AA SLICE REFINEMENT OF B ¨OKSTEDT PERIODICITY
YURI J. F. SULYMA
Abstract.
Let R be a perfectoid ring. Hesselholt and Bhatt-Morrow-Scholze have identified the Postnikovfiltration on THH( R ; Z p ): it is concentrated in even degrees, generated by powers of the B¨okstedt generator σ , generalizing classical B¨okstedt periodicity for R = F p . We study an equivariant generalization of thePostnikov filtration, the regular slice filtration , on THH( R ; Z p ). The slice filtration is again concentratedin even degrees, generated by RO ( T )-graded classes which can loosely be thought of as the norms of σ .The slices are expressible as RO ( T )-graded suspensions of Mackey functors obtained from the Witt Mackeyfunctor. We obtain a sort of filtration by q -factorials. A key ingredient, which may be of independent interest,is a close connection between the Hill-Yarnall characterization of the slice filtration and Ansch¨utz-le Bras’ q -deformation of Legendre’s formula. Contents
1. Introduction 21.1. q -analogues and p -typification 51.2. Overview 51.3. Acknowledgements 6 Part 1. Background
72. Homotopical background 72.1. Equivariant stable homotopy theory 72.2. THH and friends 112.3. Mackey functors 132.4. RO ( G )-graded homotopy 142.5. Representations of T and C p n q -analogues and p -typification 183.2. Perfectoid rings 213.3. Prisms 24 Part 2. Results RO ( T ) calculations 294.1. Mackey functors 294.2. Tools 314.3. Computations 335. Slices of THH 355.1. The slice tower 355.2. The slice filtration 375.3. The slice spectral sequence 386. Further questions 45References 46 Sulyma was supported in part by NSF grants DMS-1564289 and DMS-1151577. a r X i v : . [ m a t h . A T ] J u l . Introduction
In [BMS19], the authors construct “motivic” filtrations on THH and its variants, applying this to construct(completed) prismatic cohomology equipped with the Nygaard filtration. Their construction works by quasi-syntomic descent to the case of perfectoid rings, where the filtration is given by the Postnikov filtration. Inthis case, they show that for a perfectoid ring R , π ∗ THH( R ; Z p ) = R [ σ ] , | σ | = 2 , generalizing earlier work of Hesselholt for the case R = O C p [Hes06] and B¨okstedt’s foundational calculationfor R = F p [B¨85].Their work depends on the work of Nikolaus-Scholze [NS18], who give (in the bounded below case) adescription of cyclotomic spectra in terms of Borel equivariant homotopy theory. However, for most ofhistory cyclotomic spectra were studied via genuine equivariant homotopy theory. In the genuine toolbox,a powerful way to study G -spectra is via the (regular, equivariant) slice filtration . Modelled on the slicefiltration from motivic homotopy theory, this was developed by Dugger [Dug05] in the case G = C , and latergeneralized to finite groups by Hill-Hopkins-Ravenel as one of the main tools in their solution of the Kervaireinvariant one problem [HHR16a]. We will be interested in the “regular” slice filtration, first described byUllman [Ull09], which has better multiplicative properties than the “classical” version used by HHR.A simple but crucial point is that the slice filtration restricts to the Postnikov filtration on underlyingspectra, and thus constitutes an equivariant generalization of the Postnikov filtration—one which is howeverquite different from the Postnikov t -structure on G -spectra, as it mixes in more of the representation theoryof G . In 2018, Hill asked what happens if, in the BMS construction, one replaces the Postnikov filtrationwith the slice filtration. This gives a filtration which is sensitive to the genuine structure of THH but notits cyclotomic structure, so is in some sense intermediary between the BMS filtration and the cyclotomicfiltration constructed by Antieau-Nikolaus [AN18].In this paper we carry out the local calculation needed to answer Hill’s question, identifying the slicefiltration on THH( R ; Z p ) for a perfectoid ring R . On the arithmetic side, we present evidence that this isstrongly related to q -divided powers. Before stating the results, let us say more about the problem from ahomotopical point of view.Our understanding of the slice filtration has come a long way since [HHR16a], but is still in its infancy.In particular, most slice computations to date have been of spectra closely related to MU R , or of Mackeyfunctors; THH is of a quite different flavor than these, yet is still a very reasonable spectrum. It is fairto be skeptical that methods in the highly specialized setting of cyclotomic spectra will work for general G -spectra, but aside from the calculational foothold afforded by cyclotomicity, our arguments only relyon connectivity of geometric fixed points and the Segal conjecture. Thus, while our investigations weremotivated by arithmetic considerations, we hope that they will shed light on the slice filtration in general.One novelty is that this is the first time the slice filtration has been considered for a compact Lie group(which we define so that it restricts to the slice filtration on all finite subgroups). This causes some peculiari-ties: for example, we no longer get periodicity with respect to the regular representation. However, there is agood replacement: writing T for the circle group, let λ i denote the one-dimensional complex T -representationin which z ∈ T acts as multiplication by z i . Then set[ n ] λ = λ ⊕ · · · ⊕ λ n − { n } λ = λ ⊕ · · · ⊕ λ n A standard decomposition using roots of unity shows that [ n ] λ restricts to the complex regular representationof C n , while { n } λ restricts to the reduced complex regular representation of C n +1 . The { n } λ representa-tions also appear when calculating the K -theory of truncated polynomial algebras [HM97a, Spe19b] or ofcoordinate axes [Hes07, Spe19a], and many of the formulas there bear a striking resemblance to ours.To state the main theorem, we require some further notation. Let W ( R ) denote the Mackey functor of p -typical Witt vectors of R , and let tr C n W ( R ) be the sub-Mackey functor generated under transfers byrestriction to C p vp ( n ) . Quotients of Mackey functors are to be interpreted levelwise. Finally, we will assumefor the rest of the Introduction that R is a Z cycl p = Z p [ ζ p ∞ ] ∧ p -algebra; the results are valid for any perfectoidring, but the meaning of the q -analogues needs to be clarified. Recall that A inf ( Z cycl p ) = Z p [ q /p ∞ ] ∧ ( p,q − . heorem 1.1. Let R be a perfectoid Z cycl p -algebra. The slice covers of THH( R ; Z p ) are given by P n THH( R ; Z p ) = Σ [ n ] λ THH( R ; Z p ) . The slices are given by P n n THH( R ; Z p ) = Σ { n } λ tr C n W ( R )= Σ [ n ] λ W ( R ) / [ pn ] q /p . For comparison, Ullman [Ull09, Theorem 5.1] shows that P n ku G = Σ [ n ] λ ku G (for G finite cyclic), so thisis a reasonable answer. As a C n -spectrum the representation sphere S [ n ] λ is just the norm N C n e S , so wecan roughly think of the above filtration as being generated by the norms of the B¨okstedt generator σ . Wedo not know whether either of the two expressions for P n n THH( R ; Z p ) is more natural. Remark 1.2.
For a thorough comparison with [BMS19] and prismatic cohomology, one should additionallycompute the slice filtration on the T -spectra THH( R ; Z p ) E T + , (cid:103) E T ⊗ THH( R ; Z p ) E T + , and (cid:103) E T ⊗ THH( R ; Z p ).We will address this in future work.As the Theorem indicates, in order to work with the slice filtration it is necessary to have a handle onthe RO ( T )-graded homotopy π (cid:70) THH( R ; Z p ). For virtual representations of the form (cid:70) = ∗ − V with ∗ ∈ Z and V an actual representation, these groups come up when studying THH of monoid algebras, andcan already be found in [HM97b, Proposition 9.1]. However, the slice filtration tends to demand study ofthe degrees (cid:70) = ∗ + V . Gerhardt [Ger08] computed π (cid:70) TR n ( F p ) additively for general (cid:70) ∈ RU ( T ) (i.e.,even-dimensional virtual representations), and Angeltveit-Gerhardt [AG11] extended this to all (cid:70) ∈ RO ( T )(as well as to THH of Z and the Adams summand (cid:96) ). Their method is an RO ( T )-graded version of theisotropy separation sequence, which they slickly package as the “homotopy orbits to TR spectral sequence”(HOTRSS).In the course of applying their method to the slice filtration and generalizing it to perfectoid rings, wediscovered several enhancements, incorporating recent advances in equivariant homotopy theory.(1) As we learned from work of Zeng [Zen18, § gold elements a λ i , u λ j of equivariant homotopy theory [HHR16b, § n . This greatly simplifies calculations by removing a lot of distracting torsion, which can beput back in later (if desired). For example, there is a sharp distinction between even- and odd-dimensional TF groups, but both of these contribute to the even-dimensional TR n groups studiedby Gerhardt.(3) In order to generalize the method to perfectoid rings, our key lemma is a q -deformation of the goldrelation of [HHR16b, Lemma 3.6(vii)], describing the interaction of the aforementioned gold elements a λ i , u λ j . Our “ q -gold relation” (Lemma 4.9) says that, for 0 ≤ i < j , σa λ i = [ p i +1 ] q /p u λ i a λ j u λ i = φ i +1 ([ p j − i ] q /p ) a λ i u λ j . The case of a torsionfree perfectoid ring is in fact easier than the case of F p (except notationally),since [ p ] q /p and [ p ] q do not interfere with one another, which removes another source of torsion.The HOTRSS played a central role in our early investigations; in the final product, we have reduced toconsidering a few very nice virtual representations, which can be computed in an ad hoc way. Thus, we willuse Angeltveit-Gerhardt’s insights to derive the q -gold relation, but then use cell structures to carry out theactual calculations, avoiding the HOTRSS. In [Sul], we will deploy the HOTRSS (or rather the HOTFSS)to compute the entire RO ( T )-graded ring π (cid:70) TF( R ; Z p ).The q -gold relation together with some knowledge of π (cid:70) TF( R ; Z p ) allows us to read off the effect ofthe slice filtration on homotopy. We describe the filtration on both π i TF( R ; Z p ) and π [ i ] λ TF( R ; Z p ), sinceTheorem 1.1 as well as the answer for π i make π [ i ] λ seem more natural. heorem 1.3. The slice filtration takes the following form on homotopy. When j ≤ or i = 0 , F j S π [ i ] λ TF( R ; Z p ) is all of π [ i ] λ TF( R ; Z p ) . Otherwise, F j S π [ i ] λ TF( R ; Z p ) is generated by [ p ( i + j − q /p ![ p ( i − q /p ! . When j ≤ or i = 0 , F j S π i TF( R ; Z p ) is all of π i TF( R ; Z p ) . Otherwise, F j S π i TF( R ; Z p ) is generated by [ p ( i + j − q /p ![ p r ] i − q /p φ r (cid:18)(cid:104)(cid:106) i + j − p r − (cid:107)(cid:105) q /p ! (cid:19) , where r = (cid:6) log p (cid:0) i + ji (cid:1)(cid:7) .In particular, taking i = 1 in either case gives F j S π TF( R ; Z p ) = ([ pj ] q /p !)= ([ p ] jq /p [ j ] q !) . Remark 1.4.
The arithmetic significance of this filtration is currently mysterious, but we will share somespeculation. As explained in [Hil20, § norm functors scale filtration by the index of the subgroup. It is not clear what this means inthe case of a compact Lie group, but a good place to start is the Witt vector norm maps W n ( R ) N −→ W n +1 ( R );these are natural from the point of view of equivariant homotopy, but have received little attention in numbertheory (aside from the Teichm¨uller lift, which is a special case). The description of the Witt vector norm isdue to Angeltveit and Borger [Ang15], but they are working with W ( R ) rather than A inf ( R ); although thenorm maps can be lifted to A inf ( R ), it is not clear if there is a canonical way to do so. However, a preferredlift is available in the q -crystalline case: the q -Pochhammer symbols( x, − y ; q ) n := ( x − y )( x − qy ) · · · ( x − q n − y )are lifts of N ( x − y ), at least for x and y of rank one (Proposition 3.32). In particular, our filtration on π TF( R ; Z p ) should be compared with [AB19, Proposition 4.9]. We discovered this connection as the paperwas being finalized, and have not yet really explored it.We can also understand this filtration via the regular slice spectral sequence (RSSS). Here, for the firsttime, we see different behavior depending on the nature of R . When R is p -torsionfree, the E page isconcentrated in even degrees, so the RSSS collapses at E . However, when R is a perfect F p -algebra, thereare differentials on every page arising from the “collision” of ξ and φ ( ξ ) ([ p ] q /p and [ p ] q ). For a Mackeyfunctor M , we let Φ C n M denote the cokernel of tr C n M → M . Theorem 1.5.
The homotopy Mackey functors of the slices are given in even degrees by π i P n n THH = W i = nR < i = n Φ C pm W / [ p h +1 ] q /p < i < n where R is the constant Mackey functor on R , and m = (cid:6) log p ( n/i ) (cid:7) − , h = (cid:40) min { v p ( n ) , (cid:4) log p ( n/i ) (cid:5) } n/i not a power of p (cid:4) log p ( n/i ) (cid:5) n/i a power of p. If R is p -torsionfree, then these are the only non-vanishing homotopy Mackey functors. If R is a perfect F p -algebra, then π i +1 P n n THH = (cid:40) tr C pm + h +1 Φ C pm W n/i not a power of p tr C pm + h +1 Φ C pm +1 W n/i a power of p. roposition 1.6. The homotopy Mackey functors π [ i ] λ of the slices are π [ i ] λ P n n THH = W i = nW / [ pn ] q /p < i = n Φ C p(cid:96) ( i,n ) W / [ pn ] q /p < i < n where (cid:96) ( i, n ) = max { v p ( i ) , . . . , v p ( n − } = min { r | (cid:100) n/p r (cid:101) = (cid:100) i/p r (cid:101)} . We provide several charts illustrating these filtrations at the end of § F p -algebra is very complicated, the E ∞ page can be inferred from Theorem 1.3, since we have explicitlyidentified the slice tower. Corollary 1.7.
