects v -periodicity in the sense that ℓ k = v k is nontrivial inthe image of the Hurewicz map π ∗ S → π ∗ H F q . Therefore, we have observed an instance The proof of this theorem stretches over several papers by M. Rost and V. Voevodsky. There is currently abook in progress by C. Haesemeyer and C. Weibel [18], which provides a self contained reference for the proof.For a published account see [35]. Also, see Chapter VI Theeorem 4.1 and Historical Remark 4.4 in Weibel [37]for further discussion of the state of affairs.
ETECTING THE β -FAMILY IN ITERATED ALGEBRAIC K-THEORY OF FINITE FIELDS 3 where algebraic K-theory of a spectrum that detects v -periodic elements detects v -periodicelements. One goal of this introduction is to formulate a precise conjecture about a higherchromatic height generalization of this phenomena. The main theorem of this paper is evidencefor this conjecture at a higher chromatic height.In chromatic stable homotopy theory, we study periodic families of elements in the homo-topy groups of spheres. The first such family, due to J.F. Adams [1] and H. Toda [34], is the α -family, which consists of maps α k defined as the composites α k : Σ (2 ℓ − k S i / / Σ (2 ℓ − k S/ℓ v k / / S/ℓ δ / / Σ S where ℓ is an odd prime. The elements α k are ℓ -torsion elements in the groups π ℓ − k − S .These elements are in the image of the J-homomorphism at odd primes ℓ and as discussedearlier they are also detected in algebraic K-theory of finite fields of order q when q gener-ates ( Z /ℓ Z ) × . In particular, they correspond to certain special values of the Riemann zetafunction. Now, consider the cofiber of the periodic self-map v : Σ p − S/ℓ → S/ℓ denoted V (1) . When ℓ ≥ , there exists a periodic self-map v : Σ ℓ − V (1) → V (1) and there is anassociated periodic family of elements in the homotopy groups of spheres called the β -family.In particular, L. Smith [30] proved that the maps β k : Σ (2 ℓ − k S i i / / Σ (2 ℓ − k V (1) v k / / V (1) δ δ / / Σ ℓ S are nontrivial. This family of elements also has a deep connection to number theory by workof Behrens [9]. In particular, Behrens showed that the (divided) β -family is related to a familyof modular forms satisfying certain congruences [9, Thm. 1.3].In the language of chromatic homotopy theory, the α -family is a periodic family of heightone and the β -family is a periodic family of height two. There are a family of homology theories K ( n ) ∗ called Morava K-theory which are useful for detecting periodicity of chromatic height n in the homotopy groups of spheres. The coefficients of Morava K-theory are K ( n ) ∗ ∼ = F ℓ [ v ± n ] for n ≥ and K (0) ∗ is rational homology. We say a ℓ -local finite cell S -module V has type n if the groups K ( n ) ∗ V = 0 and the groups K ( n − ∗ V vanish. By the celebrated periodicitytheorem of Hopkins-Smith [20], any ℓ -local finite spectrum V of type n admits a periodic selfmap v mn : Σ (2 ℓ n − m V → V. We can therefore define v − n V in the same way that we defined v − S/ℓ . We can also constructthe n -th Greek letter family by including into the bottom cell, iterating v mn k -times, andthen projecting onto the top cell. However, it is highly non-trivial to prove that Greek letterelements that are constructed in this way are actually nonzero.The study of Greek letter family elements was significantly expanded by the groundbreakingwork of Miller-Ravenel-Wilson [23] using the chromatic spectral sequence(2) E ∗ , ∗ = M i ≥ Ext ∗ , ∗ BP ∗ BP ( BP ∗ , v − i BP ∗ / ( ℓ ∞ , v ∞ , . . . , v ∞ i − )) ⇒ Ext ∗ , ∗ BP ∗ BP ( BP ∗ , BP ∗ ) which converges to the input of the BP -Adams spectral sequence. If the class v kn /ℓ i v i . . . v i n − n − ∈ Ext BP ∗ BP ( BP ∗ , v − i BP ∗ / ( ℓ, v , . . . , v i − )) in the E -page of (2) survives the chromatic spectral sequence, we will write α ( n ) k/ ( i n − ,i n − ,...i ) ∈ Ext ∗ , ∗ BP ∗ BP ( BP ∗ , BP ∗ ) GABE ANGELINI-KNOLL for its image in the abutment of the chromatic spectral sequence. We will refer to the collectionof all such elements for a fixed n as the n -th divided (algebraic) Greek letter family and whenany of the i j for ≤ j ≤ n − are we omit them from the notation. If the elements α ( n ) k survive the BP -Adams spectral sequence, then we will refer to the collection as the n -th Greekletter family . The advantage of this approach is that the elements in the input of the chromaticspectral sequence always exist. The question of whether or not certain Greek letter elementsexist in homotopy can then be approached by determining whether certain elements in thechromatic spectral sequence and the BP -Adams spectral sequence are permanent cycles.We will say that a (commutative) ring spectrum R detects the n -th Greek letter family inthe homotopy groups of spheres if each element α ( n ) k is non-trivial in the image of the unitmap π ∗ S −→ π ∗ R. We conjecture the following higher chromatic height analogue of the Lichtenbaum-Quillenconjecture, which is in the same spirit as the red-shift conjectures of Ausoni-Rognes [6]. Forthe following conjecture, suppose the n -th and the n + 1 -st Greek letter family are nontrivialelements in π ∗ S for a given prime ℓ . Conjecture 1.1. If R is a commutative ring spectrum that detects the n -th Greek letterfamily, then K ( R ) detects the n + 1 -st Greek letter family.We can now state the main theorem of this paper. As discussed earlier, the spectrum K ( F q ) ℓ detects the α -family for ℓ ≥ and q a prime power that generates ( Z /ℓ Z ) × . Themain theorem of this paper is a proof of Conjecture 1.1 in the case n = 1 where R = K ( F q ) ℓ .For the following theorem, let ℓ ≥ be a prime and q a prime power that generates ( Z /ℓ Z ) × .One can easily check that ℓ = 5 and q = 2 is an example of such ℓ and q . Theorem 1.2.
The commutative ring spectrum K ( K ( F q ) ℓ ) detects the β -family.In particular, the method of proof also provides the following higher Lichtenbaum-Quillen-type result about iterated algebraic K-theory of the integers. Corollary 1.3.
The commutative ring spectrum K ( K ( Z )) detects the β -family.In [9], M. Behrens gives a description of the β -family in terms of modular forms satisfyingcertain congruences. From this point of view, our main result may be viewed as a higherchromatic height version of the Lichtenbaum-Quillen conjecture. It is therefore a step towardsthe larger program of understanding the arithmetic of commutative ring spectra.The β -elements β k that we detect only agree with the divided β -family elements β k/i,j ofM. Behrens [9] when i = j = 1 . To make the connection to arithmetic more tight, it would bedesirable to detect the entire divided β -family in iterated algebraic K-theory of finite fields. Itis a long term goal of the author’s to show that, in fact, the entire divided β -family is detectedin iterated algebraic K-theory of finite fields and consequently iterated algebraic K-theory ofthe integers.1.1. Conventions.
Let S be the category of symmetric spectra in pointed simplicial setswith the positive flat stable model structure. Most of the results here can also be provenfor other models of the stable homotopy category since they depend only on the homotopycategory, but the proof relies on the author’s joint paper with A. Salch [3] which uses thismodel for the stable homotopy category.Co-modules M over a Hopf algebroid ( E ∗ , E ∗ E ) will always be considered with left co-action ψ EM : M → E ∗ E ⊗ E ∗ M ETECTING THE β -FAMILY IN ITERATED ALGEBRAIC K-THEORY OF FINITE FIELDS 5 and we will simply write ψ when the module M and the Hopf algebroid ( E, E ∗ E ) is understoodfrom the context. The main examples of interest are E = H F p , where E ∗ E is the dual Steenrodalgebra A ∗ , and E = BP . We write ∆ E : E ∗ E → E ∗ E ⊗ E ∗ E ∗ E for the co-product of the Hopf-algebroid E ∗ E or simply ∆ when E is understood from thecontext. When E = H F p , this is the co-product in the dual Steenrod algebra A ∗ ∼ = P ( ¯ ξ i | i ≥ ⊗ E (¯ τ i | i ≥ which is defined on each algebra generator by the formulas ∆( ¯ ξ n ) = P i + j = n ¯ ξ i ⊗ ¯ ξ p i j ∆(¯ τ n ) = 1 ⊗ ¯ τ n + P i + j = n ¯ τ i ⊗ ¯ ξ p i j . Here, by ¯ x we mean χx where χ : A ∗ → A is the antipode structure map of the Hopf-algebra A ∗ . When E = BP , the co-product on elements of BP ∗ BP ∼ = Z ( p ) [ v , v , . . . ] ⊗ Z ( p ) [ t , t , . . . ] is defined by the formula ∆( t n ) = P Fi + j = n t i ⊗ t p i j where F is the formal group law of BP associated to the complex orientation M U → BP ,which equips BP with the universal p -typical formal group law. Throughout, we will write H ∗ ( − ) for H ∗ ( − ; F p ) ; i.e, homology with F p -coefficients. Also, throughout we will work at aprime p ≥ and we will let q be a prime power that topologically generates Z × p or equivalentlygenerates the units in Z /p Z . We will write Z p for p -complete integers and X p for the p -completion of a spectrum, which agrees with the Bousfield localization L S/p X at the mod p Moore spectrum
S/p . We will write ˙= to indicate that an equality holds up to multiplicationby a unit in F p . In the introduction, we used ℓ to denote our fixed prime because that is moreclosely aligned with conventions in étale cohomology, but the author is a homotopy theoristat heart and therefore can’t resist using p to denote our fixed prime, which is more commonin chromatic homotopy theory.1.2. Acknowledgements.
This paper grew out of the author’s Ph.D. thesis. The authorwould like to thank Andrew Salch for many discussions on the material in this paper and forhis constant support and encouragement. Also, the author would like to thank Bob Brunerfor offering his insight about the homological homotopy fixed point spectral sequence.2.
Overview of the toolkit
The THH-May spectral sequence for K ( F q ) p . We recall necessary results and def-initions from the author’s paper [2] and the author’s joint paper with A. Salch [3] since theywill be cited later.
Definition 2.1.
A filtered commutative ring spectrum I is a cofibrant object in Comm S N op where Comm S N op has the model structure created by the forgetful functor to S N op and S N op has the projective model structure. (See [3, Sec. 4.1] for a discussion of why these modelstructures exist and have the desired properties). We write I i for I evaluated on the naturalnumber i . The associated graded of I is defined as a commutative ring spectrum E I in [3]and it is defined so that, after forgetting the commutative monoid structure, it is the spectrum ∨ i ≥ I i /I i +1 where I i /I i +1 is the cofiber of the map I i +1 → I i . (Note that since I is cofibrant,the map I i +1 → I i is a cofibration and I i is cofibrant for each i so the cofiber agrees with thehomotopy cofiber.) GABE ANGELINI-KNOLL
Remark 2.2.
This definition differs slightly from that in [3, Def 3.1.2]. A cofibrant objectin
Comm S N op is always a decreasingly filtered commutative monoid in S in the sense of[3, Def 3.1.2] the converse is not always true. We will therefore work with a smaller categoryof filtered commutative ring spectra then in [3], but it will be sufficient for our purposes. Example 2.3.
As was proven in [3, Thm 4.2.1], an example of a filtered commutative ringspectrum associated to a connective commutative ring spectrum R is the Whitehead filtration · · · → τ ≥ R → τ ≥ R → τ ≥ R which is equipped with structure maps ρ i,j : τ ≥ i R ∧ τ ≥ j R → τ ≥ i + j R . Here τ ≥ s R is a spectrumwith π i ( τ ≥ s R ) ∼ = 0 for i < S that is equipped with a map τ ≥ s R → R that induces an isomor-phism on homotopy groups π i for i ≥ s . We write simply τ ≥• R for the filtered commutativering spectrum constructed in [3, Thm 4.2.1] as a cofibrant object in Comm S N op . Theorem 2.4 (Theorem 3.4.8 [3]) . There is a spectral sequence associated to a filteredcommutative ring spectrum I in topological Hochschild homology for any connective spectrumhomology theory E ∗ E ∗ , ∗ = E ∗ , ∗ ( T HH ( E I )) ⇒ E ∗ ( T HH ( I )) which we call the E -THH-May spectral sequence. Remark 2.5.
