aa r X i v : . [ m a t h . K T ] M a r ON CURVES IN K-THEORY AND TR
JONAS MCCANDLESS
Abstract.
We prove that TR is corepresentable by the reduced topological Hochschild ho-mology of the flat affine line S [ t ] as a functor defined on the ∞ -category of cyclotomic spectrataking values in the ∞ -category of cyclotomic spectra with Frobenius lifts, refining a resultof Blumberg–Mandell. To that end, we define the notion of an integral topological Cartiermodule using Barwick’s formalism of Mackey functors on orbital ∞ -categories, extending thework of Antieau–Nikolaus in the p -typical case. As an application, we show that TR evaluatedon a connective E -ring admits a description in terms of the spectrum of curves on algebraicK-theory generalizing the work of Hesselholt and Betley–Schlichtkrull. Contents
1. Introduction 11.1. Statement of results 21.2. Methods 32. Cyclotomic spectra with Frobenius lifts and TR 62.1. The epicyclic category 62.2. Spaces with Frobenius lifts 112.3. Cyclotomic spectra with Frobenius lifts 152.4. Topological restriction homology 183. Comparison with genuine TR 203.1. Equivariant stable homotopy theory 203.2. Topological Cartier modules 243.3. Genuine cyclotomic spectra and genuine TR 304. Applications to curves on K-theory 344.1. Topological Hochschild homology of truncated polynomial algebras 344.2. Curves on K-theory 42References 451.
Introduction
Topological cyclic homology provides an effective tool for studying the invariant determinedby algebraic K-theory by means of the cyclotomic trace. The classical construction of topologicalcyclic homology using equivariant stable homotopy theory given by B¨okstedt–Hsiang–Madsen [19]is facilitated by another invariant called TR together with the additional structure of an operatorreferred to as the Frobenius. In [59], Nikolaus–Scholze demonstrate that the construction oftopological cyclic homology admits a drastic simplification bypassing the use of equivariant stable homotopy theory, and Bhatt–Morrow–Scholze [13] use the foundational work of Nikolaus–Scholzeto construct motivic filtrations on topological Hochschild homology and its variants identifyingthe graded pieces with completed prismatic cohomology in the sense of Bhatt–Scholze [14]. Theinvariant given by TR carries important arithmetic information by itself. The calculations of the p -adic K-theory of local fields obtained by Hesselholt–Madsen [42, 43] and Geisser–Hesselholt [30]rely on the relationship between TR and the de Rham–Witt complex with log poles. In particular,Hesselholt–Madsen [42] confirmed the Lichtenbaum–Quillen conjecture for p -adic fields prior tothe work of Rost–Voevodsky on the Bloch–Kato conjecture [65, 66]. These calculations are inturn based on the previous work of Hesselholt [39] and Hesselholt–Madsen [41].1.1. Statement of results. If I ⊆ A is a two-sided ideal of a ring A , then the relative algebraicK-theory spectrum K ( A, I ) is defined as the fiber of the map of spectra K ( A ) → K ( A / I ) inducedby the canonical ring homomorphism A → A / I . The starting point of this paper is the followingresult of Hesselholt [39, Theorem 3.1.10], which asserts that if R is a commutative Z / p j -algebrafor some integer j ≥
1, then there is a natural equivalence of p -complete spectraTR ( R ) ≃ lim ←Ð Ω K ( R [ t ]/ t n , ( t )) . The inverse limit appearing on the right hand side of the equivalence is referred to as the spectrumof curves on the algebraic K-theory of R , and was defined by Hesselholt [39] based on previouswork of Bloch [15] on the relationship between crystalline cohomology and algebraic K-theory.Betley–Schlichtkrull [12, Theorem 1.3] extend the result above to associative rings after profinitecompletion, where TR is replaced by TC. In this case, the inverse limit defining the spectrum ofcurves on K-theory is replaced by a limit over a diagram which additionally encodes the transfermaps R [ t ]/ t m → R [ t ]/ t mn determined by t ↦ t n . We recall the following notation: Notation 1.1.1.
In the following, we will let S [ t ] denote the E ∞ -ring defined by S [ t ] = Σ ∞+ Z ≥ .Note that S [ t ] is not the free E ∞ -ring on one generator, but the underlying E -ring of S [ t ] is thefree E -ring on one generator since Z ≥ is the free E -monoid on one generator in the ∞ -categoryof spaces. Let S [ t ]/ t n denote the E ∞ -ring defined by the following pushout of E ∞ -rings S [ t ] S [ t ] S S [ t ]/ t n ← → t ↦ t n ←→ t ↦ ←→ ← → If R is a connective E -ring, then we define the E -ring R [ t ]/ t n by R [ t ]/ t n = R ⊗ S [ t ]/ t n , andwe have that π ∗ ( R [ t ]/ t n ) ≃ ( π ∗ R )[ t ]/ t n . We obtain a map of connective E -rings R [ t ]/ t n → R such that the kernel of the induced ring homomorphism π ( R [ t ]/ t n ) ≃ ( π R )[ t ]/ t n → π R is given by the nilpotent ideal ( t ) . If E ∶ Alg cn → Sp is a functor, then we will let E ( R [ t ]/ t n , ( t )) denote the fiber of the induced map of spectra E ( R [ t ]/ t n ) → E ( R ) . In the following, we will beinterested in situation where E = K or E = TC.
Remark 1.1.2. If R is a discrete H Z -algebra, then R ⊗ S [ t ]/ t n is discrete with π ( R ⊗ S [ t ]/ t n ) ≃ R [ t ]/ t n . In general, if R is a connective E -ring, then R ⊗ S [ t ]/ t n is not necessarily discrete. N CURVES IN K-THEORY AND TR 3
As an application of the formalism developed in this paper, we obtain a common generalizationof the results of Hesselholt and Betley–Schlichtkrull discussed above (see Corollary 4.2.4).
Theorem A. If R is a connective E -ring, then there is a natural equivalence of spectra TR ( R ) ≃ lim ←Ð Ω K ( R [ t ]/ t n , ( t )) . Note that in the setting of Theorem A, there is an equivalence of spectralim ←Ð Ω K ( R [ t ]/ t n , ( t )) ≃ lim ←Ð Ω TC ( R [ t ]/ t n , ( t )) by virtue of the Dundas–Goodwillie–McCarthy theorem [28]. The results of Betley–Schlichtkrulland Hesselholt discussed above rely on this equivalence combined with an analysis of the fixedpoints of THH ( R [ t ]/ t n ) by finite cyclic groups using [41]. The proof of Theorem A proceeds bycompletely different methods which we summarize below.1.2. Methods.
The main technical contribution of this paper is a construction of TR whichbypasses the use of equivariant stable homotopy theory. This is analogous to the construction oftopological cyclic homology recently given by Nikolaus–Scholze [59] under suitable boundednessassumptions. We will begin by briefly reviewing the classical construction of TR following [41, 18],and refer the reader to § Construction 1.2.1.
A genuine cyclotomic spectrum is a genuine T -spectrum X with respectto the family of finite cyclic subgroups C k of T together with compatible equivalences X Φ C k ≃ X for every k ≥
1, where X Φ C k denotes the geometric fixedpoints for the action of C k on X . Forinstance, if R is a connective E -ring, then THH ( R ) admits the structure of a genuine cyclotomicspectrum [19, 41]. If ( m, n ) is a pair of positive integers with m = ln , then the restriction map R ∶ X C m → X C n is the map of genuine T -spectra defined by X C m ≃ ( X C l ) C n → ( X Φ C l ) C n ≃ X C n , where the final equivalence is induced by the genuine cyclotomic structure of X . We have thatTR ( X ) = lim n,R X C n , where the limit is indexed over the set of positive integers considered as a divisibility poset.The collection of genuine cyclotomic spectra assemble into an ∞ -category CycSp gen (see Defini-tion 3.3.1), and there is a canonical functor of ∞ -categories CycSp gen → CycSp which restricts toan equivalence on the full subcategories of those objects whose underlying spectrum is boundedbelow [59]. Consequently, if X is a cyclotomic spectrum whose underlying spectrum is boundedbelow, then we may evaluate TR ∶ CycSp gen → Sp on X using this equivalence.Throughout this paper, we will be interested in the reduced topological Hochschild homologyof the E ∞ -ring S [ t ] , whose construction we briefly recall. The map of E ∞ -rings S [ t ] → S givenby t ↦ ( S [ t ]) → S , where the sphere spectrum S isequipped with the trivial cyclotomic structure. The reduced topological Hochschild homology ofthe E ∞ -ring S [ t ] is the cyclotomic spectrum defined by ̃ THH ( S [ t ]) = fib ( THH ( S [ t ]) → S ) . There is an equivalence of cyclotomic spectra THH ( S [ t ]) ≃ S ⊕ ̃ THH ( S [ t ]) . With these notionsin place we state the following result (see Theorem 3.3.12). JONAS MCCANDLESS
Theorem B.
There is a natural equivalence of spectra TR ( X ) ≃ map CycSp ( ̃
THH ( S [ t ]) , X ) for every cyclotomic spectrum X whose underlying spectrum is bounded below. We remark that a variant of Theorem B above was previously obtained by Blumberg–Mandell [18]using point-set models for genuine cyclotomic spectra.
Remark 1.2.2.
Theorem B can be regarded as an analogue of the result of Nikolaus–Scholze [59]which asserts that there is an equivalence of spectraTC ( X ) ≃ map CycSp ( S , X ) for every cyclotomic spectrum X whose underlying spectrum is bounded below. This corepre-sentability result for topological cyclic homology was conjectured by Kaledin [47] and establishedby Blumberg–Mandell [18] after p -completion prior to the work of Nikolaus–Scholze.The cyclotomic structure of the reduced topological Hochschild homology of the E ∞ -ring S [ t ] admits a more refined structure, namely the structure of a cyclotomic spectrum with Frobeniuslifts. Informally, this amounts to the datum of a T -equivariant map of spectra ψ k ∶ ̃ THH ( S [ t ]) → ̃ THH ( S [ t ]) hC k for every positive integer k , where the target carries the residual T / C k ≃ T -action, and the maps ψ k are compatible in a highly coherent sense. The maps ψ k refine the cyclotomic structure of thereduced topological Hochschild homology of S [ t ] in the sense that the p th cyclotomic Frobeniusof ̃ THH ( S [ t ]) is canonically equivalent to the following T -equivariant map of spectra ̃ THH ( S [ t ]) ψ p Ð→ ̃ THH ( S [ t ]) hC p can ÐÐ→ ̃ THH ( S [ t ]) tC p for every prime number p , where can ∶ (−) hC p → (−) tC p denotes the canonical map. The notionof a p -typical cyclotomic spectrum with Frobenius lift was introduced by Nikolaus–Scholze [59]and Antieau–Nikolaus [3]. In the p -typical situation we only require the existence of the map ψ p in sharp contrast to the integral situation, where we also require coherences between thesemaps. Furthermore, the equivalence of spectra appearing in Theorem B canonically refines toan equivalence of cyclotomic spectra with Frobenius lifts, where each Frobenius lift of TR ( X ) isgiven by the T -equivariant map of spectra TR ( X ) ≃ TR ( X ) C k → TR ( X ) hC p for every k ≥
1. Asubstantial amount of this paper is devoted to showing that the right hand side of the equivalenceappearing in Theorem B admits the structure of a cyclotomic spectrum with Frobenius lifts. Tothat end, we introduce the following notions: ● The Witt monoid is the E -monoid defined by the semidirect product W = T ⋊ N × , where themultiplicative monoid N × acts on the circle T by covering maps of positive degree (see Con-struction 2.1.1). Let B W denote the ∞ -category with one object and W as endomorphisms.We will define the notion of a cyclotomic spectrum with Frobenius lifts as a spectrum with aright action of the Witt monoid W which in turn is the datum of a functor of ∞ -categoriesB W op → Sp. This precisely encodes the datum of a spectrum X with an action of the circle T together with compatible T -equivariant maps ψ k ∶ X → X hC k for every integer k ≥
1. Thecollection of cyclotomic spectra with Frobenius lifts assemble into an ∞ -category CycSp FrN CURVES IN K-THEORY AND TR 5 and we prove that the reduced topological Hochschild homology of the E ∞ -ring S [ t ] refinesto an object of this ∞ -category. The insight that the Witt monoid parametrizes Frobeniuslifts is present in the unpublished work of Goodwillie [35] on the cyclotomic trace. ● We prove that B W is an orbital ∞ -category in the sense of [10], and we define the notion ofan integral topological Cartier module as a spectral Mackey functor on B W in the sense ofBarwick [9]. This precisely encodes the datum of a spectrum M with an action of the circle T together with compatible T -equivariant factorizations M hC k → M → M hC k of the norm map for the cyclic group C k for every integer k ≥
1. This extends the definitiongiven by Antieau–Nikolaus [3] to the integral situation. The collection of topological Cartiermodules assemble into an ∞ -category TCart, and we will prove that TR refines to a functorwith values in this ∞ -category. Remark 1.2.3.