Let R be a perfect F p -algebra, and define h = h ( n, i ) as in Theorem 1.3. The entry on the E ∞ page of the RSSS corresponding to π i P n n THH( R ) is Φ C pf +1 W /p h ( n,i )+1 , where f = (cid:80) i ≤ m Part 1 collects the needed technical background and notation. Unfortunately, there is a lot;we have tried to make the paper accessible (and self-contained) to both arithmetic geometers unfamiliar withthe pre-Nikolaus-Scholze formulation of cyclotomic spectra, and to homotopy theorists who are interestedin slice computations but who have not studied [BMS19] extensively. Homotopical background is presented One could avoid this discrepancy by using the classical slice filtration, but the regular slice filtration is more natural. n § 2, and arithmetic background is presented in § 3. We hope that collecting all of this information in oneplace will be helpful to the field.Our notation is recapitulated at the beginning of Part 2, so one may begin there and refer back as needed.However, even experts should read § § § 4, we compute certain RO ( T )-graded homotopy Mackey functors of THH, taking a geodesic route tothe computations which are needed for describing the slice filtration. The main theorems are proved in § § 6, we propose some natural followup questions.1.3. Acknowledgements. We are grateful to Mike Hill for suggesting this problem; to Andrew Blumbergfor supervision and guidance; to Aaron Royer for patiently fielding many questions about equivariant stablehomotopy theory; to Teena Gerhardt for clarifying a confusion about the Tate spectral sequence; to BenAntieau and Thomas Nikolaus for help sorting out a confusion about the A -algebra structure on TR n ; and toDylan Wilson for the crucial suggestion to use the gold elements. Figures 1 and 2 were originally created forKate Stange’s Number Theory and Friends group as a part of ICERM’s Illustrating Mathematics semester.Additionally, this paper owes a tremendous intellectual debt to [AG11] and [Zen18].Preliminary versions of the results in this paper appeared as part of the author’s PhD thesis at theUniversity of Texas at Austin. art Background Homotopical background In this section we give a crash course in equivariant stable homotopy theory, with an eye towards com-putational aspects. We do not intend to give a complete or fully rigorous account of this subject—whichwould be almost impossible in the given space—but simply to recall the main points and indicate some ofthe pitfalls. For a comprehensive introduction that covers everything we need here (except for §§ § λ -rings.In § § § § RO ( G )-graded homotopy Mackeyfunctor π (cid:70) X of a G -spectrum X . In particular, we discuss the equivariant Euler classes a V and theirrelation to isotropy separation squares. As we are interested in the groups T and C p n , we review in § § Equivariant stable homotopy theory. At the beginning of this project, the author found equivari-ant homotopy theory extremely confusing. Now, the author finds equivariant homotopy theory only veryconfusing. We have tried to explain things in the way that made the subject make sense to us.We start in § two types of fixed points in equivariant unstable homotopy theory: the homotopy fixed points X hG and the categorical fixed points X G .There are then four types of fixed points once we pass to equivariant stable homotopy theory: • the homotopy fixed points X hG ; • the categorical fixed points X G ; • the Tate fixed points X tG ; • the geometric fixed points X Φ G .The categorical and geometric fixed points arise in passing from naive to genuine equivariant homotopytheory, while the Tate and geometric fixed points have to do with passing from unstable to stable homotopytheory. Categorical fixed points are to homotopy fixed points as geometric fixed points are to Tate fixedpoints, in a sense made precise by the isotropy separation square (2). Ultimately, the difference between theunstable and stable cases comes from the existence of the trace (often called the norm).2.1.1. The unstable theory. Let S denote the ∞ -category of spaces, and let Top denote the (a) topologicalcategory of spaces. Given a compact Lie group G , we wish to define the ∞ -category G S of G -spaces. As wewill see, there are several different options for what this might mean. We will examine how to do this usingthe point-set model, then explain what this means ∞ -categorically. Definition 2.1. A G -topological space is a topological space X equipped with a continuous left action of G .The topological category of G -topological spaces is denoted G Top.If G is finite, then we may identify G Top with Top BG . However, it is very much not the case that G S = S BG . If we want to do homotopy theory, it is not enough to specify the category of G -topological spaces: wemust also specify the weak equivalences. Here, there is a choice to be made. Given a G -topological space X and a subgroup H ≤ G , write X H = { x ∈ X | h · x = x ∀ h ∈ H } for the subspace of H -fixed points, and observe that the functor ( − ) H is representable by G/H . efinition 2.2. A map X f −→ Y of G -topological spaces is called • a naive weak equivalence if the underlying map X e f e −→ Y e is a weak equivalence; • a genuine weak equivalence if X H f H −→ Y H is a weak equivalence for all subgroups H ≤ G .More generally, given a family F of subgroups of G closed under conjugation and passage to subgroups, an F -weak equivalence is a map f such that f H is a weak equivalence for all H ∈ F .We write G S naive , G S , G S F for the ∞ -categories obtained by localizing G Top with respect to the naive,genuine, or F -weak equivalences; G S naive = G S { } and G S = G S { all } . We call these the ∞ -categories of naive G -spaces , genuine G -spaces , and F -genuine G -spaces .In other words, each subgroup H of G gives a functor G Top ( − ) H −→ Top, and by specifying F we are decidingwhich of these should be homotopically meaningful, descending to a functor G S F → S . It turns out thatthis is essentially all of the homotopical data contained in a G -space. Definition 2.3. Let G be a finite group. The orbit category O ( G ) of G is the full subcategory of S BG spanned by the nonempty transitive G -sets. O ( G ) is equivalent to its full subcategory { G/H } H ≤ G . Remark 2.4. A family F of subgroups of G closed under conjugation and passage to subgroups is essentiallythe same thing as a downwards-closed subcategory of O ( G ). Theorem 2.5 ([Elm83]) . The restricted Yoneda embedding G Top → Top O ( G ) op gives an equivalence of ∞ -categories G S naive = S BG G S = S O ( G ) op G S F = S F op In particular, G/H represents the functor G S ( − ) H −→ S (now basically by fiat). In fact, this factors through S Aut( G/H ) = S W G ( H ) , where W G ( H ) = N G ( H ) /H is the Weyl group.In summary, a G -space has two different notions of fixed point: • the homotopy fixed points X hH , obtained by restricting X to Sp BH and then right Kan extendingalong BH → ∗ ; • the categorical fixed points X H .2.1.2. The stable theory. The first guess for the stable theory is to simply stabilize the unstable theory fromthe previous subsection. This gives two options: • the stabilization Sp( S BG ) of naive G -spaces is equivalent to the functor category Sp BG . These arevariously called Borel G -spectra , coarse G -spectra , FS- G -spectra , naive G -spectra , or doubly naive G -spectra in the literature. We shall use the term naive G -spectra . • the stabilization Sp( S O ( G ) op ) of genuine G -spaces is equivalent to the functor category Sp O ( G ) op .These are sometimes called naive G -spectra in the literature (clashing with the above); we shall usethe nonstandard term ersatz G -spectra . One can of course replace O ( G ) with F .Naive G -spectra are useful, but ersatz G -spectra, as the name suggests, are the wrong notion to consider.The reason they are wrong is the same reason we get more fixed point functors.If K → H is an inclusion (or subconjugacy relation) of subgroups of G , then there is a restriction map X H → X K between the fixed points. In the unstable setting, this is the only relation we should expectbetween fixed point spaces, in general. But in the presence of addition, there is an easy way to produce H -fixed points from K -fixed points: simply sum over conjugates, indexed by H/K ; this is known as a transfer map. Thus, in the stable setting, we should require transfers tr HK : X K → X H in addition to restriction mapsres HK : X H → X K . These are not present in Sp O ( G ) op .Instead, recalling that O ( G ) was defined as the subcategory of S BG spanned by the nonempty transitive G -sets (equivalently, by the orbits G/H ), we define A G to be the subcategory of Sp BG spanned by X + , where is a nonempty transitive G -set, equivalently by the orbits G/H + . A G is called the Burnside ∞ -category of G ; the classical (or algebraic) Burnside category is B G = π A G . A Mackey functor is an additive functor M : B op G → Ab; Mackey functors are reviewed in § A ( G ) op → Sp turns out to give the correct notion of genuine G -spectrum:that is, genuine G -spectra are spectral Mackey functors . This is revisionist: traditionally, genuine G -spectraare defined by starting with genuine G -spaces and inverting all representation spheres (ersatz G -spectra onlyinvert the ordinary spheres). The spectral Mackey functor formulation is due to Guillou-May [GM17], andwas used by Barwick [Bar17] to provide a fully ∞ -categorical treatment of G -spectra. Kaledin also has ahomological analogue [Kal11b].Summing over conjugates provides easy examples of fixed points; we would like to distill the interestingones. The Tate fixed point spectrum is defined as the cofiber of the trace map T from the homotopy orbitsto the homotopy fixed points: X hG T −→ X hG −→ X tG which on π is T ( x ) = (cid:80) g ∈ G x . The theory of the Tate spectrum is due to Greenlees and May [GM95]. Remark 2.6. T is often written as N , and called the norm map; however, Hill has pointed out that thatis more properly reserved for the multiplicative notion. Example 2.7. Let M be the free Z -module Z (cid:104) x, y (cid:105) , and let C act on M ⊗ in the obvious way. Then both x and xy + yx are in H ( C , M ), but only x is nonzero in (cid:98) H ( C , M ) ∼ = Z / (cid:104) x , y (cid:105) .The geometric fixed points X Φ G are a “genuine” version of the Tate fixed points X tG . To define them,let F be a family of subgroups of G , and let E F be a G -space with( E F ) H = (cid:40) ∅ H / ∈ F∗ H ∈ F (Such a G -space exists by Elmendorf’s theorem, and E F is in fact determined by this condition.) Then let (cid:103) E F be the space given by the cofiber sequence E F + → S → (cid:103) E F . For any spectrum X , we get a diagram E F + ⊗ X (cid:47) (cid:47) (cid:15) (cid:15) X (cid:47) (cid:47) (cid:15) (cid:15) (cid:103) E F ⊗ X (cid:15) (cid:15) E F + ⊗ X E F + (cid:47) (cid:47) X E F + (cid:47) (cid:47) (cid:103) E F ⊗ X E F + (1)with the following properties: • it can be shown that the left vertical map is an equivalence, so the right square is a pullback/pushout; • the terms in the left column are concentrated on the subgroups in F ; • the terms in the right column vanish on the subgroups in F ; • the bottom row depends only on the restriction of X to F .There are two cases of particular interest. When F = { e } , we write EG for E F , and set X h := EG + ⊗ XX h := X EG + X t := (cid:103) EG ⊗ X EG + This is because one can show that ( X h ) H = X hH for subgroups H ≤ G , where the right-hand side is a prioridefined as orbits for the restriction of X to a Borel spectrum X ∈ Sp BH , and similarly for X h and X t .The other particular case is when F = P is the family of all proper subgroups. In this case, we define the geometric fixed points X Φ G by X Φ G := ( (cid:103) E P ⊗ X E P + ) G . More accurately of the generalized Tate construction, c.f. [AMGR17]. his has the effect of destroying all transfers from subgroups, deleting the “cheap” fixed points. For asubgroup H ≤ G , we define X Φ H by first restricting X to H Sp, then applying the above construction. Theformal properties of the different fixed-point functors are:Ω ∞ ( X H ) = (Ω ∞ X ) H Σ ∞ ( X H ) = (Σ ∞ X ) Φ H ( X ⊗ Y ) Φ H = X Φ H ⊗ Y Φ H In a slogan, the spectra X Φ H sort fixed points according to their stabilizers, while the spectra X H mix themall together. Remark 2.8. Using isotropy separation inductively, one can describe G -spectra in terms of their geometricfixed points rather than their categorical fixed points. In place of the diagram perspective of spectral Mackeyfunctors, this describes G -spectra in terms of a stratification , making it look somewhat like the category ofcontructible sheaves on some (mythical) stack. This is due to Glasman [Gla17] and Ayala–Mazel-Gee–Rozenblyum [AMGR17]; an excellent exposition is [Wil17, §§ G = C p , these two distinguished families coincide. Consequently, when G is C n or T , wecan rewrite ( (cid:103) EG ⊗ X EG + ) G as ( X Φ C p ) G/C p . This gives a square X hG (cid:47) (cid:47) X G (cid:47) (cid:47) (cid:15) (cid:15) ( X Φ C p ) G/C p (cid:15) (cid:15) X hG T (cid:47) (cid:47) X hG (cid:47) (cid:47) X tG (2)which is the basis for many THH calculations. Remark 2.9. It may also be useful to consider the fiber of the vertical maps in (2), which is ( X (cid:103) EG ) G . Ourinterest in this comes from the following observation. The map THH = THH Φ C p ϕ −→ THH tC p is a spectralversion of the derived Frobenius A ϕ −→ A//p := A ⊗ LZ F p . If A is a perfectoid ring which is either p -torsion or p -torsionfree, containing an element π such that π p = pu for a unit u (we allow π = 0), then A π −→ A ϕ −→ A//p is an exact triangle [BMS18, Lemma 3.10]. In general, the fiber of A ϕ −→ A//p , which is also the fiberof W ( A ) F −→ A , “contains all information about p -divided powers in A ”. More generally, the kernel ofFrobenius on W ( − ) is the divided-power completion of G a at the origin; this is at the basis of Drinfeld’sapproach to crystalline cohomology through stacks [Dri].Now we define the homotopy theories of interest to us. Definition 2.10. A cyclonic spectrum is a T -spectrum genuine for the finite subgroups of T . We denotethe ∞ -category of cyclonic spectra by Sp ξ . Definition 2.11. A cyclotomic spectrum is a naive T -spectrum X together with T -equivariant maps X ϕ p −→ X tC p for all p , where X tC p has the T (cid:39) T /C p action. These are not required to be compatible for varying p .We denote the ∞ -category of cyclotomic spectra by Sp ϕ . We also write Sp ϕp for the ∞ -category of p -typical cyclotomic spectra, where we only ask for a single ϕ p .Classically [HM94], a cyclotomic spectrum was defined as a cyclonic spectrum X equipped with equiva-lences X ∼ −→ X Φ C p which now are required to be compatible. The homotopy theory of cyclotomic spectra was first constructedin [BM15]; the definition above is due to [NS18]. In the bounded below case, this agrees with the classicalnotion. This terminology is due to Barwick-Glasman [BG16]. efinition 2.12. A cyclotomic spectrum with Frobenius lifts is a naive T -spectrum X together with com-patible T -equivariant maps X ψ −→ X hC p for all p . We denote the ∞ -category of cyclotomic spectra with Frobenius lifts by Sp ψ . We also write Sp ψp for the ∞ -category of p -typical cyclotomic spectra with Frobenius lifts, where we only ask