When I = τ ≥• R we simply write Hπ ∗ R for E I . It is a generalized Eilenberg-Maclane spectrum so whenever π k R is a finitely generated abelian group for all k and E = S/p , H F p , V (1) , or BP ∧ V (1) , then E ∗ T HH ( E I ) is a graded H F p -algebra and we can apply thefollowing lemma to compute the input.The following lemma is a consequence of the fact that all H F p -modules are equivalent toa wedge of suspensions of H F p and an Adams spectral sequence argument, see [7] for analternate proof. Lemma 2.6.
Let M be an H F p -algebra. Then M is equivalent to a wedge of suspensions of H F p , and the Hurewicz map π ∗ M −→ H ∗ M induces an isomorphism between π ∗ M and the subalgebra of A ∗ -co-module primitives con-tained in H ∗ M .Using the lemma above, we can compute the E -page of the H F p ∧ V (1) -THH-May spectralsequence. For details see [2]. Proposition 2.7.
There is an isomorphism of A ∗ -comodule algebras ( H F p ∧ V (1)) ∗ T HH ( Hπ ∗ K ( F q ) p ) ∼ = A ∗ ⊗ E ( ǫ ) ⊗ P (˜ v ) ⊗ E ( σ ¯ ξ , σ ˜ v ) ⊗ P ( µ ) ⊗ HH ∗ ( S/p ∗ ( Hπ ∗ K ( F q ) p )) where the A ∗ -co-action is the usual one, that is the coproduct in A ∗ , on elements in A ∗ andthe remaining co-actions are primitive.We can compute the input of the V (1) -THH-May spectral sequence using Lemma 2.6. Proposition 2.8 ( Proposition 3.6 [2]) . There is an isomorphism of graded F p -algebras V (1) ∗ T HH ( Hπ ∗ K ( F q ) p )) ∼ = E ( λ , ǫ , σ ˜ v ) ⊗ P ( µ , ˜ v ) ⊗ HH ∗ ( S/p ∗ ( Hπ ∗ K ( F q ) p )) where | ǫ | = | λ | = | σ ˜ v | = 2 p − , | α | = 2 p − , | µ | = 2 p , | ˜ v | = 2 p − , and | σα | = 2 p − .Our computations build on the computation of homology of topological Hochschild homol-ogy of K ( F q ) p due to Angeltveit-Rognes [4]. ETECTING THE β -FAMILY IN ITERATED ALGEBRAIC K-THEORY OF FINITE FIELDS 7 Theorem 2.9 (Theorem 7.13 and Theorem 7.15 [4]) . There is an isomorphism of A ∗ -comodulealgebras H ∗ K ( F q ) p ∼ = P ( ˜ ξ p , ˜ ξ , ¯ ξ , ... ) ⊗ E (˜ τ , ¯ τ , ... ) ⊗ E ( b ) ∼ = ( A //A (1)) ∗ ⊗ E ( b ) where all the elements in ( A //A (1)) ∗ besides ˜ τ , ˜ ξ p , and ˜ ξ , and b have the usual A ∗ -co-actionand the co-action on the remaining elements ˜ τ , ˜ ξ p , ˜ ξ , and b are ψ ( b ) = 1 ⊗ bψ ( ˜ ξ p ) = 1 ⊗ ˜ ξ p − τ ⊗ b + ¯ ξ p ⊗ ψ ( ˜ ξ ) = 1 ⊗ ˜ ξ + ¯ ξ ⊗ ˜ ξ p + τ ⊗ b + ¯ ξ ⊗ ψ (˜ τ ) = 1 ⊗ ˜ τ + ¯ τ ⊗ ˜ ξ p + ¯ τ ⊗ ˜ ξ − τ τ ⊗ b + ¯ τ ⊗ . There is also an isomorphism H ∗ T HH ( K ( F q ) p ) ∼ = H ∗ K ( F q ) p ⊗ E ( σ ˜ ξ p , σ ˜ ξ ) ⊗ P ( σ ˜ τ ) ⊗ Γ( σb ) of A ∗ -co-modules and H ∗ K ( F q ) p -algebras. The A ∗ -co-action is given by using the formula ψ ( σx ) = (1 ⊗ σ ) ◦ ψ ( x ) and the previously stated co-actions.We now recall the computation of V (1) -homotopy of topological Hochschild homology of K ( F q ) p where p ≥ and q is a prime power that generates ( Z /p Z ) × in [2]. Theorem 2.10 (Theorem 1.3 [2]) . There is an isomorphism of graded F p -algebras V (1) ∗ T HH ( K ( F q ) p ) ∼ = P ( µ ) ⊗ Γ( σb ) ⊗ F p { , α , λ ′ , λ α , λ λ ′ , λ λ ′ α } . where α · ( λ λ ′ ) = λ ′ · ( λ α ) = λ λ ′ α .2.2. The generalized homological homotopy fixed point spectral sequence.
In thissection, we summarize and extend results from Sections 2-4 of [13]. The main generalizationis from H F p to a connective homology theory E ∗ such that E is a ring spectrum and E ∗ is agraded F p -algebra.Let T ⊂ C × be the circle group, and let X be a T -spectrum. We let E T = S ( C ∞ ) , the unitsphere in C ∞ , where T acts on C by rotation and on C ∞ coordinate-wise. It is well knownthat there is a T -equivariant filtration of E T + (3) ∅ ֒ → S ( C ) + ֒ → S ( C ) + ⊂ . . . ֒ → E T + . such that the cofiber of each map S ( C n ) + ֒ → S ( C n +1 ) + is T + ∧ S n for n ≥ . We mayproduce a tower of cofiber sequences by applying the functor F ( − , X ) T to the tower of T -equivariant cofibrations (3) and, since we will take F ( E T + , X ) T as our model for X h T , wehave X h T = lim k F ( S ( C n ) + , X ) T . Since, by adjunction, F ( T + ∧ S n , X ) T ∼ = F ( S n , X ) = Σ − n X, we can apply a connective homology theory E ∗ ( − ) to the tower of cofiber sequences above toproduce the unrolled exact couple of E ∗ E co-modules . . . / / E ∗ F ( S ( C n +1 ) + , X ) T i / / E ∗ F ( S ( C n ) + , X ) T i / / j (cid:15) (cid:15) E ∗ F ( S ( C n ) + , X ) T / / j (cid:15) (cid:15) . . .. . . E ∗ Σ − n X k j j ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ k i i ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ . . . Theorem 1.3 in [2] was also proven concurrently using a different method by Eva Höning in her Ph.D. thesis.
GABE ANGELINI-KNOLL where j shifts degree by − .Then there exists a spectral sequence with input(4) E s,t ∼ = ( E t X if s = − n otherwisewhich conditionally converges to E cs + t X h T := lim E s + t F ( S ( C k ) , X ) T . which we will refer to as continuous E -homology of X h T . In order to identify the E -page ina particularly nice way, we will use an extra assumption on E . Lemma 2.11.
Suppose E ∗ is a graded F p -algebra, then the E page of (4) can be identifiedas follows: E ∗ , ∗ = H ∗ gp ( T , F p ) ⊗ E ∗ X. where H ∗ gp ( T , F p ) = H ∗ ( B T , F p ) ∼ = P ( t ) where | t | = − . Proposition 2.12 (c.f. Proposition 2.1 and Proposition 4.1 in [13]) . Let E be a ring spectrumsuch that E ∗ a connective graded F p -algebra. There is a natural homological spectral sequenceof E ∗ E co-modules E ∗ , ∗ = P ( t ) ⊗ E ∗ ( X ) which strongly converges to E c ∗ ( X h T ) when E ∗ X is finite type or the spectral sequence col-lapses at the E N ∗ , ∗ -page for some N ≥ , and conditionally converges otherwise. If, in addition, X is a commutative ring spectrum, then this is a spectral sequence of E ∗ E -comodule algebraswhere E ∗ X has the Pontryagin product. Proof.
The proof is the same as that of [13] and therefore we omit it here. (cid:3)
Proposition 2.13 (c.f Section 3 in [13]) . Suppose E ∗ T HH ( R ) is a non-negatively gradedgraded F p -vector space. The d differentials in the generalized homological homotopy fixedpoint spectral sequence associated to T HH ( R ) are of the form d ( x ) = tσx. where t is the generator of H −∗ gp ( T ; F p ) ∼ = P ( t ) in degree − . Proof.
The proof is essentially the same as that of Bruner-Rognes and therefore we omitit. (cid:3)
In the sequel, we will write T k ( R ) for F ( S ( C k ) + , T HH ( R )) T . Note that there is a truncatedhomotopy fixed point spectral sequence with k columns converging to E ∗ ( T k ( R )) and lim E ∗ T k ( R ) = E c ∗ ( T HH ( R ) h T ) . Classically, negative cyclic homology HC −∗ ( A ) of a commutative ring A is π ∗ ( B cy ⊗ ( A ) h T where π ∗ B cy ⊗ ( A ) is the usual Hochschild homology of A . This lead Hesselholt [19] to coin theterm topological negative cyclic homology for the T -homotopy fixed points T HH ( R ) h T oftopological Hochschild homology of a commutative ring spectrum R and use notation T C − ( R ) to denote this object. We will continue to follow this convention and also write E c ∗ ( T C − ( R )) for E c ∗ ( T HH ( R ) h T ) . ETECTING THE β -FAMILY IN ITERATED ALGEBRAIC K-THEORY OF FINITE FIELDS 9 Detecting the β -family in iterated algebraic K-theory of finite fields Detecting v and β . The mod p Moore spectrum
S/p and the Smith-Toda complex V (1) are defined so that they fit into exact triangles S p / / S i / / S/p j / / Σ S and Σ p − S/p v / / S/p i / / V (1) j / / Σ p − S/p in the stable homotopy category of spectra. We will abuse notation and write i : π ∗ S/p → π ∗ V (1) and i i : π ∗ S → V (1) for the maps induced by i , i and i ◦ i respectively.In the proof of the following proposition, we will will make use of differentials in boththe generalized homological homotopy fixed point spectral sequence and the Adams spectralsequence. To differentiate between the two, we use notation d r for differentials in the gen-eralized homological homotopy fixed point spectral sequence and we use the notation d r fordifferentials in the Adams spectral sequence. The following argument is inspired an argumentof Ausoni-Rognes [5, Prop. 4.8]. Proposition 3.1.
The classes v , i i β , and i β ′ in V (1) ∗ map nontrivially to the classes tµ , tσb , and tσ ˜ ξ p respectively in V (1) ∗ T C − ( K ( F q ) p ) . Proof.