In this paper, the ∞ -category of topological Cartier modules will mostly functionas a convenient formalism for proving Theorem B. However, we believe that the notion of atopological Cartier module is important in its own right. For instance, the notion of a topologicalCartier module formalizes additional structure present on the rational Witt vectors and cyclicK-theory. An extensive treatment of the ∞ -category of topological Cartier modules extendingthe result of Antieau–Nikolaus [3] to the integral situation will appear in forthcoming work.We prove Theorem B by showing that the functors given by the constructions X ↦ TR ( X ) and X ↦ map CycSp ( ̃
THH ( S [ t ]) , X ) both determine right adjoints of the canonical functorCycSp Fr → CycSpwhen restricted to the full subcategories of those objects whose underlying spectrum is boundedbelow. The mapping spectrum map
CycSp ( ̃
THH ( S [ t ]) , X ) refines to a cyclotomic spectrum withFrobenius lifts since the reduced topological Hochschild homology of S [ t ] admits the structureof a bimodule over S [ W ] by left and right multiplication (see Example 2.2.11). The proof ofTheorem B relies on a genuine version of the Tate orbit lemma [59, Lemma I.2.1] established byAntieau–Nikolaus [3] together with an explicit description of the free topological Cartier moduleon a cyclotomic spectrum with Frobenius lifts. The latter we deduce from a general version of theSegal–tom Dieck splitting for Mackey functors on orbital ∞ -categories (see Proposition 3.2.12).We deduce Theorem A from the celebrated Dundas–Goodwillie–McCarthy theorem [28] and thefollowing result (see Theorem 4.2.3). Theorem C.
There is a natural equivalence of spectra TR ( X ) ≃ lim ←Ð Ω TC ( X ⊗ ̃ THH ( S [ t ]/ t n )) for every cyclotomic spectrum X whose underlying spectrum is bounded below. The crucial observation is that there is an equivalence of spectra with T -actionmap Sp ( ̃ THH ( S [ t ]) , S ) ≃ ∏ n ≥ Σ ∞− + ( S / C n ) , JONAS MCCANDLESS where the mapping spectrum on the left carries the residual T -action, and S / C n denotes thespace with T -action whose underlying space is given by S , and whose T -action is obtained byrestriction of the multiplication action of the circle T on S along the n th power map. UsingTheorem B combined with a careful analysis of the cyclotomic structure of lim ←Ð ̃ THH ( S [ t ]/ t n ) ,we deduce Theorem A by exploiting the equalizer formula for the mapping spectrum in the ∞ -category of cyclotomic spectra obtained by Nikolaus–Scholze [59]. This observation is due toAchim Krause. Acknowledgements.
It is a pleasure to thank Stefano Ariotta, Elden Elmanto, Markus Land,Malte Leip, and Jay Shah for very useful discussions related to this work. The author would alsolike to thank Elden Elmanto, Lars Hesselholt, Liam Keenan, Markus Land, and Thomas Nikolausfor very valuable comments on a previous version. Most importantly, the author is extremelygrateful to Achim Krause and Thomas Nikolaus for their constant support and many insightswhich made this project possible. Funded by the Deutsche Forschungsgemeinschaft (DFG, Ger-man Research Foundation) under Germany’s Excellence Strategy EXC 2044–390685587, Math-ematics M¨unster: Dynamics–Geometry–Structure and the CRC 1442 Geometry: Deformationsand Rigidity. 2.
Cyclotomic spectra with Frobenius lifts and TR
The main goal of this section is to construct the reduced topological Hochschild homologyof the E ∞ -ring S [ t ] as a cyclotomic spectrum with Frobenius lifts, and provide an alternativeconstruction of TR bypassing the otherwise instrumental use of equivariant stable homotopytheory. In § ∞ -category, andprove that the geometric realization of a presheaf on the epicyclic category with values in an ∞ -category which admits geometric realizations refines to an object with Frobenius lifts. In § ∞ -category of spaces with Frobenius lifts, and construct a refinement of the cyclicbar construction which automatically carries Frobenius lifts. We use this to produce examples ofspaces with Frobenius lifts which will play an essential role throughout this exposition. In § ∞ -category of cyclotomic spectra with Frobenius lifts, and construct the canonicalfunctor from the ∞ -category of cyclotomic spectra with Frobenius lifts to the ∞ -category ofcyclotomic spectra. Finally, in § § The epicyclic category.
We introduce the ∞ -category of objects with Frobenius lifts inan ∞ -category, and describe these in terms of presheaves on the epicyclic category. The followingconstruction will play an important role throughout this paper. Construction 2.1.1.
We define a monoid which we will refer to as the Witt monoid following [5].(1) The multiplicative monoid N × of positive integers acts on the circle group T by the assign-ment x ↦ x k for every positive integer k , and the Witt monoid is the topological monoiddefined by W = T ⋊ N × , where N × is regarded as a discrete topological monoid. Theunderlying space of the Witt monoid is given by T × N × , and the multiplication is given by ( λ, k ) ⋅ ( µ, l ) = ( λ l µ, kl ) N CURVES IN K-THEORY AND TR 7 for every pair of elements ( λ, k ) and ( µ, l ) of the underlying space of W . It will be convenientto regard the Witt monoid as an object of the ∞ -category Alg ( S ) of E -monoids. We willlet B W denote the ∞ -category with one object and the Witt monoid W as endomorphisms.(2) For every prime p , the multiplicative monoid µ p ∞ = { , p, p , . . . } acts on the circle group T by restriction along the canonical inclusion of multiplicative monoids µ p ∞ ↪ N × , and the p -typical Witt monoid is the topological monoid defined by W p ∞ = T ⋊ µ p ∞ , where µ p ∞ isregarded as a discrete topological monoid. We will let B W p ∞ denote the ∞ -category withone object and the p -typical Witt monoid W p ∞ as endomorphisms.The Witt monoid also appears in the unpublished work of Goodwillie [35] on the cyclotomictrace, where the insight that it parametrizes Frobenius lifts was already present. The followingdefinition also appears in the work of Ayala–Mazel-Gee–Rozenblyum [7, 5, 6]. Definition 2.1.2.
The ∞ -category of objects with Frobenius lifts in an ∞ -category C is definedby C Fr = Fun ( B W op , C ) = P C ( B W ) . Remark 2.1.3.
There is a fiber sequence of ∞ -categoriesB T ≃ ( B T ) op → ( B W ) op → ( B N × ) op , and the functor B W → B N × is a cocartesian fibration. Consequently, we see that an object withFrobenius lifts in an ∞ -category C which admits finite limits is given by an object X of C withan action of the circle group T together with a compatible system of T -equivariants maps ψ k ∶ X → X hC k , for every positive integer k , where the target carries the residual T / C k ≃ T -action. For instance, if ( k, l ) is a pair of positive integers, then the following diagram of spectra with T -action commutes X X hC k X hC l ( X hC k ) hC l ( X hC l ) hC k ← → ψ k ←→ ψ l ←→ ψ hCkl ← → ψ hClk ← → ≃ In § § ∞ -category of spaces with Frobenius lifts and the ∞ -categoryof cyclotomic spectra with Frobenius lifts in more detail. Presently, we discuss Goodwillie’sepicyclic category which provides a formalism for constructing objects with Frobenius lifts in thesense of Definition 2.1.2. We begin by reviewing the cyclic category following [59, Appendix B]. Definition 2.1.4.
A parasimplex is a nonempty linearly ordered set I equipped with an actionof the group of integers Z denoted + ∶ I × Z → I , such that the following conditions are satisfied:(1) For every pair of elements λ and λ ′ of I , the set { µ ∈ I ∣ λ ≤ µ ≤ λ ′ } is finite.(2) If λ is an element of I , then λ < λ + f ∶ I → J which is nondecreasing and Z -equivariant.The paracyclic category Λ ∞ is defined as the category whose objects are given by the parasim-plices and whose morphisms are given by the paracyclic morphisms. JONAS MCCANDLESS
Example 2.1.5.
For every positive integer n , the set n Z = { mn ∣ m ∈ Z } can be regarded as aparasimplex with respect to its usual ordering and the action of Z given by addition. We willdenote this particular parasimlex by [ n ] Λ . Conversely, if I is a parasimplex, then there is anisomorphism of parasimplices I ≃ [ n ] Λ for a uniquely determined positive integer n which, canbe characterized as the cardinality of the set { x ∈ I ∣ y ≤ x < y + } for any element y of I . Example 2.1.6. If S is an object of the simplex category ∆, then the product S × Z can beregarded as a parasimplex with respect to the lexicographic ordering and the action of Z given bythe formula m + ( s, n ) = ( s, m + n ) . The construction S ↦ S × Z defines a faithful functor ∆ → Λ ∞ which is essentially surjective since there is an isomorphism of parasimplices [ n ] × Z ≃ [ n + ] Λ for every integer n ≥ Definition 2.1.7.
The cyclic category Λ is the category whose objects are parasimplices, wherethe set of morphisms between a pair of parasimplices I and J is defined byHom Λ ( I, J ) = Hom Λ ∞ ( I, J )/ Z , where Z acts on the set of paracyclic morphisms Hom Λ ∞ ( I, J ) by the formula ( f + n )( λ ) = f ( λ ) + n . Remark 2.1.8.
The action of the group Z on the set of paracyclic morphisms Hom Λ ∞ ( I, J ) inthe definition of the cyclic category determines a strict action of the simplicial abelian group B Z on the simplicial set N ( Λ ∞ ) , which in turn induces an action of the circle T on N ( Λ ∞ ) . Notethat N ( Λ ) ≃ N ( Λ ∞ ) B Z ≃ N ( Λ ∞ ) h T since the action of the simplicial abelian group B Z on N ( Λ ∞ ) is free. Consequently, there is a fiber sequenceN ( Λ ∞ ) → N ( Λ ) → B T , where the map N ( Λ ) → B T is both a cartesian and cocartesian fibration.We will now define the epicyclic category. The construction I ↦ I / Z determines a functorΛ → C at , where the parasimplex I is regarded as a poset, and I / Z denotes the quotient in C at, which weregard as a 1-category. Let [ n ] ˜Λ denote the category defined by [ n ] ˜Λ = [ n ] Λ / Z for every n ≥ Definition 2.1.9.