for a single ψ p .There are forgetful functors Sp ψ → Sp ϕ → Sp ξ → Sp B T → Sp . THH and friends. The author has found that the proliferation of T acronyms in this subjectconfuses a great number of people—arithmetic geometers and homotopy theorists alike—so in this sectionwe document them all. (To keep the net confusion constant, we also suggest some new ones.) This is merelya collection of definitions; for an introduction to the subject we suggest [KN], [HN20]; the modern referenceis [NS18]. The pre-Nikolaus-Scholze surveys [May] and [Mad95] are also highly recommended.Recall that ordinary Hochschild homology HH( A/k ) gives an object in Mod B T k . Topological Hochschildhomology THH( A ) := HH( A/ S ) has more structure, and gives an object of Sp ϕ . To begin, we denoteHC := HH h T cyclic homologyHC − := HH h T negative cyclic homologyHP := HH t T periodic cyclic homologyThese are shown to be equivalent to the classical description in terms of bicomplexes in [Hoy18]. Analogously,we make the following definitions in the topological case:TC − := THH h T topological negative cyclic homologyTP := THH t T topological periodic cyclic homologySee Remark 2.20 below. There are exact trianglesΣHC → HC − → HPΣTHH h T → TC − → TPThe shift comes from working in the compact Lie case. Warning 2.13. Despite the name, TP ∗ ( A ) is not periodic in general. But this is the case when A lives overa single prime.The TR n spectra are defined using the categorical fixed points:TR n +1 := THH C pn length n + 1 topological Restriction homology Remark 2.14. For any ring A , TR n +10 ( A ) = W n +1 ( A ) [HM97b, Theorem 3.3]. Remark 2.15. This is more properly called p -typical TR, denoted TR n +1 ( − ; p ), whereas “big TR n +1 ” isTHH C n +1 . We will only use p -typical TR n +1 in this paper.The numbering of p -typical TR n +1 is chosen to agree with Witt vectors: for any ring A , TR n +10 ( A ) = W n +1 ( A ). As explained in [Bor11, 2.5], the mismatch in indexing is thus the number theorists’ fault.The TR n spectra are related by various maps mimicking the structure of Witt vectors:TR n +1 F −→ TR n TR n V −→ TR n +1 TR n +1 R −→ TR n In fact, it even gives a cyclonic k -module, but we do not know how to access the fixed-point information in the absence ofcyclotomicity. ere F is the equivariant restriction, V is the equivariant transfer, and R comes from the cyclotomic structure.This is interesting: classically one thinks of the R map on Witt vectors as the “easy” one and the F map asthe “exotic” one, whereas the opposite is true from the perspective of equivariant homotopy theory.With these notations, the isotropy separation sequence (2) takes the formTHH hC pn (cid:47) (cid:47) TR n +1 R (cid:47) (cid:47) (cid:15) (cid:15) TR nϕ (cid:15) (cid:15) THH hC pn (cid:47) (cid:47) THH hC pn (cid:47) (cid:47) THH tC pn We can thus express TR n +1 in Nikolaus-Scholze language as the iterated pullbackTR n +1 (cid:47) (cid:47) (cid:15) (cid:15) (cid:121) . . . (cid:47) (cid:47) (cid:15) (cid:15) (cid:121) TR R (cid:47) (cid:47) (cid:15) (cid:15) (cid:121) TR R (cid:47) (cid:47) (cid:15) (cid:15) (cid:121) (cid:47) (cid:47) TR ϕ (cid:15) (cid:15) ... (cid:15) (cid:15) ... (cid:15) (cid:15) THH hC p can (cid:47) (cid:47) ϕ hCp (cid:15) (cid:15) THH tC p ... (cid:15) (cid:15) ... (cid:15) (cid:15) THH hC p can (cid:47) (cid:47) (cid:15) (cid:15) THH tC p ... (cid:15) (cid:15) ... (cid:15) (cid:15) (cid:47) (cid:47) ...THH hC pn can (cid:47) (cid:47) THH tC pn This should be compared to the Cartier-Dieudonn´e-Dwork lemma, and with Borger’s perspective on Wittvectors [Bor11]. Remark 2.16. Angeltveit shows there is also an N map, reflecting the Tambara functor structure [Ang15].We make some remarks about this in § ←− n,R TR n topological Restriction homologyTF := lim ←− n,F TR n topological Frobenius homologyMore conceptually, THH (cid:55)→ TR is the right adjoint to the forgetful functor Sp ψp → Sp ϕp , applied toTHH [KN, Proposition 10.3]. This is analogous to W ( R ) being the cofree δ -ring on R . Furthermore,TF = Hom Sp ξ ( S , THH). At infinite level, the isotropy separation sequence (2) becomesΣTHH h T (cid:47) (cid:47) TF R (cid:47) (cid:47) (cid:15) (cid:15) TF ϕ (cid:15) (cid:15) ΣTHH h T (cid:47) (cid:47) TC − can (cid:47) (cid:47) TP Remark 2.17. The upper-right copy of TF behaves differently from the upper-left copy, and is in somesense a Frobenius untwist of it. In calculations we have found it convenient to write the upper-right copy asTΦ, but we fear that trying to introduce this notation would provoke outrage. Remark 2.18. It would be natural to defineTV = lim −→ n,V TR n topological Verschiebung homology hich is a topological analogue of the unipotent coWitt vectors [Fon77, Chapitre II]. To our knowledge thishas not yet been exploited. Open Problem 2.19. Come up with better names for TR, TF, and TV.Finally, TC := Hom Sp ϕ ( S , THH) is obtained by trivializing all the cyclotomic structure. This definitionis really a theorem, conjectured by Kaledin [Kal11a] and proven by Blumberg-Mandell [BM15]. There arevarious equivalent descriptions of this using the above spectra:TC := Hom Sp ϕ ( S , THH) topological cyclic homology= fib(TR − F −−−→ TR)= fib(TF − R −−−→ TF)= fib(TC − can − ϕ −−−−→ TP ∧ ) “Nikolaus-Scholze formula” Remark 2.20. It is really THH h T which deserves to be called topological cyclic homology, and denoted TC.We have advertised the idea of calling Hom Sp ϕ ( S , THH) topological motivic cohomology (TM); however, weshall not indulge that here.2.3. Mackey functors. If M is an abelian group acted on by G , there are several maps relating thefixed-point modules M H for subgroups H ≤ G . • The Weyl group W G ( H ) = Aut( G/H ) = N G ( H ) /H acts on each M H . • If K ≤ H , there is a restriction map res HK : M H → M K . • If K ≤ H , there is a transfer map tr HK : M H → M K .The notion of a Mackey functor axiomatizes this structure. In other words, a Mackey functor M assignsto each subgroup H ≤ G an abelian group M ( G/H ), with an action of W G ( H ), along with restriction andtransfer maps, satisfying appropriate compatibilities. We will not spell these out here; the most importantone for us is that res HK tr HK ( x ) = (cid:88) g ∈ W H ( K ) g · x A complete, explicit definition can be found in [Maz13, Definition 1.1.2]. More abstractly, a Mackey functor isan additive functor B op G → Ab. Mackey functors are to G -modules are to abelian groups as genuine G -spectraare to naive G -spectra are to spectra. Example 2.21. For any genuine G -spectrum X and n ∈ Z , the homotopy Mackey functor π n ( X ) is definedby π n ( X )( G/H ) = [ S n ⊗ G/H + , X ] . Example 2.22. The Burnside Mackey functor A is the representable functor B G ( − , G/G ). For any finitegroup G , A ( X ) is K of the category of finite G -sets. In fact, A = π S . Example 2.23. Given a G -module M , the fixed point Mackey functor M is defined by M ( G/H ) = M H .Restrictions are inclusions of fixed points, and transfers are summations over cosets. This is a Mackeyanalogue of the right adjoint to the forgetful functor G Sp → Sp BG . Example 2.24. Given a G -module M , the orbit Mackey functor O ( M ) is defined as O ( M )( G/H ) = M/H ,the orbit of H ⊂ G . Transfers are quotient maps, and restrictions are summations over representatives. Thisis a Mackey analogue of the left adjoint to the forgetful functor G Sp → Sp BG .All of our C p n -representations will be restricted from T -representations. Since T is connected, this meansall the Weyl actions will be trivial, and the Mackey functor condition amounts to res GH ◦ tr GH = | G : H | .There is a closed symmetric monoidal structure on Mackey functors, which we do not discuss. Themonoidal unit is A (and not Z ). A Mackey functor is a Z -module if and only if it satisfies the conditiontr HK res HK = | H : K | [TW95, Proposition 16.3]. xample 2.25. For any ring R , the Witt Mackey functor W ( R ) is a C p n -Mackey functor (for any n ),whose value on C p n /C p k is W k +1 ( R ). Restrictions are given by F and transfers are given by V . In fact, W ( R ) = π THH( R ) [HM97b, Theorem 3.3]. W ( R ) should not be confused with the constant Mackey functor W ( R ). If R is an F p -algebra, then V F = p and W ( R ) is a Z -module, but this is not true for general R .A common way of describing a Mackey functor is by means of a Lewis diagram . If M is a Mackey functor,we draw the modules M ( G/H ) with M ( G/G ) on top and M ( G/e ) on the bottom. Restrictions are indicatedby maps going downward, and transfers by maps going upward. For example, a Lewis diagram for C p lookslike this: M ( C p /C p ) res Cpe (cid:25) (cid:25) M ( C p /e ) tr Cpe (cid:88) (cid:88) Weyl group action (cid:87) (cid:87) All of the Mackey functors we encounter will have trivial Weyl action.2.4. RO ( G ) -graded homotopy. Let α and β be actual representations of G . If X is a genuine G -spectrum,we define the RO ( G ) -graded homotopy π Hα − β ( X ) of X as π Hα − β ( X ) = [ S α ⊗ G/H + , S β ⊗ X ] Warning 2.26. This notation is abusive: it depends on the choice of α and β , not only on α − β . This isdiscussed at length in [Ada84, § S | V | with S V for an abstract real vector space V .We address this by choosing a specific irreducible representation in each isomorphism class of irreduciblerepresentations: the λ i introduced in § RO ( T ) we really mean Z [ λ i | i ≥ Z ( p ) [ λ i | i ≥ 0] when we work p -locally. Similarly, RO ( C p n ) really means Z [ λ i | ≤ i ≤ p n − ], or Z ( p ) [ λ i | ≤ i ≤ n − 1] when we work p -locally; when p = 2, we must also include the sign representation ς n − .Varying H , these fit into a Mackey functor π α − β ( X ). Varying the representation, we obtain the RO ( G ) -graded homotopy Mackey functor π (cid:70) ( X ), which is the fundamental computational invariant of the G -spectrum X . It is conventional to use ∗ for Z -grading, and (cid:70) for RO ( G )-grading.A very important source of RO ( G )-graded classes is the Euler classes a V . Definition 2.27. Let V be an actual representation of G . Suspending the inclusion { } (cid:44) → V gives a map S → S V , which we denote by a V . If V G (cid:54) = 0, then a V is nulhomotopic. Otherwise, a V refines to a class a V ∈ π G − V S , and we continue to denote by a V its Hurewicz image in π G − V X for any G -spectrum X .The significance of these classes is as follows. Consider the isotropy family of V , F V = { H ≤ G | V H (cid:54) = 0 } . Then S ( ∞ V ) is a model of E F V , while S ∞ V is a model of (cid:93) E F V . Note that S ∞ V = lim −→ (cid:16) S a V −→ S V a V −→ S V a V −→ · · · (cid:17) , so smashing with S ∞ V has the effect of inverting a V . Consequently, the isotropy separation square (1) canbe viewed as an arithmetic square for a V .2.5. Representations of T and C p n . Let λ i be the T -representation where z ∈ T ⊂ C × acts as z i . Theseexhaust the nontrivial irreducible real representations of T . The representation spheres S λ i and S λ j are allintegrally inequivalent [Kaw80], but in the p -local setting we have S λ i ∼ = S λ j whenever the p -adic valuationsof i and j agree. Thus, let λ r = λ p r .Note that 1 ∈ R ( T ) corresponds to 2 ∈ RO ( T ); to avoid confusion, we will often write λ ∞ for theone-dimensional complex representation with trivial action. e use the same notation for the restriction of λ i to a subgroup C p n . When p (cid:54) = 2 these exhaust theirreducible real representations of C p n , and λ r is trivial for r ≥ n . When p = 2, there is additionally thesign representation of C n , which we denote ς n − , satisfying λ n − = 2 ς n − . The regular representations thendecompose as ρ C pn = 1 + pn − (cid:77) i =1 λ i p (cid:54) = 2= 1 + n − (cid:88) i =0 p n − − i ( p − λ i p -locally ρ C n = 1 + ς n − + n − − (cid:77) i =1 λ i p = 2= 1 + ς n − + n − (cid:88) i =0 n − − i λ i Warning 2.28. In the literature, one sometimes sees the notation λ d for what we call { d } λ .As C n spectra, the representation spheres have cell structures S (cid:47) (cid:47) S (cid:7) (cid:47) (cid:47) (cid:15) (cid:15) S λ r (cid:15) (cid:15) S ⊗ C p n /C p r + (cid:47) (cid:47) ( . . . ) (cid:15) (cid:15) S ⊗ C p n /C p r + The attaching map S ⊗ C p n /C p r + → S ⊗ C p n /C p r + is given by 1 − γ , where γ is a chosen generatorof C p n . The one-skeleton S (cid:7) is not actually a representation sphere, except in the case G = C n , when S λ n − / = S ς n − . Taking duals gives a dual cell structure S − ⊗ C p n /C p r + (cid:47) (cid:47) ( . . . ) (cid:15) (cid:15) (cid:47) (cid:47) S − λ r (cid:15) (cid:15) S − ⊗ C p n /C p r + (cid:47) (cid:47) S − (cid:7) (cid:15) (cid:15) S on S − λ r . Tensoring these together for various values of r , and using the fact that S λ s ⊗ C p n /C p r + = S ⊗ C p n /C p r + for r ≤ s , gives cellular structures for all α ∈ RO ( C p n ), and hence spectral sequencesto compute π α X for any C p n -spectrum X . This is discussed in [HHR17, § α is an actualrepresentation (or the negative of one), the spectral sequences become chain complexes of Mackey functors,which are determined by the underlying chain complex of abelian groups. xample 2.29. Let M be a C p -Mackey functor, and write M ( C p /C p ) = M , M ( C p /e ) = M , and trivialWeyl action on M . Then π λ + ∗ M is the homology of the following complex: M (cid:19) (cid:19) res (cid:47) (cid:47) M (cid:19) (cid:19) (cid:47) (cid:47) M (cid:19) (cid:19) p (cid:47) (cid:47) M (cid:19) (cid:19) (cid:47) (cid:47) M (cid:19) (cid:19) M (cid:83) (cid:83) ∆ (cid:47) (cid:47) M [ C p ] ∇ (cid:83) (cid:83) − γ (cid:47) (cid:47) M [ C p ] T (cid:47) (cid:47) ∇ (cid:83) (cid:83) M [ C p ] ∇ (cid:83) (cid:83) − γ (cid:47) (cid:47) M [ C p ] ∇ (cid:83) (cid:83) − − − − Example 2.30. Let M be a C p -Mackey functor, and write M ( C p /C p ) = M , M ( C p /C p ) = M , M ( C p /e ) = M , all with trivial Weyl action. Then π ∗− λ − λ M is the homology of the following com-plex: M (cid:19) (cid:19) (cid:47) (cid:47) M (cid:19) (cid:19) tr (cid:47) (cid:47) M (cid:19) (cid:19) (cid:47) (cid:47) M (cid:19) (cid:19) tr (cid:47) (cid:47) M (cid:19) (cid:19) M [ C p /C p ] ∆ (cid:19) (cid:19) ∇ (cid:84) (cid:84) − γ (cid:47) (cid:47) M [ C p /C p ] ∇ (cid:84) (cid:84) ∆ (cid:19) (cid:19) tr (cid:47) (cid:47) M [ C p /C p ] ∇ (cid:84) (cid:84) res (cid:19) (cid:19) − γ (cid:47) (cid:47) M [ C p /C p ] ∇ (cid:84) (cid:84) res (cid:19) (cid:19) ∇ (cid:47) (cid:47) M (cid:84) (cid:84) res (cid:19) (cid:19) M [ C p ] ∇ (cid:83) (cid:83) − γ (cid:47) (cid:47) M [ C p ] ∇ (cid:83) (cid:83) ∇ (cid:47) (cid:47) M [ C p /C p ] tr (cid:83) (cid:83) − γ (cid:47) (cid:47) M [ C p /C p ] tr (cid:83) (cid:83) ∇ (cid:47) (cid:47) M (cid:84) (cid:84) These cell structures imply the following very useful bound on homology. Lemma 2.31. Let M be a T -Mackey functor and let α be a fixed-point-free virtual representation of T . Write α = β − γ for actual representations β , γ (still assumed to be fixed-point-free). Then π ∗ + α M is concentratedin ∗ ∈ [ − d ( β ) , d ( γ )] . An important observation, which we learned from Mike Hill, is that these have simpler cell structures as T -spectra . Observation 2.32. In Sp ξ there is an exact triangle T /C p r + → S → S λ r . The slice filtration. The equivariant slice filtration is a filtration on equivariant spectra first intro-duced by Dugger in the C case [Dug05], then generalized to finite