First, v is represented by ¯ τ ⊗ , β ′ is represented by ¯ ξ p ⊗ and β is represented by(5) b , = Σ p − i =1 p (cid:18) pi (cid:19) ¯ ξ i ⊗ ¯ ξ j ⊗ , in the E -page of the Adams spectral sequence that converges to π ∗ V (1) by [22] (cf. Section9 of [23]). We consider the map of Adams spectral sequencesExt ∗ , ∗ A ∗ ( F p , H ∗ V (1)) −→ Ext ∗ , ∗ A ∗ ( F p , H ∗ V (1) ⊗ T ( K ( F q ) p )) induced by the unit map V (1) ∧ S V (1) ∧ η / / V (1) ∧ T ( K ( F q ) p )) . We see that ¯ τ ⊗ , ¯ ξ p ⊗ , and b , are permanent cycles in the source, which map to classes ofthe same name in the target. Since the elements in the source are infinite cycles, this impliesthat the elements that they map to are infinite cycles as well. We then have to check thatthese classes are not boundaries.We can eliminate the possibility of a d differential with ¯ τ ⊗ as a co-boundary by com-puting the differential in the cobar complex for H ∗ V (1) ⊗ H ∗ T ( K ( F q ) p ) on each class ofthe correct degree. If the ¯ τ is an element in H ∗ ( T ( K ( F q )) with ψ (¯ τ ) = ¯ τ ⊗ ⊗ ¯ τ ,then d (¯ τ ) = ¯ τ ⊗ . However, the two-column homological homotopy fixed point spectralsequence computing H ∗ T ( K ( F q ) p ) has a differential d (¯ τ ) = tµ , by Proposition 2.13 andthe fact that µ = σ ¯ τ . Therefore, the class ¯ τ does not survive to H ∗ T ( K ( F q ) p ) .The only other classes in the the right degree in H ∗ V (1) ⊗ H ∗ T ( K ( F q ) p ) to be the sourceof a d hitting ¯ τ ⊗ are σ ˜ ξ and ¯ τ σb . However, σb is primitive so d ( σb ) = 0 . Also, we cancompute directly d ( σ ˜ ξ ) = ¯ ξ ⊗ σ ˜ ξ p + ¯ τ ⊗ σb and d (¯ τ ) = ¯ τ ⊗ τ ⊗ ¯ ξ . Therefore, d ( α ¯ τ σb + βσ ˜ ξ ) = α (¯ τ ⊗ σb + ¯ τ ⊗ ¯ ξ σb ) + β ( ¯ ξ ⊗ σ ˜ ξ p + ¯ τ ⊗ σb ) = ¯ τ ⊗ for any α, β ∈ F p . Therefore, ¯ τ ⊗ survives to the E -page. There are no possible longerdifferentials hitting ¯ τ ⊗ because ¯ τ ⊗ is in Adams filtration one; hence, it is a permanentcycle.We eliminate the possibility that the class ¯ ξ p ⊗ is a boundary of a d by the same method.As in the previous argument, the truncated homotopy fixed point spectral sequence convergingto H ∗ T ( K ( F q ) p ) has a differential d ( ¯ ξ p ) = σ ¯ ξ p by Proposition 2.13, so the class ¯ ξ p does notsurvive to become a class in H ∗ T ( K ( F q ) p ) . Therefore, the only classes that are in the rightdegree in H ∗ V (1) ∧ T ( K ( F q ) p ) to have ¯ ξ p ⊗ as their co-boundary are { ¯ τ σ ¯ ξ p , σb. } However, d ( σb ) = 0 , since it is a co-module primitive, and d (¯ τ σ ˜ ξ p ) = 1 ⊗ ¯ τ σ ¯ ξ p − ¯ τ ⊗ σ ˜ ξ p − ⊗ ¯ τ σ ˜ ξ p = ¯ ξ p ⊗ modulo boundaries. The class ¯ ξ p ⊗ is in Adams filtration one so it cannot be the target ofa longer differential, therefore it is a permanent cycle.For b , , we need to check that it is not the boundary of a d or a d , because it is inAdams filtration two. We first need to check that it is not a boundary of an element in A ∗ ⊗ H ∗ V (1) ∧ T ( K ( F q ) p ) . We check the differential in the cobar complex on all the elementshere in the right degree. These classes are (cid:26) ⊗ σb, ¯ τ ⊗ ¯ τ t ˜ ξ p , ¯ ξ p − ⊗ ¯ τ ¯ τ , ¯ ξ p − ¯ τ ¯ τ ⊗ ¯ τ ¯ τ , ¯ ξ p − ¯ τ ⊗ ¯ τ , ¯ ξ p − ¯ τ ⊗ ¯ τ , ¯ ξ p − ¯ τ ¯ τ ⊗ , ¯ ξ p ⊗ (cid:27) where ˜ ξ p has a coproduct coming from H ∗ K ( F q ) and ¯ ξ p has the co-action coming from thecoproduct on A ∗ . Recall that Milnor computed the co-action of A ∗ on H ∗ ( C P ∞ , F p ) ∼ = H ∗ ( B T ; F p ) ∼ = H ∗ ( T ; F p ) , and the co-action on the class t is ψ ( t ) = Σ i ≥ ¯ ξ i ⊗ t p i , see [24]. Therefore, in the input of the truncated homotopy fixed point spectral sequencecomputing V (1) ∗ T ( K ( F q ) p ) , the A ∗ co-action on t is primitive.We compute the differential in the cobar complex on each of the elements that could possiblyhave the class representing β as a target: d (1 ⊗ σb ) = 1 ⊗ ⊗ σbd (¯ τ ⊗ ¯ τ t ˜ ξ p ) = ¯ τ ⊗ ¯ τ ⊗ t ˜ ξ p + ¯ τ ⊗ ¯ ξ p ⊗ t ¯ τ + ¯ τ ⊗ ¯ τ ⊗ ¯ τ tbd ( ¯ ξ p − ⊗ ¯ τ ¯ τ ) = 1 ⊗ ¯ ξ p − ⊗ ¯ τ ¯ τ − ∆( ¯ ξ p − ) ⊗ ¯ τ ¯ τ + ¯ ξ p − ⊗ ψ (¯ τ ¯ τ )= − P p − i =1 (cid:0) p − i (cid:1) ¯ ξ p − i − ⊗ ¯ ξ i ⊗ ¯ τ ¯ τ + ¯ ξ p − ⊗ ¯ τ ⊗ ¯ τ + ¯ ξ p − ⊗ ¯ τ ¯ τ ⊗ ξ p − ⊗ ¯ τ ⊗ ¯ τ + ¯ ξ p − ⊗ ¯ τ ⊗ ¯ τ ¯ ξ d ( ¯ ξ p − ¯ τ ¯ τ ⊗ ¯ τ ¯ τ ) = 1 ⊗ ¯ ξ p − ¯ τ ¯ τ − ∆( ¯ ξ p − ¯ τ ¯ τ ) ⊗ ¯ τ ¯ τ + ¯ ξ p − ¯ τ ¯ τ ⊗ ψ (¯ τ ¯ τ )= 1 ⊗ ¯ ξ p − ¯ τ ¯ τ ⊗ ¯ τ ¯ τ − P p − i =0 (cid:0) p − i (cid:1) ¯ ξ i ⊗ ¯ ξ p − i − ¯ τ ¯ τ ⊗ ¯ τ ¯ τ − P p − i =0 (cid:0) p − i (cid:1) ¯ ξ i ¯ τ ⊗ ¯ ξ p − i − ¯ τ ⊗ ¯ τ ¯ τ − P p − i =0 (cid:0) p − i (cid:1) ¯ ξ i ¯ τ ¯ τ ⊗ ¯ ξ p − i − ⊗ ¯ τ ¯ τ − P p − i =0 (cid:0) p − i (cid:1) ¯ ξ i ¯ τ ⊗ ¯ ξ p − i − ¯ τ ⊗ ¯ τ ¯ τ − P p − i =0 (cid:0) p − i (cid:1) ¯ ξ i ¯ τ ⊗ ¯ ξ p − i − ¯ ξ ¯ τ ⊗ ¯ τ ¯ τ + ¯ ξ p − ¯ τ ¯ τ ⊗ ⊗ ¯ τ ¯ τ + ¯ ξ p − ¯ τ ¯ τ ⊗ ¯ τ ⊗ ¯ τ + ¯ ξ p − ¯ τ ¯ τ ⊗ ¯ τ ¯ τ ⊗ ξ p − ¯ τ ¯ τ ⊗ ¯ τ ⊗ ¯ τ ETECTING THE β -FAMILY IN ITERATED ALGEBRAIC K-THEORY OF FINITE FIELDS 11 d ( ¯ ξ p − ¯ τ ⊗ ¯ τ ) = 1 ⊗ ¯ ξ p − ¯ τ ⊗ ¯ τ − ∆( ¯ ξ p − ¯ τ ) ⊗ ¯ τ + ¯ ξ p − ¯ τ ⊗ ψ (¯ τ )= 1 ⊗ ¯ ξ p − ¯ τ ⊗ ¯ τ − P p − i =0 (cid:0) p − i (cid:1) ¯ ξ i ⊗ ¯ ξ p − i − ¯ τ ⊗ ¯ τ − P p − i =0 (cid:0) p − i (cid:1) ¯ ξ i ¯ τ ⊗ ¯ ξ p − i − ⊗ ¯ τ + ¯ ξ p − ¯ τ ⊗ ⊗ ¯ τ + ¯ ξ p − ¯ τ ⊗ ¯ τ ⊗ d ( ¯ ξ p − ¯ τ ⊗ ¯ τ ) = 1 ⊗ ¯ ξ p − ¯ τ ⊗ ¯ τ − ∆( ¯ ξ p − ¯ τ ) ⊗ ¯ τ + ¯ ξ p − ¯ τ ⊗ ⊗ ψ (¯ τ )= 1 ⊗ ¯ ξ p − ¯ τ ⊗ ¯ τ − P p − i =0 (cid:0) p − i (cid:1) ¯ ξ i ⊗ ¯ ξ p − − i ¯ τ ⊗ ¯ τ − P p − i =0 (cid:0) p − i (cid:1) ¯ ξ i ¯ τ ⊗ ¯ ξ p − − i ¯ ξ ⊗ ¯ τ − P p − i =0 (cid:0) p − i (cid:1) ¯ ξ i ¯ τ ⊗ ¯ ξ p − − i ¯ τ ⊗ ¯ τ + ¯ ξ p − ¯ τ ⊗ ⊗ ¯ τ + ¯ ξ p − ¯ τ ⊗ ¯ τ ⊗ d ( ¯ ξ p − ¯ τ ¯ τ ⊗
1) = 1 ⊗ ¯ ξ p − ¯ τ ¯ τ ⊗ − ∆( ¯ ξ p − ¯ τ ¯ τ ) ⊗ ξ p − ¯ τ ¯ τ ⊗ ⊗
1= 1 ⊗ ¯ ξ p − ¯ τ ¯ τ ⊗ − P p − i =0 (cid:0) p − i (cid:1) ¯ ξ i ⊗ ¯ ξ p − − i ¯ τ ¯ τ ⊗ − P p − i =0 (cid:0) p − i (cid:1) ¯ ξ i ¯ τ ⊗ ¯ ξ p − − i ¯ ξ ¯ τ ⊗ − P p − i =0 (cid:0) p − i (cid:1) ¯ ξ i ¯ τ ⊗ ¯ ξ p − − i ¯ τ ¯ τ ⊗ − P p − i =0 (cid:0) p − i (cid:1) ¯ ξ i ¯ τ ⊗ ¯ ξ p − − i ¯ τ ⊗ − P p − i =0 (cid:0) p − i (cid:1) ¯ ξ i ¯ τ ¯ τ ⊗ ¯ ξ p − − i ¯ τ ⊗
1+ ¯ ξ p − ¯ τ ¯ τ ⊗ ⊗ d ( ¯ ξ p ⊗
1) = 0 . If some linear combination of these elements has b , as a boundary, then there is a solutionto the equation Σ p − i =1 1 p (cid:0) pi (cid:1) ξ i ⊗ ξ p − i ⊗ a d (1 ⊗ σb ) + a d (¯ τ ⊗ ¯ τ t ˜ ξ p ) + a d ( ¯ ξ p − ⊗ ¯ τ ¯ τ )+ a d ( ¯ ξ p − ¯ τ ¯ τ ⊗ ¯ τ ¯ τ ) + a d ( ¯ ξ p − ¯ τ ⊗ ¯ τ )+ a d ( ¯ ξ p − ¯ τ ⊗ ¯ τ ) + a d ( ¯ ξ p − ¯ τ ¯ τ ⊗ for some elements a i ∈ F p for ≤ i ≤ ; however, no such solutions to this equation exist sowe can conclude that b , is not a boundary of a d .Since b , is in Adams filtration two, we still have to check that there is no d differentialhitting it in the Adams spectral sequence,Ext ∗ , ∗ A ∗ ( F p , H ∗ ( V (1) ∧ T ( K ( F q ) p )) ⇒ V (1) ∗ T ( K ( F q ) p ) . Since a d would have to have its source on the -line in degree p − p − , it would haveto be a class in H p − p − V (1) ∧ T ( K ( F q ) p ) .We compute H p − p − V (1) ∧ T ( K ( F q ) p ) ∼ = F p { ¯ τ t ˜ ξ p } , since d ( b ) = tσb in the two column homotopy fixed point spectral sequence that computes H ∗ T ( K ( F q ) p ) . Since d (¯ τ ) = ¯ τ ⊗ , the Leibniz rule implies d (¯ τ ( t ˜ ξ p )) = (¯ τ ⊗ · d ( t ˜ ξ p ) = 0 , since d ( t ˜ ξ p ) = ¯ ξ p ⊗ t + ¯ τ ⊗ tb = 0 . So ¯ τ ( t ˜ ξ p ) does not survive to the E -page and thereforeit cannot support a differential hitting b , . Therefore, the class b , is a permanent cycle.We conclude the elements v , β ′ and β map nontrivially from V (1) ∗ S to V (1) ∗ T ( K ( F q ) p ) via map induced by the unit map S → T ( K ( F q ) p ) . In V (1) ∗ T ( K ( F q ) p ) , the only pos-sible classes in the right degree to be v , β ′ and β are tµ , tσb and tσ ¯ ξ p , respectively. The unit map factors through V (1) ∗ T C − ( K ( F q ) p ) , so these classes pull back to classes in V (1) ∗ T C − ( K ( F q ) p ) . (cid:3) Corollary 3.2.