The epicyclic category ˜Λ is the subcategory of C at given by the essentialimage of the functor Λ → C at and those functors which are essentially surjective.The epicyclic category was introduced by Goodwillie in an unpublished letter to Waldhausen,and Burghelea–Fiedorowicz–Gajda [21] give a combinatorial description of the epicyclic cate-gory similar to the combinatorial description of the cyclic category [25]. Kaledin [48] studies avariant of the epicyclic category which is referred to as the cyclotomic category. Ayala–Mazel-Gee–Rozenblyum [5] describe the epicyclic category in terms of stratified 1-manifolds and diskrefinements. The definition of the epicyclic category given in Definition 2.1.9 is due to Nikolaus.Note that the functor Λ → ˜Λ is essentially surjective and faithful by construction. To see thatthis functor is not full, we give a geometric description of the morphisms in Λ and ˜Λ in thefollowing examples. We may regard the object [ n ] Λ of Λ as a cyclic graph with n vertices, orequivalently as a configuration of n marked points on S . N CURVES IN K-THEORY AND TR 9
Example 2.1.10.
The datum of a morphism [ m ] Λ → [ n ] Λ in the cyclic category Λ is equivalentto the datum of a self-map of S of degree 1 which preserves markings and satisfies that theinduced self-map on universal covers is non-decreasing. For example, the morphism τ n ∶ [ n ] Λ → [ n ] Λ defined by τ n ( i ) = i + S given by rotation with 2 π / n , wherethe configuration of S is given by { , ζ, . . . , ζ n − } for a primitive n th root of unity ζ . Note thatthe morphism τ n is an automorphism of [ n ] Λ , and there is a group isomorphism Z /( n + ) Z → Aut Λ ([ n ] Λ ) given by sending the generator of the cyclic group Z /( n + ) Z to the automorphism τ n . Inconclusion, the group of automorphisms of [ n ] Λ in Λ is cyclic of order n + τ n . Incontrast, the group of automorphisms of an object of the simplex category is trivial. Example 2.1.11.
The datum of a morphism [ m ] ˜Λ → [ n ] ˜Λ in the epicyclic category ˜Λ is equiv-alent to the datum of a self-map of S of positive degree which preserves markings and satisfiesthat the induced self-map on universal covers is non-decreasing. If α is a morphism in ˜Λ, thenthe degree of α is defined as the degree of the induced self-map of S . There is a mapdeg ∶ N ( ˜Λ ) → B N × given by sending a morphism of ˜Λ to its degree, and this is a cartesian fibration which classifiesthe canonical action of the multiplicative monoid N × on the cyclic category Λ. As an example,the map p k ∶ S → S given by α k ( x ) = x k corresponds to a morphism α k ∶ [ kn ] ˜Λ → [ n ] ˜Λ in ˜Λof degree k , where the source of p k carries the configuration consisting of kn preimages of the n marked points on the target. The morphism α k does not exist in the cyclic category, whichmeans that the functor Λ → ˜Λ is not full. Furthermore, for every morphism f ∶ [ n ] ˜Λ → [ n ] ˜Λ in ˜Λof degree k , there exists a morphism f ′ ∶ [ n ] ˜Λ → [ kn ] ˜Λ in ˜Λ of degree 1 such that f = α k ○ f ′ .The epicyclic category ˜Λ admits an alternative description in terms of a right action of the Wittmonoid on the paracyclic category which we now address. We use the notation of Example 2.1.11. Construction 2.1.12.
There is a functor of ∞ -categoriesN ( ˜Λ ) → B W which is uniquely determined by sending the morphism α k to the element ( , k ) , and the mor-phism τ n to the element ( ζ n , ) , where ζ n = exp ( πin ) . Note that the morphism N ( ˜Λ ) → B W is acartesian fibration, and that the morphism deg ∶ N ( ˜Λ ) → B N × from Example 2.1.11 factors overN ( ˜Λ ) → B W . Lemma 2.1.13.
There is a fiber sequence of ∞ -categories N ( Λ ∞ ) → N ( ˜Λ ) → B W .Proof. It follows from Remark 2.1.8, that there is a fiber sequence of ∞ -categoriesN ( Λ ∞ ) → N ( Λ ) → B T Combining this with Construction 2.1.12, we conclude that every square in the following diagramN ( Λ ∞ ) N ( Λ ) N ( ˜Λ ) ∆ B T B W ∆ B N × ← → ←→ ← → ←→ ←→ ← → ← → ←→ ←→ ← → is a pullback, which shows what we wanted. (cid:3) Remark 2.1.14.
It follows from Lemma 2.1.13 that the cartesian fibration N ( ˜Λ ) → B W classifiesa right action of the Witt monoid on N ( Λ ∞ ) . Informally, this action is determined by the strictaction of the simplicial abelian group B Z on N ( Λ ∞ ) as in Remark 2.1.8, and the action of N × on N ( Λ ∞ ) given by the assignment I ↦ nI for every n ≥
1, where nI denotes the parasimplexwith the same underlying linearly ordered set as I but whose Z -action is given by restricting the Z -action on I along the homomorphism Z → Z given by multiplication by n .Recall that the geometric realization of a cyclic object admits a canonical action of the circlegroup as observed by Connes [25]. Similarly, we introduce the formalism of epicyclic objects andshow that the geometric realization of an epicyclic object canonically admits Frobenius lifts, thatis a right action of the Witt monoid. Definition 2.1.15.
The ∞ -category of epicyclic objects in an ∞ -category C is defined by P C ( ˜Λ ) = Fun ( N ( ˜Λ op ) , C ) . The ∞ -category of epicyclic objects in the ∞ -category of spaces is denoted by P ( ˜Λ ) .Assume that X is an epicyclic object in an ∞ -category C which admits geometric realizations.The geometric realization of X is defined as the colimit of the underlying simplicial objectN ( ∆ op ) → N ( Λ op ∞ ) → N ( Λ op ) → N ( ˜Λ op ) → C . Let ∣ X ∣ denote the geometric realization of X , and note that there is a functor of ∞ -categories P C ( ˜Λ ) → C determined by the construction X ↦ ∣ X ∣ . Burghelea–Fiedorowicz–Gajda [21] prove that thegeometric realization of an epicyclic object admits the structure of an object with Frobenius liftsusing an explicit combinatorial description of the epicyclic category. We present a proof of thisresult using the language employed in this paper. To that end, we will use the formalism of laxlimits and colimits in the sense of Gepner–Haugseng–Nikolaus [32]. Recall that if F ∶ B M op → C at ∞ denotes a functor which defines a right action of an E -monoid M on an ∞ -category C , then thelax coinvariants is the cartesian fibration p ∶ C ℓM → B M classified by F . The lax fixed points C ℓM is the ∞ -category of sections of p which carry every arrow in B M to a p -cartesian edge. N CURVES IN K-THEORY AND TR 11
Proposition 2.1.16. If C is an ∞ -category which admits geometric realizations, then the con-struction X ↦ ∣ X ∣ admits a canonical refinement to a functor of ∞ -categories P C ( ˜Λ ) → C Fr such that the following diagram commutes P C ( ˜Λ ) C Fr P C ( Λ ) C B T ← → ←→ ←→ ← → Proof.
The functor N ( ∆ op ) → N ( Λ op ∞ ) is cofinal by virtue of [59, Theorem B.3], hence the geo-metric realization of an epicyclic object of C is equivalent to the colimit of the paracyclic objectN ( Λ op ∞ ) → N ( Λ op ) → N ( ˜Λ op ) → C . There is an adjunction of ∞ -categories P C ( Λ ∞ ) C ← → ∣−∣ ←→ δ ∗ where the right adjoint δ ∗ is given by precomposition with the terminal functor δ ∶ N ( Λ op ∞ ) → ∆ which is equivariant with respect to the left W -action on N ( Λ op ∞ ) obtained in Lemma 2.1.13,and the trivial left W -action on ∆ . This in turn means that the right adjoint δ ∗ is equivariantwith respect to the induced right action of W on P C ( Λ ∞ ) and the trivial right action of W on C . Consequently, the left adjoint canonically refines to a functor of ∞ -categories ∣ − ∣ ∶ P C ( Λ ∞ ) ℓ W → C Fr , where ( − ) ℓ W denotes the lax fixed points of the right action of the Witt monoid on P C ( Λ ∞ ) .Additionally, we have used that C Fr ≃ C ℓ W since the action of the Witt monoid on C is trivial.The desired functor is given by composing the functor above with the canonical functor P C ( ˜Λ ) → P C ( Λ ∞ ) ℓ W , where we have used Lemma 2.1.13 to identify the lax coinvariants of the right action of theWitt monoid on N ( Λ ∞ ) with N ( ˜Λ ) . The final assertion follows by a similar argument using theequivalences ( − ) ℓ T ≃ ( − ) h T and ( − ) ℓ T ≃ ( − ) h T since T is a group. (cid:3) Spaces with Frobenius lifts.
Using the formalism of epicyclic spaces discussed in § ∞ -category of spaces with Frobenius lifts. Weconstruct an epicyclic variant of the cyclic bar construction and use this to produce examples ofspaces with Frobenius lifts which will play an important role throughout this paper. Recall thatthe ∞ -category of spaces with Frobenius is defined by S Fr = P ( B W ) . We summarize the salientfeatures of the ∞ -category of spaces with Frobenius lifts. Proposition 2.2.1.
The ∞ -category of spaces with Frobenius lifts is a presentable ∞ -category,and the forgetful functor S Fr → S B T is conservative and preserves small limits and colimits. Alternatively, the ∞ -category of spaces with Frobenius lifts is equivalent to the full subcategoryof S / B W spanned by those maps over B W which are right fibrations [54, Section 2.2]. In practice,we will make use of the formalism of epicyclic spaces as discussed in § Example 2.2.2.
Let X denote an object of the ∞ -category of spaces. The free loop space L ( X ) of X is the geometric realization of the epicyclic space given by the assignment [ n ] ˜Λ ↦ L ( X ) .The self-map of L ( X ) determined by the cyclic operator τ n is induced by the self-map of S given by rotation with 2 π / n . The self-map of L ( X ) determined by the epicyclic operator α k isinduced by the self-map of S given by the k th power map. Consequently, we may regard thefree loop space L ( X ) as a space with Frobenius lifts by Proposition 2.1.16.An important class of examples of spaces with Frobenius lifts arises by evaluating the cyclicbar construction on an E -monoid. Using the combinatorial description of the epicyclic category,Burghelea–Fiederowicz–Gajda [21] prove that the cyclic space whose geometric realization definesthe cyclic bar construction further refines to an epicyclic space. A similar result was obtainedby Schlichtkrull [60]. More recently, Nikolaus–Scholze [59] construct the individual Frobeniuslifts on the cyclic bar construction using the space-valued diagonal. For instance, if M is the E -monoid given by Ω X for a connected pointed space X , then B cyc M ≃ L ( X ) . In fact, thisequivalence refines to an equivalence of spaces with Frobenius lifts, where we regard L ( X ) as aspace with Frobenius lifts as in Example 2.2.2 (see Corollary 2.2.8). We proceed by introducingan epicyclic bar construction following an idea of Nikolaus. Construction 2.2.3.
