Algebraic K -theory of THH( F p )
aa r X i v : . [ m a t h . A T ] S e p ALGEBRAIC K -THEORY OF THH ( F p ) HALDUN ¨OZG ¨UR BAYINDIR, TASOS MOULINOS
Abstract.
In this work we study the E ∞ -ring THH( F p ) from various perspectives.Following an identification at the level of E -algebras of THH( F p ) with F p [Ω S ],the group ring of the E -group Ω S over F p , we use trace methods to compute itsalgebraic K -theory. We also show that as an E H F p -ring, THH( F p ) is uniquelydetermined by its homotopy groups. These results hold in fact for THH( k ), where k is any perfect field of characteristic p . Along the way we expand on some of themethods used by Hesselholt-Madsen and later by Speirs to develop certain tools tostudy the THH of graded ring spectra and the algebraic K -theory of formal DGAs. Introduction
In recent years, a rich interplay of higher categorical structures and classical ho-motopy theoretic calculations have led to a wealth of progress in our computationalunderstanding of algebraic K -theory. A central ingredient to the story, at least inthe characteristic p setting, is the computation due to B¨okstedt of the homotopygroups of the topological Hochschild homology of F p as a graded polynomial algebrain degree 2; this has come to be known as B¨okstedt periodicity. As the term maysuggest, whenever R is linear over F p , this allows for the periodicity operator in Tatecohomology t ∈ ˆ H ( B T , π ∗ THH( R )))to survive to the E ∞ -page of the spectral sequence computing the Tate fixed pointspectrum THH( R ) t T . Working in the modern formulation of cyclotomic spectra dueto Nikolaus-Scholze, this often allows for a streamlined computation of the topologicalcyclic homology TC( R ).In this paper we turn this phenomenon of B¨okstedt periodicity on its head by usingit to compute the topological cyclic homology of THH( k ) itself, for k any perfect fieldof characteristic p . Before stating our main result, we recall the following celebratedtheorem of Dundas-Goodwillie-McCarthy: Theorem 1.1 (Dundas-Goodwillie-McCarthy) . Let A ! B be a map of connective S -algebras such that π A ! π B has nilpotent kernel. Then the square K ( A ) T C ( A ) K ( B ) T C ( B ) trtr is a homotopy pullback square. With this result at our disposal, the main theorem in this paper takes on thefollowing form.
Theorem 1.2.
For every perfect field of positive characteristic k , there are isomor-phisms K r +1 (THH( k ) , k )) ∼ = W r ( k ) where W r ( k ) denotes the Big Witt vectors of length r in k . The relative algebraic K -theory groups are trivial in even degrees. Our starting point for this calculation is the observation from [KN19] that THH( F p )is equivalent as an E -algebra to F p [Ω S ], the algebra of chains on the free E -groupΩΣ S on S , viewed as a pointed space. As topological Hochshild homology is sym-metric monoidal, one then obtains an equivalence of T -equivariant spectraTHH(THH( F p )) ≃ THH( F p [Ω S ]) ≃ THH( F p ) ∧ THH( S [Ω S ])We improve upon this with the following result, which serves as a key input in theensuing calculation: Theorem 1.3.
For every perfect field k of positive characteristic, there is an equiv-alence of E Hk -algebras THH( k ) ≃ k ∧ S [Ω S ] . Indeed, we prove a stronger statement. We show that there is a unique E k -DGAwith homology ring k [ x ] for every positive and even | x | , see Theorem 2.1. We provethis result using the E n Postnikov extension theory of Basterra and Mandell [BM13].
Corollary 1.4.
There is an equivalence of E -algebras K(THH( k )) ≃ K( k ∧ S [Ω S ]) . Among other things, the splitting in Theorem 1.3 provides the multiplicative struc-ture on the (second) iterated topological Hochschild homology groups of k . Namely,we obtain a ring isomorphismTHH ∗ (THH( k )) ∼ = π ∗ (THH( k ) ∧ THH( S [Ω S ])) ∼ = THH ∗ ( k ) ⊗ HH k ∗ ( Hk ∧ S [Ω S ]) ∼ = k [ x ] ⊗ k [ y ] ⊗ Λ k ( z )(1.5)where subscripts denote the internal degrees and the last factor denotes the exterioralgebra over k on a single generator. The third isomorphism is described in (5.9).Note the isomorphism in (1.5) for odd p and k = F p is due to Veen [Vee18]. Indeed,Veen calculates the n -iterated topological Hochschild homology groups of F p for n ≤ p when p ≥ n ≤ p = 3. His methods are different than ours and Veendoes not cover the p = 2 case.With this at our disposal, we proceed as in [HM97a]. In [HM97a], Hesselholtand Madsen compute the algebraic K -theory of truncated polynomial algebras byconsidering them as monoid rings. This provides a decomposition of the cyclic barconstruction as a cyclotomic spectrum. Later, these methods are used by Speirs toreproduce these computations using the Nikolaus Scholze approach to topologicalcyclic homology [Spe20].In this work, we generalize this approach to formal DGAs whose homology is agraded monoid ring. In particular, we obtain a decomposition of the cyclic bar con-struction on THH( k ), which is moreover a specific instance of a functorial decomposi-tion on the THH of a graded ring spectrum. This decomposes the Tate and homotopy LGEBRAIC K -THEORY OF THH( F p ) 3 fixed points spectral sequences, and ultimately, the computation of TC(THH( k ) , k )into simpler computationally accessible pieces. Equipped with both the Dundas-Goodwillie-McCarthy theorem, and Quillen’s computation of the algebraic K -theory,this results in a complete understanding of the homotopy groups of K(THH( k )) forevery finite field k . Remark . It is possible to compute K ∗ (THH( k ) , k ) without using Theorem 1.3.Since THH( k ) is equivalent to k [Ω S ] as an E -algebra these K -theory groups aregiven by K ∗ ( k [Ω S ] , k ) and our arguments actually provide a computation of thesegroups. However, with the identification in Theorem 2.1, the computation of the K-theory groups K ∗ (THH( k ) , k ) flows in a more direct and natural way. Furthermore,Theorem 2.1 is of interest on its own. For instance, it results in Corollary 1.4 andprovides the multiplicative structure on the second iterated topological Hochschildhomology groups of k .1.1. Applications.
We now state some applications of these calculations. Note thatone has the following sequence of mapsK(K( F p )) ! K(TC( F p )) ! K(THH( F p )) , obtained by applying the K -theory functor to the Dennis and cyclotomic trace maps.Theorem 1.2 identifies the right hand side. We also identify the middle term by show-ing that TC( F p ) as a Z p -algebra is quasi-isomorphic to the cochain algebra C ∗ ( S , Z p )of the 1-sphere with coefficients in Z p , see Proposition 7.1. It follows by the theoremof the heart [AGH19, 4.8] that these groups are given by K( Z p ). Indeed, the sequenceabove is given by K(K( F p )) ! K( Z p ) ! K(THH( F p )) . where the second map is induced by a map H Z p ! THH( F p ) of E ∞ -ring spectra[NS18, IV.4.10] given by the composite of the connective cover map of TC( F p ) andthe map TC( F p ) ! THH( F p ). Remark . It is also important to note that the map K( Z p ) ! K(THH( F p )) as amap of E -ring spectra, may or may not be the canonical map Z p ! F p ! THH( F p ) . For instance, it follows from [NS18, IV.4.8] that as a T -equivariant map of E ∞ -ringspectra, the map H Z p ! THH( F p ) obtained from the connective cover of TC( F p )does not factor through F p .It is interesting to view these maps through the lens of chromatic redshift. In-deed, the expected idea is that the iterated K -theory spectrum K(K( F p )) exhibits v -periodic phenomena. This is made somewhat more precise in [AK18] where it isshown that the β -family in the homotopy groups of spheres is detected by K(K( F p )).This means that under the unit map S ! K(K( F p )) , there are classes β k ∈ π ∗ S which map to nonzero elements of π ∗ K K( F p ). This familyof elements in the stable homotopy groups of spheres are significant in that they arisefrom v self maps, which is a chromatic height 2 phenomenon. HALDUN ¨OZG ¨UR BAYINDIR, TASOS MOULINOS
In addition, THH( F p ), being an H Z -algebra is K (1)-acyclic, hence by Mitchell’stheorem its K -theory is K (2)-acyclic. Thus, the mapK(K( F p )) ! K(THH( F p ))annihilates these elements in homotopy. Moreover, note that by recent work in[LMT20], K (1)-localized K-theory is truncating on H Z -algebras. In particular L K (1) K(THH( F p )) ≃ L K (1) K( F p )so one can say that K (1)-localization is insensitive to the difference between the K -theory of F p and of THH( F p ). In other words, there is an equivalence L K (1) K(THH( F p )); F p ) ≃ L K (1) K(TC( F p )); F p ) ≃ K (1)-local category of spectra.Another point of view on Theorem 1.2 is through what is sometimes called the fun-damental theorem of algebraic K -theory. This theorem states that for a Noetherianregular ring R , the K -theory groups of R and R [ x ] agree [Wei13, V.3.3,V.6.2]. Recallthat THH( k ) is the formal DGA with homology k [ x ]. Therefore, Theorem 1.2 showsthat the fundamental theorem of algebraic K -theory does not generalize to formalDGAs.1.2. Future directions.