The classes tµ , tσb , and tσ ˜ ξ p are permanent cycles in the generalized ho-mological homotopy fixed point spectral sequence H ∗ ( T , V (1) ∗ T HH ( K ( F q ) p )) ⇒ V (1) ∗ T HH ( K ( F q )) h T . in particular, d p − ( tµ ) = 0 , d p − ( tσb ) = 0 and d p − ( tσ ˜ ξ p ) = 0 . Lemma 3.3.
There is a differential d p − ( t ) = t p α in the homotopy fixed point spectralsequence H ∗ ( T , V (1) ∗ T HH ( K ( F q ) p )) ⇒ V (1) ∗ T HH ( K ( F q )) h T . Proof.
First, we can show that there is a differential d p − ( t ) = t p α in the homotopy fixedpoint spectral sequence H ∗ ( T , V (1) ∗ K ( F q ) p )) ⇒ V (1) ∗ K ( F q ) h T p . where K ( F q ) p has trivial T -action, because α is an attaching map in B T . This has alreadybeen proven in [12, Theorem 3.5], so we omit the details. There is an T -equivariant map ofcommutative ring spectra T HH ( K ( F q ) p ) → K ( F q ) p which induces a map of homotopy fixed point spectral sequences and since this map sends t to t and α to α , the differential d p − ( t ) = α t p also occurs in the homotopy fixed pointspectral sequence H ∗ ( T , V (1) ∗ T HH ( K ( F q ) p )) ⇒ V (1) ∗ T HH ( K ( F q )) h T . Note that we could have also proven this directly by examining T -equivariant attaching mapsin E T , but for the sake of brevity we give the simpler proof. (cid:3) Corollary 3.4.
There are differentials d p − ( µ ) = − t p − α µ and d p − ( σb ) = − t p − α σb . Proof.
This is immediate from Lemma 3.3 and Corollary 3.2. (cid:3)
Now, the classes β k have the property that in the BP -Adams spectral sequence for V (1) they are represented by the classes (cid:18) i (cid:19) v i − k + iv i − b , , where k = 2 t p ⊗ t ⊗ − t p ⊗ t p +11 ⊗ − t p ⊗ t ⊗ which are in BP -Adams filtration two. We will therefore give a similar argument to the onein the proof of Proposition 3.1, except that we will work in the BP -Adams spectral sequencein order to use the fact that the classes representing β k are in low BP -Adams filtration. Todo this we must compute BP ∧ V (1) ∗ T k ( K ( F q ) p ) up to possible d differentials or longer.3.2. The BP ∧ V (1) -THH-May spectral sequence. In this section, we begin by computingthe input of the BP ∧ V (1) -THH-May spectral sequence. Lemma 3.5.
There is an isomorphism of ( BP ∧ V (1)) ∗ ( Hπ ∗ ( K ( F q ) p )) -algebras(6) ( BP ∧ V (1)) ∗ T HH ( Hπ ∗ K ( F q ) p ) ∼ = P ( t , t , . . . ) ⊗ E ( ǫ , λ , σv ) ⊗ P ( v , µ ) ⊗ HH ∗ ( S/p ∗ ( Hπ ∗ K ( F q ) p )) , ETECTING THE β -FAMILY IN ITERATED ALGEBRAIC K-THEORY OF FINITE FIELDS 13 and the Hurewicz map ( BP ∧ V (1)) ∗ T HH ( E ( τ ≥• K ( F q ) p )) → ( H F p ∧ BP ∧ V (1)) ∗ T HH ( E ( τ ≥• K ( F q ) p )) sends t to ¯ ξ − ˆ ξ , where ¯ ξ is the generator in degree p − of H ∗ BP and ˆ ξ is the generatorin degree p − of H ∗ ( V (1) ∧ T HH ( E ( τ ≥• K ( F q ) p )) . Proof.
Recall that V (1) ∧ T HH ( Hπ ∗ ( K ( F q ) p )) is a V (1) ∧ Hπ ∗ ( K ( F q ) p ) -algebra, and hence an H F p algebra, since V (1) ∧ Hπ ∗ ( K ( F q ) p ) is itself an H F p -algebra. Thus, there is an equivalence BP ∧ V (1) ∧ T HH ( Hπ ∗ ( K ( F q ) p )) ≃ BP ∧ H F p ∧ H F p V (1) ∧ T HH ( Hπ ∗ ( K ( F q ) p )) and by the collapse of the Künneth spectral sequence, the isomorphism (6) holds.Since BP ∧ V (1) ∧ T HH ( Hπ ∗ ( K ( F q ) p )) is an H F p -module we can use Lemma 2.6, whichstates that ( BP ∧ V (1)) ∗ T HH ( Hπ ∗ ( K ( F q ) p )) includes as the co-module primitives inside of ( H F p ∧ BP ∧ V (1)) ∗ T HH ( Hπ ∗ ( K ( F q ) p )) . We recall that by the Künneth isomorphism and Proposition 2.7 there is an isomorphismof graded rings ( H F p ∧ BP ∧ V (1)) ∗ T HH ( Hπ ∗ ( K ( F q ) p )) ∼ = H ∗ ( BP ) ⊗ E (¯ τ , ¯ τ , λ , σv ) ⊗ ( A//E (0)) ∗ ⊗ P ( v , µ ) ⊗ HH ∗ ( S/p ∗ ( Hπ ∗ ( K ( F q ) p ))) where we use the notation ( A //E (0)) ∗ ∼ = P ( ˆ ξ , ˆ ξ , . . . ) ⊗ E (ˆ τ , ˆ τ , . . . ) and H ∗ ( BP ) ∼ = P ( ¯ ξ , ¯ ξ , . . . ) to distinguish the two sets of generators. We also write E (¯ τ , ¯ τ ) for the homology of V (1) .The co-action on ¯ ξ i , ¯ τ i , ˆ τ i and ˆ ξ i are the same as the coproduct in the dual Steenrod algebra,and hence for example ¯ ξ − ˆ ξ is a co-module primitive, since ψ ( ¯ ξ − ˆ ξ ) = 1 ⊗ ¯ ξ + ¯ ξ ⊗ − ⊗ ˆ ξ − ¯ ξ ⊗ ⊗ ¯ ξ − ⊗ ˆ ξ . The co-action on the remaining elements in degrees less than p − is ψ ( α ) = 1 ⊗ α ψ ( σv ) = 1 ⊗ σv + ¯ τ ⊗ σα ψ ( σα ) = 1 ⊗ σα ψ ( λ ) = 1 ⊗ λ ψ ( γ p k ( σα )) = 1 ⊗ γ p k ( σα ) ψ ( µ ) = 1 ⊗ µ + ¯ τ ⊗ λ ψ ( v ) = 1 ⊗ v + ¯ τ ⊗ α . and we may observe that there are no other A ∗ co-module primitives in degree p − otherthan ¯ ξ − ˆ ξ so t must map to ¯ ξ − ˆ ξ . (cid:3) Proposition 3.6.
There is an isomorphism of ( BP ∗ , BP ∗ BP ) -co-modules ( BP ∧ V (1)) ∗ T HH ( K ( F q ) p ) ∼ = P ( t p , t , ... ) ⊗ E ( b ) ⊗ E ( σ ¯ ξ p , σ ¯ ξ ) ⊗ P ( µ ) ⊗ Γ( σb ) where the co-action is given by ψ ( t p ) = 1 ⊗ t p + t p ⊗ ψ ( µ ) = 1 ⊗ µ ψ ( t n ) = ∆( t n ) for n ≥ ψ ( γ p k ( σb )) = 1 ⊗ γ p k ( σb ) ψ ( b ) = 1 ⊗ b ψ ( σx ) = (1 ⊗ σ ) ∗ ψ ( x ) Proof.
We need to compute differentials in the BP ∧ V (1) -THH-May spectral sequence E ∗ , ∗ = ( BP ∧ V (1)) ∗ , ∗ T HH ( Hπ ∗ ( K ( F q ) p ))) ⇒ BP ∧ V (1) ∗ T HH ( K ( F q ) p ) so we examine the map of spectral sequences ( BP ∧ V (1)) ∗ , ∗ T HH ( Hπ ∗ ( K ( F q ) p ))) + h (cid:15) (cid:15) BP ∧ V (1) ∗ T HH ( K ( F q ) p ) (cid:15) (cid:15) ( H F p ∧ BP ∧ V (1)) ∗ , ∗ T HH ( Hπ ∗ ( K ( F q ) p )) + ( H F p ∧ BP ∧ V (1)) ∗ T HH ( K ( F q ) p ) . induced by the Hurewicz map BP → H F p ∧ BP.