G by Hill-Hopkins-Ravenel as a key toolin their solution of the Kervaire invariant one problem [HHR16a]. It is modeled on the motivic slice filtra-tion of Voevodsky, hence the qualifier “equivariant”. For an introduction to the slice filtration, the original[HHR16a, § 4] is still highly recommended, as well as the survey [Hil12]. However, significant advances havebeen made since then: [HY18] gave a much easier characterization of slice connectivity, and [Wil17] providedan algebraic description of categories of slices, as well as a general recipe for computing slices. A thoroughtreatment of the slice spectral sequence is given in [Ull09], and of course there are many computations inthe literature one can learn from, such as [HHR17] or [Yar15].We caution the reader that there is no single slice filtration; [HHR16a] and [Hil12] use the classical slicefiltration, whereas [HY18] and [Ull09] treat the regular slice filtration. We shall be concerned with the regularslice filtration. A general framework for slice filtrations is given in [Wil17, § Definition 2.33. Let G be a finite group. A regular slice cell of dimension n is a G -spectrum of the form ↑ GH S kρ H , where H is a subgroup of G , ρ H is the regular representation of H , and k | H | = n . The regular slice filtration is the filtration generated by the regular slice cells. More explicitly, We say that a G -spectrum X is slice n -connective , and write X ≥ n , if X is in the localizingsubcategory generated by the regular slice cells of dimension ≥ n . Equivalently [HY18], X is slice n -connective if and only if for all subgroups H ≤ G , the geometric fixed points X Φ H are in thelocalizing subcategory of ordinary spectra generated by S (cid:100) n/ | H |(cid:101) .We write G Sp ≥ n for the slice n -connective spectra. This is a coreflective subcategory, and wedenote by P n X → X the coreflection, the slice n -connective cover of X . • We say that a G -spectrum X is slice n -truncated and write X ≤ n if Map G Sp ( Y, X ) = ∗ whenever Y ≥ n + 1. It suffices to check this for Y a regular slice cell of dimension ≥ n + 1.The slice n -truncated spectra are reflective, and we denote by X → P n X the localization functor,the slice n -truncation of X . • We say that a G -spectrum X is an n -slice if X ≥ n and X ≤ n . There is an exact triangle P n +1 X → X → P n X , whence a canonical equivalence P n P n X = P n P n X . We write P nn X for eitherof these, the n -slice of X . • The slice spectral sequence takes the form E s,t = π t − s P tt X ⇒ π t − s X. This follows Adams grading, so the E rs,t term is placed in the plane in position ( t − s, s ). The d r differential has bidegree ( r, r − − , r ) in terms of the plane display. There is also an RO ( G )-graded version E s,t = π α − s P dim α dim α X ⇒ π α − s X. The slice filtration is generally not associated to a t -structure, but rather a sequence of t -structures [Wil17,Definition 1.43].The above definition is only for finite groups, and it is not immediately clear how to interpret the slicefiltration for cyclonic or cyclotomic spectra. We have chosen to interpret this as the C p n -slice filtration onthe restriction to a C p n -spectrum for all n , but we do not claim this is the only or best option. Warning 2.34. The slice covers P n give a descending filtration, while the slice truncations P n give an ascending filtration. This is opposed to the usual super/subscript convention for filtrations. Remark 2.35. In [AN18], Antieau and Nikolaus produce a t -structure on Sp ϕ by stipulating that theforgetful functor Sp ϕ → Sp reflect n -connective objects. A natural idea is to transport the slice filtration alongSp ϕ → Sp ξ in the same way. However, the Hill-Yarnall characterization implies that the slice connectivityof a cyclotomic spectrum is equal to the connectivity of its underlying spectrum, so this reproduces thecyclotomic t -structure. . Arithmetic background In this section we collect the needed background from number theory. In § q -analogues,and survey several key instances where these appear; in particular, we repeatedly encounter the problemof p -typification. This discussion is the core of the narrative of the paper. In § A inf , followed by the homotopical properties of perfectoid rings. Finally,in § q -analogues and p -typification.Definition 3.1. Let n ∈ N . The q -analogue of n is the formal expression[ n ] q := q n − q − · · · + q n − ∈ Z [ q ] . These give a one-parameter deformation of the natural numbers, and many notions in mathematics admitso-called q -deformations , recovering the classical notion when q = 1. Such q -deformations arise naturally incounting problems over finite fields, combinatorics, and a wide variety of other contexts. The relevance toTHH is ultimately that TP can be used to construct q -de Rham cohomology, as envisioned in [Sch17] andcarried out in [BMS19].For us, the key feature of Z [ q ] is that it is a Λ -ring : we have commuting ring endomorphisms ψ n ( q ) = q n for n ∈ N × , which are Frobenius lifts in the sense that ψ p ( x ) = x p + pδ p ( x )for some (unique, since Z [ q ] is torsionfree) δ p ( x ) ∈ Z [ q ], for all primes p . This gives an action n (cid:55)→ ψ n of N × on Z [ q ] in the category of commutative rings. This action interacts with q -analogues in two ways:(1) The map N × → ( Z [ q ] , × ) sending n (cid:55)→ [ n ] q is not a map of commutative monoids, but rather issemi-multiplicative with respect to this action, in the sense that[ mn ] q = [ m ] q ψ m ([ n ] q ) = [ n ] q ψ n ([ m ] q )for m, n ∈ N × . Note that [ m ] q still divides [ mn ] q .(2) For m, n, k ∈ N × , there is a congruence ψ mk ([ n ] q ) ≡ n mod [ m ] q since q m ≡ m ] q . We will explore the significance of this congruence in § n (cid:55)→ [ n ] q deviates from being truly multiplicative.We will mostly restrict attention to the submonoid p N ⊂ N × and write φ := ψ p , δ := δ p (although it wouldbe good to formulate some of this more globally). Note that φ n thus means ψ p n . Prisms can be viewed asan axiomatization and generalization of the p -typical part of the above structure.We now examine several case studies involving q -analogues.3.1.1. Perfectoid algebras. This concerns the specialization Z [ q ] → Z p [ q /p ∞ ] ∧ ( p,q − ;the reference for this section is [BMS18, Example 3.16] and the next few propositions.The starting point is the observation that [ p ] q is the minimal polynomial of ζ p , and thus Z p [ q ] / [ p ] q = Z p [ ζ p ].We think of this as “characteristic close to p ”; note that ζ p = 1 implies p = 0. Similarly, φ n ([ p ] q ) is theminimal polynomial of ζ p n +1 , so Z p [ q ] /φ n ([ p ] q ) = Z p [ ζ p n +1 ]. Thus, taking perfections yields Z p [ q ] (cid:15) (cid:15) φ (cid:47) (cid:47) Z p [ q ] (cid:15) (cid:15) φ (cid:47) (cid:47) · · · (cid:15) (cid:15) (cid:47) (cid:47) Z p [ q /p ∞ ] (cid:15) (cid:15) Z p [ ζ p ] (cid:47) (cid:47) Z p [ ζ p ] (cid:47) (cid:47) · · · (cid:47) (cid:47) Z p [ ζ p ∞ ] e want q = 1 to be the “base case”, so we complete at q − q /p , rather than q , be a primitive p th root of unity. Thus we set A = Z p [ q /p ∞ ] ∧ ( p,q − [ n ] A = [ n ] q /p R = A/ [ p ] A We warn that A/φ n ([ p ] A ) is now isomorphic to Z cycl p = Z p [ ζ p ∞ ] ∧ p for all n ; however, these have different A -algebra structures, namely q (cid:55)→ ζ p n − .What about the other q -analogues? It turns out that there are canonical identifications A/ [ p n ] A ∼ −→ W n ( R )such that the projections A/ [ p n +1 ] A → A/ [ p n ] A are identified with the Frobenius maps F : W n +1 ( R ) −→ W n ( R ).(We would get a different A -algebra structure if we wanted the R maps to be A -linear.) Since the A -algebrastructure is important to us, we will almost exclusively write A/ [ p n ] A rather than W n ( R ).The ring R is an example of a perfectoid ring and the triple ( A, φ, [ p ] A ) is the corresponding perfect prism .It suffices to think of this example for the remainder of the paper (except when we explicitly restrict toperfect F p -algebras).3.1.2. Legendre’s formula. Throughout this paper we will need to count the number of integers in { , . . . , n } having a particular p -adic valuation. One instance of this is the following classical formula for the p -adicvaluation of a factorial. Proposition 3.2 (Legendre) . The p -adic valuation of n ! is given by v p ( n !) = ∞ (cid:88) r =1 (cid:22) np r (cid:23) The q -factorial is defined by [ n ] q ! := [1] q · · · [ n ] q . Ansch¨utz-le Bras have supplied a q -deformation of Legendre’s formula in their work identifying the cyclotomictrace in degree 2 as a q -logarithm. Lemma 3.3 ([AB19, Lemma 4.8]; “ q -Legendre formula”) . In the specialization Z [ q ] → Z p [[ q − , we have [ n ] q ! = u ∞ (cid:89) r =1 φ r − ([ p ] q ) (cid:98) n/p r (cid:99) = u ∞ (cid:89) r =1 [ p r ] (cid:98) n/p r (cid:99)− (cid:98) n/p r +1 (cid:99) q for a unit u ∈ Z p [[ q − × . In what follows, we will apply this to [ n ] A ! rather than [ n ] q !.An illustration of the q -Legendre formula is given in Figures 1 and 2. In the classical Legendre formula, allbars would be the same color. However, as we have already noted, q -analogues are only semimultiplicative:[ p n ] q = [ p ] q · · · φ n − ([ p ] q ) rather than [ p ] nq .In § (cid:24) n + 1 p k (cid:25) − (cid:22) np k (cid:23) is useful for translating between the two. Remark 3.4. Bhargava [Bha00] has an interesting framework for generalized factorials. Figure 1. The q -Legendre formula at p = 2 Figure 2. The q -Legendre formula at p = 33.1.3. Circle representations. This concerns the specialization Z [ q ] → Z [ λ ± ] = R ( T ) , where R ( T ) is the complex representation ring of the circle group T . In this case, λ -analogues are somethingfamiliar: ↓ T C n [ n ] λ = C [ C n ] = ↑ C n e C is another name for the complex regular representation of C n . We prefer the notation [ n ] λ as it fits with theoverall theme of q -analogues. An important consequence is that ↓ T C n S [ n ] λ = N C n e S , so S [ n ] λ is ≥ n by [Ull09, Corollary I.5.8]; this is ultimately the reason for our interest in λ -analogues. Wewill also use the (admittedly atrocious) notation { n } λ := λ + · · · + λ n = [ n + 1] λ − λ ∞ = λ [ n − λ which has the property that ↓ T C n +1 { n } λ is the complex reduced regular representation of C n +1 .As noted earlier, when working p -locally it suffices to consider the representations λ i := λ p i . bservation 3.5. Determining the decomposition of { n } λ into λ i ’s is isomorphic to the problem consideredin the q -Legendre formula. More precisely, it corresponds to decomposing [ pn ] A ! = [ p ] nA φ ([ n ] A !), since wealways pick up a new irreducible representation when passing from { n } λ to { n + 1 } λ , whereas we only pickup new (non-unit) factors in [ n ] A ! when we hit multiples of p .For example, with p = 3 the representation { } λ = λ + λ + λ + λ has p -typical decomposition { } λ = 3 λ + λ and dimension-sequence d • ( { } λ ) = (4 , . This parallels the decomposition [4 p ] A ! = u [ p ] A [ p ] A = u [ p ] A φ ([ p ] A ) . Corollary 3.6. The dimension-sequences of the representations [ n ] λ and { n } λ are d s ([ n ] λ ) = (cid:24) np s (cid:25) , d s ( { n } λ ) = (cid:22) np s (cid:23) , and their p -typical irreducible decompositions are [ n ] λ = λ ∞ + ∞ (cid:88) s =0 (cid:18)(cid:24) np s (cid:25) − (cid:24) np s +1 (cid:25)(cid:19) λ s , { n } λ = ∞ (cid:88) s =0 (cid:18)(cid:22) np s (cid:23) − (cid:22) np s +1 (cid:23)(cid:19) λ s . Remark 3.7. The representation [ n ] λ is also familiar to homotopy theorists as the Bott cannibalistic class ρ n ( λ ) arising in topological K -theory. In fact, since the Bott element is given by β = λ − 1, we can write π ∗ ku = Z [[ λ − λ -analogues and λ -deformations is that we are deformingthe trivial line bundle to a general line bundle.This is related to the present situation as follows. By [Sus83] we have K( O C p ; Z p ) = ku p , and thecyclotomic trace π ku p = K ( O C p ; Z p ) → TP ( O C p ; Z p )sends λ to (a possible choice of) q by [Hes06, Lemma 3.2.3]. This can be generalized to Z cycl p algebras, see[Mat20, Example 5.5].3.1.4. Tate cohomology. Our final example is more philosophical in nature. It concerns the specialization Z [ q ] → Z [ q ] / ( q n − 1) = Z [ C n ]in which q is viewed as a generator of C n . In this case [ n ] q gets sent to the C n - trace , so that quotienting by[ n ] q essentially amounts to taking Tate cohomology. This suggests yet another perspective on q -deformations,one which may appeal to the equivariant homotopy theorist: we are deforming from a trivial action to anon-trivial action, such that multiplication by n gets deformed to a transfer for a subgroup of index n .One would then hope that the equivariant norm for a subgroup of index n would correspond to some q -deformation of raising to the power n . At present, we are only able to relate the norm to existing notionsof q -powers on elements of rank one. We record this in § Perfectoid rings. .2.1. Arithmetic aspects. The reference for this section is [BMS18, §§ Definition 3.8. Let R be a p -complete ring. The tilt R (cid:91) of R is the inverse limit perfection of R/p : R (cid:91) := lim ←− φ R/p. There is a multiplicative bijection R (cid:91) ∼ = lim ←− x (cid:55)→ x p R, and we write ( − ) (cid:93) : R (cid:91) → R for the multiplicative map ( x , x , . . . ) (cid:55)→ x . Example 3.9. If R = Z cycl p := Z p [ ζ p ∞ ] ∧ p , then R/p = F p [ x /p ∞ ] /x and R (cid:91) = F p [[ x /p ∞ ]]. The (cid:93) map is givenby x (cid:93) = ( ζ p − p .We get the same R/p and R (cid:91) by taking instead R = Z p [ p /p ∞ ] ∧ p . In this case the (cid:93) map is given by x (cid:93) = p .Tilting is right adjoint to the functor of ( p -typical) Witt vectors: { perfect F p -algebras } W (cid:47) (cid:47) ⊥ { p -complete Z p -algebras } ( − ) (cid:91) (cid:111) (cid:111) Morally, we think of this adjunction as extension/restriction of scalars along the Teichm¨uller lift F p → Z p (which is a map of multiplicative monoids, but not a map of rings). Definition 3.10. We write A inf ( R ) := W ( R (cid:91) ), and θ : A inf ( R ) → R for the counit of this adjunction.By functoriality of Witt vectors, A inf ( R ) inherits a Frobenius φ from R (cid:91) (even though there is usually noFrobenius on R ).Explicitly, every element of A inf ( R ) has the form x = ∞ (cid:88) i =0 [ a i ] p i with a i ∈ R (cid:91) , and the maps φ and θ are given by φ ( x ) = ∞ (cid:88) i =0 [ a pi ] p i , θ ( x ) = ∞ (cid:88) i =0 a (cid:93)i p i . Example 3.11. If R = Z cycl p , then A inf ( R ) = Z p [ q /p ∞ ] ∧ ( p,q − is the ring we encountered in the previoussection. The element q is constructed as follows. Fix a compatible choice of { ζ p i } , and let (cid:15) = (1 , ζ p , ζ p , . . . ) ∈ R (cid:91) q = [ (cid:15) ] ∈ A inf ( R ) = W ( R (cid:91) )We see that θ ( q /p ) = ζ p , so θ ( q ) = 1 and ker θ is generated by [ p ] q /p = q − φ − ( q − .In particular, A inf ( R ) is a Z p [ q /p ∞ ] ∧ ( p,q − -algebra for any Z cycl p -algebra R . Traditionally q − µ , [ p ] q /p is denoted ξ , and [ p ] q is denoted ˜ ξ . Remark 3.12. q is roughly a p -adic analogue of e πi . Note that e πi = 1, but we can formally write( e πi ) /p = e πi/p = ζ p whereas 1 /p = 1. Keeping track of an inverse system of p -power roots allows usto make this formal manipulation precise, and interpret e πi as an