A potentially fruitful next step would be to generalize theseresults to the setting where the perfect field k is replaced by a perfectoid ring R .Starting off with the fact that π ∗ THH( R ) ∼ = R [ v ]for such rings, one would expect the calculations involving the Tate and homotopyfixed point spectral sequences to proceed in a similar fashion. Alternatively, oneway to bypass this issue is to use an alternate argument to compute the Tate andhomotopy fixed points, using explicit models for the summands that appear in thedecomposition for THH( R [Ω S ]), as they appear in [Spe19, Section 3.3] and [Rig20],and are in fact exploited in the perfectoid setting in the latter paper.Our methods, together with the constructions in [Bay19b], provide a general frame-work to study the algebraic K -theory of formal DGAs whose homology is a connectedgraded monoid. For such DGAs, this results in a decomposition of the cyclic bar con-struction. We plan to use these ideas to generalize the calculation of the K -theory oftruncated polynomial algebras, due to Hesselholt and Madsen, to formal DGAs withtruncated polynomial algebra homology.Another potential application of the methods used in this paper lies in the realm of A -theory calculations. Using the succinct description of TC( S [Ω S ]) from [NS18] as ahomotopy Cartesian square with terms THH( S [Ω S ]) and Σ THH( S [Ω S ]) h T togetherwith our decomposition of the relevant cyclic bar construction, one can compute ex-plicitly the integral homology of A ( S ) = K ( S [Ω S ]). We expect to return to theseideas and their extensions in future work. Outline:
Section 2 is devoted to the proof of Theorem 1.3 and may be viewedas independent from the rest of this work. Section 3 consists of a brief overview ofcyclotomic spectra. Next in section 4, we introduce a graded variant of THH anduse it to obtain functorial decompositions of THH( A ) when A is equipped with a LGEBRAIC K -THEORY OF THH( F p ) 5 grading. In section 5 we study in greater depth the cyclotomic structure of the spec-trum THH( F p [Ω S ]). The pieces are all tied together in section 6 where we computeTP(THH( F p )) , TC − (THH( F p )) and finally TC(THH( F p )). Last but not least, weinclude a short argument computing K(TC( F p )) in section 7. Conventions:
We use ∞ -categorical language in some of our constructions. How-ever we resort to the occasional point-set level argument so we have added a shortappendix with relevant rectification results. Acknowledgments
This work owes an obvious debt to the work of Speirs in[Spe19, Spe20]. Furthermore, we are grateful to Martin Speirs for answering ourquestions on his work. The second author was supported by grant NEDAG ERC-2016-ADG-741501 during the writing of this work.2.
A new description of THH ( F p ) as an E -algebra In this section, we prove Theorem 1.3. In other words, we show that there is anequivalence of E Hk -algebras THH( k ) ≃ k ∧ S [Ω S ]whenever k is a perfect field of positive characteristic. Note that S is a group objectin spaces and therefore is an E -algebra. This makes Ω S an E -algebra in spaces.Together with the fact that the infinite suspension functor is a symmetric monoidalfunctor, this provides the E -algebra structure on the right hand side.We prove the following stronger statement. Theorem 2.1.
There is a unique connective E Hk -algebra with homotopy ring k [ x m ] for every field k where | x m | = m is even. To prove this result, we use the theory of k -invariants for Postnikov extensions of E n -algebras developed by Basterra and Mandell[BM13]. We start with the followinglemma. Lemma 2.2.
There is a unique E n Hk -algebra with homotopy ring Λ[ x m ] (exterioralgebra over k with a single generator) for every even m > and for every n ≥ including n = ∞ .Proof. For n = 0, the result follows by the fact that k -chain complexes are determinedby their homology. For n >
0, we show that there is a unique Postnikov extension X ! Hk where X is an E n Hk -algebra with homotopy ring Λ[ x m ]. The k -invariants of suchextensions are classified by the Andr´e-Quillen Cohomology groups H m +1 C n ( Hk, Hk )of E n Hk -algebras [BM13, 4.2]. Since the input in this cohomology theory is actuallyour base ring, these groups are trivial. Indeed, to calculate these cohomology groups,one starts with the fiber of the augmentation map of Hk . This is the identity mapof Hk and therefore this fiber is trivial. It follows by Definition 2.6 of [BM13] thatthese cohomology groups are also trivial. (cid:3) HALDUN ¨OZG ¨UR BAYINDIR, TASOS MOULINOS
Let X denote the E ∞ Hk -algebra corresponding to the formal E ∞ k -DGA withhomology k [ x m ] and let Y be an E Hk -algebra with homotopy ring k [ x m ]. We doinduction on the Postnikov towers of X and Y in E Hk -algebras to show that X and Y are weakly equivalent as E Hk -algebras. Let X [ l ] and Y [ l ] denote the degree l Postnikov sections of X and Y respectively. It is sufficient to show that there is amap of Postnikov towers that is a weak equivalence for each l . Since all the non-trivialhomotopy groups are in degrees that are multiples of m , we only need consider thePostnikov sections at multiples of m . Note that each Postnikov section of X couldalso be calculated in E ∞ DGAs and therefore each X [ l ] also comes from a formal E ∞ k -DGA. By Lemma 2.2 above, there is a weak equivalence of E Hk -algebras, X [ m ] ˜ ! Y [ m ] . Since X [0] = Y [0] = Hk , we can set the degree zero map in the Postnikov tower of Y to be Y [ m ] ˜ ! X [ m ] ! Hk.
This shows that we have a map of Postnikov section up to degree m .Now we proceed inductively, assume that we have compatible weak equivalences Y [ l ] ˜ ! X [ l ]for every l ≤ m ( e −
1) where 2 ≤ e . It is sufficient to show that there is a weakequivalence Y [ me ] ˜ ! X [ me ] such that the following diagram commutes.(2.3) Y [ me ] X [ me ] Y [ m ( e − X [ m ( e − ≃≃ The obstructions to existence of such a lift lie in the Andr´e Quillen cohomology group H me +1 C ( Y [ me ] , Hk ) . Indeed, this obstruction class is given by the pull back of the k -invariant of thePostnikov extension X [ me ] ! X [ m ( e − H me +1 C ( Y [ m ( e − , Hk ) ! H me +1 C ( Y [ me ] , Hk )is trivial. Note we have H me +1 C ( Y [ m ( e − , Hk ) = k due to Lemma 2.5. The k -invariant of the Postnikov extension Y [ me ] ! Y [ m ( e − k -invariant is non-trivial; otherwise, the homotopyring of Y [ me ] would be a square-zero extension[BM13, 4.2]. Let x ∈ k denote this k -invariant. LGEBRAIC K -THEORY OF THH( F p ) 7 There is a pullback square Y [ me ] Y [ m ( e − Y [ m ( e − Y [ m ( e − ∨ Σ me +1 Hk. ix where i denotes the trivial derivation. This shows that the left vertical map in bothdiagrams above pulls back x to the trivial element. In other words, x is mapped to 0by the map given in (2.4). Since this is a map of k -modules and x = 0 in the field k ,we deduce that the map in (2.4) is trivial. Hence, the obstruction to the lift shownin Diagram (2.3) is trivial.This shows that there is a map Y [ me ] ! X [ me ] that makes Diagram (2.3) commuteup to homotopy. By inspection on homotopy groups, it is clear that this map is aweak equivalence.In order to obtain a weak equivalence Y [ me ] ˜ ! X [ me ] that makes Diagram (2.3)commute strictly, we could start with Postnikov towers where the objects are cofibrantand the section maps are fibrations, see the proof of Theorem 4.2 in [BM13]. In thissituation, one argues as in the proof of Proposition A.2 in [Bay18] to show that Y [ me ]and X [ me ] are weakly equivalent in the category of E Hk -algebras over X [ m ( e − Y [ me ] is cofibrant and X [ me ]is fibrant in this category.What is left to prove is the following lemma. Lemma 2.5.
Let Z be the E Hk -algebra corresponding to the formal E ∞ k -DGAwith homology k [ x m ] / ( x em ) . In this situation, we have H me +1 C ( Z, Hk ) ∼ = k Proof.