Recall from Lemma 3.5 that ( BP ∧ V (1)) ∗ T HH ( Hπ ∗ ( K ( F q ) p )) ∼ = P ( ξ , ξ , . . . ) ⊗ E ( ǫ , λ , σv ) ⊗ P ( v , µ ) ⊗ HH ∗ ( S/p ∗ ( Hπ ∗ ( K ( F q ) p ))) . We know that in the H F p ∧ BP ∧ V (1) -THH-May spectral sequence the classes ¯ ξ i for i ≥ and ¯ τ j for j = 0 , survive to E ∞ , since the output of the spectral sequence is known to be ( H F p ∧ BP ∧ V (1)) ∗ T HH ( K ( F q ) p ) ∼ = P ( ¯ ξ , ¯ ξ , . . . ) ⊗ E (¯ τ , ¯ τ ) ⊗ H ∗ ( K ( F q ) p ) ⊗ E ( σ ¯ ξ p , σ ¯ ξ ) ⊗ P ( σ ¯ τ ) ⊗ Γ( σb ) by Theorem 2.9 and the Künneth isomorphism. This forces the same d differentials thatoccur in the H F p ∧ V (1) -THH-May spectral sequence and consequently there is an additiveisomorphism E ∗ , ∗ = P ( ξ , ξ , . . . ) ⊗ E ( ǫ , λ , σv , α ) ⊗ P ( v , µ ) ⊗ Γ( σα ) . The map of spectral sequences is therefore again injective on E -pages. In the H F p ∧ V (1) -THH-May spectral sequence there are differentials d p − ( ˆ ξ ) = α d r ( ¯ ξ i ) = 0 d p − ( λ ) = σα d p − (ˆ τ ) = v d p − ( µ ) = σv d r (¯ τ i ) = 0 for i = 0 , d r (ˆ τ i ) = 0 for i > for r ≥ and no further differentials. Since the Hurewicz map h is injective and it sends t to ¯ ξ − ˆ ξ , the differential d p − ( t ) in the top spectral sequence can be computed using theformula d p − ( t ) = d p − ( h − ( ¯ ξ − ˆ ξ )) = h − d p − ( ¯ ξ − ˆ ξ ) = h − ( α ) = α . Similarly, ǫ maps to ¯ τ − ˆ τ implying d p − ( ǫ ) = v . Hence, in the BP ∧ V (1) -THH-Mayspectral sequence there are differentials d p − ( t ) = α , d p − ( λ ) = σα , d p − ( ǫ ) = v , d p − ( µ ) = σv . On E -pages the map of spectral sequences induced by the Hurewicz map is again injective.Since E ∼ = E ∞ in the target spectral sequence, the same is true in the source. This impliesthat the BP ∧ V (1) -THH-May spectral sequence collapses at the E -page.By examining the long exact sequence BP ∗ V (1) ∧ K ( F q ) p → BP ∗ V (1) ∧ ℓ → BP ∗ V (1) ∧ Σ p − ℓ we can determine that the co-action on t p and t i for i ≥ in BP ∗ K ( F q ) p is the same as theco-action on these elements in BP ∗ V (1) ∧ ℓ ∼ = P ( t , t , . . . ) . Note that there is no hiddencomultiplication on t p since there are no classes in degrees p − p − (2 p − or lower andthe lowest degree element in BP ∗ BP is in degree p − . The class b is the class in lowestdegree and therefore it is primitive. This produces the co-action on b, t p , t i for i ≥ in BP ∗ V (1) ∧ T HH ( K ( F q ) p ) , by using the splitting of BP ∗ BP -co-modules BP ∗ V (1) ∧ T HH ( K ( F q ) p ) ∼ = BP ∗ V (1) ∧ K ( F q ) p ⊕ BP ∗ V (1) ∧ T HH ( K ( F q ) p ) ETECTING THE β -FAMILY IN ITERATED ALGEBRAIC K-THEORY OF FINITE FIELDS 15 induced by the splitting T HH ( K ( F q ) p ) ≃ K ( F q ) p ∨ T HH ( K ( F q ) p ) , which we have because K ( F q ) p is a commutative ring spectrum.The co-action on µ is primitive because | µ | = 2 p and there are no classes in degrees p − p + 2 or p − p + 4 or lower and the classes in BP ∗ BP are in degrees congruent tozero mod p n − for some n . Similarly, the co-action on λ ′ is primitive because there are noclasses in degree p − p + 1 − (2 p − or lower.To determine the co-action on λ , note that there is an isomorphism BP ∗ V (1) ∧ K ( F q ) p ∼ = P ( ¯ ξ p , ¯ ξ , . . . ) ∼ = P ( t p , t , . . . ) so ¯ ξ and t are two names for the same basis element up to multiplication by a unit. Similarly, ¯ ξ p and t are two names for the same basis element up to multiplication by a unit. The opera-tion σ gives λ = σ ¯ ξ ˙= σt and λ ′ ˙= σt p and we can therefore compute the co-action on λ usingthe formula ψ ( λ ) = (1 ⊗ σ )∆( t ) , due to [4]. In other words, in ( BP ∧ V (1)) ∗ T HH ( K F q ) p ) , ψ ( λ ) ˙=1 ⊗ λ + t ⊗ λ ′ . This just leaves the classes γ p k ( σb ) for k > . Note that we already showed that inthe input of BP ∧ V (1) -THH-May spectral sequence the classes γ p k +1 ( σα ) = γ p k ( σb ) areprimitive. Therefore, it suffices to check that there is not a hidden co-action in the THH-Mayspectral sequence. If the co-action contains terms of the form x ⊗ m where | m | < | γ p k ( σα ) | ,then the May filtration of m must be greater or equal to the May filtration of γ p k ( σb ) .Suppose the May filtration of m is greater or equal to p k +1 , the May filtration of γ p k ( σb ) .Then, since the only classes with positive May filtration are γ p j ( σb ) , b , λ ′ , and λ , the class m must be of the form ( γ p j ( σb )) ℓ b ǫ λ ′ ǫ λ ǫ z, for some possibly zero element z , where ≤ ℓ < p and ǫ , ǫ , ǫ ∈ { , } . Write mfilt( x ) forthe May filtration of an element, then mfilt( γ p j ( σb )) = p j +1 mfilt( b ) = 1mfilt( λ ′ ) = p − λ ) = 1 . so j, ℓ, ǫ , ǫ , and ǫ must satisfy(7) ℓp j +1 + ǫ + ǫ ( p −
1) + ǫ ≥ p k +1 . We split into cases. If k = 1 , then j ≥ k − , and if j = k − , then the inequality (7) onlyholds if ℓ = p − . In that case, ǫ must be and either ǫ or ǫ must be . Thus, | ( γ p j ( σb )) ℓ b ǫ λ ′ ǫ λ ǫ | ≥ (2 p − p )( p −
1) + 2 p − p + 1 + 2 p − p − p − p But, p − p > p − p = | γ p ( σb ) | contradicting the assumption that | m | < | γ p ( σb ) | . Inthe case k > , then the inequality (7) only holds if j ≥ k , but if j ≥ k , then | ( γ p j ( σb )) ℓ | ≥ p k +2 − p k +1 = | γ p k ( σb ) | so again m does not satisfy | m | < | γ p k ( σb ) | . Thus, no such m such that | m | < | γ p k ( σb ) | and mfilt( m ) ≥ mfilt( γ kp ( σb )) exists. This implies that there are no hidden co-actions and γ p k ( σb ) remains a co-module primitive. (cid:3) Corollary 3.7.
In the generalized homological homotopy fixed point spectral sequence H ∗ ( T , ( BP ∧ V (1)) ∗ T HH ( K ( F q ) p )) ⇒ ( BP ∧ V (1)) c ∗ T HH ( K ( F q ) p ) there are differentials d ( t p ) ˙= tλ ′ d ( t ) ˙= tλ d ( b ) = tσb and no further d differentials besides those generated from these d differentials using theLeibniz rule. Proof.
This follows from Proposition 2.13 and the fact that λ ˙= σt and λ ′ ˙= σt p as discussedin the proof of Proposition 3.6. (cid:3) Remark 3.8.
We will also need to know the co-action of BP ∗ BP on BP ∗ V (1) ∧ T k +1 ( K ( F q ) p ) , which is isomorphic to P ( ¯ ξ p , ¯ ξ , ... ) ⊗ E ( b ) ⊗ E ( σ ¯ ξ p , σ ¯ ξ ) ⊗ P ( µ ) ⊗ Γ( σb ) ⊗ P ( t ) /t k modulo differentials. This just amounts to describing the co-action on the class t in the inputof the generalized homological homotopy fixed point spectral sequence(8) H ∗ ( T , BP ∗ V (1) ∧ T HH ( K ( F q ) p )) , since the coaction on a subquotient of (8) is determined by the coaction on (8).Since we know that ψ H F p ( t ) = P i ≥ ¯ ξ i ⊗ t and t i ˙= ¯ ξ i in H ∗ BP , the functor that sendsthe A ∗ -comodule H ∗ ( T , F p ) to the BP ∗ BP -comodule H ∗ ( T , F p ) ⊗ BP ∗ / ( p, v ) produces thecoaction ψ BP ( t ) = X i ≥ t i ⊗ t p i H ∗ ( T , F p ) .Note that there is a truncated generalized homotopy fixed point spectral sequence withinput(9) E ∗ , ∗ = P ( t ) /t k +1 ⊗ BP ∗ V (1) ∧ T HH ( K ( F q ) p ) and abutment BP ∗ V (1) ∧ T k +1 ( K ( F q ) p ) . Corollary 3.9.
In the spectral sequence (9) computing BP ∗ V (1) ∧ T k +1 ( K ( F q ) p ) , there isan isomorphism between E ∗ , ∗ and (cid:16) P ( ¯ ξ p , ¯ ξ p , ¯ ξ , ... ) ⊗ E ( bγ p − ( σb )) ⊗ E ( σ ¯ ξ p t p − p , σ ¯ ξ t p − ) ⊗ P ( µ ) ⊗ Γ( γ p ( σb )) ⊗ P ( t ) /t k +1 (cid:17) ⊕ F p { λ ′ ( t p ) j − , λ t j − , γ s ( σb ) , t k ( t p ) j , t k ( t ) j , t k bγ s − ( σb ) | ≤ j < p, ≤ j < p, ≤ s < p }⊗ (cid:16) P ( ¯ ξ p , ¯ ξ p , ¯ ξ , ... ) ⊗ E ( bγ p − ( σb )) ⊗ E ( σ ¯ ξ p t p − p , σ ¯ ξ t p − ) ⊗ P ( µ ) ⊗ Γ( γ p ( σb )) (cid:17) of BP ∗ BP co-modules, where the coaction on an element x ∈ E ∗ , ∗ is determined multiplica-tively by the coaction of classes in BP ∗ V (1) ∧ T HH ( K ( F q ) p ) and the coaction of t fromRemark 3.8 modulo the differential d determined in Corollary 3.7. Proof.
This is a direct consequence of Corollary 3.7 and the Leibniz rule. (cid:3)
Detecting the β family in homotopy fixed points of topological Hochschildhomology. Recall that T k +1 ( R ) is defined to be the spectrum F ( S ( C k +1 ) + , T HH ( R )) T . Asnoted before Proposition 1.4 in [13], T k +1 ( R ) is a commutative ring spectrum whenever R isa commutative ring spectrum. In particular, T C − ( R ) is a commutative ring spectrum. Wenow recall a theorem, which is a consequence of computations of Ausoni-Rognes [5]. ETECTING THE β -FAMILY IN ITERATED ALGEBRAIC K-THEORY OF FINITE FIELDS 17 Theorem 3.10 (Ausoni-Rognes [5]) . The classes v k map to nonzero classes ( tµ ) k under theunit map V (1) ∗ S → V (1) ∗ T k +1 ( ℓ p ) . Remark 3.11.
Since we showed v maps to tµ under the unit map V (1) ∗ S → V (1) ∗ T ( K ( F q ) p ) and the maps V (1) ∗ S → T k +1 ( K ( F q ) p ) → V (1) ∗ T k +1 ( ℓ p ) are ring maps for k ≥ , the classes v k also map to ( tµ ) k under the unit map V (1) ∗ S → V (1) ∗ T k +1 ( K ( F q ) p ) . We therefore know that ( tµ ) k are permanent cycles in the BP -Adams spectral sequence andhomotopy fixed point spectral sequences computing V (1) ∗ T k +1 K ( F q ) p .We will continue to use notation d r for differentials in the generalized homological homotopyfixed point spectral sequence and d r for differentials in the BP -Adams spectral sequence todifferentiate the two. Theorem 3.12.
The elements β i in π ∗ S are detected by a unit times the class (cid:18) i (cid:19) ( tµ ) i − tλ λ + i ( tµ ) i − tσb in V (1) ∗ T C − ( K ( F q ) p ) ; i.e., the elements β i ∈ π (2 p − i − p S map to the nonzero elements (cid:0) i (cid:1) ( tµ ) i − tλ λ + i ( tµ ) i tσb in V (1) ∗ T C − ( K ( F q ) p ) up to multiplication by a unit. Proof.
Due to the length of this proof, we will break it into steps.