expression (like q ) whose “underlyingelement” is 1, but which contains the information of the ζ p ∞ . Definition 3.13 ([BMS18, Definition 3.5]) . A perfectoid ring is a ring R satisfying:(a) there is some π ∈ R such that π p divides p ;(b) R is π -complete;(c) the Frobenius R φ −→ R/p is surjective; d) the kernel of A inf ( R ) θ −→ R is principal.If R is p -torsionfree, the last condition may be replaced by:(d’) if x ∈ R [ p ] with x p ∈ R , then x ∈ R . Example 3.14. An F p -algebra is a perfectoid ring if and only if it is perfect; in this case π = 0, and A inf ( R ) = W ( R ). Z cycl p and Z p [ p /p ∞ ] ∧ p are both perfectoid. If C is a complete algebraically closed extensionof Q p , then O C is a perfectoid ring; in particular, O C is a Z cycl p -algebra.Some examples of rings which are not perfectoid are Z p (a), Z p [ p /p ] (c), and Z p [[ x /p ∞ ]] /x (d’). Remark 3.15. We are mainly interested in perfectoid rings which are either p -torsion or p -torsionfree.There is a fracture theorem expressing any perfectoid ring as a (homotopy) pullback of such perfectoid rings[Bha18, Proposition IV.3.2], so we are justified in restricting to such. Remark 3.16. If R is perfectoid, then A inf ( R ) θ −→ R is the universal pro-infinitesimal formal p -adic thick-ening of R [Fon94, Th´eor`eme 1.2.1]. This is the reason for the notation A inf .Assume from now on that R is perfectoid. In this case, there is an alternative description of A inf whichis very useful. Although by definition A inf ( R ) := lim ←− n,R W n ( R (cid:91) ) , it turns out [BMS18, Lemma 3.2] that there is also a canonical isomorphism A inf ( R ) ∼ = lim ←− n,F W n ( R ) . (3)This is very important for the topological story: TR ( R ) = W ( R ), but TF ( R ) = A inf ( R ).Under the isomorphism (3), we define ˜ θ n : A inf ( R ) −→ W n ( R ) to be the projection. Explicitly, for x =( x (0) , x (1) , . . . ) ∈ R (cid:91) , we have ˜ θ n ([ x ]) = [ x ( n ) ]. The map θ introduced above is the same as ˜ θ φ , and in factwe will mainly be concerned with ˜ θ n φ (which is relevant to TF) rather than ˜ θ n (which is relevant to TP).We also will not use the maps θ n := ˜ θ n φ n (for n > F , R , V operators on W n ( R ). Lemma 3.17 ([BMS18, Lemma 3.4]) . The following diagrams commute: A inf ( R ) ˜ θ n +1 (cid:47) (cid:47) φ − (cid:15) (cid:15) W n +1 ( R ) R (cid:15) (cid:15) A inf ( R ) ˜ θ n (cid:47) (cid:47) W n ( R ) A inf ( R ) ˜ θ n +1 (cid:47) (cid:47) W n +1 ( R ) F (cid:15) (cid:15) A inf ( R ) ˜ θ n (cid:47) (cid:47) W n ( R ) A inf ( R ) ˜ θ n +1 (cid:47) (cid:47) W n +1 ( R ) A inf ( R ) ˜ θ n (cid:47) (cid:47) ˜ λ n +1 (cid:79) (cid:79) W n ( R ) V (cid:79) (cid:79) Here ˜ λ n +1 is an element of A inf ( R ) satisfying ˜ θ n +1 ( λ n +1 ) = V (1) ∈ W n +1 ( R ) . By definition, ker θ is a principal ideal; we let [ p ] A be a choice of generator. (This is not a Teichm¨ullerrepresentative.) Then the elements [ p n ] A := [ p ] A φ ([ p ] A ) · · · φ n − ([ p ] A ) φ ([ p n ] A ) = φ ([ p ] A ) · · · φ n ([ p ] A )generate ker(˜ θ n φ ) and ker ˜ θ n , respectively [BMS18, Lemma 3.12]. Note in particular that for n ≤ m ,[ p m ] A = φ n ([ p m − n ] A )[ p n ] A . This gives the following, which will turn out to be related to the cell structure of S λ n − ( § emma 3.18 ([BMS18, Remark 3.19]) . For n < m , the bottom row of the following commutative diagramis exact: A inf ( R ) ˜ θ n φ (cid:15) (cid:15) (cid:47) (cid:47) A inf ( R ) ˜ θ m φ (cid:15) (cid:15) [ p n ] A (cid:47) (cid:47) A inf ( R ) ˜ θ m φ (cid:15) (cid:15) A inf ( R ) ˜ θ n φ (cid:15) (cid:15) (cid:47) (cid:47) W n ( R ) V m − n (cid:47) (cid:47) W m ( R ) (cid:47) (cid:47) W m ( R ) F m − n (cid:47) (cid:47) W n ( R ) (cid:47) (cid:47) Topological aspects. The fundamental theorem of topological Hochschild homology is that THH ∗ ( F p ) = F p [ σ ], with | σ | = 2. This is due to B¨okstedt [B¨85], but for several decades his paper was available only viaclandestine channels; fortunately, a public proof is now available, see [HN20, § § Theorem 3.19 ([Hes06], [BMS19, Theorem 6.1]) . Let R be a perfectoid ring. Then THH ∗ ( R ; Z p ) = R [ σ ] ,for some choice of σ ∈ THH ( R ; Z p ) . Proposition 3.20 ([BMS19, Propositions 6.2 and 6.3]) . Let R be a perfectoid ring, and let A = A inf ( R ) .We can choose generators [ p ] A ∈ ker θ , σ ∈ TC − ( R ; Z p ) , t ∈ TC −− ( R ; Z p ) , and τ ∈ TP − ( R ; Z p ) to giveidentifications TC −∗ ( R ; Z p ) = A [ σ, t ] σt − [ p ] A TP ∗ ( R ; Z p ) = A [ τ ± ] such that TC − ( R ; Z p ) can (cid:47) (cid:47) ϕ (cid:47) (cid:47) TP( R ; Z p ) act as can( σ ) = [ p ] A τ − ϕ ( σ ) = τ − can( t ) = τ ϕ ( t ) = φ ([ p ] A ) τ Since we will mainly be concerned with the canonical map, we will make a slight abuse of notation andwrite t rather than τ .3.3. Prisms. The classical definition of perfectoid rings given in the previous section is most useful forrecognizing perfectoid rings in the wild. However, for our purposes it is more natural to view perfectoid ringsas equivalent to perfect prisms , as the emphasis will be on A inf ( R ) rather than R . We hope to convince thereader that prisms are natural from the perspective of equivariant homotopy theory. Although we will onlyuse perfect prisms in the remainder of the paper, the general setting is illuminating; we conjecture that ourresults hold for general prisms, but this is nontrivial to prove as TR and TF are not yet understood in thegeneral case. References for this section are [BS19], [Bha18], and [AB19]. Definition 3.21. A δ -structure on a ring A is a ring homomorphism φ : A → A which is a derived lift ofFrobenius . When A is p -torsionfree, this just means that φ ( x ) ≡ x p mod p ; in this case, we can uniquelysolve for the element “witnessing” this congruence, and thus define a function δ : A → A such that φ ( x ) = x p + pδ ( x )for all x ∈ A . “Derived lift” means that for general A , we specify δ rather than φ , where δ is required tosatisfy whatever identies are needed for φ to be a ring homomorphism.A δ -ring is a ring together with a δ -structure. A δ -ring is perfect if φ is an isomorphism. Remark 3.22. A δ -structure on A is equivalent to a ring section of W ( A ) R −→ A , given in Witt coordinatesby x (cid:55)→ ( x, δ ( x )) and in ghost coordinates by x (cid:55)→ ( x, φ ( x )). This can be seen from the pullback diagram W ( A ) R (cid:47) (cid:47) F (cid:15) (cid:15) (cid:121) A ϕ (cid:15) (cid:15) A can (cid:47) (cid:47) A//p := A ⊗ LZ F p gain, this pullback diagram follows from injectivity of the ghost map in the torsionfree case, and in generalby Kan extending. This perspective is apparently due to Rezk [Rez19]. Definition 3.23 ([BS19, Definition 3.2]) . A prism consists of a δ -ring A together with an ideal I such that: • I defines a Cartier divisor on Spec A ; • A is derived ( p, I )-complete; • (“prism condition”) p ∈ I + φ ( I ) A .We write R = A/I . A prism is: • perfect if φ is an isomorphism; • crystalline if I = ( p ); • orientable if I is principal, in which case a choice of generator is called an orientation . We denotean orientation by [ p ] A ; • transversal if it is orientable and ( p, [ p ] A ) is a regular sequence. (This implies that R is p -torsionfree.)It is usually safe to assume all prisms are orientable. In this case, the first condition requires that [ p ] A isa non-zerodivisor, and the prism condition becomes φ ([ p ] A ) ≡ up mod [ p ] A for a unit u ∈ A × , or equivalently δ ([ p ] A ) ∈ A × . Example 3.24. Here are examples of prisms. • A crystalline prism is simply a δ -ring which is p -torsionfree and p -complete. • The category of perfectoid rings is equivalent to the category of perfect prisms via R (cid:55)→ ( A inf ( R ) , ker θ )and ( A, I ) (cid:55)→ A/I . Perfect F p -algebras correspond to perfect crystalline prisms, while torsionfreeperfectoid rings correspond to perfect transversal prisms. Different untilts of a perfect F p -algebra k correspond to different prism structures on W ( k ). • Let K/ Q p be a finite extension with residue field k and fixed uniformizer π . Let S = W ( k )[[ z ]], with δ -struture given by the usual Frobenius on W ( k ) and φ ( z ) = z p . There is a surjection S → O K given by z (cid:55)→ π , with kernel generated by an Eisenstein polynomial E ( z ). Then ( S , ( E ( z ))) is aprism, said to be of “Breuil-Kisin type”. Different local fields K with residue field k correspond todifferent prism structures on S , so we can think of K as an untilt (in the sense of “characteristiczero incarnation”) of k (( z )). • Let A = Z p [[ q − φ ( q ) = q p , and let I = ([ p ] q ) with orientation [ p ] A := [ p ] q . This “ q -crystalline”prism has the interesting property that δ ([ p ] A ) ≡ p ] A .Our main goal in this section is to explain the significance of the prism condition. First, we need anelaboration of it. We again set [ p n ] A = [ p ] A φ ([ p ] A ) · · · φ n − ([ p ] A ). The i = 1 case of the following lemmaappears as [AB19, Lemma 3.5]. Proposition 3.25. For i ≤ j , there is a congruence φ j ([ p ] A ) ≡ u i,j p mod [ p i ] A for some unit u i,j ∈ A × .Proof. The case ( i, j ) = (1 , 1) is the prism condition, with u , = δ ([ p ] A ). To induct in the direction( i, i ) = ⇒ ( i + 1 , i + 1), we write φ i +1 ([ p ] A ) = φ i ([ p ] A ) p + pφ i u , = ( u i,i p + x [ p i ] A ) p − φ i ([ p ] A ) + pφ i u , ≡ ( u p − i,i p p − φ i ([ p ] A ) + φ i u , ) p mod [ p i +1 ] A . The final parenthesized expression is a unit because φ i ([ p ] A ) ∈ rad( A ). To induct in the ( i, j ) = ⇒ ( i, j + 1)direction, we write φ j +1 ([ p ] A ) = φ j ([ p ] A ) p + pφ j u , ≡ ( u pi,j p p − + φ j u , ) p mod [ p i ] A . The final parenthesized expression is a unit because p ∈ rad( A ). (cid:3) orollary 3.26. For i ≤ min { j, r } there is a congruence φ r ([ p j − i ] A ) ≡ up j − i mod [ p i ] A for some unit u ∈ A × . Remark 3.27 (Algebro-geometric interpretation of the prism condition) . Geometrically, the prism conditionsays that the closed subschemes of Spec( A ) cut out by any two φ -iterates φ i ([ p ] A ) and φ j ([ p ] A ) intersectonly in characteristic p : V ( φ i [ p ] A ) ∩ V ( φ j [ p ] A ) ⊂ V ( p ) . We imagine shining a beam of characteristic p into Spec A , refracting it into beams of characteristic φ • ([ p ] A ),which can then be studied one at a time; this is the reason for the name “prism”. Our proof of Proposition3.31 is an example of this idea. Spec A V ( p ) V ([ p ] A ) V ( φ ([ p ] A )) V ( φ ([ p ] A )) . . .. . .. . .There is also an equivariant interpretation of the prism condition: it is just what we need to build aMackey functor (specifically, to satisfy the axiom res ◦ tr = p ). Corollary 3.28 (Equivariant interpretation of the prism condition) . The prescription W ( T /C p n ) = A/ [ p n +1 ] A defines a Mackey functor W , with restriction maps given by the natural projections, by defining the transfersas indicated: ... (cid:24) (cid:24) A/ [ p ] A (cid:24) (cid:24) (cid:90) (cid:90) A/ [ p ] A (cid:25) (cid:25) φ ([ p ] A ) u − , (cid:89) (cid:89) A/ [ p ] Aφ ([ p ] A ) u − , (cid:89) (cid:89) In subsequent Lewis diagrams we will omit the u − i,i to save space (and since we will be working up tounits anyway), but they do matter. Remark 3.29. Proposition 3.25 is not optimal: already in the base case we in fact have φ ([ p ] A ) ≡ u , p mod[ p ] pA . The congruences we have given suffice to construct the Mackey structure, but it seems that sharperstatements are needed to study the interaction of the norm maps with the slice filtration.Let us explain the connection to topology. Assume there exists a connective E ∞ -ring S A such that S A ⊗ S Z = A ; this can be constructed by hand in all the examples we have given, starting from Lurie’sspherical Witt vectors [Lur18, Example 5.2.7]. In the case of a perfectoid ring R , the canonical mapTHH( R ; Z p ) → THH( R/ S A )is an equivalence ([KN19, Proposition 3.5], [Zho20, Lemma 14]); for general prisms, THH( R/ S A ) is what oneshould study. The Mackey functor W arises as π THH( R/ S A ) E T + , which agrees with π THH( R/ S A ) when A is perfect. The reader should consult [Zho20] for more in this direction, such as a Hopkins-Mahowaldresult for perfectoid rings and complete regular local rings. he multiplication on A descends to W , making it a Green functor . In the situation of the precedingparagraph, the framework of equivariant homotopy theory implies that W has even more structure: it is a Tambara functor . We refer to [HM19, Definition 2.11] for a full definition, but essentially this means that W comes with norm maps which are multiplicative analogues of transfers. In the remainder of this section wewill try to identify the norm algebraically. This material is not used elsewhere in the paper, but we expectit to be important to further developments. Definition 3.30. Let A be a perfect prism. A function A N −→ A is a lift of the norm N p n p n − if A N (cid:47) (cid:47) (cid:15) (cid:15) A (cid:15) (cid:15) A/ [ p n ] A N (cid:47) (cid:47) A/ [ p n +1 ] A commutes, where A/ [ p n ] A = W n ( A/ [ p ] A ) N −→ W n +1 ( A/ [ p ] A ) = A/ [ p n +1 ] A is Angeltveit’s norm map for Witt vectors [Ang15]. We also say that N ( x ) lifts N p n p n − ( x ) if this diagramcommutes for a particular x ∈ A . Proposition 3.31. Let A be a perfect prism. N ( x ) is a lift of N p n p n − ( x ) if and only if N ( x ) ≡ φ ( x ) mod φ n ([ p ] A ) N ( x ) ≡ x p mod [ p n ] A Proof. By functoriality, we may assume that A is transversal, so that A/ [ p ] A is p -torsionfree. This has theadvantage that the ghost map gh : W n ( A/ [ p ] A ) → ( A/ [ p ] A ) n is injective. The identification A/ [ p n ] A ∼ −→ W n ( A/ [ p ] A )is given in ghost coordinates by x (cid:55)→ ( φ − ( n − ( x ) mod [ p ] A , . . . , x mod [ p ] A ) . By [Ang15, Theorem 1.4], the norm is given in ghost coordinates by W n ( A/ [ p ] A ) N −→ W n +1 ( A/ [ p ] A )( w , w , . . . , w n − ) (cid:55)→ ( w , w p , w p , . . . , w pn − ) . Thus we get the congruences N ( x ) ≡ φ ( x ) mod φ n ([ p ] A ) N ( x ) ≡ x p mod φ i ([ p ] A ) , ≤ i < n. Using transversality again, the second line is equivalent to N ( x ) ≡ x p mod [ p n ] A by [AB19, Lemma 3.6]. (cid:3) Since φ and x (cid:55)→ x p are both multiplicative, we immediately get N ( xy ) ≡ N ( x ) N ( y ) mod [ p n +1 ] A . We donot expect there to be lifts of the norm which are multiplicative as maps A → A , however. Proposition 3.32. Let A be a prism over ( Z p [ q /p ∞ ] ∧ ( p,q − , [ p ] q /p ) , and let x, y ∈ A with δ ( x ) = δ ( y ) = 0 .Then the q -Pochhammer symbol ( x, − y ; q p n − ) p := p − (cid:89) i =0 ( x − q ip n − y ) is a lift of N p n p n − ( x − y ) .Proof. This follows from q p n − ≡ ζ p mod φ n ([ p ] q /p ), q p n − ≡ p n ] q /p , and φ ( x − y ) = x p − y p . (cid:3) The following is an adaptation of Borger’s formula for the norm [Ang15, Definition 1.1]. roposition 3.33. The function N n ( x ) = φ ( x ) − φ n ([ p ] A ) u n,n δ ( x ) is a lift of the norm N p n p n − .Proof. Certainly N n ( x ) ≡ φ ( x ) mod φ n ([ p ] A ), and N n ( x ) ≡ x p mod [ p n ] A by Proposition 3.25. (cid:3) Warning 3.34. The q -Pochhammer symbol ( x, − y ; q p n − ) p does not agree with our handicrafted norm N n ( x − y ) as functions A → A unless n = 1 , p = 2. It seems highly non-trivial to write down lifts of thenorm that make sense for arbitrary prisms and which specialize to the