Using the universal coefficient spectral sequence of [BM13, 3.1], one sees thatthis group is the k -dual of the Andre Quillen homology group H C n me +1 ( Z, Hk ). There-fore, it is sufficient to show that H C me +1 ( Z, Hk ) = k. By Theorem 3.2 of [BM13], there is spectral sequence calculating B ∗ = k ⊕ H C ∗− ( Z, k ) . whose second page is given by E ∗ , ∗ = HH k ∗ , ∗ ( k [ x m ] / ( x e ) , k ) . The identification of the second page follows from the description of the bar construc-tion given in [BM11, Section 6]. We have E ∗ , ∗ = Λ( σx m ) ⊗ Γ( ϕ e x m )where deg( σx m ) = (1 , m ), deg( ϕ e x m ) = (2 , em ) and Γ( ϕ e x m ) denotes the dividedpower algebra over k on a single generator, see [Bay19a, 5.9]. The differentials inthis spectral sequence are trivial for degree reasons. We obtain an isomorphism of k -modules B ∗ = Λ( y ) ⊗ Γ( z )where | y | = m + 1, | z | = me + 2. HALDUN ¨OZG ¨UR BAYINDIR, TASOS MOULINOS
Note that Z is an E ∞ Hk -algebra by hypothesis because it corresponds to a com-mutative k -DGA. Therefore, B ∗ is a graded commutative ring [BM13, Section 3].From this, we obtain that y = 0.There is another spectral sequence with E ∗ , ∗ = Tor B ∗ ∗ , ∗ ( k, k )converging to B ∗ = k ⊕ H C ∗− ( Z, k ) , see [BM13, 3.3]. We only care about this page up to total degree me + 3. Thereforewe can safely assume a ring isomorphism B ∗ ∼ = Λ( y ) ⊗ k [ z ] . We obtain that E ∗ , ∗ = Tor Λ( y ) ∗ , ∗ ( k, k ) ⊗ Tor k [ z ] ∗ , ∗ ( k, k ) ∼ = Γ( σy ) ⊗ Λ( σz ) . where deg( σy ) = (1 , m + 1) and deg( σz ) = (1 , me + 2). Note that all divided powersof σy are in even total degrees and therefore they do not contribute to total degree me + 3. This is the degree we care about and the only contribution to this degreecomes from σz . Furthermore, σz survives to the E ∞ page because of degree reasons.Therefore we have either H me +1 C ( Z, Hk ) = k or H me +1 C ( Z, Hk ) = 0 . If the second option is true, then we would conclude that Z has a unique Postnikovextension with homotopy groups isomorphic to k [ x m ] / ( x e +1 ) as a k -module. Thisfollows by Theorem 4.2 of [BM13]. We know that there are at least two distinctPostnikov extensions of Z of this type. One is the square-zero extension which leadsto the square-zero extension ring k [ x m ] / ( x em ) ⊕ Σ me k in homotopy. The other Postnikov extension of Z is the E Hk -algebra correspondingto the formal E ∞ k -DGA with homology k [ x m ] / ( x e +1 ). Note that this Postnikovextension has a homotopy ring that is not isomorphic to the homotopy ring of thesquare-zero extension. This shows that the second option above is not true. (cid:3) Trace methods
We give in this section a quick recap on the Nikolaus-Scholze approach to cyclotomicspectra.3.1.
Cyclotomic spectra and topological cyclic homology.
As discussed ear-lier, our K -theory calculations reduce to topological cyclic homology calculations asa consequence of the results of Dundas, Goodwillie and McCarthy.For our calculations of topological cyclic homology, we use the recent Nikolaus-Scholze definition of topological cyclic homology and cyclotomic spectra. Note thatfor a spectrum X with a C p -action, the Tate construction X tC p denotes the cofiberof the normalization map X hC p ! X hC p , see [NS18, I.1]. LGEBRAIC K -THEORY OF THH( F p ) 9 Definition 3.1. [NS18, 1.3] A cyclotomic spectrum is a spectrum X with a T -actionand T -equivariant maps ϕ p : X ! X tC p for every prime p where the T -action on theright hand side is given by the residual T /C p ∼ = T -action. These maps are called theFrobenius maps of X .This definition agrees with the older B¨okstedt–Hsiang–Madsen definition of cyclo-tomic spectra for cyclotomic spectra that are bounded below in homotopy. This isthe case for all the cyclotomic spectra that we consider in this work.The main examples of cyclotomic spectra are the topological Hochschild homologyof ring spectra. Indeed, THH( − ) is a symmetric monoidal functor(3.2) Alg E (Sp) ! Cyc Sp,see [NS18, Section IV.2].With this structure, the topological cyclic homology of a connective ring spectrum A is defined via the following fiber sequence.TC( A ) ! THH( A ) h T Q p ∈ P ( ϕ h T p − can ) −−−−−−−−−! Y p ∈ P (THH( A ) tC p ) h T Here, can is the canonical map obtained by the identification of the middle term as(THH( A ) hC p ) h T /C p .Note that if p is invertible on A , then THH( A ) tC p = 0. This simplifies the definitionof cyclotomic spectra as well as the definition of topological cyclic homology in varioussituations. For instance, only one of the factors of the product above is possiblynontrivial when A is an H F p -module.4. THH of graded ring spectra The results of this section help explain and place in a general context one of the keyingredients to our computation of K (THH( F p )), namely that of a weight decomposi-tion of THH(THH( F p )) into homologically more tractable T -equivariant spectra, thusfacilitating the trace methods used here and in [HM97a, Spe19, Spe20] to compute K -theory in various settings.In particular, we introduce the ∞ -category of Z -graded spectra and describe variantof THH for E -algebras in this category. We then use it to show that when A is aring spectrum admitting a grading, then THH( A ) inherits a canonical grading. Definition 4.1.
Let C be an ∞ -category. The category of graded objects of C is C Z := Fun( Z , C )where Z is the integers regarded as a discrete ∞ -groupoid. Definition 4.2.
Let C ⊗ be a symmetric monoidal ∞ -category. Then C Z inherits asymmetric monoidal structure, the so-called Day convolution product from that of C ⊗ and from the abelian group structure on Z . Concretely, if X, Y ∈ C Z , then is givenby ( X ⊗ Y )( n ) ≃ ⊕ i + j = n X ( i ) ⊗ Y ( j )(see [Lur15, Section 2.3] for a more comprehensive account.) In order to prove the main proposition of this section, we briefly introduce themonoidal envelope construction appearing in [NS18] (see also [Lur16, 2.2.4.1])
Definition 4.3.
Let C ⊗ be a symmetric monoidal ∞ -category. The monoidal enve-lope of C ⊗ is the fiber product C ⊗ act := C ⊗ × Fin ∗ Finwhere Fin ⊂ Fin ∗ is the subcategory of the category of based finite sets consisting ofthe active morphisms. In this setting, a morphism f : [ n ] ! [ m ] of based finite setsis active if f − ( ∗ ) = ∗ .Note that the fiber of C ⊗ act over a finite set I is just C I , so that objects may bethought of as lists of objects of C . We remark that this comes equipped with acanonical symmetric monoidal structure ⊕ : C ⊗ act ⊗ C ⊗ act ! C ⊗ act (( X i ) i ∈ I , ( X j ) j ∈ J ) ( X k ) k ∈ I ⊔ J , together with a natural symmetric monoidal functor ι C : C ⊗ ! C ⊗ act . By [NS18, Remark 3.4], this admits a left adjoint ⊗ : C ⊗ act ! C ⊗ Proposition 4.4.
The functor
THH : Alg Sp ! Sp B T may be promoted to a functor ] THH : Alg Sp Z ! Sp Z × B T , where Sp Z × B T denotes the ∞ -category of graded spectra with a T -action. Moreover,there exists a natural equivalence colim Z ^ THH( − ) ≃ THH( − ) Proof.
We briefly recall the construction of THH as it appears in [NS18]. Givenan associative algebra A , the spectrum THH( A ) is defined as the realization of thefollowing cyclic object Λ op ! Ass ⊗ act A ⊗ −−! Sp ⊗ act ⊗ −! SpThe first functor is defined in [NS18, Proposition B.1]. Now let ( A ( n )) n ∈ Z be a graded E -algebra (an object in Alg Sp Z ) such that colim Z A ( n ) ≃ W n ∈ Z A ( n ) ≃ A . We define ^ THH( A ) to be the realization of the cyclic object in Sp Z classified by the followingdiagram Λ op ! Ass ⊗ act ( A ( n )) ⊗ n ∈ Z −−−−−−! Sp Z act ⊗ ! Sp Z . We now show that the following diagram is commutative.(4.5) Λ op Ass ⊗ act Sp Z act ⊗ Sp Z Λ op Ass ⊗ act Sp ⊗ act Sp . = = (colim Z ) ⊗ act colim Z LGEBRAIC K -THEORY OF THH( F p ) 11 Keeping in mind the fact that colim Z : Sp Z ! Spis symmetric monoidal, the commutativity of the middle square follows by applyingthe pullback along the map Fin ! Fin ∗ to the commutative square of ∞ -operadsAss Sp Z Ass Sp ⊗ , A ( n )= colim Z A classifying the compatible associative algebra structure on ( A ( n )) n ∈ Z and A ; the com-mutativity of the right square follows from Lemma 4.6 below. Hence, we have shownthat one can apply the “cyclic bar construction” to a graded monoid in spectra toobtain a cyclic object in graded spectra. We remark that Sp Z is a product in d Cat ∞ (in fact, for each n ∈ Z , one has a retract Sp Z ! Sp see e.g. [Mou19, Proof ofProposition 4.2]); hence a cyclic object in Sp Z is equivalent to a cyclic object in Spfor every integer n . (cid:3) We have used the following lemma in the above proof:
Lemma 4.6.
Suppose F : C ⊗ ! D ⊗ is a symmetric monoidal functor. Then thefollowing diagram commutes C ⊗ act D ⊗ act C ⊗ D ⊗ F act ⊗ ⊗ F Proof.