Let T
Ass denote the Lawvere theory of monoids [52]. By definition, theopposite of T
Ass is the full subcategory of the category of monoids spanned by those monoidswhich are free on a finite set. We construct a functor j ∶ ˜Λ → T opAss which informally is given bysending an object [ n ] ˜Λ of the epicyclic category to the free monoid on the set { , . . . , n } . Firstnote that there is a functor of ∞ -categoriesN ( ˜Λ ) → C at ∆ ∞ cofib ÐÐ→ ( C at ∞ ) ∗ determined by the construction which sends an object [ n ] ˜Λ of the epicyclic category ˜Λ to thecofiber of the canonical functor [ n ] ds˜Λ ↪ [ n ] ˜Λ which exhibits the source as the discrete categoryon the set of objects of [ n ] ˜Λ . The space of endomorphisms of the canonical basepoint of thecofiber [ n ] ˜Λ /[ n ] ds˜Λ is equivalent to a discrete monoid which is free on the finite set { , . . . , n } , sowe obtain a functor j ∶ N ( ˜Λ ) → N ( T opAss ) defined by j ([ n ] ˜Λ ) = End [ n ] ˜Λ /[ n ] ds˜Λ ( ∗ ) , where ∗ denotes the canonical basepoint of [ n ] ˜Λ /[ n ] ds˜Λ .The datum of an E -monoid in the sense of [55] is equivalently specified by the datum of afunctor N ( T Ass ) → S which preserves finite products as explained in [26] or [31, Appendix B].We define the epicyclic bar construction of an E -monoid using Construction 2.2.3. Definition 2.2.4.
The epicyclic bar construction B epi M of an E -monoid M is defined byforming the geometric realization of the epicyclic space given byN ( ˜Λ op ) j op ÐÐ→ N ( T Ass ) M Ð→ S . N CURVES IN K-THEORY AND TR 13
The construction M ↦ B epi M determines a functor of ∞ -categoriesB epi ∶ Alg ( S ) → S Fr . Remark 2.2.5. If M is an E -monoid, then the geometric realization of the cyclic spaceN ( Λ op ) → N ( ˜Λ op ) j op ÐÐ→ N ( T Ass ) M Ð→ S is canonically equivalent to the cyclic bar construction B cyc M regarded as an object of S B T .We introduce an epicyclic variant of topological Hochschild homology of a small ∞ -categoryas a refinement of the unstable topological Hochschild homology introduced by Nikolaus [44]. Definition 2.2.6.
The epicyclic topological Hochschild homology THH epi ( C ) of a small ∞ -category C is defined as the geometric realization of the epicyclic space given by the assignment [ n ] ˜Λ ↦ Fun ([ n ] ˜Λ , C ) ≃ . The construction C ↦ THH epi ( C ) determines a functor of ∞ -categoriesTHH epi ∶ C at ∞ → S Fr . It will be convenient to work with both the epicyclic bar construction and the epicyclic topo-logical Hochschild homology, so we show that they coincide as spaces with Frobenius lifts for E -monoids. The author learned the proof of the following result from Nikolaus, and a proof ofa similar result appears in Krause–Nikolaus [50, Proposition 8.1]. Proposition 2.2.7.
Let M be an E -monoid, and let B M denote the ∞ -category with one objectand M as endomorphisms. There is an equivalence of spaces with Frobenius lifts B epi M → THH epi ( B M ) . Proof.
Let C n denote the cofiber of the functor ([ n ] ds˜Λ ) + ↪ ([ n ] ˜Λ ) + in ( C at ∞ ) ∗ , and note that C n is equivalent to the pointed ∞ -category B j ([ n ] ˜Λ ) , where j ([ n ] ˜Λ ) dentotes the monoid discussedin Construction 2.2.3. There is a fiber sequence of epicyclic spacesFun ∗ ( C n , B M ) ≃ → Fun ([ n ] ˜Λ , B M ) ≃ → ( B M × ) n , where M × denotes the group of units of M , and the epicyclic space on the left is equivalent tothe epicyclic space defining B epi M since Fun ∗ ( C n , B M ) ≃ ≃ Map S ([ n ] ˜Λ , M ) . As a consequence,we obtain a map of spaces with Frobenius liftsB epi M → THH epi ( B M ) obtained as the geometric realization of the first map in the fiber sequence above, and we showthat this map is an equivalence. Note that the space Fun ∗ ( C n , B M ) ≃ admits a canonical actionof the epicyclic group given by G ● ∶ [ n ] ˜Λ ↦ ( M × ) n , and the first map in the fiber sequenceexhibits Fun ([ n ] ˜Λ , B M ) ≃ as the homotopy coinvariants of this action in the ∞ -category P ( ˜Λ ) .In other words, the map B epi M → THH epi ( B M ) above is equivalent to the following compositeB epi M → ( B epi M ) h ∣ G ● ∣ ≃ Ð→ THH epi ( B M ) , where ∣ G ● ∣ denotes the geometric realization of the epicyclic group G ● . Note that the geometricrealization ∣ G ● ∣ of G ● is contractible since the underlying simplicial object of G ● admits an extradegeneracy, so the map B epi M → ( B epi M ) h ∣ G ● ∣ is an equivalence, which proves the claim. (cid:3) Consequently, we obtain the following result which allow us to explicitly identify the Frobeniuslifts on the epicyclic bar construction in certain examples. This result was first obtained byGoodwillie [34], Burghelea–Fiedorowicz [20], and Jones [46] ignoring the epicyclic structure, andby Burghelea–Fiedorowicz–Gajda [21] in the epicyclic case.
Corollary 2.2.8.
Let X denote an object of the ∞ -category of spaces regarded as an ∞ -category.There is an equivalence of spaces with Frobenius lifts THH epi ( X ) ≃ L ( X ) , where L ( X ) is regarded as a space with Frobenius lifts as in Example 2.2.2.Proof. For every integer n ≥
1, there is an equivalence of spacesFun ([ n ] ˜Λ , X ) ≃ ≃ Map S (∣[ n ] ˜Λ ∣ , X ) ≃ L ( X ) , which shows that the underlying simplicial object of THH epi ( X ) is constant with value L ( X ) .The assertion now follows from Example 2.2.2. (cid:3) Finally, using the material discussed above we present further examples of spaces with Frobe-nius lifts which will play an important role throughout this paper.
Notation 2.2.9.
Let i ≥ S / C i denote the space with T -action whoseunderlying space is given by S , and whose T -action is defined by x ↦ λ i x for every element λ ∈ T . Let S / C denote the space given by S regarded as a space with trivial T -action. Example 2.2.10.
There is an equivalence of spaces with Frobenius lifts B epi Z ≃ L ( S ) obtainedby combining Proposition 2.2.7 and Corollary 2.2.8. Furthermore, the map S × Z → L ( S ) defined by ( x, n ) ↦ ( t ↦ t n x ) is an equivalence of spaces with T -action provided that we regardthe left hand side as a space with T -action given by the formula λ ⋅ ( x, n ) = ( λ n x, n ) . Additionally,the p th Frobenius lift of L ( S ) ≃ S × Z is given by the assignment ( x, n ) ↦ ( x p , pn ) . In conclusion,there is an equivalence of spaces with T -actionB epi Z ≃ ∐ i ∈ Z S / C ∣ i ∣ , where the p th Frobenius lift is determined by the T -equivariant map S / C i → ( S / C pi ) hC p induced by the p th power map. Furthermore, since the inclusion of monoids Z ≥ ↪ Z induces amap of spaces with Frobenius lifts B epi Z ≥ → B epi Z which is a monomorphism in the ∞ -category S , we conclude that there is an equivalence of spaces with T -actionB epi Z ≥ ≃ ∗ ∐ ∐ i ≥ S / C i , where the Frobenius lifts are described as above. This identification is originally due to Hessel-holt [39, Lemma 2.2.3]. We will let ̃ B epi Z ≥ denote the space with Frobenius lifts given by ̃ B epi Z ≥ ≃ ∐ i ≥ S / C i , and we refer to ̃ B epi Z ≥ as the reduced epicyclic bar construction of Z ≥ . N CURVES IN K-THEORY AND TR 15
Example 2.2.11.
We show that the underlying space of the Witt monoid W refines to aspace with Frobenius lifts. Indeed, the underlying space of the Witt monoid is the geometricrealization of the epicylic space given by the assignment [ n ] ˜Λ ↦ W . The self-map of W inducedby the morphism τ n ∶ [ n ] ˜Λ → [ n ] ˜Λ is given by left multiplication by ( ζ n , ) , and the self-mapof W induced by the morphism α k ∶ [ kn ] ˜Λ → [ n ] ˜Λ is given by right multiplication by ( , k ) .Consequently, we may regard the underlying space of the Witt monoid as a space with Frobeniuslifts by Proposition 2.1.16, and there is an equivalence of spaces with Frobenius lifts ̃ B epi Z ≥ ≃ W by Example 2.2.10. It follows that the Yoneda embedding B W → S Fr is given by the construction ∗ ↦ ̃ B epi Z ≥ . The induced self-map of ̃ B epi Z ≥ is given by the assignment ( x, n ) ↦ ( λ kn x k , kn ) for every element ( λ, k ) of the Witt monoid.2.3. Cyclotomic spectra with Frobenius lifts.
We discuss the ∞ -category of cyclotomicspectra with Frobenius lifts in further detail, and construct the canonical functor from the ∞ -category of cyclotomic spectra with Frobenius lifts to the ∞ -category of cyclotomic spectra asan instance of the stable nerve-realization adjunction. Definition 2.3.1.
The ∞ -category of cyclotomic spectra with Frobenius lifts is defined byCycSp Fr = P Sp ( B W ) . We summarize the features of the ∞ -category of cyclotomic spectra with Frobenius lifts. Proposition 2.3.2.
The ∞ -category CycSp Fr of cyclotomic spectra with Frobenius lifts is astable and presentable ∞ -category, and the forgetful functor CycSp Fr → Sp B T is conservative andpreserves small limits and colimits. Remark 2.3.3.
The notion of a cyclotomic spectrum with Frobenius lifts in the sense of Defini-tion 2.3.1 is equivalent to the notion considered by Krause–Nikolaus [50]. Additionally, a variantof Definition 2.3.1 appears in the work of Ayala–Mazel-Gee–Rozenblyum [7, 5, 6].
Remark 2.3.4.
Recall that a p -typical cyclotomic spectrum with Frobenius lift consists of aspectrum X with T -action together with a T -equivariant map of spectra ψ p ∶ X → X hC p where the target carries the residual T / C p ≃ T -action, and these assemble into an ∞ -categoryCycSp Fr p which is defined by a lax equalizer [59, Section II.1]. There is a canonical functorCycSp Fr → P Sp ( B W p ∞ ) → CycSp Fr p , where the first functor is induced by the canonical map of Witt monoids W p ∞ → W , and thesecond functor is induced by the universal property of CycSp Fr p as a lax equalizer. Informally,this functor is given by only remembering the p th Frobenius lift.In practice, it is difficult to specify the datum of a cyclotomic spectrum with Frobenius liftsdue to the infinite hierarchy of coherences that needs to be specified. However, there is a functor S Fr → CycSp
Fr6 JONAS MCCANDLESS given by postcomposition with Σ ∞+ ∶ S → Sp, and we will mostly be interested in cyclotomicspectra with Frobenius lifts contained in the essential image of this functor. In this situation, theformalism of epicyclic spaces as discussed in § Example 2.3.5.