q -Pochhammer symbol. The best wehave found in this direction is the identity( x, − y ; q ) = ψ ( z ) − [3] q δ ( z ) − [3] q ([2] q − zδ ( z ) − δ ( z )) , where z = x − y , obtained through trial and error with the help of Sage. art Results Notation Throughout R is a fixed perfectoid ring, A = A inf ( R ) equipped with its lift of Frobenius φ , [ p ] A is agenerator of ker( A θ −→ R ), and [ p n ] A = [ p A ] φ ([ p A ]) · · · φ n − ([ p ] A )[ n ] A := [ p v p ( n ) ] A [ n ] A ! = [1] A [2] A · · · [ n ] A . (This definition of [ n ] A does not agree with [ n ] q /p , but it does up to units, which is enough for our purposes.)We will simply write THH, TC − , etc. for THH( R ; Z p ), TC − ( R ; Z p ), etc. We caution the reader thatwe write THH for both the cyclonic spectrum and its underlying Borel T -spectrum; the intended meaningshould always be clear from context. We also write TC −∗ := π ∗ TC − = π ∗ TC − ( R ; Z p ), etc.The circle group is denoted by T . The ring of real representations of T is denoted RO ( T ), while thering of complex representations is denoted by R ( T ). We write ∗ for Z -grading and (cid:70) for RO ( T )-grading.Occasionally we may write (cid:70) where only an actual representation would make sense; we hope this is clearfrom context. We write a V ∈ π G − V S for the Euler class associated to an actual representation V . λ n is the one-dimensional complex T -representation where z ∈ T acts by z n , λ i := λ p i , and λ ∞ := λ .We also define [ n ] λ = 1 + λ + . . . + λ n − { n } λ = λ + . . . + λ n . Given α ∈ RO ( T ), we write α ( r ) for the fixed space α C pr pulled back along the root isomorphism T ∼ −→ T /C p r . We then set d r ( α ) = dim C ( α ( r ) ). Explicitly, λ (cid:48)∞ = λ ∞ , λ (cid:48) i = λ i − for i > 1, and λ (cid:48) = 0. Thusfor a representation α = k λ + · · · + k n λ n + k ∞ λ ∞ , we get d r ( α ) = k r + k r +1 + · · · + k n + k ∞ . When a single representation α is in play, we may abbreviate d r ( α ) to d r .The (equivariant) sphere spectrum is denoted by either S or S . The smash product of spectra, including G -spectra, is denoted by ⊗ . We do not distinguish between an abelian group (or Mackey functor) and itsassociated Eilenberg-Mac Lane (equivariant) spectrum, nor between a G -space and its suspension spectrum.We write P n X for the n th slice cover of a G -spectrum X , and P nn X for its n -slice.4. RO ( T ) calculations In this section we study the portion of the RO ( T )-graded homotopy π (cid:70) THH that will be needed for theslice computations in § 5. The Mackey functors we will be using are introduced in § § q -gold relation (Lemma 4.9). Our actualcalculations are carried out in § Mackey functors.Definition 4.1. The Witt Mackey functor W (c.f. § T -Mackey functor given on objects by W ( T /C p k ) = A/ [ p k +1 ] A . Restrictions are given by the natural quotient maps, and for i ≤ j the transfer is defined as multiplicationby [ p j +1 ] A [ p i +1 ] A u i,j = φ i +1 ([ p j − i ] A ) u i,j , where u i,j ∈ A × is an appropriate unit (c.f. Proposition 3.25). We will enerally suppress this unit from the notation. We also write W ( n ) := ↓ T C pn W for the restriction of W to a C p n -Mackey functor. Abstractly we have W ( T /C p k ) ∼ = W k +1 ( R ), but the A -algebra structure will beimportant for us. Definition 4.2. Let M be a G -Mackey functor, and let H ≤ G be a subgroup. Then tr H M is defined tobe the sub-Mackey functor of M generated under transfers by ↓ GH M , while Φ H M is defined as the quotient0 → tr H M → M → Φ H M → . Importantly, we have Φ K tr H M = 0 for H ≤ K . Remark 4.3. If G is finite, then tr H M is just the image of the canonical map ↑ GH ↓ GH M → M . Example 4.4. For G = C p , these sequences are0 (cid:47) (cid:47) (cid:25) (cid:25) A/ [ p ] Ap (cid:24) (cid:24) φ ([ p ] A ) (cid:47) (cid:47) A/ [ p ] A (cid:24) (cid:24) (cid:47) (cid:47) A/φ ([ p ] A ) (cid:24) (cid:24) (cid:47) (cid:47) (cid:25) (cid:25) (cid:47) (cid:47) (cid:90) (cid:90) (cid:26) (cid:26) A/ [ p ] Ap (cid:25) (cid:25) (cid:88) (cid:88) φ ([ p ] A ) (cid:47) (cid:47) A/ [ p ] A (cid:25) (cid:25) φ ([ p ] A ) (cid:88) (cid:88) (cid:47) (cid:47) A/φ ([ p ] A ) (cid:26) (cid:26) φ ([ p ] A ) (cid:88) (cid:88) (cid:47) (cid:47) (cid:90) (cid:90) (cid:26) (cid:26) (cid:91) (cid:91) (cid:47) (cid:47) A/ [ p ] A (cid:89) (cid:89) (cid:47) (cid:47) A/ [ p ] Aφ ([ p ] A ) (cid:89) (cid:89) (cid:47) (cid:47) (cid:89) (cid:89) (cid:47) (cid:47) (cid:91) (cid:91) (cid:47) (cid:47) tr e W (2) (cid:47) (cid:47) W (2) (cid:47) (cid:47) Φ e W (2) (cid:47) (cid:47) (cid:47) (cid:47) (cid:25) (cid:25) A/ [ p ] Ap (cid:24) (cid:24) φ ([ p ] A ) (cid:47) (cid:47) A/ [ p ] A (cid:24) (cid:24) (cid:47) (cid:47) A/φ ([ p ] A ) (cid:25) (cid:25) (cid:47) (cid:47) (cid:25) (cid:25) (cid:47) (cid:47) (cid:90) (cid:90) (cid:26) (cid:26) A/ [ p ] A (cid:25) (cid:25) (cid:88) (cid:88) (cid:47) (cid:47) A/ [ p ] A (cid:25) (cid:25) φ ([ p ] A ) (cid:88) (cid:88) (cid:47) (cid:47) (cid:26) (cid:26) (cid:88) (cid:88) (cid:47) (cid:47) (cid:90) (cid:90) (cid:26) (cid:26) (cid:91) (cid:91) (cid:47) (cid:47) A/ [ p ] Aφ ([ p ] A ) (cid:89) (cid:89) (cid:47) (cid:47) A/ [ p ] Aφ ([ p ] A ) (cid:89) (cid:89) (cid:47) (cid:47) (cid:91) (cid:91) (cid:47) (cid:47) (cid:91) (cid:91) (cid:47) (cid:47) tr C p W (2) (cid:47) (cid:47) W (2) (cid:47) (cid:47) Φ C p W (2) (cid:47) (cid:47) (cid:47) (cid:47) (cid:25) (cid:25) A/ [ p ] Ap (cid:24) (cid:24) φ ([ p ] A ) (cid:47) (cid:47) A/ [ p ] Ap (cid:24) (cid:24) (cid:47) (cid:47) A/φ ([ p ] A ) p (cid:24) (cid:24) (cid:47) (cid:47) (cid:25) (cid:25) (cid:47) (cid:47) (cid:90) (cid:90) (cid:26) (cid:26) A/ [ p ] Ap (cid:25) (cid:25) (cid:88) (cid:88) φ ([ p ] A ) (cid:47) (cid:47) A/ [ p ] A (cid:25) (cid:25) (cid:88) (cid:88) (cid:47) (cid:47) A/φ ([ p ] A ) (cid:26) (cid:26) (cid:88) (cid:88) (cid:47) (cid:47) (cid:90) (cid:90) (cid:26) (cid:26) (cid:91) (cid:91) (cid:47) (cid:47) A/ [ p ] A (cid:89) (cid:89) (cid:47) (cid:47) A/ [ p ] Aφ ([ p ] A ) (cid:89) (cid:89) (cid:47) (cid:47) (cid:89) (cid:89) (cid:47) (cid:47) (cid:91) (cid:91) (cid:47) (cid:47) tr e W (2) (cid:47) (cid:47) tr C p W (2) (cid:47) (cid:47) Φ e tr C p W (2) (cid:47) (cid:47) . Abusive Notation 4.5. In our present case G = C p ∞ , we will write tr C n when we really mean tr C pvp ( n ) .We expect that our results hold as stated without this abuse when THH is regarded as an integral cyclonicspectrum, but we have not carefully verified this. .2. Tools. The most powerful tool for computing RO ( T )-graded THH is the isotropy separation square.This takes the form π (cid:70) ΣTHH h T (cid:47) (cid:47) TF (cid:70) R (cid:47) (cid:47) (cid:15) (cid:15) TF (cid:70) (cid:48) ϕ (cid:15) (cid:15) π (cid:70) ΣTHH h T (cid:47) (cid:47) TC − (cid:70) (cid:47) (cid:47) TP (cid:70) (4)at infinite level, and π (cid:70) THH hC pn (cid:47) (cid:47) TR n +1 (cid:70) R (cid:47) (cid:47) (cid:15) (cid:15) TR n (cid:70) (cid:48) ϕ (cid:15) (cid:15) π (cid:70) THH hC pn (cid:47) (cid:47) π (cid:70) THH hC pn (cid:47) (cid:47) π (cid:70) THH tC pn (5)for finite subgroups.We will understand TF (cid:70) by relating it to TC − (cid:70) and TP (cid:70) . In doing so, we will rely heavily on the following RO ( T )-graded version of Tsalidis’ theorem [Tsa98, Theorem 2.4]. Our reliance on this result is the maindifficulty in generalizing our results to imperfect prisms. Theorem 4.6 ([AG11, Theorem 5.1]) . Write (cid:70) = α + ∗ for ∗ ∈ Z and α ∈ RO ( T ) . The vertical maps inthe isotropy separation square (5) are isomorphisms for ∗ ≥ {− d ( α ) , . . . , − d n ( α ) } . Consequently, the vertical maps in (4) are isomorphisms for ∗ ≥ {− d i ( α ) | i ≥ }≥ . To begin, we had better know the Z -graded homotopy of TF and TR n , which was not spelled out in[BMS19]. Abstractly the following tells us that TR n ∗ = W n ( R )[ σ ]; this was already known (at least on π )by [HM97b, Theorem 3.3]), but the A -module structure is crucial for what follows. Proposition 4.7. The Z -graded homotopy of TF and TR n +1 are TF ∗ = A [ σ ]TR n ∗ = ( A/ [ p n ] A )[ σ ] Proof. We already know that TP ∗ = A [ t ± ], and we have by [BMS19, Remark 6.6] that the Tate cohomologyat finite level is π ∗ THH tC pn = ( A/φ ([ p n ] A ))[ t ± ] . The result now follows from Tsalidis’ theorem (in Z degrees). Note the map ϕ is φ -linear with respect to A -module structures, which is why we get ( A/ [ p n ] A )[ σ ] instead of ( A/φ ([ p n ] A ))[ t − ]. (cid:3) Our next task is to understand the bottom row of the isotropy separation square. As in the Z -gradedcase, this can be computed by a spectral sequence H ∗ ( B T , TR α + ∗ ) ⇒ TC − α + ∗ . On the underlying, we have TR α + ∗ = TR | α | + ∗ , and we let u α ∈ TR | α |− α be the image of 1 ∈ TR ∼ −→ TR | α |− α .The action of T on this group is necessarily trivial, and the spectral sequence collapses for degree reasons, so u α lifts to TC −| α |− α ; by construction, it is invertible. A similar discussion applies to TP, so the whole bottomrow of the isotropy separation sequence is u λ i -periodic. We also recall ( § a λ -periodic. As Zeng [Zen18] puts it, the clash between the u λ i -periodicity of ΣTHH h T and the a λ -periodicityof the rightmost TF produces a lot of classes in the middle TF. Note that the classes u λ i lift to TF − λ i byTsalidis’ theorem, but are no longer invertible there. emark 4.8. We explain the relation between our account in terms of the gold elements and that givenby Angeltveit-Gerhardt. Their analysis of the Tate spectral sequence begins by observing that the E pageof the α -graded HOSS, HFPSS, or TSS is simply a shift by 2 d ( α ) of the usual one: this is u (cid:70) -periodicity.Then they point out that while the Tate spectral sequence obviously depends on d ( α ), the Tate spectrum THH t only depends on α (cid:48) (which is not true of THH h or THH h ). To exploit this, they reindex the α -gradedTate spectral sequence to make it isomorphic to the usual one, but with a different meaning of “first/secondquadrant”: this trick is essentially a λ -periodicity. The correspondence between our names and Angeltveit-Gerhardt’s is u − α ←→ t d ( α ) [ − α ].In principle—although it is not the strategy we will use here—, this allows one to completely computeTF (cid:70) , using a λ -periodicity for the induction and u λ i -periodicity for the base case. The remaining ingredientneeded is to understand the relation between the various classes a λ i , u λ i , σ , and t . For this, we will bring inthe other tool we have for computing RO ( T )-graded homotopy (and the one we will use in this paper): cellstructures, c.f. § C n -spectra, the representation spheres S λ j have cell structures of length 2, whichwe call the “long cell structure”. However, as pointed out to us by Hill, these have “short cell structures” as T -spectra : there is an exact triangle T /C p j + → S → S λ j . The following is a q -deformation of the “gold relation” from [HHR16b, Lemma 3.6(vii)]. Lemma 4.9 ( q -gold relation) . The following relations hold in TF (cid:70) . For j ≥ , σa λ j = [ p j +1 ] A u λ j . For ≤ i < j , a λ j u λ i = tr C pj C pi (1) a λ i u λ j = [ p j +1 ] A [ p i +1 ] A a λ i u λ j = φ i +1 ([ p j − i ] A ) a λ i u λ j . Proof. By Tsalidis’ theorem, we have that TF λ j = TC − λ j , and TC − λ j = A (cid:104) σu − λ j (cid:105) since TC − is u λ j -periodic.On the other hand, mapping the short cell structure into THH gives a short exact sequence0 (cid:47) (cid:47) TF λ j a λj (cid:47) (cid:47) TF (cid:47) (cid:47) TR j +10 (cid:47) (cid:47) (cid:47) (cid:47) A (cid:104) σu − λ j (cid:105) (cid:47) (cid:47) A (cid:47) (cid:47) A/ [ p j +1 ] A (cid:47) (cid:47) σu − λ j a λ j = [ p j +1 ] A , which proves the first claim. Now write σ = [ p i +1 ] A u λ i a λ i = [ p j +1 ] A u λ j a λ j . Since [ p j +1 ] A = φ i +1 ([ p j − i ] A )[ p i +1 ] A , multiplying out gives the second claim. (cid:3) In particular a λ j = φ ([ p j ] A ) a λ u λ j u − λ ,t = a λ u − λ , so from now on we will stop writing t . We also see that when working in TC − or TP (where the u λ j areinvertible), it suffices to consider only the classes σ , a λ , and u λ j . emark 4.10. The two forms of the q -gold relation, σa λ j = [ p j +1 ] A u λ j a λ j u λ i = φ i +1 ([ p j − i ] A ) a λ i u λ j , can be combined by formally setting u λ − := σ , a λ − := 1. This is one possible explanation of the ‘philo-sophical’ role of the B¨okstedt generator.4.3. Computations.Lemma 4.11. For j < n , TR n +1 λ j + ∗ = tr C pn C pj (1) u − λ j ∗ = − σ i u − λ j ∗ = 2 i ≥ elseThese generate the Mackey functors tr C pj W ( n ) for ∗ = − and W ( n ) for ∗ = 2 i ≥ .Proof. Since TR n +1 ∗ is even, the long cell structures give long exact sequences0 → TR j +12+ ∗ V n − j −−−→ TR n +1 λ j + ∗ a λj −→ TR n +1 ∗ F n − j −−−→ TR j +1 ∗ → , described algebraically by Lemma 3.18. When ∗ = − 2, this says that the transfer TR j +10 → TR n +1 λ j − is anisomorphism. The case ∗ = 2 i ≥ (cid:3) While the statement of the next lemma is imposing, its content is very simple. To compute TF i − α , westart with TF i generated by σ i , and exchange σ s for u λ j s, starting with the largest value of j . If we run outof σ s, we continue by adding a λ j s to land in the correct RO ( T )-graded degree. Alternatively, we can startwith TF − α generated by a α , and exchange a λ j s for u λ j s, or σ s once we run out of a λ j s. For example:TF = A (cid:104) σ (cid:105) TF − λ − λ = A (cid:104) a λ u λ u λ (cid:105) TF − λ = A (cid:104) σu λ (cid:105) TF − λ − λ = A (cid:104) a λ u λ (cid:105) TF − λ − λ = A (cid:104) u λ u λ (cid:105) TF − λ − λ = A (cid:104) a λ a λ (cid:105) Lemma 4.12. The portion of TF (cid:70) of the form (cid:70) = ∗ − α , with ∗ ∈ Z and α an actual representation, is A [ σ, a λ i , u λ i ] . Explicitly, let α = k λ + · · · + k n − λ n − be an actual, fixed-point free representation of T . Write α [ s, t ) := k s λ s + · · · + k t − λ t − . Then for i ≥ , TF i − α = (cid:40) A (cid:104) a α [0 ,r − a d r − ( α ) − iλ r − u i − d r ( α ) λ r − u α [ r,n ) (cid:105) d r ( α ) ≤ i < d r − ( α ) A (cid:104) σ i − d ( α ) u α (cid:105) d ( α ) ≤ i TR n +12 i − α = (cid:40) A/φ ([ p n − r ] A ) (cid:104) a α [0 ,r − a d r − ( α ) − iλ r − u i − d r ( α ) λ r − u α [ r,n ) (cid:105) d r ( α ) ≤ i < d r − ( α ) A/ [ p n +1 ] A (cid:104) σ i − d ( α ) u α (cid:105) d ( α ) ≤ i When d ( α ) ≤ i , the corresponding Mackey functor is W . When d r ( α ) ≤ i < d r − ( α ) , the correspondingMackey functor is Φ C pr − W .Proof. We proceed by induction on ( k , . . . , k n − ) ∈ N n in dictionary order. Let β be an actual T -representation whose restriction to C p r is trivial, and let α = β + λ r . The short cell structure gives0 (cid:47) (cid:47) TF i − β a λr (cid:47) (cid:47) TF i − α (cid:47) (cid:47) TR r +12 i − d r ( α ) (cid:47) (cid:47) (cid:47) (cid:47) A (cid:104) g β (cid:105) (cid:47) (cid:47) A (cid:104) g α (cid:105) (cid:47) (cid:47) A/ [ p r +1 ] A (cid:104) σ i − d r ( α ) (cid:105) (cid:47) (cid:47) or some generators g β , g α . From this we see that g α = (cid:40) g β a λ r d r ( α ) > ig β u λ r σ − d r ( α ) ≤ i which implies the description of the generators. The A -module structure and Mackey structure follow fromthe fact that a λ r kills transfers from C p r , along with TR n +1 = TF /a λ n (the short cell structure). (cid:3) In the case of a perfect F