As described in [NS18, Proposition III.3.2], there is an equivalenceFun lax ( C , D ) ≃ Fun ⊗ ( C ⊗ act , D )where the left hand side denotes the ∞ -category of operad maps and the equivalenceis induced by restriction along the canonical lax monoidal functor ι C : C ⊗ ! C ⊗ act .Moreover, the symmetric monoidal left adjoint ⊗ : C ⊗ act ! C can be characterizedas the image of the identity functor id ∈ Fun lax ( C , C ) under this equivalence ([NS18,Remark III.3.4]) From this we deduce that(4.7) ⊗ ◦ ι ≃ id for any symmetric monoidal category C . Hence, the diagram in the statement com-mutes if and only if it commutes upon precomposition by ι C : C ⊗ ! C ⊗ act : ⊗ D ◦ F act ◦ ι C ≃ F ⊗ C ◦ ι C . This is true since ⊗ C ◦ ι C ≃ id C , ⊗ D ◦ ι D ≃ id D (by 4.7), together with the fact that F act ◦ ι C ≃ ι D ◦ F (by naturality of ι C ). (cid:3) As a corollary, we obtain, a canonical splitting on the topological Hochschild ho-mology of a graded algebra A . Corollary 4.8.
Let A be a graded E -algebra, so that there exists an algebra ( A ( n )) n ∈ Z in Sp Z with colim Z ( A ( n )) ≃ A . Then THH( A ) admits a canonical T -equivariantdecomposition THH( A ) ≃ _ m ∈ Z B ( m ) Proof.
This follows from Proposition 4.4 since the cyclic object realizing to THH( A )splits as a cyclic object in spectra for each n ∈ Z , together with the fact (eg. [NS18,Proposition B.5]) that upon applying realization, this results in splitting of objectsin Sp B T . (cid:3) The decomposition is compatible with the Frobenius maps ϕ : THH( A ) ! THH( A ) tC p in the following manner: Proposition 4.9.
Let A be a graded E -algebra. For every prime p, the p -typicalFrobenius map on THH( A ) restricts to maps ϕ m : B ( m ) ! B ( pm ) tCp on the summands.Proof. We recall from [NS18] the construction of the maps THH( A ) ! THH( A ) tC p .The key ingredient is a natural transformation, from the functor I : N (Free Cp ) × N (Fin) Sp ⊗ act ! Spgiven heuristically by ( S, ( X ) s ∈ S/C p ) O s ∈ S/C p X s to the functor ˜ T p : N (Free Cp ) × N (Fin) Sp ⊗ act ! Spgiven by ( S, ( X ) s ∈ S/C p ) ( O s ∈ S X s ) tC p . This makes precise the following natural transformation of cyclic objects:(4.10) ... A ∧ A ∧ A... ( A ∧ p ) tC p ( A ∧ p ) tC p ( A ∧ p ) tC p ∆ p ∆ p ∆ p The existence of this natural transformation follows from [NS18, Lemma III.3.7].Note that in each cyclic degree, the map is precisely the Tate diagonal. By naturalityof the Tate diagonal map the following diagram commutes
LGEBRAIC K -THEORY OF THH( F p ) 13 (4.11) B n ( m ) W m B n ( m ) ≃ A ∧ n ( B n ( m ) ⊗ p ) tC p (( W m B n ( m )) ⊗ p ) tC p ≃ ( W m B np ( m )) tC p ≃ ( A ∧ np ) tC p ∆ ∆ where B n ( m ) denotes the n th simplicial level of the cyclic object B • ( m ). The top hor-izontal row represents the inclusion, in simplicial degree n of the summand B ( m ) ! THH( A ). In particular, the right hand map is the natural transformation I ! ˜ T p aluded to above in simplicial degree n −
1. Finally, the bottom horizontal map is theTate fix points functor ( − ) tC p applied to the inclusion map of the summand B n ( m ) ∧ p ! B np ( mp ) = _ i + ...i p B n ( i ) ⊗ ...B n ( i p ) , where i = ... = i p = m . Here, B np ( mp ) may be understood as the np -th level of thecyclic object which upon taking realization recovers B ( pm ). We remark that B n ( m ) ∧ p has to map into the summand B np ( pm ). Indeed, as the multiplicative structure on A is induced from that of an algebra in graded spectra (see section 4), the p -foldmultiplication of summands in weight m of A ∧ n sends these to summands in weight pn in A ∧ pn . Hence we obtain, for each m , a morphism of cyclic objects φ : B • ( m ) ! B • ( pm ) tC p Taking into account the compatibility map between the realization of the simplicialdiagram of Tate fix points to the Tate fix points of the realization | B • ( pm ) tC p | ! B ( pm ) tC p , we obtain the desired description of the restriction of the Frobenius ϕ m : B ( m ) ! B ( pm ) tC p . (cid:3) Compatibility with the pointed cyclic bar construction.
We also considermonoid objects in the ∞ -category of based spaces S ∗ . To these, one may apply apointed version of the cyclic bar construction, as it appears in [Spe19, Section 3.1].In the situations that arise in this paper, these pointed monoids will have an additionalstructure, that of a wedge decomposition A ≃ _ A ( n )giving them the structure of a graded object in pointed spaces. Applying the cyclicbar construction to A results in a graded cyclic object in pointed spaces. Let b • denotethe cyclic bar construction of A and assume that b • is proper. In this situation, the“weight” m part of b • at cyclic degree s is given by b s ( m ) = _ n + n + ··· + n s +1 = m A ( n ) ∧ · · · A ( n s +1 ) . We have the following compatibility result between the pointed cyclic bar constructionof graded monoids in pointed topological spaces and the THH of graded monoids inthe ∞ -category of spectra. Proposition 4.12.
Following the notation above, we have the following equivalencein the ∞ -category of T -spectra. Σ ∞ | b • ( m ) | ≃ ] THH(Σ ∞ A )( m ) Here, Σ ∞ − denotes the infinite suspension functor from pointed spaces to the ∞ -category of spectra and |−| denotes the geometric realization functor in pointed topo-logical spaces.Proof. Let A be an E -monoid in the ∞ -category of graded pointed spaces, givingrise via infinite suspension to the graded E -algebra Σ ∞ A . Since Σ ∞ ( − ) is compat-ible with the relevant symmetric monoidal structures and preserves coproducts, oneobtains a decompositionTHH(Σ ∞ A ) ≃ Σ ∞ ( ∨ m | b • ( m ) | ) ≃ ∨ m Σ ∞ | b • ( m ) | ≃ ∨ m | Σ ∞ b • ( m ) | , where the first equivalence follows for example from [HM97b, Theorem 7.1]. Byinspection, the cyclic objects Σ ∞ b • ( m ) are precisely the weight m cyclic summandrealizing to ] THH(Σ ∞ A )( m ) in the decomposition of corollary 4.8 since infinite sus-pension at the level of cyclic objects preserves internal weight m . (cid:3) Weight decomposition on topological Hochschild homology
Let X denote THH( F p ) for the rest of this section. We describe in more detail theweight decomposition on THH( X ) and describe its equivariant properties. This isanalogous to the weight splitting used by Hesselholt and Madsen for the Hochschildhomology of quotients of discrete polynomial rings on a single generator [HM97a].5.1. A T -equivariant splitting of THH ( THH ( F p )) . We start by describing theweight decomposition for THH( X ). Due to Theorem 1.3, there is an equivalence of E -algebras. X ≃ k ∧ S [Ω S ] . This, together with (3.2) shows that we have the following splitting of E -algebras in T -spectra. THH( X ) ≃ THH( k ) ∧ THH( S [Ω S ])Indeed, we construct a weight decomposition on THH( S [Ω S ]). For this, note thatthere is an equivalence of E -algebras (see for example [CMT78])(5.1) S [Ω S ] ≃ _ ≤ m Σ m S , making S [Ω S ] into a graded spectrum where the degree m part is given by Σ m S .Here, Σ m S denotes (Σ S ) ∧ m for m ≥ S for m = 0 and Σ S denotes the doublesuspension of S . With these identifications, the product on the right hand side isgiven by the concatenation of the smash factors.One may also obtain this graded spectrum from the monoid in graded pointedspaces given by _ ≤ m Σ m S LGEBRAIC K -THEORY OF THH( F p ) 15 via applying the infinite suspension functor. Here, Σ S denotes the 2-sphere and therest of the cofactors are given by smash powers as before.By Proposition 4.4, THH( S [Ω S ]) is canonically a graded object in T -spectra. Weanalyze this splitting more closely. Let B = THH( S [Ω S ]) and B ( m ) = ] THH( S [Ω S ])( m ) . Recall that the standard complex realizing B is given by B s = S [Ω S ] ∧ s +1 at simplicial degree s . Therefore, we have B s ∼ = _ ≤ m _ ≤ m · · · _ ≤ m s +1 Σ m S ∧ Σ m S ∧ · · · ∧ Σ m s +1 S . With the previous identifications, B ( m ) is given by the realization of the subcomplex B • ( m ) consisting of the cofactors with m + m + · · · + m s +1 = m. Therefore, we obtain the following splitting of THH( S [Ω S ]) as a spectrum with T -action. THH( S [Ω S ]) ≃ _ ≤ m B ( m )This gives the following splitting of spectra with T -action.(5.2) THH( X ) ≃ THH( F p ) ∧ _ ≤ m B ( m ) Remark . We remark that splittings for THH( S [ΩΣ U ]) for U a connected spacehave been previously obtained. For example, by Cohen in [Coh87], there is a decom-position Σ ∞ L Σ U ≃ _ n ≥ Σ ∞ ( U n ∧ C n T + )It is not clear to the authors whether or not this splitting is T -equivariant and thereforewhether it coincides with the decomposition described above.5.2. Homology of the summands.