The suspension spectrum Σ ∞+ ̃ B epi Z ≥ admits the structure of a cyclotomicspectrum with Frobenius lifts. There is an equivalence of spectra with T -actionΣ ∞+ ̃ B epi Z ≥ ≃ ⊕ i ≥ Σ ∞+ ( S / C i ) , and the k th Frobenius lift is induced by the T -equivariant map of spaces S / C i → ( S / C ki ) hC k .The coherences are encoded by the epicyclic bar construction as in § X denotes a cyclotomic spectrum with Frobenius lifts, then the underlyingspectrum with T -action admits the structure of a cyclotomic spectrum, where the p th cyclotomicFrobenius is given by the T -equivariant composite map of spectra X → X hC p → X tC p for everyprime p . We show that this construction refines to a functor of ∞ -categoriesCycSp Fr → CycSpas an instance of the stable nerve-realization adjunction. The stable nerve-realization adjunctionappears in the work of Dwyer–Kan [29] in the unstable context of G -spaces, and we refer thereader to [4, Appendix A] for a systematic treatment phrased in the language of ∞ -categories. Definition 2.3.6.
Let C be a small ∞ -category. The stable realization of a functor F ∶ C → D with values in a stable and presentable ∞ -category D is the functor ∣ − ∣ F ∶ P Sp ( C ) → D defined by the left Kan extension of F along the stable Yoneda embedding y st ∶ C → P Sp ( C ) . Proposition 2.3.7.
Let C be a small ∞ -category, and let F ∶ C → D be a functor with values ina stable and presentable ∞ -category D . The stable realization of F admits a right adjoint N F ∶ D → P Sp ( C ) determined by the construction d ↦ map D ( F ( − ) , d ) .Proof. Let L y F denote the left Kan extension of F along the Yoneda embedding y ∶ C → P ( C ) ,and note that it suffices to show that L y F admits a right adjoint determined by the construction d ↦ Map D ( F ( − ) , d ) . It follows from [54, Theorem 5.1.5.6] that there is an adjoint equivalenceFun L ( P ( C ) , D ) Fun ( C , D ) ← → y ∗ ←→ where the right adjoint is determined by the assignment F ↦ L y F , so we conclude that L y F admits a right adjoint by [54, Corollary 5.5.2.9]. There is a natural equivalence ( L y F )( X ) ≃ ∫ c ∈ C Map P ( C ) ( y ( c ) , X ) ⊗ F ( c ) , N CURVES IN K-THEORY AND TR 17 for every presheaf X on C , where we have used that D is canonically tensored over the ∞ -categoryof spaces. For every object d of D , there is a sequence of natural equivalencesMap D (( L y F )( X ) , d ) ≃ Map D ( ∫ c ∈ C Map P ( C ) ( y ( c ) , X ) ⊗ F ( c ) , d ) ≃ ∫ c ∈ C Map D ( X ( c ) ⊗ F ( c ) , d ) ≃ ∫ c ∈ C Map S ( X ( c ) , Map D ( F ( c ) , d )) ≃ Map P ( C ) ( X, Map D ( F ( − ) , d )) which finishes the proof. (cid:3) We construct a functor B W → CycSp which encodes the cyclotomic structure of Σ ∞+ ̃ B epi Z ≥ ,which will allow us to define the canonical functor from the ∞ -category of cyclotomic spectrawith Frobenius lifts to the ∞ -category of cyclotomic spectra. Construction 2.3.8.
Consider the following sequence of functors of ∞ -categories S Fr → ∏ p ( S B T ) ∆ Σ ∞+ ÐÐ→ ∏ p ( Sp B T ) ∆ can ÐÐ→ ∏ p ( Sp B T ) ∆ , where the first functor is determined by the construction X ↦ ( ψ p ∶ X → X hC p ) for every prime p ,where the target is equipped with the residual T / C p ≃ T -action. If X is a space with Frobeniuslifts, then this composite carries X to the T -equivariant map of spectraΣ ∞+ X → Σ ∞+ ( X hC p ) → ( Σ ∞+ X ) hC p can ÐÐ→ ( Σ ∞+ X ) tC p for every prime number p . By the construction of the ∞ -category of cyclotomic spectra as alax equalizer [59, Definition II.1.6], we obtain a functor S Fr → CycSp, and combining this withExample 2.2.11, we obtain a functor B W → S Fr → CycSp which regards the underlying spectrumwith T -action of Σ ∞+ ̃ B epi Z ≥ as a cyclotomic spectrum whose p th cyclotomic Frobenius is givenby the T -equivariant map of spectraΣ ∞+ ̃ B epi Z ≥ → Σ ∞+ (̃ B epi Z ≥ ) hC p → ( Σ ∞+ ̃ B epi Z ≥ ) hC p can ÐÐ→ ( Σ ∞+ ̃ B epi Z ≥ ) tC p . The stable realization of the functor B W → CycSp defines a functor CycSp Fr → CycSp which wewill refer to as the canonical functor from the ∞ -category of cyclotomic spectra with Frobeniuslifts to the ∞ -category of cyclotomic spectra.The canonical functor CycSp Fr → CycSp does not exhibit the ∞ -category of cyclotomic spec-tra with Frobenius lifts as a subcategory of the ∞ -category of cyclotomic spectra, that is theFrobenius lifts are structure and not a property. We establish the main result of this section. Theorem 2.3.9.
There is an adjunction of ∞ -categories CycSp Fr CycSp ← →←→ where the right adjoint is determined by the construction X ↦ map CycSp ( ̃
THH ( S [ t ]) , X ) , andthe left adjoint is given by the canonical functor constructed above.Proof. There is an equivalence of cyclotomic spectra ̃ THH ( S [ t ]) ≃ Σ ∞+ ̃ B epi Z ≥ , so the claimfollows by combining Proposition 2.3.7 and Construction 2.3.8. (cid:3) Topological restriction homology.
We present an alternative definition of TR as a func-tor defined on the ∞ -category of cyclotomic spectra with values in the ∞ -category of cyclotomicspectra with Frobenius lifts inspired by a result of Blumberg–Mandell [18]. Additionally, westudy TR as a localizing invariant in the sense of Blumberg–Gepner–Tabuada [16], and discussvarious descent properties of TR. Definition 2.4.1. If X is a cyclotomic spectrum, then TR ( X ) is defined byTR ( X ) = map CycSp ( ̃
THH ( S [ t ]) , X ) ≃ map CycSp ( ⊕ i ≥ Σ ∞+ ( S / C i ) , X ) . The construction X ↦ TR ( X ) determines a functor of ∞ -categories TR ∶ CycSp → Sp.
Notation 2.4.2. If C is a stable ∞ -category, then TR ( C ) = TR ( THH ( C )) . In particular, if R isan E -ring, then TR ( R ) = THH ( Perf R ) ≃ TR ( THH ( R )) .Definition 2.4.1 is based on a result of Blumberg–Mandell [18, Theorem 6.11] which asserts thatthe classical construction of TR as considered by [19, 41, 18] is corepresentable by the reducedtopological Hochschild homology of the flat affine line S [ t ] as a functor on the homotopy categoryof genuine cyclotomic spectra with values in the homotopy category of spectra. In §
3, we provethat the definition of TR given above recovers the classical definition of TR in the boundedbelow case (see Theorem 3.3.12). The following result is now an immediate consequence of theformalism developed in § Proposition 2.4.3.
The functor TR ∶ CycSp → Sp refines to a functor of ∞ -categories TR ∶ CycSp → CycSp Fr which is a right adjoint of the canonical functor CycSp Fr → CycSp .Proof.
Combine Theorem 2.3.9 and Definition 2.4.1. (cid:3)
Krause–Nikolaus [50, Proposition 10.3] show that the construction given by p -typical TRdetermines a right adjoint of the canonical functor CycSp Fr p → CycSp p , and Proposition 2.4.3extends this result to the integral situation. Remark 2.4.4.
Nikolaus–Scholze [59] prove that for every cyclotomic spectrum whose under-lying spectrum is bounded below, there is a natural equivalence of spectraTC ( X ) ≃ map CycSp ( S , X ) , where TC ( X ) denotes Goodwillie’s integral topological cyclic homology. This was conjectured byKaledin [47] and proven by Blumberg–Mandell [18] after p -completion. Definition 2.4.1 providesa similar description of TR in the bounded below case removing the otherwise instrumental useof equivariant stable homotopy theory in the construction of TR. Remark 2.4.5.
An advantage of Definition 2.4.1 is that we obtain an explicit equalizer formulafor TR. Let X be a cyclotomic spectrum whose underlying spectrum is bounded below, and notethat X hC i ≃ map Sp B T ( Σ ∞+ ( S / C i ) , X ) for every i ≥
1. Consequently, we obtain thatTR ( X ) ≃ Eq ( ∏ i ≥ X hC i ∏ p ∏ i ≥ ( X tC p ) hC i ← →← → ) , N CURVES IN K-THEORY AND TR 19 where the top map is induced by the canonical map X hC i → X tC i and the bottom map isinduced by the cyclotomic Frobenius map X → X tC p . This is a consequence of the formula forthe mapping spectrum in the ∞ -category CycSp obtained in [59, Proposition II.1.5].There is a construction of p -typical TR due to Nikolaus–Scholze [59] which only relies on theBorel equivariant homotopy theory of cyclotomic spectra, which we recall for completeness. Remark 2.4.6.
For every p -typical cyclotomic spectrum X , we let TR n + ( X, p ) denote thefollowing iterated pullback in the ∞ -category of spectra with T -action X hC pn × ( X tCp ) hCpn − X hC pn − × ( X tCp ) hCpn − ⋯ × ( X tCp ) hCp X hC p × X tCp X for each n ≥
0. The maps from the left factors to the right factors are induced by the canonicalmap X hC p → X tC p , and the maps from the right factors to the left factors are induced bycyclotomic Frobenius X → X tC p . In fact, if the underlying spectrum of X is bounded below,then the underlying spectrum of X admits the structure of a genuine T -spectrum with respect tothe family of finite p -subgroups of T , such that there is an equivalence TR n + ( X, p ) ≃ X C pn foreach n ≥
0. For each n ≥
1, there is a map of spectra with T -action R ∶ TR n + ( X, p ) → TR n ( X, p ) induced by forgetting the first factor in the iterated pullback defining TR n + ( X, p ) , and there isan equivalence of spectra with T -actionTR ( X, p ) ≃ lim ←Ð n TR n + ( X, p ) . Krause–Nikolaus [50, Definition 9.5] give a description of integral TR by similar methods asabove. However, it becomes complicated to specify coherences between the Frobenius lifts usingthis description. By similar methods as used in this paper, it is possible to show that there is anatural equivalence of p -typical cyclotomic spectra with Frobenius liftTR ( X, p ) ≃ map CycSp p ( ⊕ i ≥ Σ ∞+ ( S / C p i ) , X ) for every p -typical cyclotomic spectrum X whose underlying spectrum is bounded below.As an application, we describe TR as a localizing invariant in the sense of Blumberg–Gepner–Tabuda [16]. Let C at perf ∞ denote the ∞ -category of small stable ∞ -categories which are idem-potent complete and exact functors among them. Recall that an exact sequence in C at perf ∞ isa sequence which is both a fiber sequence and a cofiber sequence. A localizing invariant withvalues in a stable ∞ -category D is a functor E ∶ C at perf ∞ → D which carries exact sequences in C at perf ∞ to fiber sequences in D . Corollary 2.4.7.