p -algebra, these groups were already known by [HM97b, Proposition 9.1] and[AG11, Theorem 8.3]. Our result is finer, as it identifies the multiplicative and Mackey structure. Remark 4.13. We point out some important structural features of TF (cid:70) that are implicit in the precedingcomputations; proofs will be given in [Sul]. For α ∈ RU ( T ), one can show that the (nonzero) groups TF α are all isomorphic to A , and vanish for d ∞ ( α ) < 0; one can even determine the generators relatively quickly.On the other hand, the groups TF α − are all torsion, need not be cyclic, and are more difficult to determine.This is why we used the long cell structure in the proof of Lemma 4.11 rather than the short cell structure:TF λ j − is zero , and instead TR n +1 λ j − = TF λ j + λ n − . It is easier to calculate the TR n +1 group directly than tocalculate the latter TF group. This problem does not arise in Lemma 4.12 since there are no odd-dimensionalTF classes in that range.There is also a “gap” phenomenon present in Lemma 4.11. For k ≥ kλ j − (2 k − , . . . , TF kλ j − are nonzero, and yet TF kλ j − = 0. This is essential to the proof of Theorem 5.1, in order to be able to isolatea single Mackey functor by “clipping off the Tsalidis part”. It also allows us, for an actual representation V , to read off the homotopy Mackey functor π V + ∗ THH as quotients of TF V + ∗ for ∗ ≥ ∗ ≥ § α = k λ + . . . + k n − λ n − , this is the spectral sequence computing TF α + ∗ from π α + ∗ (ΣTHH h T ) a ± λ π α (cid:48) + ∗ (ΣTHH h T )... a ± λ · · · a ± λ n − π α ( n − + ∗ (ΣTHH h T ) a ± λ · · · a ± λ n − TF ∗ The integer degrees of the orbit terms depend on the dimension sequence of α , but the bottom term willalways start in degree 0. For virtual representations that are not too wild (in particular, those of the form ∗ + V for an actual representation V ), this kills any potential contributions to degree − . Slices of THH In this section, we study the slice filtration on THH and prove the main theorems. In § RO ( T )-graded suspensions (Theorem 5.1). In § § The slice tower. We begin by explaining the idea. Non-equivariantly, B¨okstedt periodicity impliesthat the Whitehead tower of THH isTHH Σ THH σ (cid:111) (cid:111) Σ THH σ (cid:111) (cid:111) · · · (cid:111) (cid:111) Since the slice filtration restricts to the Postnikov filtration on the underlying spectrum, it is reasonable toguess that equivariantly, the slice covers will be given by P n THH = Σ V n THHfor appropriate T -representations V n . Since( S V ⊗ THH) Φ H = S V H ⊗ THHby cyclotomicity, we are reduced to finding T -representations V n with d ( V n ) = n and2 d k ( V n ) ≥ (cid:24) np k (cid:25) for all k ≥ . Using the irreducible representations λ i , these are uniquely determined p -locally. From the p -typicaldescription it is not so easy to see a pattern, but contemplating the q -Legendre principle (Observation 3.5),we find that V n = [ n ] λ . For example, with p = 3 we get2 d ( V ) = 82 d ( V ) ≥ d r ( V ) ≥ ∀ r ≥ V = 2 λ + λ + λ ∞ , which we recognize as p -locally equivalent to λ + λ + λ + λ = [4] λ .Educated guesses like this are actually a standard way to compute slices, thanks to the recognitionprinciple [HHR16a, Lemma 4.16]. To verify it, we must produce maps Σ [ n +1] λ THH → Σ [ n ] λ THH restrictingto σ on underlying spectra, and check that the cofibers are n -slices. Theorem 5.1. There is a canonical identification of exact triangles P n +2 THH (cid:47) (cid:47) P n THH (cid:47) (cid:47) P n n THHΣ [ n +1] λ THH (cid:47) (cid:47) Σ [ n ] λ THH (cid:47) (cid:47) Σ { n } λ tr C n W The bottom row is identified with Σ { n } λ (cid:18) Σ λ ∞ THH σu − λn −−−→ Σ λ ∞ − λ n THH → tr C n W (cid:19) or equivalently with Σ [ n ] λ (cid:18) Σ λ n THH σu − λn −−−→ THH → W / [ pn ] A (cid:19) Proof. There are four steps.(1) Show that Σ [ n ] λ THH is slice 2 n -connective.(2) Produce the map Σ [ n +1] λ THH −→ Σ [ n ] λ THH with cofiber Σ { n } λ tr C n W .(3) Check that Σ { n } λ tr C n W is ≥ n .(4) Check that Σ { n } λ tr C n W is ≤ n . e have already observed (Corollary 3.6) that d s ([ n ] λ ) = (cid:108) np s (cid:109) , which verifies (1) by the preceding discussiontogether with the inequality 2 (cid:108) np s (cid:109) ≥ (cid:108) np s (cid:109) . The exact triangleΣ λ ∞ THH σu − λn −−−→ Σ λ ∞ − λ n THH → tr C n W follows from Lemma 4.11, which takes care of (2).The alternative expression for the cofiber sequence can be obtained either from Lemma 4.12, or by usingcell structures (Example 2.30) to see thatΣ λ n − λ ∞ tr C n W = W / [ pn ] A . Now we check that Σ { n } λ tr C n W ≥ n . Let us write conn X for the connectivity of an ordinary spectrum X . We must show that conn Φ C pk (cid:16) Σ { n } λ tr C n W (cid:17) ≥ (cid:24) np k (cid:25) . We have conn Φ C pk S { n } λ = 2 (cid:24) n + 1 p k (cid:25) − C pk tr C n W = (cid:40) k ≤ v p ( n ) ∞ k > v p ( n )which proves (3) since k ≤ v p ( n ) implies that 2 (cid:108) n +1 p k (cid:109) − np k = (cid:108) np k (cid:109) .Finally, (4) follows from Lemma 5.2 below. (cid:3) Lemma 5.2. If M is any T -Mackey functor, then Σ { n } λ M ≤ n .Proof. This requires showing that [ S sρ pk , ↓ T C pk Σ { n } λ M ] = 0for all sp k > n . We write rρ p k = r [ p k ] λ , and note that ↓ T C pk S { n } λ M = (cid:22) np k (cid:23) λ ∞ + k − (cid:88) r =0 (cid:18)(cid:22) np r (cid:23) − (cid:22) np r +1 (cid:23)(cid:19) λ r so we need to compute [ S V , M ] where V = (cid:18) s − (cid:22) np k (cid:23)(cid:19) λ ∞ + s α − β,α = k − (cid:88) r =0 ( p k − r − p k − ( r +1) ) λ r ,β = k − (cid:88) r =0 (cid:18)(cid:22) np r (cid:23) − (cid:22) np r +1 (cid:23)(cid:19) λ r . Our assumption sp k > n gives sp k − r > np r ≥ (cid:22) np r (cid:23) , which implies that in the irreducible decomposition V = k λ + · · · + k k − λ k − + k ∞ λ ∞ , we have k i ≥ ≤ i < ∞ and k ∞ > 0. The desired result then follows from Lemma 2.31. (cid:3) xample 5.3. When p = 3, the regular slice tower up to P is P = Σ λ +2 λ + λ ∞ THH (cid:47) (cid:47) (cid:15) (cid:15) Σ λ +2 λ + λ ∞ tr C p W = P P = Σ λ + λ + λ ∞ THH (cid:47) (cid:47) (cid:15) (cid:15) Σ λ + λ tr e W = P P = Σ λ +2 λ + λ ∞ THH (cid:47) (cid:47) (cid:15) (cid:15) Σ λ +2 λ tr e W = P P = Σ λ + λ ∞ THH (cid:15) (cid:15) (cid:47) (cid:47) Σ λ + λ tr C p W = P P = Σ λ +2 λ + λ ∞ THH (cid:47) (cid:47) (cid:15) (cid:15) Σ λ +2 λ tr e W = P P = Σ λ + λ ∞ THH (cid:15) (cid:15) (cid:47) (cid:47) Σ λ tr e W = P P = Σ λ + λ + λ ∞ THH (cid:47) (cid:47) (cid:15) (cid:15) Σ λ +2 λ tr C p W = P P = Σ λ ∞ THH (cid:15) (cid:15) (cid:47) (cid:47) Σ λ tr e W = P P = Σ λ + λ + λ ∞ THH (cid:47) (cid:47) Σ λ + λ tr e W = P P = THH (cid:47) (cid:47) W = P Remark 5.4. The slice filtration is not a filtration by cyclotomic spectra: instead, we haveΦ C pk P n THH = P (cid:100) n/p k (cid:101) THH . The slice filtration. In this section we work out the filtration induced on homotopy groups. Wewill treat both π [ i ] λ TF and π i TF; the latter is what one would be probably interested in a priori, but theformer appears to be more natural. The filtration is defined asF j S π [ i ] λ TF = im( π T [ i ] λ P i + j ) THH → π [ i ] λ TF)F j S π i TF = im( π T i P i + j ) THH → π i TF)Since TF is even in this range, our formulas will apply just as well to π [ i ] λ THH and π i THH. Theorem 5.5. The slice filtration takes the following form on homotopy. When j ≤ or i = 0 , F j S π [ i ] λ TF is all of π [ i ] λ TF . Otherwise, F j S π [ i ] λ TF is generated by [ p ( i + j − A ![ p ( i − A ! . When j ≤ or i = 0 , F j S π i TF is all of π i TF . Otherwise, F j S π i TF is generated by [ p ( i + j − A ![ p r ] i − A φ r (cid:16)(cid:104)(cid:106) i + j − p r − (cid:107)(cid:105) A ! (cid:17) , where r = (cid:6) log p (cid:0) i + ji (cid:1)(cid:7) . In particular, taking i = 1 in either case gives F j S π TF = [ pj ] A ! π TF . Remark 5.6. These formulas are easier to understand from our illustration of the q -Legendre formula(Figures 1 and 2). The generator of F j S π [ i ] λ corresponds to taking the first i + j − i − j S π i corresponds to taking the first i + j − i − tallest columns first . Proof. The key identity, which follows from the q -gold relations and the q -Legendre principle, is σ k a { k } λ u − { k } λ = [ pk ] A ! . Starting with the case of π [ i ] λ , we want to identify the image of π T [ i ] λ P i + j ) THH = π [ i ] λ − [ i + j ] λ TF σ i + j u − i + j ] λ −−−−−−−→ π [ i ] λ TF . he terms are π [ i ] λ TF = A (cid:104) σ i u − i ] λ (cid:105) by Tsalidis’ theorem π [ i ] λ − [ i + j ] λ TF = (cid:40) A (cid:104) σ − j u − i ] λ u [ i + j ] λ (cid:105) j ≤ A (cid:104) a { i + j − } λ a − { i − } λ (cid:105) j > u [ i ] λ = u { i − } λ , this gives all the claims about π [ i ] λ TF.For the case of π i , we must identify the image of π T i P i + j ) THH = π i − [ i + j ] λ TF σ i + j u − i + j ] λ −−−−−−−→ π i TF = A (cid:104) σ i (cid:105) Let α = { i + j − } λ with p -typical decomposition α = k λ + · · · + k n − λ n − . By Lemma 4.12, π i − [ i + j ] λ TF = π i − − α TF= A (cid:104) a α [0 ,r − a d r − ( α ) − i +1 λ r − u i − − d r ( α ) λ r − u α [ r,n ) (cid:105) d r ( α ) ≤ i − < d r − ( α ) A (cid:104) σ i − − d ( α ) u α (cid:105) d ( α ) ≤ i − d r ( α ) = (cid:108) i + jp r (cid:109) − 1, so the dimension conditions become( p r − − i < j ≤ ( p r − i or j ≤ , equivalently r = (cid:6) log p (cid:0) i + ji (cid:1)(cid:7) . This takes care of the case j ≤ j > σ i + j u − α a α [0 ,r − a d r − ( α ) − i +1 λ r − u i − − d r ( α ) λ r − u α [ r,n ) = σ i + j − a α u − α σ i − a i − − d r ( α ) λ r − u − ( i − − d r ( α )) λ r − a α [ r,n ) u − α [ r,n ) σ i = [ p ( i + j − A ![ p r ] i − A ( a − λ r − u λ r − ) d r ( α ) a α [ r,n ) u − α [ r,n ) σ i To deal with the remaining term on the bottom, we write a α [ r,n ) u − α [ r,n ) = n − (cid:89) s = r ( a λ s u − λ s ) k s ( a − λ r − u λ r − ) d r ( α ) a α [ r,n ) u − α [ r,n ) = n − (cid:89) s = r φ r ([ p s − r +1 ] A ) k s = φ r n − r (cid:89) (cid:96) =1 [ p (cid:96) ] k (cid:96) + r − A Since k s = (cid:106) i + j − p s (cid:107) − (cid:106) i + j − p s +1 (cid:107) , and (cid:106) (cid:98) x/m (cid:99) n (cid:107) = (cid:4) xmn (cid:5) , we can apply the q -Legendre formula (Lemma 3.3) toconclude that this final expression is φ r (cid:16)(cid:104)(cid:106) i + j − p r − (cid:107)(cid:105) A ! (cid:17) . (cid:3) The slice spectral sequence. Finally, we interpret the slice filtration using the regular slice spectralsequence (RSSS). The RSSS for a G -spectrum X has signature E s,α = π α − s P | α || α | X ⇒ π α − s X for α ∈ RO ( G ). We draw this in the plane using Adams indexing, so E s,α + t is placed at ( α + t − s, s ). Thedifferentials go d r : E s,αr → E s + r,α +( r − r , or in terms of the plane display, translate by ( − , r ). We will againtreat the two cases π ∗ THH and π [ ∗ ] λ THH. We suggest that the reader study the charts at the end beforedigesting the proofs in this section. t will be useful to recall that the Mackey functors tr C pr W and Φ C pr W are given explicitly by(tr C pr W )( T /C p k ) = (cid:40) A/ [ p k +1 ] A ≤ k ≤ rA/ [ p r +1 ] A r < k (Φ C pr W )( T /C p k ) = (cid:40) ≤ k ≤ rA/φ r +1 ([ p k − r ] A ) r < k Theorem 5.7. The homotopy Mackey functors of the slices are given in even degrees by π i P n n THH = W i = nR < i = n Φ C pm W / [ p h +1 ] A < i < n where R is the constant Mackey functor on R , and m = (cid:6) log p ( n/i ) (cid:7) − , h = (cid:40) min { v p ( n ) , (cid:4) log p ( n/i ) (cid:5) } n/i not a power of p (cid:4) log p ( n/i ) (cid:5) n/i a power of p. If R is p -torsionfree, then these are the only non-vanishing homotopy Mackey functors. If R is a perfect F p -algebra, then π i +1 P n n THH = (cid:40) tr C pm + h +1 Φ C pm W n/i not a power of p tr C pm + h +1 Φ C pm +1 W n/i a power of p. Proof. We use the exact sequence0 → π i +1 P n n THH → π i P n +2 THH σu − λn −−−→ π i P n THH → π i P n n THH → . By Lemma 4.12 and Corollary 3.6, we have π i P n THH = W if i = n , and π i P n THH = Φ C pm W if0 < i < n . Let us write g for the generator of π i P n THH and g for the generator of π i P n +2 THH. Thedegrees are related by | g | = | g | − λ n = | g | − λ v p ( n ) . Note that m is the highest value of j such that a λ j appears in g .If we write e k ( j ) for the exponent of a λ j in g k , then there is a unique h such that e ( h ) = e ( h ) + 1, e ( j ) = e ( j ) for j (cid:54) = h . The map π i P n +2 THH → π i P n THH will then hit [ p h +1 ] A times g . If i = n ,then h = 0 and we get W / [ p ] A = R . Otherwise, there are three possibilities: • If v p ( n ) ≤ m , then h = v p ( n ). • If v p ( n ) > m and n/i is not a power of p , then h = m . • If v p ( n ) > m and n/i is a power of p , then h = m + 1.Some examples of the three cases with p = 3 areΦ C p W (cid:104) a λ a λ (cid:105) [ p ] A (cid:47) (cid:47) Φ C p W (cid:104) a λ a λ (cid:105) (cid:47) (cid:47) π P THHΦ e W (cid:104) a λ u λ (cid:105) [ p ] A (cid:47) (cid:47) Φ e W (cid:104) a λ u λ (cid:105) (cid:47) (cid:47) π P THHΦ C p W (cid:104) a λ a λ u λ (cid:105) [ p ] A (cid:47) (cid:47) Φ e W (cid:104) a λ u λ (cid:105) (cid:47) (cid:47) π P THHWe then combine these cases into the stated formula for h by noting that (cid:4) log p ( n/i ) (cid:5) = (cid:6) log p ( n/i ) (cid:7) if n/i isa power of p , and m = (cid:4) log p ( n/i ) (cid:5) otherwise.When R is p -torsionfree, the extended prism condition (Proposition 3.25) shows that the above maps areinjective, so the odd homotopy Mackey functors vanish. When R is a perfect F p -algebra, there are two cases.When n/i is not a power of p , we get( π i +1 P n n THH)( T /C p k ) = ≤ k ≤ mA/p k − m < k − m < h + 1 p k − m − ( h +1) A/p k − m A h + 1 ≤ k − m nd thus π i +1 P n n THH = tr C pm + h +1 Φ C pm W . When n/i is a power of p , we instead get( π i +1 P n n THH)( T /C p k ) = ≤ k ≤ m + 1 A/p k − ( m +1) < k − ( m + 1) < hp k − ( m +1) − h A/p k − ( m +1) A h ≤ k − ( m + 1)and thus π i +1 P n n THH = tr C pm + h +1 Φ C pm +1 W . (cid:3) The RSSS thus collapses at E when R is p -torsionfree. When R is a perfect F p -algebra, the RSSS is verycomplicated; however, since we identified the entire slice tower (i.e. the maps between the slice covers, notjust the slices), the E ∞ page can be read off from Theorem 5.5. Corollary 5.8. Let R be a perfect F p -algebra, and define h = h ( n, i ) as in Theorem 5.7. The entry on the E ∞ page of the RSSS corresponding to π i P n n THH( R ) is Φ C pf +1 W /p h ( n,i )+1 , where f = (cid:80) i ≤ m The homotopy Mackey functors π [ i ] λ of the slices are π [ i ] λ P n n THH = W i = nW / [ pn ] A < i = n Φ C p(cid:96) ( i,n ) W / [ pn ] A < i < n where (cid:96) ( i, n ) = max { v p ( i ) , . . . , v p ( n − } = min { r | (cid:100) n/p r (cid:101) = (cid:100) i/p r (cid:101)} .Proof. We use the exact sequence π [ i ] λ P n +2 THH σu − λn −−−→ π [ i ] λ P n THH → π [ i ] λ P n n THH → . By Lemma 4.12, the terms are π [ i ] λ − [ n +1] λ THH = i < n Φ C n W (cid:104) a λ n (cid:105) < i = n Φ C p(cid:96) ( i,n +1) W (cid:104) a { n } λ a − { i − } λ (cid:105) < i < nπ [ i ] λ − [ n ] λ THH = (cid:40) W ≤ i = n Φ C p(cid:96) ( i,n ) W (cid:104) a { n − } λ a − { i − } λ (cid:105) < i < n so the result follows by the q -gold relation. (cid:3) We provide charts of the RSSS below. It is customary to use hieroglyphics to denote Mackey functors inspectral sequences. Our notations are listed in