For our calculations, we need to understandthe homology of B and its summands. Note that we have H Z ∧ B ≃ H Z ∧ THH( S [Ω S ]) ≃ HH( H Z ∧ S [Ω S ])Due to equation (5.1), this is the Hochschild homology of the free DGA with a singlegenerator in degree 2. The E page of the standard spectral sequence calculating theseHochschild homology groups is given by E ∼ = tor Z [ x ] e ∗ , ∗ ( Z [ x ] , Z [ x ])where Z [ x ] e is the enveloping algebra Z [ x ] ⊗ Z [ x ] and the subscript denotes the in-ternal degree [EKMM97, IX.2.8]. The standard argument is to use the automorphismof Z [ x ] e given by x ⊗ ! x ⊗ − ⊗ x and 1 ⊗ x ! ⊗ x . Precomposing with this automorphism makes the action of the first factor of Z [ x ] e trivial and the action of the second factor the canonical non-trivial action on Z [ x ].We obtain E ∼ = Z [ x ] ⊗ tor Z [ x ] ( Z , Z ) ∼ = Z [ x ] ⊗ Λ( σx )where deg( σx ) = (1 , H Z ∗ B ∼ = Z [ x ] ⊗ Λ( y )where the | y | = 3. Remark . Alternatively, one may deduce the above by using the well-known iden-tification THH( S [Ω S ]) ≃ Σ ∞ + L X togethere with the fact that S is an H -space (it is in fact a Lie group) which givesa splitting of the canonical loop space fiber sequenceΩ S ! L S ! S Hence, L S ≃ Ω S × S and so, there is an equivalence H Z [ L S ] ≃ H Z [Ω S ] ⊗ H Z [ S ]Upon taking homotopy, groups, and using the fact that the integral homology of Ω S is polynomial in degree 2, one obtains the above isomorphism.By inspection of the simplicial resolution defining this spectral sequence, H Z ∗ B ( m )is given by the contribution that comes from the internal degree 2 m elements in thespectral sequence above. We obtain the following. Lemma 5.5.
The H Z homology of B ( m ) is given by Z concentrated in degrees m and m + 1 . We now turn our attention to Hk ∗ B ( m ) and its multiplicative structure. This willin turn, be used to recover the multiplicative structure on the topological Hochschildhomology groups of THH( k ), see (1.5).Before we proceed, we prove the following compatibility statement between THH( A )and HH k ( A ∧ Hk ). Surely this is well known, but we were unable to find a reference. Proposition 5.6.
Let A be an E n -ring spectrum. There is a natural equivalence of E n − -algebras: THH( A ) ∧ Hk ≃ HH k ( A ∧ Hk ) in Mod k . We shall need the following lemma:
Lemma 5.7.
Let F be a symmetric monoidal left adjoint functor Sp ! Mod k . Thereis an induced natural equivalence of symmetric monoidal functors colim ∆ op ◦ F ! F ◦ colim ∆ op : Sp ∆ op ! Mod k LGEBRAIC K -THEORY OF THH( F p ) 17 Proof.
The map realizing this equivalence is the standard natural transformation(5.8) colim ∆ op F ◦ ( X • ) ! F ◦ colim ∆ op ( X • ) , of functors Sp ∆ op ! Sp induced by the universal property of the colimit. Since F commutes with colimits, this is an equivalence. We now show that this is a multiplica-tive natural transformation. To do this we interpret it as a map in the ∞ -category offunctors Fun(Sp ∆ op , Sp). Recall that the category of lax symmetric monoidal functorsis identified with the categoryCAlg(Fun(Sp ∆ op , Mod k ))of commutative algebra objects with respect to the Day convolution monoidal struc-ture on Fun(Sp ∆ op , Mod k ). Hence the problem reduces to displaying (5.8) as a mapof commutative algebra objects in this category. For X • ∈ Sp ∆ op , this is a colimit ofmaps F ( ev n ( X • )) ! F (colim ∆ op X • ) , each of which is a map of commutative algebra objects in Fun(Sp ∆ op , Mod k ). Sincesifted colimits of (commutative) algebra objects are detected in the underlying ∞ -category (see [Lur16, Corollary 3.2.3.2]) and since F is itself symmetric monoidal,we may identify (5.8) as the map induced by the universal property of the colimitin CAlg(Fun(Sp ∆ op , Mod k )) . Hence, this is a map of commutative algebra objects.Note that F ◦ colim ∆ op is symmetric monoidal because the smash product in spectracommutes with geometric realizations of simplicial objects. We may now concludethat the map (5.8) is a morphism of symmetric monoidal functors, and in fact anequivalence. (cid:3) Furthermore, both THH( − ) and HH k ( − ), being symmetric monoidal, send E n -algebras to E n − -algebras. Hence we obtain the following: Proof of Proposition 5.6.
Let B Rcyc ( − ) • denote the simplicial object whose realizationis THH R ( − ) for a commutative ring spectrum R which is the sphere spectrum whenit is omitted. If A is an E n -algebra in spectra, then B Rcyc ( A ∧ Hk ) • will be an E n − -algebra in Sp ∆ op . To see this, recall that an E n algebra is the same thing as an E -algebra in the symmetric monoidal ∞ -category Alg E n − (Sp). Hence the cyclic barconstruction B Rcyc ( A ) • may be viewed as a simplicial object in E n − algebras; giventhe pointwise symmetric monoidal structure on Sp ∆ op , this is the same thing as an E n − algebra in this category. We now apply the previous lemma with F = − ∧ Hk to conclude that ρ : | B Hkcyc ( A ∧ Hk ) • | ≃ −! | B cyc ( A ) • | ∧ Hk is a natural equivalence of E n − -algebras. (cid:3) Applying the above discussion to our current setting, we have an equivalence of E -algebras. Hk ∧ THH( S [Ω S ]) ≃ HH k ( Hk ∧ S [Ω S ])Arguing as before, we obtain that the second page of the spectral sequence calculatingthese Hochschild homology groups is given by E ∼ = k [ x ] ⊗ Λ( σx ) . The differentials are again trivial by degree reasons. We obtain that(5.9) Hk ∗ B ∼ = HH k ∗ ( Hk ∧ S [Ω S ]) ∼ = k [ x ] ⊗ Λ( y ) . We claim that this is a ring isomorphism. By Theorem 2.1, we know that Hk ∧ S [Ω S ]is equivalent to the E Hk -algebra corresponding to the formal E ∞ k -DGA withhomology k [ x ]. Therefore we can consider Hk ∧ S [Ω S ] as an E ∞ Hk -algebra.This equips the spectral sequence above with a multiplicative structure that givesthe multiplicative structure on the target. The only multiplicative extension problemis resolved once we show that y = 0. This follows by the fact that the target isgraded commutative. We obtain the following lemma. Lemma 5.10.
The Hk homology of B ( m ) is given by k concentrated in degrees m and m + 1 . Frobenius morphism on the weight decomposition.
In this section, weprove key lemmas regarding the Frobenius map on B . By the formalism of Section4, the Frobenius structure map ϕ : THH( B ) ! THH( B ) tC p decomposes, for every m , into maps ϕ m : B ( m ) ! B ( pm ) tC p . Lemma 5.11.
The maps ϕ m above are p -completions for every ≤ m . Furthermore, B ( m ) tC p is contractible as a spectrum whenever p ∤ m .Proof. First, we need to establish the behaviour of the weight decomposition withrespect to the Tate construction. For this, we claim that the coproduct defining theweight splitting is indeed a product, i.e. the canonical map B ≃ _ ≤ m B ( m ) ! Y ≤ m B ( m )is an equivalence. Each B ( m ) is connective by construction. This, together withLemma 5.5 shows that each B ( m ) is indeed 2 m − B tCp = Y ≤ m B ( m ) tCp . By Lemma 4.9, the Frobenius ϕ : B ! B tCp is given by Y ≤ m ϕ m : Y ≤ m B ( m ) ! Y ≤ m B ( m ) tCp . First, note that B ( m ) tC p is p -complete due to [NS18, I.2.9]. Therefore, it is sufficientto show that each map ϕ m is a S /p -local equivalence where S /p denotes the Moorespectrum of Z /p . Since B = Σ ∞ + L S , it follows from Theorem IV.3.7 of [NS18] that LGEBRAIC K -THEORY OF THH( F p ) 19 the Frobenius ϕ on B is a p -completion. Therefore, it is a homtopy isomorphismafter applying the functor S /p ∧ − . Since S /p is dualizable, this functor preserves allhomotopy limits as well as all homotopy colimits. We deduce that the map Y ≤ m π ∗ ( S /p ∧ ϕ m ) : Y ≤ m π ∗ ( S /p ∧ B ( m )) ! Y ≤ m π ∗ ( S /p ∧ B ( m ) tCp )is an isomorphism. Therefore, it is an isomorphism at each component and we havethat B ( m ) tC p is S /p -locally contractible whenever p ∤ m . Since B ( m ) tC p is already S /p -local, i.e. p-complete, we deduce that B ( m ) tC p is contractible as a spectrumwhenever p ∤ m . (cid:3) Lemma 5.12.