The functor C at perf ∞ THH
ÐÐÐ→
CycSp TR ÐÐ→
CycSp Fr is a localizing invariant.Proof. The functor THH ∶ C at perf ∞ → CycSp is a localizing invariant by Blumberg–Mandell [17],and the functor TR ∶ CycSp → CycSp Fr preserves limits by virtue of Proposition 2.4.3. (cid:3) Remark 2.4.8.
We extend the definition of TR to schemes. Let Perf X denote the ∞ -categoryof perfect O X -modules for a scheme X , and define TR ( X ) = TR ( Perf X ) . Consequently, theconstruction X ↦ TR ( X ) satisfies Nisnevich descent on quasi-compact quasi-separated schemesby a result of Thomason [63] since TR is a localizing invariant. Localizing invariants are additionally required to preserve filtered colimits in [16].
We end by discussing various descent properties for TR building on the work of [13, 49, 3, 24,22]. In the following, we will let CAlg ♡ denote the category of discrete commutative rings. Corollary 2.4.9.
The construction R ↦ TR ( R ) determines a functor of ∞ -categories TR ∶ CAlg ♡ → CycSp Fr which is a sheaf for the fpqc topology on CAlg ♡ .Proof. We first show that the functor THH ∶ CAlg ♡ → CycSp is an fpqc sheaf on CAlg ♡ . As aconsequence of the construction of the ∞ -category CycSp as a lax equalizer it suffices to showthat the functors THH ∶ CAlg ♡ → Sp B T and THH tC p ∶ CAlg ♡ → Sp B T are fpqc sheaves for everyprime p . This follows from [13, Corollary 3.4 and Remark 3.5] since limits in Sp B T are computedpointwise. This shows the wanted since the functor TR ∶ CycSp → CycSp Fr preserves limits. (cid:3) The proof of Corollary 2.4.9 above shows that every descent result for THH regarded as afunctor with values in the ∞ -category of cyclotomic spectra yields a corresponding descent state-ment for TR. In [49], Keenan extends the result of Bhatt–Morrow–Scholze [13] on fpqc descentfor THH to connective E ∞ -rings, and Antieau–Nikolaus [3] prove that THH is a hypercompletesheaf with values in the ∞ -category of cyclotomic spectra for the pro-´etale topology on CAlg ♡ .In [22], Clausen–Mathew prove that THH is a Postnikov complete sheaf for the ´etale topologyon E -rings extending their previous work with Naumann and Noel in [24].3. Comparison with genuine TR
In this section, we establish the main technical result of this paper which asserts that the clas-sical construction of TR agrees with the construction of TR considered in § § ∞ -categories following Barwick [9], and em-ploy this to construct the ∞ -category of genuine T -spectra following Barwick–Glasman [11]. In § ∞ -categories. In § § Equivariant stable homotopy theory.
In this section, we briefly review the notionsfrom equivariant stable homotopy theory that will play an important role in this paper. In [38],Guillou–May establish a model for the homotopy theory of G -spectra in the sense of [53, 56, 45]for a finite group G , in terms of spectral Mackey functors, and Barwick [9] developed an ∞ -categorical approach to Mackey functors on orbital ∞ -categories. Barwick–Dotto–Glasman–Nardin–Shah revisit the result of Guillou–May in the general context of parametrized homotopytheory [10, 57]. Our goal in this section is to briefly review the formalism of Mackey functorsfollowing [9], and use this to construct the ∞ -category of genuine T -spectra following Barwick–Glasman [11]. Definition 3.1.1.
The finite coproduct completion Fin T of a small ∞ -category T is the smallestfull subcategory of P ( T ) which contains the essential image of the Yoneda embedding and which is N CURVES IN K-THEORY AND TR 21 closed under finite coproducts. A small ∞ -category T is orbital if the finite coproduct completionof T admits pullbacks. Remark 3.1.2.
Definition 3.1.1 was introduced by Barwick–Dotto–Glasman–Nardin–Shah in [10].In [33], Glasman introduced the notion of an epiorbital ∞ -category and observed that these pro-vide examples of orbital ∞ -categories. Finally, we note that the finite coproduct completion ofan orbital ∞ -category is disjunctive in the sense of Barwick [9].For instance, if G is a finite group, then the finite coproduct completion of the orbit categoryof G is equivalent to the category of finite G -sets, since every finite G -set admits a uniquedecomposition as a disjoint union of orbits. Remark 3.1.3.
The finite coproduct completion of a small ∞ -category T is characterized by auniversal property: For every ∞ -category D which admits finite coproducts, the Yoneda embed-ding j ∶ T → Fin T induces an equivalence of ∞ -categoriesFun ∐ ( Fin T , D ) → Fun ( T, D ) , where Fun ∐ ( Fin T , D ) denotes the full subcategory of Fun ( Fin T , D ) spanned by those functorswhich preserve finite coproducts. This follows from [54, Proposition 5.3.6.1] by taking K = Finand R = ∅ . Note that the inverse is determined by forming the left Kan extension along theYoneda embedding.Let T be an orbital ∞ -category, and let Span ( Fin T ) denote the ∞ -category of spans in thefinite coproduct completion of T . Concretely, the objects of Span ( Fin T ) are given by the objectsof Fin T , and a morphism from X to X ′ in Span ( Fin T ) is given by a span YX X ′ ←→ ←→ in Fin T . Additionally, if X ′ ← Y ′ → X ′′ is a morphism from X ′ to X ′′ in Span ( Fin T ) , thencomposition is defined by forming a pullback in Fin T PY Y ′ X X ′ X ′′ ←→ ←→ ←→ ←→ ←→ ← → which exists by virtue of the assumption that T is an orbital ∞ -category. The reader is invited toconsult [9, Proposition 3.4] for a precise construction of Span ( Fin T ) as a complete Segal space. Construction 3.1.4.
Let T be an orbital ∞ -category. There is a functor i ∶ Fin op T → Span ( Fin T ) which is the identity on objects, and determined by the following construction on morphisms ( X → X ′ ) ↦ ( X ′ ← X id Ð→ X ) Consequently, we obtain a functor T op → Span ( Fin T ) given by precomposing the functor i with the opposite of the canonical functor T → Fin T . Similarly, there is a functor i ′ ∶ Fin T → Span ( Fin T ) which is the identity on objects, and determined by the following construction onmorphisms ( X → X ′ ) ↦ ( X id ←Ð X → X ′ ) We obtain a functor T → Span ( Fin T ) given by precomposing the functor i ′ above with thecanonical functor T → Fin T . See [9, Notation 3.9] for a formal description of these functors.If T is an orbital ∞ -category, then the ∞ -category Span ( Fin T ) is semiadditive and the sumis given by the coproduct in Fin T (cf. [9, Proposition 4.3]). Consequently, the ∞ -category ofMackey functors on an orbital ∞ -category is defined as follows (cf. [9, Definition 6.1]): Definition 3.1.5.
Let T denote an orbital ∞ -category, and let D denote an additive ∞ -category.The ∞ -category of D -valued Mackey functors on T is defined byMack D ( T ) = Fun × ( Span ( Fin T ) , D ) , where Fun × ( Span ( Fin T ) , D ) denotes the full subcategory of Fun ( Span ( Fin T ) , D ) spanned bythose functors which preserve products. If D = Sp is the ∞ -category of spectra, then we writeMack ( T ) instead of Mack Sp ( T ) , and refer to the former as the ∞ -category of spectral Mackeyfunctors on T . Example 3.1.6.
The orbit category of a finite group G is an example of an orbital ∞ -category.The ∞ -category of spectral Mackey functors on Orb G is equivalent to the ∞ -category of genuine G -spectra by the work of Guillou–May [38]. By the ∞ -category of genuine G -spectra we meanthe underlying ∞ -category of the category of orthogonal G -spectra equipped with the modelstructure established by Mandell–May in [56]. Alternatively, we refer to Nardin [57, Theorem A.4]or Clausen–Mathew–Naumann–Noel [23, Appendix A] for a direct comparison using Barwick’smodel of spectral Mackey functors.We construct the ∞ -category of genuine T -spectra with respect to the family of finite cyclicsubgroups of T using the formalism of spectral Mackey functors on orbital ∞ -categories followingBarwick–Glasman [11]. An alternative construction has been obtained by Ayala–Mazel-Gee–Rozenblyum [7], and we refer the reader to the foundational work of Lewis–May–Steinberger [53]and Mandell–May [56] for the classical approach. Definition 3.1.7.
The orbit ∞ -category Orb T of the circle is defined as the full subcategory ofthe ∞ -category S B T spanned by those spaces with T -action of the form T / C n for every n ≥ ∞ -category of the circle is equivalent to the cyclonic category of Barwick–Glasman [11,Definition 1.10] as explained in [11, Remark 1.13], so we conclude that the orbit ∞ -category Orb T is orbital by virtue of Barwick–Glasman [11, Proposition 1.25.1]. Definition 3.1.8.
The ∞ -category of genuine T -spectra is defined by Sp T = Mack ( Orb T ) . Remark 3.1.9.
The ∞ -category of genuine T -spectra is equivalent to the underlying ∞ -categoryof orthogonal T -spectra with respect to the family of finite subgroups of T , equipped with thestable model structure established by Mandell–May [56]. This is proved by Barwick–Glasman [11,Theorem 2.8], where Sp T is referred to as the ∞ -category of cyclonic spectra. N CURVES IN K-THEORY AND TR 23
We construct an action of the multiplicative monoid N × on the ∞ -category Sp T of genuine T -spectra which in turn is determined by an action of N × on the orbit ∞ -category Orb T . Construction 3.1.10.
Let
Orb T denote the category whose objects are given by the orbits T / C n for every integer n ≥
1, and whose morphisms are given by T -equivariant maps. For everypair of positive integers m and n , there is a canonical bijectionHom Orb T ( T / C m , T / C n ) ≃ ( T / C n ) C m , where the right hand side is equipped with the canonical topology. As a consequence, we mayregard Orb T as a topological category. The underlying ∞ -category of Orb T is equivalent to theorbit ∞ -category Orb T as defined in Definition 3.1.7. For every integer k ≥
1, the construction T / C n ↦ T / C kn determines a functor of topological categories i k ∶ Orb T → Orb T which in turndefines an action of the multiplicative monoid N × on Orb T . This defines an action of N × onthe orbit ∞ -category Orb T which is determined by a functor of ∞ -categories B N × → C at ∞ .The following result is due to Barwick–Glasman [11, Lemma 3.2]. Lemma 3.1.11.
For every positive integer k , there is an adjunction of ∞ -categories Fin
Orb T Fin
Orb T ← → Fin ( i k ) ←→ p k where the left adjoint Fin ( i k ) additionally preserves finite coproducts. The action of the multi-plicative monoid N × on the ∞ -category Orb T extends to an action on the ∞ -category Sp T .Proof. See [11, Lemma 3.2]. (cid:3)
We have that the action of the multiplicative monoid N × on the ∞ -category Sp T of genuine T -spectra afforded by Lemma 3.1.11 determines a functor of ∞ -categories F Ψ ∶ B N × → C at ∞ which is determined by the construction which sends k ∈ N × to the endofunctor of the ∞ -categorySp T determined by the construction X ↦ X C k . Furthermore, there is an action of the multiplica-tive monoid N × on the ∞ -category Sp T given by the geometric fixed points construction ( − ) Φ C k .We briefly discuss this, and refer the reader to [11, Notation 3.4] for a complete treatment. Example 3.1.12.