Figure 5.3; Lewis diagrams for these can be found in Example4.4. Colors correspond to vanishing lines: for example, after restricting to { e } we would only see the classesin red, after restricting to C p we would only see the red and orange classes, etc. In particular, we see thatafter restricting to any finite subgroup, the spectral sequence is bounded in each degree.The E page of the Z -graded RSSS, in both the p -torsionfree and torsion cases, is depicted in Figures4–7. We indicate the E ∞ page of the Z -graded RSSS for a perfect F p -algebra in Figure 8. Here, an entry( f h ) means that the entry in plane coordinate (2 i, s ) is Φ C pf +1 W /p h . Finally, the filtration on π [ i ] λ THH(when R is p -torsionfree) is depicted in Figures 9 and 10; here the group π [ i ] λ appears in the 2 i column, sothis is not really a spectral sequence chart. [ p ] A / [ p ] A / [ p ] A / [ p ] A / [ p ] A / [ p ] A / [ p ] A W (cid:70) ♠ ♣ Φ e W (cid:70) ♠ ♣ Φ C p W (cid:70) ♠ ♣ Φ C p W (cid:70) ♠ ♣ Φ C p W (cid:70) ♠ ♣ Φ C p W (cid:70) ♠ ♣ Φ C p W (cid:70) ♠ ♣ tr C p ∗ +1 tr C p ∗ +2 tr C p ∗ +3 tr C p ∗ +4 tr C p ∗ +5 Φ e W Φ C p W Φ C p W Φ C p W Φ C p W Φ C p W Figure 3. Hieroglyphics for Mackey functors (cid:70) (cid:70) ♠ Figure 4. E page of the RSSS for THH( Z cycl2 ; Z ) (cid:70) (cid:70) (cid:57) ♠ Figure 5. E page of the RSSS for THH( F ) Figure 6. E page of the RSSS for THH( Z cycl3 ; Z ) Figure 7. E page of the RSSS for THH( F ) p = 2 π π π s = 2 0 2 0 1 0 1 s = 4 2 1 1 2 1 1 s = 6 3 3 3 1 2 2 s = 8 6 1 4 2 4 1 s = 10 7 2 6 1 5 2 s = 12 9 1 7 3 7 1 s = 14 10 4 10 1 8 2 s = 16 14 1 11 2 10 1 s = 18 15 2 13 1 11 3 s = 20 17 1 14 3 14 1 s = 22 18 3 17 1 15 2 s = 24 21 1 18 2 17 1 s = 26 22 2 20 1 18 3 s = 28 24 1 21 4 21 1 s = 30 25 5 25 1 22 2 s = 32 30 1 26 2 24 1 s = 34 31 2 28 1 25 3 s = 36 33 1 29 3 28 1 p = 3 π π π s = 2 0 1 0 1 0 1 s = 4 1 2 1 1 1 1 s = 6 3 1 2 1 2 1 s = 8 4 1 3 2 3 1 s = 10 5 2 5 1 4 1 s = 12 7 1 6 1 5 2 s = 14 8 1 7 2 7 1 s = 16 9 3 9 1 8 1 s = 18 12 1 10 1 9 2 s = 20 13 1 11 2 11 1 s = 22 14 2 13 1 12 1 s = 24 16 1 14 1 13 2 s = 26 17 1 15 2 15 1 s = 28 18 2 17 1 16 1 s = 30 20 1 18 1 17 2 s = 32 21 1 19 3 19 1 s = 34 22 3 22 1 20 1 s = 36 25 1 23 1 21 2 Figure 8. E ∞ page of the RSSS for THH( F ) and THH( F ). (cid:70) ♠ (cid:70) ♠ (cid:70) ♠ (cid:70) ♠ (cid:70) ♠ (cid:70) ♠ (cid:70) ♠ (cid:70) ♠ (cid:70) ♠ (cid:70) ♠ (cid:70) ♠ (cid:70) ♠ (cid:70) ♠ (cid:70) ♠ (cid:70) ♠ (cid:70) ♠♠ (cid:70) ♠ (cid:70) ♠ (cid:70) ♠ (cid:70) ♠ (cid:70) ♠ (cid:70) ♠ (cid:70) ♠ (cid:70) ♠ (cid:70) ♠ (cid:70) ♠ (cid:70) ♠ (cid:70) ♠ (cid:70) ♠ (cid:70) ♠ (cid:70) ♠ (cid:70)(cid:70) ♣ Figure 9. Filtration on π [ ∗ ] λ THH( Z cycl2 ; Z ) Figure 10. Filtration on π [ ∗ ] λ THH( Z cycl3 ; Z ) . Further questions We close with some questions to stimulate ideas. Question 6.1. Let G be a finite cyclic group, and let X be a G -spectrum whose underlying is 2-polynomial.By this we mean that π e ∗ ( X ) = π e ( X )[ x ] for a class x ∈ π e ( X ), so that x n exhibits Σ n X as the 2 n th Postnikov cover of X e . Under what conditions is X “slice 2-polynomial”, in the sense that the odd slicesvanish and P n X = Σ [ n ] λ X ?For example, Ullman [Ull09, Theorem 5.1] shows that P n KU G = Σ [ n ] λ ku G , so ku G is slice 2-polynomial (and KU G is “slice 2-periodic”). What should we expect for higher periodicities? Question 6.2. If C n acts on R , then the fixed point Mackey functor R is a C n -Tambara functor. When n is prime to p , Angeltveit-Blumberg-Gerhardt-Hill-Lawson-Mandell show that the “relative THH” N T C n R is a p -cyclotomic spectrum [ABG + 18, Theorem 8.6]. For example, R could be a perfectoid Z cycl p -algebra, and C n a finite quotient of Gal( Q cycl p / Q p ) = Z × p . What does the slice filtration on ( N T C n R ) ∧ p look like in this case? Question 6.3. Is there a formal proof of Theorem 5.1, relating the slice filtration of any cyclotomic spectrumto its Postnikov filtration? Question 6.4. The q -gold relation provides a dictionary between representations and q -analogues; forexample, { n } λ corresponds to [ pn ] q !. (Bhargava’s perspective [Bha00] makes the additional factor of p less distressing.) What q -analogues do other families of representations (symmetric powers, wedge powers,cannibalistic classes, . . . ) correspond to? What representations correspond to q -multinomial coefficients, q -Catalan numbers, . . . ? Are these useful for THH? Question 6.5. We have used the regular slice filtration due to its superior multiplicative properties, butthere are other versions of the slice filtration. For example, a back-of-the-envelope calculation suggests thatthe classical slice filtration is given by P cls2 n THH = Σ { n } λ −{ n } λ THH = Σ [2 n +1] λ − [ n +1] λ THH . Wilson describes a general framework for slice filtrations in [Wil17, § arbitrary dimensionfunction ν ? If so, what is the arithmetic interpretation of ν ? Question 6.6. In the cyclotomic t -structure [AN18], the “cyclotomic homotopy groups” π cyc i of THH areessentially given by the ordinary homotopy groups π i of TR. Slices are generalizations of homotopy groups,and TR is again a cyclotomic spectrum. Do the slices of TR, or the RO ( T )-graded homotopy groups of TR,correspond to something in the cyclotomic t -structure? Question 6.7. Wilson [Wil17] has given algebraic descriptions of the category of n -slices, as well as analgorithm for computing slices. What does this look like when G = T , and how does it compare to ourmethod? eferences [AB19] Johannes Ansch¨utz and Artur C´esar-Le Bras, The p -completed cyclotomic trace in degree 2 , https://arxiv.org/abs/1907.10530v1 . 1.4, 1, 3.3, 3.3, 3.3, 3.3[ABG + 18] Vigleik Angeltveit, Andrew J. Blumberg, Teena Gerhardt, Michael A. Hill, Tyler Lawson, and Michael A. Mandell, Topological cyclic homology via the norm , Documenta mathematica (2018). 6.2[Ada84] J. F. Adams, Prerequisites (on equivariant stable homotopy) for Carlsson’s lecture , Algebraic Topology Aarhus 1982(Berlin, Heidelberg) (Ib H. Madsen and Robert A. Oliver, eds.), Springer Berlin Heidelberg, 1984, pp. 483–532. 2.26[AG11] Vigleik Angeltveit and Teena Gerhardt, RO ( S ) -graded TR-groups of F p , Z and (cid:96) , Journal of Pure and AppliedAlgebra (2011), no. 6, 1405–1419. 1, 1.3, 4.6, 4.3, 4.13[AMGR17] David Ayala, Aaron Mazel-Gee, and Nick Rozenblyum, A naive approach to genuine G -spectra and cyclotomicspectra , https://arxiv.org/abs/1710.06416 . 2, 2.8[AN18] Benjamin Antieau and Thomas Nikolaus, Cartier modules and cyclotomic spectra , https://arxiv.org/abs/1809.01714 . 1, 2.35, 6.6[Ang15] Vigleik Angeltveit, The norm map of Witt vectors , Comptes Rendus Mathematique (2015), no. 5, 381–386.1.4, 2.16, 3.30, 3.3, 3.3[B¨85] M. B¨okstedt, Topological Hochschild homology of Z and Z /p . 1, 3.2.2[Bar17] Clark Barwick, Spectral Mackey functors and equivariant algebraic K-theory (I) , Advances in Mathematics (2017), 646 – 727. 2.1.2[BD17] Andrew Blumberg and Arun Debray, The Burnside category: Notes for a class on equivariant stable homotopytheory , https://web.ma.utexas.edu/users/a.debray/lecture_notes/m392c_EHT_notes.pdf . 2[BG16] Clark Barwick and Saul Glasman, Cyclonic spectra, cyclotomic spectra, and a conjecture of Kaledin , https://arxiv.org/abs/1602.02163 . 3[Bha00] Manjul Bhargava, The Factorial Function and Generalizations , The American Mathematical Monthly (2000),no. 9, 783–799. 3.4, 6.4[Bha18] Bhargav Bhatt, Prismatic cohomology , . 3.2.1, 3.15, 3.3[BM15] Andrew J. Blumberg and Michael A. Mandell, The homotopy theory of cyclotomic spectra , Geom. Topol. (2015),no. 6, 3105–3147. 2.1.2, 2.2[BMS18] Bhargav Bhatt, Matthew Morrow, and Peter Scholze, Integral p -adic Hodge theory , Publications math´ematiquesde l’IH´ES (2018), no. 1, 219–397. 2.9, 3.1.1, 3.2.1, 3.13, 3.2.1, 3.17, 3.2.1, 3.18[BMS19] , Topological Hochschild homology and integral p -adic Hodge theory , Publications math´ematiques de l’IH´ES (2019), no. 1, 199–310. 1, 1.2, 1.2, 3.1, 3.2.2, 3.19, 3.20, 4.2, 4.2[Bor11] James Borger, The basic geometry of Witt vectors, I The affine case , Algebra Number Theory (2011), no. 2,231–285. 2.15, 2.2[BS19] Bhargav Bhatt and Peter Scholze, Prisms and prismatic cohomology , https://arxiv.org/abs/1905.08229v1 . 3.3,3.23[Dri] Crystallization of the affine line , https://math.uchicago.edu/~drinfeld/Seminar-2019/Winter/Crystallization%20of%20affine%20line.pdf . 2.9[Dug05] Daniel Dugger, An Atiyah-Hirzebruch spectral sequence for K R -Theory , K-Theory (2005), no. 3-4, 213–256. 1,2.6[Elm83] A. D. Elmendorf, Systems of fixed point sets , Transactions of the American Mathematical Society (1983), no. 1,275–284. 2.5[Fon77] Jean-Marc Fontaine, Groupes p -divisibles sur les corps locaux , Ast´erisque, no. 47-48, Soci´et´e math´ematique deFrance, 1977 (fr). MR 498610 2.18[Fon94] , Expos´e II : Le corps des p´eriodes p -adiques , P´eriodes p -adiques - S´eminaire de Bures, 1988 (Jean-MarcFontaine, ed.), Ast´erisque, no. 223, Soci´et´e math´ematique de France, 1994, talk:2, pp. 59–101 (fr). MR 12939713.16[Ger08] Teena Gerhardt, The R ( S ) -graded equivariant homotopy of THH( F p ), Algebraic & Geometric Topology (2008),no. 4, 1961–1987. 1[Gla17] Saul Glasman, Stratified categories, geometric fixed points and a generalized Arone-Ching theorem , https://arxiv.org/abs/1507.01976 . 2.8[GM95] J. P. C. Greenlees and J. P. May, Generalized Tate cohomology , Memoirs of the American Mathematical Society (1995), no. 543, 0–0. 2.1.2[GM17] Bert Guillou and Peter May, Models of G -spectra as presheaves of spectra , https://arxiv.org/abs/1110.3571 .2.1.2[Hes06] Lars Hesselholt, On the topological cyclic homology of the algebraic closure of a local field , An Alpine Anthology ofHomotopy Theory (Dominique Arlettaz and Kathryn Hess, eds.), American Mathematical Society, 2006, pp. 133–162. 1, 3.7, 3.19[Hes07] Lars Hesselholt, On the K -theory of the coordinate axes in the plane , Nagoya Math. J. (2007), 93–109. 1[HHR16a] M.A. Hill, M.J. Hopkins, and D.C. Ravenel, On the nonexistence of elements of Kervaire invariant one , Annals ofMathematics (2016), no. 1, 1–262. 1, 2, 2.6, 5.1[HHR16b] , The slice spectral sequence for the C analog of real K -theory , https://arxiv.org/abs/1502.07611 . 1, 3,4.2 HHR17] Michael A. Hill, Michael J. Hopkins, and Douglas C. Ravenel, The slice spectral sequence for certain RO ( C p n ) -graded suspensions of H Z , Boletn de la Sociedad Matemtica Mexicana (2017), no. 1, 289–317. 2.5, 2.6[Hil12] Michael A. Hill, The equivariant slice filtration: a primer , Homology, Homotopy and Applications (2012), no. 2,143–166. 2.6[Hil20] Michael A. Hill, Equivariant stable homotopy theory , Handbook of Homotopy Theory (Haynes Miller, ed.), Chapmanand Hall/CRC, January 2020, pp. 699–756. 1.4, 2[HM94] Lars Hesselholt and Ib Madsen, Topological cyclic homology of perfect fields and their dual numbers , InstitutMittag-Leffler, Djursholm, 1994. 2.1.2[HM97a] , Cyclic polytopes and the K -theory of truncated polynomial algebras , Inventiones mathematicae (1997),no. 1, 73–97. 1[HM97b] , On the K -theory of finite algebras over Witt vectors of perfect fields , Topology (1997), no. 1, 29 – 101.1, 2.14, 2.25, 4.2, 4.3[HM19] Michael A. Hill and Kristen Mazur, An equivariant tensor product on Mackey functors , Journal of Pure and AppliedAlgebra (2019), no. 12, 5310–5345. 3.3[HN20] Lars Hesselholt and Thomas Nikolaus, Topological cyclic homology , Handbook of Homotopy Theory (Haynes Miller,ed.), Chapman and Hall/CRC, Jan 2020, pp. 619–656. 2.2, 3.2.2[Hoy18] Marc Hoyois, The homotopy fixed points of the circle action on Hochschild homology , https://arxiv.org/abs/1506.07123 . 2.2[HY18] Michael A. Hill and Carolyn Yarnall, A new formulation of the equivariant slice filtration with applications to C p -slices , Proceedings of the American Mathematical Society (2018), no. 8, 3605–3614. 1.1, 2.6, 2.33[Kal11a] D. Kaledin, Motivic structures in non-commutative geometry , Proceedings of the International Congress of Mathe-maticians 2010 (ICM 2010), Published by Hindustan Book Agency (HBA), India. WSPC Distribute for All MarketsExcept in India, Jun 2011. 2.2[Kal11b] Dmitry Borisovich Kaledin, Derived Mackey functors , Moscow Mathematical Journal (2011), no. 4, 723–803.2.1.2[Kaw80] Katsuo Kawakubo, Equivariant homotopy equivalence of group representations , J. Math. Soc. Japan (1980),no. 1, 105–118. 2.5[KN] Achim Krause and Thomas Nikolaus, Lectures on topological Hochschild homology and cyclotomic spectra , . 2.2, 2.2[KN19] , B¨okstedt periodicity and quotients of DVRs , https://arxiv.org/abs/1907.03477 . 3.2.2, 3.3[LMS86] L. Gaunce Lewis Jr., J. Peter May, and Mark Steinberger, Equivariant stable homotopy theory , Lecture Notes inMathematics, no. 1213, Springer Berlin Heidelberg, 1986. 2[Lur18] Jacob Lurie, Elliptic cohomology II: Orientations , .3.3[Mad95] Ib Madsen, Algebraic K -theory and traces , Current Developments in Mathematics (1995), no. 1, 191–321. 2.2[Mat20] Akhil Mathew, On K (1) -local TR, https://arxiv.org/abs/2005.08744 . 3.7[May] Peter May, Topological Hochschild and cyclic homology and algebraic K -theory , https://pdfs.semanticscholar.org/1b7d/a48f625142b3b472ce7856d2a4bdbe1e9933.pdf . 2.2[Maz13] Kristen Luise Mazur, On the structure of Mackey functors and Tambara functors , https://sites.lafayette.edu/mazurk/files/2013/07/Mazur-Thesis-4292013.pdf . 2.3[NS18] Thomas Nikolaus and Peter Scholze, On topological cyclic homology , Acta Math. (2018), no. 2, 203–409. 1,2.1.2, 2.2[Rez19] Charles Rezk, ´Etale extensions of λ -rings , https://faculty.math.illinois.edu/~rezk/etale-lambda.pdf . 3.22[Sch17] Peter Scholze, Canonical q -deformations in arithmetic geometry , Annales de la Facult´e des sciences de Toulouse :Math´ematiques Ser. 6, 26 (2017), no. 5, 1163–1192. 3.1[Spe19a] Martin Speirs, On the K -theory of coordinate axes in affine space , https://arxiv.org/abs/1901.08550 . 1[Spe19b] , On the K -theory of truncated polynomial algebras, revisited , https://arxiv.org/abs/1901.10602 . 1[Sul] Yuri J.F. Sulyma, RO ( T ) -graded TF of perfectoid rings , In preparation. 1, 4.13[Sus83] A. Suslin, On the K -theory of algebraically closed fields , Inventiones Mathematicae (1983), no. 2, 241–245. 3.7[tD79] Tammo tom Dieck, Transformation groups and representation theory , Springer Berlin Heidelberg, 1979. 2[Tsa98] Stavros Tsalidis, Topological Hochschild homology and the homotopy descent problem , Topology (1998), no. 4,913 – 934. 4.2[TW95] Jacques Th´evenaz and Peter Webb, The structure of Mackey functors , Transactions of the American MathematicalSociety (1995), no. 6, 1865–1961. 2.3[Ull09] John Richard Ullman, On the regular slice spectral sequence , 2009, http://jrullman.co.nf/thesis.pdf . 1, 1, 2.6,3.1.3, 6.1[Wil17] Dylan Wilson, On categories of slices , https://arxiv.org/abs/1711.03472 . 2.8, 2.6, 2.6, 6.5, 6.7[Yar15] Carolyn Yarnall, The slices of S n ∧ H Z for cyclic p -groups , https://arxiv.org/abs/1510.02077 . 2.6[Zen17] Mingcong Zeng, RO ( G ) -graded homotopy Mackey functor of H Z for C p and homological algebra of Z -modules , https://arxiv.org/abs/1710.01769v1 . 4[Zen18] , Equivariant Eilenberg-Mac Lane spectra in cyclic p -groups , https://arxiv.org/abs/1710.01769 . 1, 1.3,4, 4.2[Zho20] Zhouhang Mao, Perfectoid rings as Thom spectra , https://arxiv.org/abs/2003.08697 . 3.3 rown University, Providence, RI 02912 E-mail address : yuri [email protected] [email protected]