As a spectrum with C p -action, B ( m ) is the suspension spectrum of afinite C p -CW space.Proof. There is an alternative construction of B ( m ) as the infinite suspension of a T -pointed space. Let U denote the free monoid on S in pointed spaces. This is givenby U = _ ≤ l S l where S denotes the standard 2-sphere with a base point and S l denotes the l -fold smash power of S with the zero smash power given by S . As before, themultiplication is given by the concatenation of the smash factors.Let b • denote the cyclic bar construction for U . In particular, b • is a cyclic pointedspace given by U ∧ ( s +1) in cyclic degree s . As before, we can decompose b • into piecesconsisting of m -fold products of S and obtain a splitting of cyclic pointed spaces. b • ∼ = _ ≤ m b • ( m )Due to Proposition 4.12, Lemma 8.5 and Lemma 8.3, there is an equivalence of T -spectra B ( m ) ≃ Σ ∞ | b • ( m ) | . Therefore, it is sufficient to show that | b • ( m ) | is equivalent as a pointed space with C p -action to a finite C p -CW complex.To understand the C p -action on | b • ( m ) | , one considers the p -subdivision of b • ( m )as a simplicial pointed space denoted by sd p b ( m ). This is described for cyclic setsand cyclic spaces in [BHM93, Section 1]. Let ∆ denote the simplex category. Thereis a functor ∆ ! ∆given by [ m − ! [ mp −
1] on objects and carries a morphism f to f ` · · · ` f . The p -subdivision of b • ( m ) is given by precomposing with this functor. For instance, wehave sd p b ( m ) s − = b ps − ( m ).There is a canonical C p -action on sd p b ( m ) s − given by cyclic permutations of blocksof s -fold smash factors. This action commutes with the face and degeneracy mapsand gives a C p -action on the simplicial space sk p b ( m ). Furthermore, the realizationof sk p b ( m ) agrees with the realization of b • ( m ) in a way that is compatible with the C p -action [BHM93, 1.11]. Therefore, it is sufficient to show that | sd p b ( m ) | is a finitepointed C p -CW complex. To achieve this, we consider a bisimplicial set b ′• ( m ). This is obtained in the sameway we obtain b • ( m ) from U except that we use the standard pointed simplicialset S instead of the pointed space S . For instance, if we apply geometric realiza-tion to b ′• ( m ) at cyclic degree s , we obtain b s ( m ); this follows by the fact that thegeometric realization functor preserves smash products. Similarly, we have a bisim-plicial set sd p b ′ ( m ) with a C p -action where the geometric realization of sd p b ′ ( m ) s is sd p b ( m ) s . Furthermore, these equivalences preserve the C p -action and therefore thereis an equivalence of spaces with C p -action | sd p b ( m ) | ≃ | sd p b ′ ( m ) | . The geometric realization on the right hand side is obtained by first applying geo-metric realization at each cyclic degree and then applying the geometric realizationfunctor from simplicial spaces to spaces. Since this is naturally equivalent to applyingthe realization functor from bisimplicial pointed sets to simplicial pointed sets throughthe cyclic direction and then applying the geometric realization functor, we can in-stead apply this procedure and obtain the same C p -space. Therefore, it is sufficientto show that after applying the realization functor to the bisimplicial set sd p b ′ ( m ) inthe cyclic direction, the C p -simplicial set we obtain has finitely many non-degeneratesimplices.By inspection, one sees that every simplex in the pointed simplicial set sd p b ′ ( m ) k for k > m is in the image of some degeneracy map from sd p b ′ ( m ) m . Thereforein the realization, one only needs to consider finitely many cyclic degrees. Indeed,the realization in this case is a coequalizer of a coproduct of finitely many pointedsimplicial sets with finitely many non-degenerate simplices and therefore results in apointed simplicial set with finitely many non-degenerate simplices. (cid:3) Remark . The above argument takes place using point-set level arguments. Werefer the reader to the appendix for the compatbility between these constructions andtheir higher categorical counterparts.Proceeding forward, we obtain the following analogue of Lemma 8 in [Spe20].
Lemma 5.14.
The map
THH( k ) tCp ∧ B ( m ) ! (THH( k ) ∧ B ( pm )) tCp is an equivalence.Proof. This map factors asTHH( k ) tCp ∧ B ( m ) id ∧ ϕ m −−−−! THH( k ) tC p ∧ B ( pm ) tC p ! (THH( k ) ∧ B ( pm )) tCp where the second map is the lax monoidal structure map of the functor ( − ) tC p . Thismap is an equivalence due to Lemma 5.12 and [Spe19, Lemma 15]. For the first map,note that S /p ∧ ϕ m is an equivalence due to Lemma 5.11. Furthermore, we haveTHH( k ) tC p = S /p ∧ THH( k ) t T , see [NS18, proof of IV.4.13], and therefore the firstmap is also an equivalence. (cid:3) The Frobenius φ m : THH( k ) ∧ B ( m ) ! (THH( k ) ∧ B ( pm )) tCp LGEBRAIC K -THEORY OF THH( F p ) 21 factors as(5.15) THH( k ) ∧ B ( m ) ! THH( k ) tC p ∧ B ( m ) ! (THH( k ) ∧ B ( pm )) tCp where the first map is induced by the Frobenius on THH( k ) and the second map isthe equivalence given in Lemma 5.14. Corollary 5.16.
The Frobenius φ m described above induces an isomorphism in ho-motopy groups for sufficiently large degrees. In particular, π ∗ φ m is an isomorphismin degree m + 1 and above. Furthermore, π ∗ φ h T m is also an isomorphism in degree m + 1 and above.Proof. It is sufficient to show that the first map of the composite in (5.15) is anequivalence in sufficiently large degrees.Let E denote the fiber of the Frobenius map of THH( k ). We need to show that E ∧ B ( m ) is bounded from above.The Frobenius on THH( k ) is an equivalence on connective covers due to Proposition6.2 in [BMS19]. Since THH( k ) is connective, this shows that τ ≤− E ≃ E .Furthermore, Lemma 5.5 implies that B ( m ) is a finite spectrum, see [Sch, II.7.4].This is also implied by Lemma 5.12. This shows that E ∧ B ( m ) is built from E by taking suspensions, completing triangles and taking retracts finitely many times.This implies that E ∧ B ( m ) is also bounded from above.To be precise, one can show using Lemma 5.5 that there is a fiber sequenceΣ m S ! B ( m ) ! Σ m +1 S which in turn gives a fiber sequenceΣ m E ! E ∧ B ( m ) ! Σ m +1 E. This shows that E ∧ B ( m ) is trivial in homotopy groups in degree 2 m and above.Therefore, π ∗ φ m is an isomorphism in degree 2 m + 1 and above.To show that π ∗ φ h T m is also an isomorphism in degree 2 m +1 and above, it is sufficientto show that ( E ∧ B ( m )) h T is bounded above degree 2 m . This follows by the factthat the homotopy fixed points spectral sequence described in Section 6 preservescoconnectivity. (cid:3) Calculation of TC ∗ (THH( F p ))In this section, we calculate TC ∗ (THH( k )). This provides the relative K -theorygroups for the map THH( k ) ! k. Our methods closely follow those of Speirs in [Spe20, Sections 5 and 6].First, we show that our weight splittings also result in splittings at the level ofnegative cyclic homology and periodic cyclic homology.Let THH( X )( m ), TC − ( X )( m ) and TP( X )( m )denote THH( k ) ∧ B ( m ), (THH( k ) ∧ B ( m )) h T and (THH( k ) ∧ B ( m )) t T respectively. Proposition 6.1.
There are equivalences TC − ( X ) ≃ Y ≤ m TC − ( X )( m ) and TP( X ) ≃ Y ≤ m TP( X )( m ) . Proof.
This is similar to the argument at the beginning of the proof of Lemma 5.11.Due to connectivity reasons, the coproduct in (5.2) is at the same time a product.Therefore it commutes with taking fixed points which is a homotopy limit. This givesthe first splitting.The coproduct in (5.2) commutes with taking homotopy orbits because homotopyorbits is a homotopy colimit. Furthermore, the homotopy orbits functor preservesconnectivity and therefore the coproduct splitting one obtains after taking homotopyorbits is at the same time a product. It is also clear that the canonical map fromthe homotopy orbits to homotopy fixed points respects this splitting. Therefore, weobtain the product splitting for the Tate construction too. (cid:3)
To calculate TC − ( X )( m ) and TP( X )( m ), we use the homotopy fixed point andthe Tate spectral sequences.For a T -spectrum E , the second page of the homotopy fixed point spectral sequenceis given by E = Z [ t ] ⊗ π ∗ E = ⇒ π ∗ E h T where deg( t ) = ( − , E = Z [ t, t − ] ⊗ π ∗ E = ⇒ π ∗ E t T where deg( t ) = ( − ,
0) as before. This is a conditionally convergent first half planespectral sequence. Furthermore, this is a multiplicative spectral sequence when E isa T -ring spectrum [HM03, 4.3.5].There is a subtlety regarding the way we incorporate the multiplicative structure ofthe Tate spectral sequence into our calculations. We use the Tate spectral sequenceto calculate the homotopy groups of( T HH ( k ) ∧ B ( m )) t T . However, B ( m ) is not a ring spectrum for m >
0; it does not contain a unit and it isnot closed under the multiplication in B .Since THH( − ) is a symmetric monoidal functor from E -ring spectra to cyclotomicspectra [NS18, Section IV.2], we obtain a splitting of T -ring spectra by using Theorem1.3. THH( X ) ≃ THH( k ) ∧ THH( S [Ω S ]) := THH( k ) ∧ B. Therefore, the Tate spectral sequence for the right hand side of this equality ismultiplicative. Indeed, this is why we needed Theorem 1.3 for our calculations.The weight splitting of THH( k ) ∧ B also splits the Tate spectral sequence forTHH( k ) ∧ B through the Tate spectral sequences calculating THH( k ) ∧ B ( m ) for all m .This can be seen by noting that each THH( k ) ∧ B ( m ) is actually a retract of THH( k ) ∧ B as a T -spectrum. Using this, we incorporate the multiplicative structure on theTate spectral sequence for THH( k ) ∧ B into the Tate spectral sequence calculatingTHH( k ) ∧ B ( m ). LGEBRAIC K -THEORY OF THH( F p ) 23 Due to Lemma 5.10, the second page of the Tate spectral sequence for THH( k ) ∧ B ( m ) is given by E = k [ t, t − , x ] { y m , z m } where deg( y m ) = (0 , m ) and deg( z m ) = (0 , m + 1) and deg( x ) = (0 , Lemma 6.2.