For every positive integer k , there is an adjunction of ∞ -categoriesSp T Sp T , ← → (−) Φ Ck ←→ Span ( p k ) ∗ where the left adjoint ( − ) Φ C k is defined by left Kan extension along Span ( p k ) ∗ , where p k denotes aright adjoint of Fin ( i k ) as in Lemma 3.1.11. Consequently, there is an action of the multiplicativemonoid N × on Sp T by functoriality, and there is a functor of ∞ -categories F Φ ∶ B N × → C at ∞ which is determined by the construction which sends k ∈ N × to the endofunctor of the ∞ -categoryof genuine T -spectra determined by the construction X ↦ X Φ C k . Topological Cartier modules.
We introduce the notion of an integral topological Cartiermodule using the formalism of Mackey functors on orbital ∞ -categories as reviewed in § p -typical situation. Furthermore, we prove ageneral version of the Segal–tom Dieck splitting for Mackey functors on orbital ∞ -categories, anduse this describe the free topological Cartier module on a cyclotomic spectrum with Frobeniuslifts. The starting point is the following result: Lemma 3.2.1.
The ∞ -category B W is orbital, and there is an equivalence of ∞ -categories B W ≃ ( Orb T ) h N × , where the multiplicative monoid N × acts on the orbit ∞ -category Orb T as in Construction 3.1.10.Proof. We first prove that the ∞ -category B W is orbital. Let Mlfd c denote the topologicalcategory whose objects are given by compact oriented 1-manifolds and whose space of morphismsbetween a pair of compact oriented 1-manifolds is given by the set of covering maps of positivedegree equipped with the compact-open topology. In the following, we will let Mlfd c denote theunderlying ∞ -category of the topological category Mlfd c . The construction ∗ ↦ S determinesa functor of ∞ -categories B W ↪ Mlfd c which is fully faithful since there is an equivalence of E -monoids W ≃ End
Mlfd c ( S ) . We obtain a coproduct-preserving functor of ∞ -categoriesFin B W → Mlfd c given by forming the left Kan extension of the functor B W ↪ Mlfd c along the Yoneda embeddingB W → Fin B W by virtue of Remark 3.1.3, and this functor is given by the construction ∗∐⋯∐∗ ↦ S ∐ ⋯ ∐ S . Consequently, it suffices to show that the ∞ -category Mlfd c admits pullbacks sincethe functor Fin B W → Mlfd c reflects pullbacks. To this end, it suffices to show that the diagram S S S ←→ α m ← → α n admits a pullback in Mfld c , where α k denotes the k th power map of S for every integer k ≥ g = gcd ( m, n ) denote the greatest common divisor of m and n . Then the following diagram S ∐ ⋯ ∐ S S S S ← → α ng ○( gα g ) ←→ α mg ○( gα g ) ←→ α m ← → α n is a pullback of the diagram above in the ∞ -category Mfld c , where gα g denotes the self-map ofthe g -fold coproduct of S with itself defined by the formula gα g = α g ∐ ⋯ ∐ α g . This proves thatthe ∞ -category B W is orbital, so it remains to prove that there is an equivalence of ∞ -categoriesB W ≃ ( Orb T ) h N × . The construction which carries an element ( λ, k ) of the Witt monoid W tothe T -equivariant self-map of T = T / C given by the multiplication by λ k determines a functorof topological categories B W → Orb T . The construction which carries a T -equivariant map T / C m → T / C n to the element ( λ, n ) of W with λC n ∈ ( T / C n ) C m , determines a functor of N CURVES IN K-THEORY AND TR 25 topological categories
Orb T → B W . It is straightforward to check these functors of topolog-ical categories induce an equivalence of ∞ -categories B W → Orb T → ( Orb T ) h N × using that ( Orb T ) h N × consists of a single object represented by the orbit T / C . (cid:3) We define the ∞ -category of topological Cartier modules as follows: Definition 3.2.2.
The ∞ -category of D -valued topological Cartier modules is defined byTCart D = Mack D ( B W ) = Fun × ( Span ( Fin B W ) , D ) , where D denotes an additive ∞ -category. If D = Sp, then we will write TCart instead of TCart Sp ,and refer to the former as the ∞ -category of topological Cartier modules. Remark 3.2.3.
The functor B W op → Span ( Fin B W ) induces a functor of ∞ -categoriesTCart → CycSp Fr which regards the underlying spectrum of a topological Cartier module as a cyclotomic spectrumwith Frobenius lifts, where the underlying spectrum of a topological Cartier module is definedas the spectrum with T -action obtained by precomposing the functor above with the forgetfulfunctor CycSp Fr → Sp B T . The functor B W → Span ( Fin B W ) induces a functor of ∞ -categoriesTCart → Fun ( B W , Sp ) , so as a consequence, if M is a topological Cartier module, then for every integer k ≥
1, there isa T -equivariant endomorphism of the underlying spectrum with T -action of M , where the T -action on the source is given by restricting the T -action along the k th power map. Equivalently,such an endomorphism is given by a T -equivariant map of spectra V k ∶ M hC k → M , which wewill refer to as the k th Verschiebung map of M . Remark 3.2.4.
Antieau–Nikolaus [3, Definition 3.1] define a p -typical topological Cartier mod-ule as a spectrum M with T -action equipped with the datum of a T -equivariant factorization M hC p V Ð→ M F Ð→ M hC p of the norm map for the cyclic group C p . In Remark 3.2.9, we make the discussion in Remark 3.2.3precise by showing that the underlying spectrum of a topological Cartier module in the senseof Definition 3.2.2 canonically admits the structure of a p -typical topological Cartier module forevery prime number p . Example 3.2.5. If M denotes a topological Cartier module, then the abelian group π M comesequipped with a pair of endomorphism V k = i ′ ( k ) and F k = i ( k ) for every positive integer k ,where i ′ and i denote the functors defined in Construction 3.1.4. Let g = gcd ( m, n ) denote thegreatest common divisor of a pair of positive integers m and n . Then the commutative diagram ∗ ∐ ⋯ ∐ ∗ ∗∗ ∗ ← → ng ○( g ⋅ id ) ←→ mg ○ ( g ⋅ id ) ←→ m ← → n is a pullback in the finite coproduct completion Fin B W , so we conclude that F m V n = gcd ( m, n ) V ng F mg .In particular, we have that F k V k = k ⋅ id for every positive integer k , and if gcd ( m, n ) =
1, then F m V n = V n F m . We will refer to this as a Cartier module structure on π M (see Zink [67]). Thisstructure arises frequently in algebra; for instance if R is a commutative ring, then the ring ofbig Witt vectors W ( R ) and the ring of rational Witt vectors W rat ( R ) both admit the structureof a Cartier module in this sense.We show that the ∞ -category of topological Cartier modules is equivalent to the ∞ -categoryof genuine topological Cartier modules as defined by Antieau–Nikolaus [3, Definition 5.1]. Definition 3.2.6.
The ∞ -category of genuine topological Cartier modules is defined byTCart gen = lim ( F Ψ ∶ B N × → C at ∞ ) = ( Sp T ) h N × , where the multiplicative monoid N × acts on the ∞ -category Sp T as in Lemma 3.1.11. Remark 3.2.7.
Unwinding the definition, we see that an object of TCart gen is given by a genuine T -spectrum M with compatible equivalences of genuine T -spectra M ≃ M C k for every k ≥ Proposition 3.2.8.
There is an equivalence of ∞ -categories TCart ≃ TCart gen .Proof.
There is a sequence of equivalences of ∞ -categoriesTCart = Mack ( B W ) ≃ Ð→ Mack (( Orb T ) h N × ) ≃ Mack ( Orb T ) h N × = TCart gen by virtue of Lemma 3.2.1 and Lemma 3.1.11 which shows the wanted. (cid:3)
Antieau–Nikolaus prove that there is an equivalence TCart p ≃ TCart gen p [3, Proposition 5.5],and Proposition 3.2.8 provides an integral version of this result. Note that these comparisonresults hold unconditionally in contrast to the comparison between the ∞ -category of cyclotomicspectra and the ∞ -category of genuine cyclotomic spectra established by Nikolaus–Scholze [59],where it is crucial to restrict to the bounded below case. Remark 3.2.9.
There is a canonical sequence of functors of ∞ -categoriesTCart → Sp T → P Sp ( Orb T ) → Sp B T , where the first functor exists by virtue of Proposition 3.2.8. In particular, if M is a topologicalCartier module, then there is a commutative diagram of spectra with T -action M hC k M M Φ C k M hC k M hC k M tC k ← → ←→ ≃ ← → ←→ ←→ ← → Nm Ck ← → can for every positive integer k , where we have used that M C k ≃ M in Sp T . This diagram wasintroduced by Hesselholt–Madsen [41] building on work of Greenlees–May [36], and is commonlyreferred to as the isotropy-separation diagram. Note that the commutative square on the left inthe diagram above exhibits the underlying spectrum with T -action of M as a p -typical topologicalCartier module for every prime number p in the sense of Antieau–Nikolaus [3]. Remark 3.2.10.
A detailed treatment of the ∞ -category of topological Cartier modules extend-ing the results of Antieau–Nikolaus [3] to the integral situation will be the content of forthcomingwork. N CURVES IN K-THEORY AND TR 27
We obtain an explicit identification of the topological Cartier module obtained from a cyclo-tomic spectrum with Frobenius lifts by freely adjoining Verschiebung maps. It follows formallyfrom [9, Proposition 6.5], that there is an adjunction of ∞ -categories(1) CycSp Fr TCart ← →
Free ←→ where Free denotes a left adjoint of the functor obtained by restricting along B W op → Span ( Fin B W ) .Antieau–Nikolaus [3] give a direct construction of the p -typical analogue of Free, but this is not aviable approach in the integral situation due to the infinite hierarchy of coherences that need tobe specified to define such a functor. We will resolve this issue by establishing a general versionof the Segal–tom Dieck splitting for spectral Mackey functors on orbital ∞ -categories. We willbegin by recalling the usual Segal–tom Dieck splitting [64, 53] in equivariant stable homotopytheory (see [61] for a classical treatment). For a finite group G , there is an adjunction(2) P Sp ( Orb G ) Sp G ← → F ←→ where the right adjoint is obtained by restricting along the functor Orb op G → Span ( Fin G ) , andthe left adjoint F is given by sending a spectral presheaf X on Orb G to the genuine G -spectrumwith F ( X ) G ≃ ⊕ ( H ) ( X H ) h W G ( H ) , where the sum is indexed by conjugacy classes of subgroups of G , and W G ( H ) is the Weyl group.In general, if Y is a spectral presheaf on Orb G , then we define Y H = Y ( G / H ) for every subgroup H of G . The idea is to regard the adjunction in (1) as an instance of this adjunction, where wereplace the orbit category of G with an arbitrary orbital ∞ -category. Notation 3.2.11.
Let T be an orbital ∞ -category, and let α denote the composite T op → Fin op T i Ð→ Span ( Fin T ) defined in Construction 3.1.4.If D is an ∞ -category which admits small colimits, then there is an adjunction of ∞ -categoriesFun ( T op , D ) Fun ( Span ( Fin T ) , D ) ← → α ! ←→ α ∗ where the left adjoint α ! is given by left Kan extension along α , and the right adjoint α ∗ is givenby restricting along α . In Proposition 3.2.12, whose statement and proof was indicated to theauthor by Jay Shah, we obtain an explicit formula for the functor α ! . Similar results have beenobtained by Glasman [33] and Bachmann–Hoyois [8, Appendix C]. In [9, Theorem A.9], Barwickproves a version of the Segal–tom Dieck splitting for coherent topoi. Proposition 3.2.12. If F is a presheaf on an orbital ∞ -category T with values in a semiadditive ∞ -category D which admits small colimits, then the functor α ! F preserves finite products, andthere is a natural equivalence ( α ! F )( t ) ≃ colim ( t ′ → t )∈( T / t ) ≃ F ( t ′ ) for every object t of T . Proof.