Step 1:
We will show that v i in the BP ∗ BP -cobar complex for V (1) maps to ( tµ ) i in the BP ∗ BP -cobar complex for V (1) ∗ T i ( K ( F q ) p )) . As discussed in Remark 3.11, we know that v i maps to ( tµ ) i under the map V (1) ∗ → V (1) ∗ T i +1 ( K ( F q ) p ) as a consequence of Theorem 3.10. By examining the map of THH-May spectral sequencesinduced by the unit map η ∧ id V (1) : S ∧ V (1) → BP ∧ V (1) and the subsequent map ofgeneralized homological homotopy fixed point spectral sequences induced by this same map,we see that ( tµ ) i maps to ( tµ ) i under the map V (1) ∗ T i +1 ( K ( F q ) p ) → ( BP ∧ V (1)) ∗ T i +1 ( K ( F q ) p ) . We also know the map π ∗ ( η ∧ id V (1) ) : π ∗ ( S ∧ V (1)) → π ∗ ( BP ∧ V (1)) sends the class v i to v i since the edge-homomorphism in the BP -Adams spectral sequence isa ring homomorphism. We then use the commutative diagram V (1) ∗ / / (cid:15) (cid:15) V (1) ∗ T i +1 ( K ( F q ) p ) (cid:15) (cid:15) BP ∗ V (1) / / ( BP ∧ V (1)) ∗ T i +1 ( K ( F q ) p ) to determine that v i ∈ BP ∗ V (1) maps to ( tµ ) i ∈ ( BP ∧ V (1)) ∗ T i +1 ( K ( F q ) p ) and also in themap of exact couples of the respective BP -Adams spectral sequences. Step 2:
We recall that the class β i is represented by(10) (cid:18) i (cid:19) v i − k + iv i − b , mod ( p, v ) in Ext ∗ , ∗ BP ∗ BP ( BP ∗ , BP ∗ ) where k = 2 t p ⊗ t ⊗ − t p ⊗ t p +11 ⊗ − t p ⊗ t ⊗ and(11) b , = p − X i =1 p (cid:18) pi (cid:19) t p − i ⊗ t i ⊗ due to Ravenel [29, Example 5.1.20]. We therefore need to check that the classes (10) map topermanent cycles in the BP ∗ BP -cobar complex for V (1) ∗ T i +1 ( K ( F q ) p )) . We begin with the element β . We observe that the element β is represented by the class b , in the E -page of the BP -Adams spectral sequence for V (1) and it maps to a class of thesame name in the cobar complex for the BP ∗ BP -co-module BP ∗ V (1) ∧ T ( K ( F q ) p ); i.e. the E - page of the BP -Adams spectral sequence for V (1) ∧ T ( K ( F q ) p ) . Let ¯ b , be the elementin the BP -Adams spectral sequence for the sphere spectrum that maps to b , . Then ¯ b , inthe BP -Adams spectral sequence for the sphere spectrum maps to (5) in the Adams spectralsequence for the sphere spectrum by [23, Thm. 9.4] and both are permanent cycles. Thereforeusing the square of spectral sequences and Proposition 3.1, we know that b , is a permamentcycle in the BP -Adams spectral sequence for V (1) ∧ T ( K ( F q ) p ) . Step 3:
The class b , v k − represents β k in the BP -Adams spectral sequence for V (1) when k ≡ mod p . It maps to b , ( tµ ) k − in the BP -Adams spectral sequence for V (1) ∧ T k ( K ( F q ) p ) up to multiplication by a unit by the argument at the beginning of the proof and the factthat the cobar complex for V (1) ∧ T k ( K ( F q ) p ) is multiplicative. Since the class representing β pm +1 is a permanent cycle in the BP -Adams spectral sequence for V (1) (this follows from[26, Lemma 5.4]), the class Σ p − i =1 1 p (cid:0) pi (cid:1) t i ⊗ t p − i ⊗ ( tµ ) pk is an infinite cycle in the BP -Adamsspectral sequence for V (1) ∧ T pk +1 ( K ( F q ) p ) , but it could still be a co-boundary. It is onthe two-line of the BP -Adams spectral sequence, so we just need to check that it is not theco-boundary of a d or d differential. Note that we will prove that, in fact, the element b , ( tµ ) k − is never a boundary for any k . This only implies that b , v k − is a permanentcycle for all k if it is already an infinite cycle for all k in the BP -Adams spectral sequence for V (1) , which to the author’s knowledge is unknown when k p . We will break thisinto two further sub-steps. Sub-step 1:
If the class Σ p − i =1 1 p (cid:0) pi (cid:1) t i ⊗ t p − i ⊗ ( tµ ) k − is the co-boundary of a d , then thereis a sum of classes P i a i ⊗ m i ∈ BP ∗ BP ⊗ BP ∗ BP ∗ V (1) ∧ T k ( K ( F q ) p ) such that d ( P i a i ⊗ m i ) ˙= b , ( tµ ) k − . Recall that the co-action on m is of the form ψ ( m ) = 1 ⊗ m + P j a j ⊗ m j where | m j | < | m | . Observe that the only elements in ( BP ∧ V (1)) ∗ T k ( K ( F q ) p ) whose co-action contains ( tµ ) k − as either m or m j for some j are classes of the form ( tµ ) k − y for some y ∈ ( BP ∧ V (1)) ∗ T k ( K ( F q ) p ) up to multiplication by a unit. The co-action of such a class is ψ (( tµ ) k − y ) = (1 ⊗ ( tµ ) k − ) ψ ( y ) , and ψ ( y ) must be of the form ψ ( y ) = 1 ⊗ y + z ⊗ X b i ⊗ y i since ψ (( tµ ) k − y ) must have ⊗ ( tµ ) k − as a term. Since the only classes in ( BP ∧ V (1)) ∗ T k ( K ( F q ) p ) that have z ⊗ as a term in their co-action for some element z = 0 are the classes t p and t i for i ≥ the class y must be a product of these. Since the internal degree of a i · ( · tµ ) k − y ETECTING THE β -FAMILY IN ITERATED ALGEBRAIC K-THEORY OF FINITE FIELDS 19 must equal (2 p − k + 2 p − p , we have | a i · ( · tµ ) k − y | = (2 p − k −
1) + | y | + | a i | = (2 p − k + 2 p − p so, the degree of | y | + | a i | must be p − p . However, the class t p is the element of lowestdegree in the set { t p , t , . . . } and | t p | = 2 p − p . Also, the only classes in degrees less than orequal to p − p in BP ∗ BP are powers of v and t . Therefore, the only options are y = t p and a i = 1 or a i = t p − j v j for some ≤ j ≤ p and y = 1 . We know that ∆( t p − j v j ) = v j ∆( t p − j ) = v j ∗ ( t ⊗ ⊗ t ) p − j and ψ ( t p ) = ∆( t p ) = t p ⊗ ⊗ t p mod p So, we compute d ( t p − j v j ⊗ ( tµ ) k − ) = 1 ⊗ t p − j v j ⊗ ( v · tµ ) k − − ∆( t p − j v j ) ⊗ ( tµ ) k − )+ t p − j v j ⊗ ψ (( tµ ) k − )= ¯∆( t p − j v j ) ⊗ ( tµ ) k − = Σ p − i =1 1 p (cid:0) pi (cid:1) t i ⊗ t p − i ⊗ ( tµ ) k − and d (1 ⊗ t p ( tµ ) k − ) = 1 ⊗ ⊗ t p ( tµ ) k − − ⊗ ⊗ t p ( tµ ) k − +1 ⊗ ψ ( t p ( µ ) k − )= 1 ⊗ t p ⊗ ( tµ ) k − + 1 ⊗ ⊗ t p ( tµ ) k − = Σ p − i =1 1 p (cid:0) pi (cid:1) t i ⊗ t p − i ⊗ ( tµ ) k − where ¯∆( a i ) = ∆( a i ) − a i ⊗ − ⊗ a i . Thus, m i ˙=( tµ ) k − for at least one i .Now, if m i ˙=( tµ ) k − for only one i , then the element a i corresponding to m i must havereduced co-product Σ p − i =1 1 p (cid:0) pi (cid:1) t i ⊗ t p − i + z for some element z ∈ BP ∗ BP ⊗ BP ∗ BP ∗ BP , up tomultiplication by a unit; i.e., ¯∆( a i ) ˙=Σ p − i =1 p (cid:18) pi (cid:19) t i ⊗ t p − i + z. The degree of a i must be p − p , so a i ˙= t j v p − j . However, ¯∆( t j v p − j ) = v p − j ¯∆( t j )= v p − j ( t ⊗ ⊗ t ) j − ⊗ t j v p − j − t j v p − j ⊗ and this does not equal b , + z for any j , and any element z ∈ BP ∗ BP ⊗ BP ∗ BP ∗ BP .Suppose that m i ˙=( tµ ) k − for i ∈ I where I contains more than one natural number. Then ψ ( P i ∈ I a i ) ˙= b , + z ′ for some possibly trivial element z ′ in BP ∗ BP ⊗ BP ∗ BP ∗ BP . However,we checked in the proof of Proposition 3.1 that no class of the form P i ∈ I a i has co-action b , + z ′ and the same proof applies here.Thus, there is no sum of classes P i a i ⊗ m i such that d ( P i a i ⊗ m i ) ˙= b , ( tµ ) k − andtherefore the class b , ( tµ ) k − survives to the E -page. Sub-step 2:
Now suppose there is a class in bidegree (2 p k − k + 2 p − p + 1 , that is thesource of a d differential hitting b , ( tµ ) k − . This class is therefore in BP p k − k +2 p − p +1 V (1) ∧ T k ( K ( F q ) p ) . Since this class is in an odd degree, we can classify all the classes that could pos-sibly be in this degree as a linear combination of elements in the three families, { λ ′ z , λ z , t k − bz } where z and z are some nontrivial product of even dimensional classes and z is somenontrivial product of even dimensional classes that does not include tσb or ( tµ ) j for any j ≥ as a factor since tσb · t k − bz = tµ · t k − bz = 0 .We can explicitly compute d ( λ ) = ¯ ξ ⊗ λ ′ modulo differentials in the generalized homo-logical homotopy fixed point spectral sequence. Therefore, by the Leibniz rule, d ( λ z ) =( ¯ ξ ⊗ λ ′ ) z + λ d ( z ) = 0 . Therefore, the classes of the form λ z do not survive to the E page and cannot be the source of a d differential hitting b , ( tµ ) k − . We therefore just need to check elements of the form t k bz or λ ′ z where z does not contain tµ or tσb as a factor. Note that the Leibniz rule implies d ( t k bz ) = d ( t k b ) z + t k bd ( z ) and similarly, d ( λ ′ z ) = d ( λ ′ ) z + λ ′ d ( z ) so we need to check if α ( d ( tb ) z + tbd ( z )) + β ( d ( λ ′ ) z + λ ′ d ( z )) = ( tµ ) k tσb for some α, β ∈ F p . However, note that the internal degree of d ( λ ) is p − p + 2 andthere are no classes in that degree in BP ∗ V (1) ∧ T k +1 ( K ( F q ) p ) ⊗ BP ⊗ j ∗ for any j ≥ . Thus, d ( λ ′ ) = 0 , and we need to check if α ( d ( tb ) z + tbd ( z )) + β ( λ ′ d ( z )) − ( tµ ) k − tσb = 0 for any α, β ∈ F p . Since z cannot contain tµ or tσb as a factor, tb is not a factor of ( tµ ) k tσb ,and λ ′ is not a factor of ( tµ ) k − tσb , there are no such α and β that make this equation hold. Step 4:
We now discuss how to detect the elements β i where i p . First, we willdiscuss how to detect β in V (1) ∗ T ( K ( F q ) p ) . The class β is represented by k + 2 b , v mod ( p, v ) in the input of the BP -Adams spectral sequence for S . It is also a permanentcycle in the BP -Adams spectral sequence for V (1) as a consequence of [26, Lemma 5.4]. Itmaps to the class k + 2 b , ( tµ ) in BP ∗ BP ⊗ BP ∗ BP ∗ BP ⊗ BP ∗ BP ∗ V (1) ∧ T ( K ( F q ) p ) under the map of E -pages of BP -Adams spectral sequences induced by the map V (1) → V (1) ∧ T ( K ( F q ) p ) , by the remarks at the beginning of the proof and the multiplicativity of E -page of the BP -Adams spectral sequence.Recall that BP ∗ V (1) ∧ T ( K ( F q ) p ) is isomorphic to H ∗ (cid:0) P ( t p , t , . . . ) ⊗ E ( b ) ⊗ E ( σt p , σt ) ⊗ P ( µ ) ⊗ Γ( σb ) ⊗ P ( t ); d ( x ) = tσx (cid:1) modulo d differentials. We can therefore check every element in degree p − p − in BP ∗ BP ⊗ BP ∗ BP ∗ V (1) ∧ T ( K ( F q ) p ) ETECTING THE β -FAMILY IN ITERATED ALGEBRAIC K-THEORY OF FINITE FIELDS 21 to see if it has the element of interest as a boundary. We therefore make a table of all elementsin this degree: t p ⊗ t v i t p +1 − i ⊗ σb t p +11 ⊗ σb v p +11 ⊗ σbt p t ⊗ v p ⊗ t t p +11 v p ⊗ v ⊗ ( t p ) t p v ⊗ v p t ⊗ t p v ⊗ v ⊗ γ ( σb ) t p +11 ⊗ v p v ⊗ t ⊗ ( t p ) v ⊗ t p σbt p t ⊗ v p +11 ⊗ t ⊗ γ ( σb ) v ⊗ t p t p v p +11 ⊗ v p t ⊗ t ⊗ t p σb v ⊗ σbt p +11 ⊗ t p v p +11 ⊗ t p t v p ⊗ v i t p +1 − i ⊗ t ⊗ t p v i t p − i ⊗ t ⊗ t σb v p ⊗ tµt ⊗ σb v i t p − i v ⊗ t p ⊗ tµ ⊗ σbtµ ⊗ bσt v i t p − i t ⊗ ⊗ t p tµ ⊗ ( tt p ) µ ⊗ t t p v i t p +1 − i ⊗ t p v i t p − i ⊗ tµ where the elements that are crossed out are elements that do not survive to the E page inthe generalized homological homotopy fixed point spectral sequence.We can immediately rule out any element of the form x ⊗ where x ∈ BP ∗ BP because if aclass of this form hit the target class then it would have also happened in the source spectralsequence. We therefore just need to check the classes: v i t p − i ⊗ tµ v ⊗ σb t p ⊗ tµ t p +11 ⊗ σb ⊗ σbtµ t ⊗ γ ( σb ) v p ⊗ tµ v p +11 ⊗ σbv i t p +1 − i ⊗ σb v ⊗ γ ( σb ) t ⊗ σb. We compute the d differential on each of these classes: d ( v i t p − i ⊗ tµ ) = − v i P p − i − j =1 (cid:0) p − i − j (cid:1) t j ⊗ t p − i − j − ⊗ tµd (1 ⊗ σbtµ ) = 1 ⊗ ⊗ σbtµd ( t p ⊗ tµ ) = − P p − i =1 (cid:0) pi (cid:1) t i ⊗ t p − i ⊗ tµ ≡ p ) d ( v p ⊗ tµ ) = v p ⊗ ⊗ tµd ( t ⊗ σb ) ≡ − t p ⊗ t ⊗ σb mod ( p, v ) d ( v ⊗ σb ) = v ⊗ ⊗ σbd ( t ⊗ γ ( σb )) = 0 d ( v ⊗ γ ( σb )) = v ⊗ ⊗ γ ( σb ) d ( t p +11 ⊗ σb ) = − P p +1 i =1 (cid:0) p +1 i (cid:1) t i ⊗ t p +1 − i ⊗ σb ≡ t ⊗ t p ⊗ σb + t p ⊗ t ⊗ σb mod ( p ) d ( v p +11 ⊗ σb ) = v p +11 ⊗ ⊗ σbd ( v i t p +1 − i ⊗ σb ) = − v i P p +1 − ij =1 (cid:0) p − i +1 j (cid:1) t j ⊗ t p − i − j +11 ⊗ σb. We observe that no linear combination these classes hits the element k + 2 b , ( tµ ) . There-fore, k + 2 b , ( tµ ) survives to the E -page of the BP -Adams spectral sequence for V (1) ∧ T ( K ( F q ) p ) .We next need to check if it is the boundary of a d . However, all of the potential elementsin BP p − p − V (1) ∧ T ( K ( F q ) p ) were killed by a d differential in the generalized homologicalhomotopy fixed point spectral sequence. Therefore, there are no elements with k + 2 b , v as a boundary.The only classes in degree p − p − in V (1) ∗ T ( K ( F q ) p ) are tλ ′ λ and tµσb . Now, byCorollary 3.4 and the Leibniz rule, there is a differential d p − ( tµσb ) = − t p α µσb in the generalized homological homotopy fixed point spectral sequence that computes V (1) ∗ T C − ( K ( F q ) p ) so the element tµσb does not survive to V (1) ∗ T C − ( j ) . Therefore, tµσb cannot be in theimage of the unit map V (1) ∗ S → V (1) ∗ T ( K ( F q ) p because this map factors through V (1) ∗ T C − ( j ) ; i.e., the diagram of ring spectra V (1) / / ' ' ❖❖❖❖❖❖❖❖❖❖❖❖❖ V (1) ∧ T C − ( j ) (cid:15) (cid:15) V (1) ∧ T ( K ( F q ) p ) commutes. Therefore, in degree V (1) p − p − T ( K ( F q ) p ) the image of the unit map is either F p generated by a linear combination of tλ ′ λ and tµσb or it is trivial in that degree. Itcannot be trivial in degree p − p − because we just showed that there is a permanentcycle in the BP -Adams spectral sequence that survives to become an element in this degreein V (1) ∗ T ( K ( F q ) p ) . Therefore, V (1) p − p − T ( K ( F q ) p ) ∼ = F p { c · tλ ′ λ + c ′ · tµσb } where c = 0 . Since d p − ( c · tλ ′ λ + c ′ · tµσb ) = c · d p − ( tλ ′ λ )+ c ′ · d p − ( tµσb ) = c · d p − ( tλ ′ λ ) − c ′ · t p α µσb where c, c ′ ∈ F p and c = 0 . We see that d p − ( tλ ′ λ ) = c − · c ′ · t p α µσb and therefore c ′ = 0 . Since we already identified that tµ is a permanent cycle and b , is apermanent cycle that represents the homotopy class tσb , we know b , tµ is represents tσbtµ if it survives. Conequently, c = 1 and c ′ = 2 (up to multiplying each of these by the sameunit). Step 5:
We now discuss how to detect β k where k p . In this case, β k is representedby (cid:0) k (cid:1) v k − k + kb , v k , which maps to ( tµ ) k − ( (cid:0) k (cid:1) k + kb , ( tµ )) so we just need to check the d differentials on classes of the form ( tµ ) k − w where w is an element in BP ∗ BP ⊗ BP ∗ BP ∗ BP .If the class ( tµ ) k (2 t p ⊗ t ⊗ − t p ⊗ t p ⊗ − t p ⊗ t ⊗ is the co-boundary of a d ,then there is a sum of classes P i a i ⊗ m i ∈ BP ∗ BP ⊗ BP ∗ BP ∗ V (1) ∧ T k ( K ( F q ) p ) such that d ( P i a i ⊗ m i ) ˙=( tµ ) k (2 t p ⊗ t ⊗ − t p ⊗ t p ⊗ − t p ⊗ t ⊗ . Sub-step 1:
Recall that the co-action on m is of the form ψ ( m ) = 1 ⊗ m + P j a j ⊗ m j where | m j | < | m | . Again, observe that the only elements in ( BP ∧ V (1)) ∗ T k ( K ( F q ) p ) whoseco-action contains ( tµ ) k − as either m or m j for some j are classes of the form ( tµ ) k − y forsome y ∈ ( BP ∧ V (1)) ∗ T k ( K ( F q ) p ) up to multiplication by a unit. The co-action of such a classis ψ (( tµ ) k − y ) = (1 ⊗ ( tµ ) k − ) ψ ( y ) , and ψ ( y ) must be of the form ψ ( y ) = 1 ⊗ y + z ⊗ P b i ⊗ y i since ψ (( tµ ) k − y ) must have ⊗ ( tµ ) k − as a term, up to multiplication by a unit. Since theonly classes in ( BP ∧ V (1)) ∗ T k ( K ( F q ) p ) that have a term z ⊗ in their co-action are the classes t p , t i for i ≥ the class y must be a product of these. Since | ( u · tµ ) k − y | = (2 p − k − | y | and the degree must equal p − p + k (2 p − , the degree of y must be p − p . However,the class t p is the element of lowest degree in the set { t p , t , . . . } and | t p | = 2 p − p and thenext lowest degree element is t with | t | = 2 p − so no product of classes in this set can bein degree p − p . Thus, m i ˙=( tµ ) k − for at least one i .Now, if m i · = ( tµ ) k − for only one i , then the element a i corresponding to m i musthave reduced co-product t p ⊗ t ⊗ − t p ⊗ t p ⊗ − t p ⊗ t ⊗ z for some class z in ETECTING THE β -FAMILY IN ITERATED ALGEBRAIC K-THEORY OF FINITE FIELDS 23 BP ∗ BP ⊗ BP ∗ BP ∗ BP ; i.e ¯∆( a i ) ˙=(2 t p ⊗ t ⊗ − t p ⊗ t p ⊗ − t p ⊗ t ⊗ . The degree of a i must be p − p , so a i ˙= t j v p − j v ǫ t ǫ .However, ∆( t j v p − j v ǫ t ǫ ) = v p − j v ǫ ( t ⊗ ⊗ t ) j ( t ⊗ ⊗ t + t p ⊗ t ) ǫ and so ¯∆( t j v p − j v ǫ t ǫ ) does not equal t p ⊗ t ⊗ − t p ⊗ t p ⊗ − t p ⊗ t ⊗ z up to multiplication by a unit for any j , and any element z ∈ BP ∗ BP ⊗ BP ∗ BP ∗ BP .Suppose that m i = ( tµ ) k − for i ∈ I where I contains more than one natural number.Then ψ ( X i ∈ I a i ) · =2 t p ⊗ t ⊗ − t p ⊗ t p ⊗ − t p ⊗ t ⊗ z ′ for some possibly trivial element z ′ in BP ∗ BP ⊗ BP ∗ BP ∗ BP . However, we checked in Step 4that no class of the form P i ∈ I a i ⊗ has co-action (2 t p ⊗ t ⊗ − t p ⊗ t p ⊗ − t p ⊗ t ⊗
1) + z ′ and the same proof applies here.Thus, there is no sum of classes P i a i ⊗ m i such that d ( X i a i ⊗ m i ) = (2 t p ⊗ t ⊗ − t p ⊗ t p ⊗ − t p ) ⊗ ( tµ ) k − and therefore the class (2 t p ⊗ t ⊗ − t p ⊗ t p ⊗ − t p ) ⊗ ( tµ ) k − survives to the E -page. Sub-step 2:
To see that there are no d differentials that hit (cid:0) k (cid:1) ( tµ ) k − k + kb , ( tµ ) k weneed to check that no elements in BP (2 p − k +4 p − p − V (1) ∧ F ( S ( C k ) + , T HH ( j )) T for k ≥ have (cid:0) k (cid:1) ( tµ ) k − k + kb , ( tµ ) k as a boundary. The only possible classes are elements in oneof the families { ( λ ′ t p − p ) z , ( λ t p − ) z , ( bγ p − ( σb )) z } where z i ∈ P ( t p , t p , t , . . . ) ⊗ E ( λ ′ t p − p , λ t p − , bγ p − ( σb )) ⊗ P ( µ ) ⊗ Γ( γ p ( σb )) ⊗ P ( t ) /t k +1 for i = 0 , , , or elements in one of the families { λ ′ y , λ y , ( t k b ) y , } where y i ∈ P ( t p , t , t , . . . ) ⊗ E ( λ ′ , λ , b ) ⊗ P ( µ ) ⊗ Γ( σb ) for i = 0 , , by Corollary 3.9.There are differentials d ( λ ′ t p − p ) = − t p − p ⊗ λ ′ and d ( λ t p − ) = ( t ⊗ λ ′ ) d ( t p − ) and d ( t p − ) = 0 , so these classes do not survive to the E page and therefore no class in thefamilies λ ′ y and λ y can be an element in the E -page that has (cid:0) k (cid:1) ( tµ ) k − k + kb , ( tµ ) k as a boundary. We know d ( bγ p − ( σb )) = 0 , so in order for bγ p − ( σb ) z to survive to the E -page, z must be a comodule primitive so that d ( z ) = 0 as well and hence by the Leibnizrule d ( bγ p − ( σb ) z ) = 0 . The only comodule primitives are products of elements in the set { µ , γ p k ( σb ) , t k | k ≥ } . Since | bγ p − ( σb ) | = 2 p − p − and the homotopy degree of (cid:0) k (cid:1) ( tµ ) k − k + kb , ( tµ ) k is (2 p − k + 2 p − p − , then we must have | bγ p − ( σb ) z | = (2 p − k + 2 p − p − so | z | = (2 p − k +2 p − p − − p +2 p +1 . However, | µ | ≡ p , | γ p k ( σb ) | ≡ p , t k ≡ − k mod 2 p , and | z | ≡ − k − p mod 2 p , so no product of elements inthe set { µ , γ p k ( σb ) , t k | k ≥ } can be congruent to | z | modulo p (note that we use thefact that ( t k ) = 0 here). Thus, there is no element that is both a comodule primitive and inthe correct degree, so element of the form bγ p − ( σb ) z can have (cid:0) k (cid:1) ( tµ ) k − k + kb , ( tµ ) k asa