In the Tate spectral sequence for
THH( k ) ∧ B ( m ) , z m is an infinitecycle. In the Tate spectral sequence for THH( k ) ∧ B , x , t and t − are infinite cycles.Proof. As described in [NS18] after Corollary IV.4.10, there is a map of T -equivariantring spectra H Z trivp ! THH( F p ) where H Z trivp is given the trivial T -structure. This,together with the ring map F p ! k gives a map of T -equivariant ring spectra H Z trivp ! THH( k ) . The Tate spectral sequence calculating H Z trivp ∧ B ( m ) has E = Z p [ t, t − ] { y m , z m } . In this spectral sequence, z m is an infinite cycle due to degree reasons. Furthermore,this spectral sequence maps into the Tate spectral sequence for THH( k ) ∧ B ( m ) in away that carries z m to z m on the E page. This shows that z m is an infinite cycle inthe Tate spectral sequence calculating THH( k ) ∧ B ( m ).Since B is a cyclotomic ring spectrum, there is a map of cyclotomic spectra S triv ! B given by the unit of B . This map induces a map of spectral sequences between theTate spectral sequence for THH( k ) and the Tate spectral sequence for THH( k ) ∧ B that carries x to x , t to t and t − to t − . In the first spectral sequence, everythingis in even degrees and therefore all the differentials are trivial. This gives the desiredresult. (cid:3) Proposition 6.3.
Let m = p v m ′ for p ∤ m ′ . There are isomorphisms π r +1 (THH( k ) ∧ B ( m )) t T = W v ( k ) and π r +1 (THH( k ) ∧ B ( m )) h T = ( W v +1 ( k ) if m ≤ rW v ( k ) if r < m for all z . The even homotopy groups in both cases are trivial. Furthermore, thecanonical map can is an isomorphism for r < m .Proof. This proof is a direct adaptation of [Spe20, Proposition 12]. We start with thecase p ∤ m . In this case, THH( k ) tC p ∧ B ( m ) tC p ≃ − ) tC p is anequivalence in this situation due to Lemma 5.12 and [Spe19, Lemma 15]. We concludethat (THH( k ) ∧ B ( m )) tC p ≃ . Since THH( k ) ∧ B ( m ) is p -complete (for every m ), (THH( k ) ∧ B ( m )) t T is also p -complete [BMS19, Section 2.3]. Therefore we have(THH( k ) ∧ B ( m )) t T ≃ ((THH( k ) ∧ B ( m )) tC p ) h T ≃ . due to Lemma of II.4.2 of [NS18].Recall that the second page of the Tate spectral sequence for (THH( k ) ∧ B ( m )) t T is given by E = k [ t, t − , x ] { y m , z m } . Considering Lemma 6.2 together with the multiplicative structure, one sees that thefirst non-trivial differential on y m determines the rest of the non-trivial differentialson this spectral sequence. In this case, we have E ∞ = 0 and therefore up to a unit,we have d y m = tz m which is the only differential that guarantees this. See the following picture of the E -page.... ...2 m + 3 t x z m tx z m x z m t − x z m m + 2 t x y m tx y m x y m t − x y m m + 1 · · · t z m tz m z m t − z m · · · m t y m ty m y m t − y m ... 0 0 0 0 · · · − · · · We showed that π k ((THH( k ) ∧ B ( m )) t T ) vanishes for each k . We now truncatethe Tate spectral sequence, removing the first quadrant to obtain the homotopy fixedpoint spectral sequence. In this case, the classes x n z m will no longer be hit by anydifferentials and so survive to the E ∞ page; hence we conclude that E ∞ = k [ x ] { z m } ,where z m is of degree 2 m + 1.Now suppose that m = p v m ′ , where ( p, m ′ ) = 1. We use an induction argument on v . Suppose then that the claim is true for all integers less than or equal to v . Recall,from Corollary 5.16 that the Frobenius φ h T m : (THH( k ) ∧ B ( p v m )) h T ! (THH( h ) ∧ B ( p v +1 m )) t T will be a π ∗ isomorphism in sufficiently high degrees. Using this fact, we may concludethat(6.4) π r +1 (THH( k ) ∧ B ( p v +1 m ′ )) t T ∼ = W v +1 ( k )for sufficiently large r and that the even homotopy groups are trivial in sufficientlylarge degrees. Since ( − ) t T is lax monoidal and THH( k ) is a T -equivariant H Z -algebra[NS18, IV.4.10], we deduce that (THH( k ) ∧ B ( p v +1 m ′ )) t T is an H Z t T -module. It followsfrom the Tate spectral sequence that π ∗ Z t T = Z [ t, t − ] LGEBRAIC K -THEORY OF THH( F p ) 25 with | t | = 2. From this, we deduce that (THH( k ) ∧ B ( p v +1 m ′ )) t T is periodic inhomotopy and this shows that the equality in 6.4 holds for every r and that the evenhomotopy groups are trivial.It now remains to compute π ∗ (THH( k ) ∧ B ( p v +1 )) h T First, we observe that in the Tate spectral sequence, d v +2 ( y p v +1 m ′ ) = t ( x t ) v z p v +1 m ′ at least up to multiplication by a unit. Since this spectral sequence is multiplicative,this is the only differential that guarantees (6.4). There are no other non-trivialdifferentials.This completes the calculation of π r +1 (THH( k ) ∧ B ( m )) h T , up to extension prob-lems. Note that we only need to do this for the homotopy fixed point calculation. Sowe reduce the homotopy fixed point case as in [Spe19, Proposition 12]. This will followfrom us showing that π r (THH( k ) ∧ B ( m )) h T is cyclic (and so completely determinedby its length, which we may extract from the E ∞ page) as a π THH( k ) h T ∼ = W ( k )-module. The argument proceeds by exhibiting a T -equivariant mapTHH( k )[2 m + 1] ! THH( k ) ∧ B ( m )which induces a map of homotopy spectral sequences. Choose an element α ∈ π r +1 (THH( k ) ∧ B ( m )) h T , and let α = t a x a z m be its image in the E ∞ page. This lies in the image of t a x a in the E ∞ page of the spectral sequence computing THH( k ) h T where the extension problemhas been solved; thus p a lifts t a x a up to a unit. Since the mapTHH( k )[2 m + 1] ! THH( k ) ∧ B ( m )corresponding to z m is W ( k ) linear, one concludes that α = p a z m .Hence we have shown that π r +1 (THH( k ) ∧ B ( m )) h T = ( W v +1 ( k ) if m ≤ rW v ( k ) if r < m It only remains to verify that can ∗ : π r +1 (THH( k ) ∧ B ( m )) h T ! π r +1 (THH( k ) ∧ B ( m )) t T is an isomorphism for r < m . This follows from the fact that can induces a morphismof spectral sequences from the homotopy fixed point to the tate spectral sequence; inthese degrees, the induced maps on the E ∞ page are equivalences as there will be nocontribution there from the 1st quadrant terms. (cid:3) The relative topological cyclic homology TC(
X, k ) is the fiber of the mapTC − ( X, k ) φ − can −−−! TP(
X, k ) . With the splitting in Proposition 6.1, this is the fiber of the following map. Y ≤ m ′ p ∤ m ′ Y ≤ v TC − ( X )( p v m ′ ) φ − can −−−! Y ≤ m ′ p ∤ m ′ Y ≤ v TP( X )( p v m ′ )Recall that φ increases the weight degree by a factor of p , see Lemma 4.9.We claim that, φ − can is surjective in homotopy groups. To see this, fix an m ′ with p ∤ m ′ and consider the degree 2 r + 1 homotopy groups of the factors correspondingto this m ′ . If r < m ′ , then can is an isomorphism due to Proposition 6.3. Thisshows that φ − can is surjective in homotopy since φ increases the weight degree bya factor of p . If m ′ ≤ r , then the surjectivity for p v m ′ ≤ r follows by the fact that φ is an isomorphism in these cases and that T P ( m ′ ) ≃
0, see Corollary 5.16. For v with p v m ′ > r , surjectivity is observed by noting that can is an isomorphism in thesecases.All the non-trivial homotopy groups are in odd degrees due to Proposition 6.3.This, together with the surjectivity of π ∗ ( φ − can ) show that for every positive m ′ with p ∤ m ′ , there is a short exact sequence0 ! TC ∗ ( X )( m ′ ) ! Y ≤ v TC −∗ ( X )( p v m ′ ) ! Y ≤ v TP ∗ ( X )( p v m ′ ) ! ∗ ( X, k ) = Y ≤ m ′ p ∤ m ′ TC ∗ ( X )( m ′ )with this notation.Restricting to degree 2 r + 1, we obtain the following diagram where the horizontallines are short exact sequences.0 Q s ≤ v W v ( k ) Q ≤ v TC − r +1 ( X )( p v m ′ ) Q ≤ v
1. Rest of this map is defined via a downwardinduction on v . Given a map to W v +1 , the map to W v is given by W s ( k ) ! W v +1 ( k ) can −−! W v ( k ) φ − −−! W v ( k ) LGEBRAIC K -THEORY OF THH( F p ) 27 where the first map is the given map. In conclusion, the contribution to TC r +1 ( X )from m ′ is given by W s ( k ).We deduce thatTC r +1 ( X, k ) = Y ≤ m ′ ≤ rp ∤ m ′ W s ( k ) and TC r ( X, k ) = 0for every integer r . It follows from Proposition Example 1.11 of [Hes15] the the righthand side is indeed the big Witt vectors W r ( k ) of length r . This finishes the proof ofTheorem 1.2. 7. Algebraic K -theory of TC( F p )We start by proving the following classification result. Proposition 7.1.
Let R be a commutative ring. There is a unique R -DGA withhomology ring Λ R ( x − ) , i.e. the exterior algebra over R with a single generator indegree − . The first author and P´eroux prove this result when R is a field [BP]. We describehow their proof generalize to commutative rings.Let X be the HR -algebra corresponding to the formal R -DGA with homologyring Λ R ( x − ) and let E be the HR -algebra corresponding to another R -DGA withhomology ring Λ R ( x − ). We need to show that E is weakly equivalent to X as an HR -algebra.For this, we use Hopkins–Miller obstruction theory which provides obstructionsto lifting a map of monoids in the homotopy category of HR -modules to a map of HR -algebras [Rez98].Since homotopy groups of E and X are free as graded R -modules, maps between E and X in the homotopy category of HR -modules is simply given by maps ofhomotopy groups that are maps of graded R -modules, see [EKMM97, IV.4.1]. Againby [EKMM97, IV.4.1], smash powers of X and E have homotopy given by tensorproducts in graded R -modules. From these, we conclude that X and E are isomorphicas monoids in the homotopy category of HR -modules.Again because π ∗ X and π ∗ E are free as R -modules, we can apply Hopkins-Millerobstruction theory to lift this isomorphism to a weak equivalence of HR -algebras.The obstructions to this lift lies in the Andr´e–Quillen cohomology groupsDer s +1 (Λ R ( x − ) , Ω s Λ R ( x − )) for s ≥ . It follows by [Qui70, 3.6] that these cohomology groups are equivalent to the Hochschildcohomology groupsExt s +2Λ R ( x − ) ⊗ Λ R ( x − ) op (Λ R ( x − ) , Ω s Λ R ( x − )) for s ≥ . There is an automorphism of Λ R ( x − ) ⊗ Λ R ( x − ) op given by x − ⊗ ! x − ⊗ ⊗ x − ! x − ⊗ − ⊗ x − . Precomposing with this automorphism makes the action of the second factor ofΛ R ( x − ) ⊗ Λ R ( x − ) op on Λ R ( x − ) trivial and the action of the first factor the canonical nontrivial one. Via base change with respect to the inclusion of the second factor ofΛ R ( x − ) ⊗ Λ R ( x − ) op , one sees that these obstructions lie in the groupsExt s +2Λ R ( x − ) ( R, Ω s ( R ⊕ Ω R )) for s ≥ R as a Λ R ( x − )-module it is clear that these groupsare trivial due to degree reasons.This shows in particular that TC( Z p ) and C ∗ ( S , Z p ) are both formal as E H Z p -algebras and therefore quasi-isomorphic to each other. It follows from the theoremof the heart in [AGH19, 4.8] that the algebraic K -theory of C ∗ ( S , Z p ) and thereforeTC( Z p ) is K( Z p ).8. Appendix A. Rectification of cyclic and graded objects
In this short appendix, we prove rectification results regarding cyclic objects invarious ∞ -categories which we use in the main part of the text.We first recall the following compatibility between realization of a cyclic objectusing point-set models and in the ∞ -categorical setting. Proposition 8.1 (Nikolaus-Scholze) . Let
T op denote the category of compactly gen-erated weak Hausdorff spaces. The diagram (8.2) N (Fun(Λ op , T op )) prop N T − T op
Fun(Λ op , S ) S B T hcolimcolim commutes.Proof. This is Corollary B.14 in [NS18]. (cid:3)
We shall need follow pointed analogue of this result.
Lemma 8.3.
Let
T op ∗ denote the category of pointed compactly generated weak Haus-dorff spaces. The diagram (8.4) N (Fun(Λ op , T op ∗ )) prop N T − T op ∗ Fun(Λ op , S ∗ ) S B T ∗ hcolimcolim commutes.Proof. This boils down to the analogous statement for unpointed spaces, togetherwith the fact that the geometric realization of a simplicial object in pointed spacesis canonically pointed, and that the geometric realization of a pointed (para)cyclicspace is the geometric realization of the underlying simplicial pointed space. (cid:3)
As we occasionally work with point-set models in the setting of graded topologicalspaces, we need to show that various point-set constructions are compatible withconstructions at the level of ∞ -categories. For this, let T op Z ∗ := Fun( Z , T op ∗ ) LGEBRAIC K -THEORY OF THH( F p ) 29 denote the category of functors F : Z ! T op ∗ . With the Day convolution product andthe projective model structure, this becomes a symmetric monoidal model category.In our constructions, we sometimes start with a monoid in this category and use thefollowing compatibility result. Lemma 8.5.
Let X str be a cofibrant monoid in T op Z ∗ and let b str • denote the corre-sponding Λ -object in T op Z ∗ given by the cyclic bar construction on X str . We denoteby X the image of X str under the canonical map N ((Alg( T op Z ∗ ) c )[ W − ] ! Alg( N (( T op Z ∗ ) c )[ W − ]) . Furthermore, the image of b str • under the map N (( T op Z × Λ op ∗ ) c )[ W − ] ! N (( T op ∗ ) c )[ W − ] Z × Λ op ≃ S Z × Λ op ∗ is denoted by b • . There is an equivalence b • ≃ b ( X ) where b ( X ) is the cyclic bar construction on X .Proof. This amounts to showing that the following diagram commutes(8.6) N (Alg( T op Z ∗ ))[ W − ] Alg( S Z ∗ ) N ( T op Z × Λ op ∗ )[ W − ] S Z × Λ op ∗≃≃ where the vertical arrows represent the strict version of the cyclic bar constructionand the ∞ -categorical version, respectively. The cyclic object b • as an object in thebottom right hand corner, obtained by applying the cyclic bar construction to X str ,is given by Λ op ! Ass ⊗ act X str −−! N ( T op Z ∗ ) ∧ act ∧ −! N ( T op Z ∗ );this follows since Alg( N ( T op Z ∗ )[ W − ]) ≃ N (Alg( T op Z ∗ )[ W − ] (see e.g. [Lur16, Theo-rem 4.1.8.4]; the category T op Z ∗ doesn’t satisfy these properties on the nose, but maybe replaced by a Quillen equivalent one which does).In the above, we are implicitly using [Lur09, Corollary 4.2.4.7] which implies thatthe diagram N (Λ op ) ! N ( T op Z ∗ )[ W − ] ≃ S Z ∗ may be straightened, in that it arisesfrom the nerve construction applied to a functor Λ ! T op Z ∗ , and it determines thisfunctor uniquely.In cyclic degree n , this cyclic object is given by the n + 1 pointed smash product X str ∧ ... ∧ X str with the standard Hochschild structure maps, encoded by the map Λ op ! Ass ⊗ act .This is none other than the cyclic objectΛ op ! Ass ⊗ act X −! S ∗∧ act ∧ −! S ∗ which is the bar construction b ( X ) on X . (cid:3) References [AGH19] Benjamin Antieau, David Gepner, and Jeremiah Heller, K -theoretic ob-structions to bounded t -structures , Invent. Math. (2019), no. 1, 241–300. MR 3935042[AK18] Gabe Angelini-Knoll, Detecting the β -family in iterated algebraic K-theory of finite fields , arXiv preprint arXiv:1810.10088 (2018).[Bay18] Haldun ¨Ozg¨ur Bayındır, Topological equivalences of E-infinity differen-tial graded algebras , Algebr. Geom. Topol. (2018), no. 2, 1115–1146.MR 3773750[Bay19a] , DGAs with polynomial algebra homology , arXiv preprintarXiv:1911.01089 (2019).[Bay19b] Haldun ¨Ozg¨ur Bayındır,
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