boundary. We now consider the elements in the second set of families of elements. Noticethat in the second set of families, none of the elements y i have t as a factor. First, we notethat d ( λ ) = t ⊗ λ ′ so λ y does not survive to the E page and therefore it cannot have (cid:0) k (cid:1) ( tµ ) k − k + kb , ( tµ ) k as a boundary. The elements t k b and λ ′ are comodule primitives, sothey survive to E . However, as we discussed before in order for λ ′ y and t k by to survive to E as well, then by the Leibniz rule y and y must be comodule primitives. | λ ′ | = 2 p − p +1 ,and we know | λ ′ y | = (2 p − k + 2 p − p − , which implies | y | = (2 p − k + 2 p − p − − (2 p − p + 1) = (2 p − k − .Note that the only comodule primitives that y could be are products of elements in the set { µ , γ p j ( σb ) | j ≥ } . We know | µ | ≡ p and | γ p j ( σb ) | ≡ p for j ≥ .Since | y | ≡ − k + 1) , the only way that a product of one of these classes could have thecorrect degree is if k + 1 ≡ p . However, that would imply that k = p ℓ + 1 for someinteger ℓ . In that case, (cid:0) ℓp +12 (cid:1) ( tµ ) ℓp k + ( p ℓ + 1) b , ( tµ ) ℓp +1 = b , ( tµ ) pm +1 for m = pℓ ,since (cid:0) ℓp +12 (cid:1) = ( p + 1)( p ) / ≡ p , in which case we already proved that b , ( tµ ) pm +1 is a permanent cycle for m ≥ an integer. Therefore, this does not occur. The last case toconsider is the family of classes t k by . In this case, | t k by | = (2 p − k + 2 p − p − implies that | y | = (2 p − k +2 p − p − − ( − k +2 p − p −
1) = 2 p k − . Again, the onlycomodule primitives in even degrees are products of elements in the set { µ , γ p j ( σb ) | j ≥ } .We observe that | y | ≡ − p , whereas | µ | ≡ p , and | γ p j ( σb ) | ≡ p so no such element y exists such that d ( t k by ) = 0 . Thus, there is no possible class in thecorrect degree at the E -page, which could have (cid:0) k (cid:1) ( tµ ) k − k + kb , ( tµ ) k as a boundary when k p , which covers all the remaining cases.We now just need to show that the class in V (1) (2 p − k − T k ( K ( F q ) p ) is represented by (cid:0) k (cid:1) ( tµ ) k − k + kb , ( tµ ) k is in fact (cid:0) k (cid:1) ( tµ ) k − ( tλ ′ λ ) + k ( tσb )( tµ ) k . To see this, note that (cid:18) k (cid:19) ( tµ ) k − k + kb , ( tµ ) k = (cid:18) k (cid:19) ( tµ ) k − ( k + 2 b , ( tµ )) + (2 k − k )( b , )( tµ ) for k ≥ so since k + 2 b , ( tµ ) survives to become tλ ′ λ + 2( tσb )( tµ ) , tµ survives to become tµ and b , survives to become tσb , we see that (cid:0) k (cid:1) ( tµ ) k − k + kb , ( tµ ) k survives to become (cid:0) k (cid:1) ( tµ ) k − ( tλ ′ λ ) + k ( tσb )( tµ ) k . (cid:3) Detecting the β -family in iterated algebraic K-theory. The goal of this sectionis to prove that the β -family is detected in the iterated algebraic K-theory of finite fields.We prove this as a Corollary to Theorem 10. The proof relies on the fact that the tracemap K ( R ) → T C − ( R ) is a map of commutative ring spectra when R is a commutative ring ETECTING THE β -FAMILY IN ITERATED ALGEBRAIC K-THEORY OF FINITE FIELDS 25 spectrum. The proof that the cyclotomic trace map K ( R ) → T C ( R ) is a map of commu-tative ring spectra when R is a commutative ring spectrum is due to Hesselholt-Geisser [16]for Eilenbrg-MacLane spectra and later Dundas [15] and Blumberg-Gepner-Tabuada [11] forcommutative ring spectra. The advantage of the approach of Blumberg-Gepner-Tabuada [11]is that they prove that the multiplicative cyclotomic trace map is also unique. This workbuilds on their proof that algebraic K-theory is the universal additive functor [10] (also seeBarwick [8]).We believe the fact that the trace map K → T C − is multiplicative is well known, but it isnot explicitly stated in the literature to our knowledge, so we include a proof. Note that bythe Nikolaus-Scholze equalizer [25, Cor. 1.5], there is a natural transformation T C → T C − and therefore there exists a natural transformation K → T C − . The proof will then followfrom the fact that algebraic K-theory is initial amongst multiplicative additive functors by[11, Cor. 7.2]. Lemma 3.13.
Suppose R is a commutative ring spectrum, then the trace map K ( R ) → T C − ( R ) , which factors through T C ( R ) , is a map of commutative ring spectra. Proof.
By [11, Cor. 7.2], it suffices to show that
T C − ( − ) is an E ∞ -object in the symmetricmonoidal ∞ -category of additive functors from Cat perf ∞ , the ∞ -category of small idempotent-complete stable infinity categories and exact functors, to the stable ∞ -category of spectra,denoted Fun add ( Cat perf ∞ , S ∞ ) ⊗ . Since E ∞ -objects in a functor category of infinity stable cat-egories are equivalent to commutative monoids in this functor category and by [17, Prop.2.12] [14, Ex. 3.2.2] an E ∞ -object in this functor category is equivalent to a lax symmetricmonoidal functor. The fact that the functor T C − is lax symmetric monoidal follows by thediagram F ( E T + , T HH ( − )) T ∧ F ( E T + , T HH ( − )) T ∧ (cid:15) (cid:15) F (( E T × E T ) + , T HH ( − ) ∧ T HH ( − )) T ) µ ∆ (cid:15) (cid:15) F ( E T + , T HH ( − )) T ) where ∆ is induced by the diagonal ∆ + : E T + → ( E T × E T ) + and µ is induced by the E ∞ -structure of T HH as an -object in
Fun add ( Cat perf ∞ , S ∞ ) ⊗ ( cf. [13, Sec. 4]). Specifically, ∆ = F (∆ + , T HH ( − ) ∧ T HH ( − )) T and µ = F ( E T + , µ ) T where µ : T HH ( − ) ∧ T HH ( − ) → T HH ( − ) is the multiplicationmap of the E ∞ -object T HH in Fun add ( Cat perf ∞ , S ∞ ) ⊗ . The fact that T C − ( − ) is an additivefunctor follows from [19, 25], or see [32, Sec. 2.2.13]. (cid:3) Remark 3.14.
The proof above implies by [11, Cor. 7.2] that there is a unique morphismfrom K → T C − in the ∞ -category of E ∞ -objects in Fun add ( Cat perf ∞ , S ∞ ) ⊗ , which is strongerthan what is needed for the statement of the lemma. We give the simpler statement becausethat is the version that will be used in the next proof. In the end, the result will be a resultabout homotopy groups, so the change of model from the model category of symmetric spectra S to the infinity category of spectra S ∞ should not be concerning. Corollary 3.15.
Let p ≥ be a prime and q be a prime power that generates ( Z /p ) × . Theclasses β i map from π ∗ S to nonzero elements in π ∗ K ( K ( F q )) under the unit map. Proof.
First, the classes β i in V (1) ∗ map to V (1) ∗ K ( K ( F q ) p ) p under the unit map since thecyclotomic trace is multiplicative and therefore the map V (1) ∗ S → V (1) ∗ T C − ( K ( F q ) p ) factors through V (1) ∗ K ( K ( F q ) p ) ; i.e, there is a commutative diagram V (1) ∗ S V (1) ∗ η TC − ( K ( F q ) p ) + + ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ V (1) ∗ η K ( K ( F q ) p ) / / V (1) ∗ K ( K ( F q ) p ) V (1) ∗ tr (cid:15) (cid:15) V (1) ∗ T C − ( K ( F q ) p ) . There is also a commuting diagram of ring spectra S ≃ S ∧ S S ∧ η / / i i ∧ S (cid:15) (cid:15) S ∧ K ( K ( F q )) i i ∧ K ( K ( F q )) (cid:15) (cid:15) i i ∧ K ( f p ) * * ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ V (1) ∧ S V (1) ∧ η / / V (1) ∧ K ( K ( F q )) V (1) ∧ K ( f p ) / / V (1) ∧ K ( K ( F q ) p ) where f p : K ( F q ) → K ( F q ) p is the p -completion map and η is the unit map. Since theclasses β i pull back to π ∗ S along the unit map and since they map nontrivially to classes in π ∗ V (1) ∧ K ( K ( F q ) p ) , they must map to nontrivial classes in π ∗ K ( K ( F q )) under the unit map π ∗ S → π ∗ K ( K ( F q )) . (cid:3) Corollary 3.16.
Let O F be the ring of integers in a number field F whose residue fieldis F q for some prime power q which generates ( Z /p Z ) × . Then the β -family is detected in K ( K ( O F )) . In particular, the β -family is detected in K ( K ( Z )) . Proof.
Let F be a number field and q a prime power satisfying the conditions in the statementof the corollary. Since O F has residue field F q , there exists a map of commutative rings O F → F q inducing a map of commutative ring spectra K ( K ( O F )) → K ( K ( F q )) . Therefore,there is a commutative diagram(12) S / / $ $ ■■■■■■■■■■ K ( K ( F q )) K ( K ( O F )) ♣♣♣♣♣♣♣♣♣♣♣ of commutative ring spectra. Since the β -family is nontrivial in the image of the unit map π ∗ S → π ∗ K ( K ( F q )) , it is also nontrivial in the image of the unit map π ∗ S → π ∗ ( K ( K ( O F ))) .In particular, let p = 5 , then q = 2 generates ( Z / Z ) × and consequently it topologicallygenerates Z × . Thus, there is a map of commutative rings Z → F inducing a map of commu-tative ring spectra K ( K ( Z )) → K ( K ( F )) . Since the β -family is detected in K ( K ( F )) , bythe same diagram (12) with O F = Z we see that the β -family is detected in iterated algebraicK-theory of the integers. (cid:3) Note that the α -family is detected in K ( Z ) . Since K ( Z ) ∼ = Z , there is a map of commu-tative ring spectra K ( Z ) → H Z . We may consider the infinite family of maps S → . . . → K ( K ( K ( Z ))) → K ( K ( Z )) → K ( Z ) ETECTING THE β -FAMILY IN ITERATED ALGEBRAIC K-THEORY OF FINITE FIELDS 27 and a specialization of the Greek-letter family red-shift conjecture is that the n -th Greekletter family is in the image of the unit map S → K ( n ) ( Z ) where K ( n ) ( Z ) is algebraic K-theory iterated n -times. As a consequence of Corollary 3.16, we have therefore proved thisversion of the conjecture for n = 2 . References [1] J. F. Adams,
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