Let F ′ denote the left Kan extension of F along T op → Fin op T , and note that the functor F ′ preserves coproducts by virtue of Remark 3.1.3. If X is an object Fin T , then ( Fin T ) / X → Span ( Fin T ) / X admits a left adjoint determined by ( Y ← Z → X ) ↦ ( Z → X ) . We conclude that the functorΦ ∶ ( Fin T ) / X × Span ( Fin T ) Fin op T → Span ( Fin T ) / X × Span ( Fin T ) Fin op T obtained as the basechange of the functor above along i also admits a left adjoint by functoriality,so Φ is cofinal. The source of Φ is equivalent to the wide subcategory of ( Fin T ) / X on thosemorphisms that are equivalences in Fin T . Combining these two observations, we conclude that ( i ! F ′ )( X ) ≃ colim ( Y → X )∈(( Fin T ) / X ) ≃ F ′ ( Y ) since the target of Φ above is the index category for computing the value of the left Kan extension F ′ along i on the object X . If X ′ is an additional object of Fin T , then the canonical functor ( Fin T ) / X × ( Fin T ) / X ′ → ( Fin T ) / X ∐ X ′ is an equivalence since T is orbital, so we have that ( i ! F ′ )( X ∐ X ′ ) ≃ colim ( Y → X ∐ X ′ )∈(( Fin T ) / X ∐ X ′ ) ≃ F ′ ( Y ) ≃ colim ( Y → X )∈(( Fin T ) / X ) ≃ colim ( Y ′ → X ′ )∈(( Fin T ) / X ′ ) ≃ F ′ ( Y ) ⊕ F ′ ( Y ′ ) ≃ ( i ! F ′ )( X ) ⊕ ( i ! F ′ )( X ′ ) , where we additionally have used that the functor F ′ preserves finite coproducts in the secondequivalence. This proves that α ! F preserves finite products since α ! F ≃ i ! F ′ . It remains to provethe final assertion. If t is an object of T , then j ∶ T → Fin T induces a functorcolim ( t ′ → t )∈( T / t ) ≃ F ( t ′ ) → colim ( Y → t )∈(( Fin T ) / t ) ≃ F ′ ( Y ) which we claim is an equivalence. First note that there is an equivalencecolim ( t ′ → t )∈( T / t ) ≃ F ( t ′ ) ≃ ⊕ ( t ′ → t )∈ T / t F ( t ′ ) h Aut ( t ′ ) , where Aut ( t ′ ) denotes the group of automorphisms of ( t ′ → t ) regarded as an object of T / t . If Y is an object of ( Fin T ) / t , then we may write Y ≃ t ∐ ⋯ ∐ t n as a finite coproduct of objects of T / t . Using that F ′ preserves finite coproducts, we obtain a map F ′ ( Y ) ≃ F ( t ) ∐ ⋯ ∐ F ( t n ) → F ( t ) h Aut ( t ) ∐ ⋯ ∐ F ( t n ) h Aut ( t n ) ↪ ⊕ ( t ′ → t )∈ T / t F ( t ′ ) h Aut ( t ′ ) which induces a map colim ( Y → t )∈(( Fin T ) / t ) ≃ F ′ ( Y ) → colim ( t ′ → t )∈( T / t ) ≃ F ( t ′ ) , and this map provides an inverse to the map that we claimed was an equivalence above. (cid:3) Example 3.2.13.
Let G be a finite group, and consider the adjunction in (2). If X ∶ Orb op G → Spis a spectral presheaf on the orbit category of G , then there is an equivalence F ( X ) G ≃ colim G / H → G / G ∈(( Orb G ) /( G / G ) ) ≃ X ( G / H ) ≃ ⊕ ( H ) ( X H ) hW G ( H ) , by Proposition 3.2.12. This is precisely the classical Segal–tom Dieck splitting as discussed above. N CURVES IN K-THEORY AND TR 29
Recall that there is an adjunction of ∞ -categoriesCycSp Fr TCart ← →
Free ←→ where Free denotes a left adjoint of the forgetful functor. As a consequence of Proposition 3.2.12,we obtain an explicit description of Free which we will use in the proof of Theorem 3.3.12, wherewe show that the classical definition of TR coincides with the definition considered in § Corollary 3.2.14. If X is a cyclotomic spectrum with Frobenius lifts, then there is an equivalenceof spectra with T -action Free ( X ) ≃ ⊕ n ≥ X hC n , The k th Verschiebung map of Free ( X ) is given by the canonical inclusion Free ( X ) hC k ≃ ⊕ n ≥ X hC kn ↪ ⊕ n ≥ X hC n ≃ Free ( X ) for every integer k ≥ , where the Verschiebung maps of X are defined in Remark 3.2.3.Proof. The first assertion follows directly from Proposition 3.2.12, so it remains to identify theVerschiebung maps of Free ( X ) . For every prime number p , we let Free p denote a left adjoint ofthe forgetful functor TCart p → CycSp Fr p . If X is a p -typical cyclotomic spectrum with Frobeniuslift, then there is an equivalence of spectra with T -actionFree p ( X ) ≃ ⊕ i ≥ X hC pi by virtue of [3, Lemma 4.1], where the p th Verschiebung of Free p ( X ) is given by the canonicalinclusion under this identification. Since the following diagram commutesTCart TCart p CycSp Fr CycSp Fr p ← → ←→ ←→ ← → we obtain a canonical map of p -typical topological Cartier modules Free p ( X ) → Free ( X ) forevery cyclotomic spectrum X with Frobenius lifts, such that the underlying map of spectra with T -action is given by the canonical inclusion. This identifies the p th Verschiebung of Free ( X ) aswanted. (cid:3) Remark 3.2.15.
The functor Free restricts to a functor of ∞ -categoriesFree ∶ CycSp Fr ♭ → TCart ♭ , where the decoration ( − ) ♭ denotes the full subcategory spanned by those objects whose underlyingspectrum is bounded below, since the functor ( − ) hC k preserves connectivity for every integer k ≥ Genuine cyclotomic spectra and genuine TR.
We recall the construction of TR basedon the notion of a genuine cyclotomic spectrum following Hesselholt–Madsen [41] and Blumberg–Mandell [18]. We establish an adjunction between the ∞ -category of topological Cartier modulesas introduced in § ∞ -category of genuine cyclotomic spectra whose right adjoint isgiven by TR. Finally, we prove Theorem 3.3.12 which asserts that the classical construction ofTR agrees with the construction discussed in § Definition 3.3.1.
The ∞ -category of genuine cyclotomic spectra is defined byCycSp gen = lim ( F Φ ∶ B N × → C at ∞ ) = ( Sp T ) h N × , where the multiplicative monoid N × acts on the ∞ -category Sp T as in Example 3.1.12. Remark 3.3.2.
Unwinding the definition, we see that an object of the ∞ -category CycSp gen isgiven by a genuine T -spectrum X with compatible equivalences of genuine T -spectra X ≃ X Φ C k for every integer k ≥ Remark 3.3.3.
There is a canonical sequence of functors of ∞ -categoriesCycSp gen → Sp T → Sp B T which extracts the underlying spectrum with T -action of a genuine cyclotomic spectrum. Let X denote a genuine cyclotomic spectrum. For every prime number p , the T -equivariant map X → X Φ C p → X tC p exhibits the underlying spectrum with T -action of X as a cyclotomic spectrum in the sense ofNikolaus–Scholze [59]. The main result of Nikolaus–Scholze [59, Theorem II.6.9] asserts that theconstruction above determines a functor of ∞ -categoriesCycSp gen → CycSpwhich restricts to an equivalence on the full subcategories of those objects whose underlyingspectrum is bounded below.
Remark 3.3.4. If R is a connective E -ring, then B¨okstedt–Hsiang–Madsen [19] and Hesselholt–Madsen [41] prove that THH ( R ) admits the structure of a genuine cyclotomic spectrum byusing the B¨okstedt construction. Angeltveit–Blumberg–Gerhardt–Hill–Lawson–Mandell [1] con-struct the genuine cyclotomic structure on THH ( R ) using the Hill–Hopkins–Ravenel norm [45],and these constructions are equivalent by the work of Dotto–Malkiewich–Patchkoria–Sagave–Woo [27]. Nikolaus–Scholze [59] construct a cyclotomic structure on THH ( R ) using the Tate-valued diagonal.We recall the classical construction of TR following Blumberg–Mandell [18]. N CURVES IN K-THEORY AND TR 31
Construction 3.3.5.
Let X be a genuine cyclotomic spectrum. For every pair of positive integer ( m, n ) with m = ln , the restriction map R l ∶ X C m → X C n is the map of genuine T -spectra givenby R l ∶ X C m ≃ ( X C l ) C n → ( X Φ C l ) C n ≃ Ð→ X C n , where the final equivalence is induced by the genuine cyclotomic structure of X . Furthermore,the construction n ↦ X C n determines a functor of ∞ -categories X C (−) ∶ ( N , div ) → Sp T , where ( N , div ) denotes set of natural numbers regarded as a poset with respect to the divisibilityrelation, which means that there is a morphism m → n precisely if n divides m . Definition 3.3.6.
Let TR gen denote the functor of ∞ -categoriesTR gen ∶ CycSp gen → Sp T determined by the construction X ↦ lim ( N , div ) X C (−) . Remark 3.3.7.
The construction X ↦ TR gen canonically refines to a functor of ∞ -categoriesTR gen ∶ CycSp gen → TCart , where we have used Proposition 3.2.8 and that the functor ( − ) C k preserves limits for every k ≥ gen ∶ CycSp gen → TCart restricts to a functorTR gen ∶ CycSp ♭ ≃ Ð→ CycSp gen ♭ → TCart ♭ , where the decoration ( − ) ♭ denotes the full subcategory spanned by those objects whose underlyingspectrum is bounded below, and the first equivalence follows from Nikolaus–Scholze [59].We recall an important construction due to Antieau–Nikolaus [3, Example 3.5]. If M is a p -typical topological Cartier module, then there is a cofiber sequence of spectra with T -action M hC p V Ð→ M → M / V, and the cofiber M / V canonically admits the structure of a p -typical cyclotomic spectrum. Theconstruction M ↦ M / V determines a functor of ∞ -categories ( − )/ V ∶ TCart p → CycSp p , which admits a right adjoint given by p -typical TR by Antieau–Nikolaus [3, Proposition 3.17]using Krause–Nikolaus [50, Proposition 10.3]. We obtain an integral version of the functor ( − )/ V .Note that if we regard M as a genuine p -typical topological Cartier module, then the underlyingspectrum with T -action of M Φ C p is equivalent to M / V by the isotropy-separation sequence. Construction 3.3.8.
Let M be a genuine topological Cartier module. For every pair of positiveintegers ( m, n ) with m = ln , consider the map of genuine T -spectra defined by M Φ C n ≃ Ð→ ( M C l ) Φ C n → M Φ C m , where the first equivalence is induced by the genuine topological Cartier module structure of M .Furthermore, the construction n ↦ M Φ C n determines a functor of ∞ -categories M Φ C (−) ∶ ( N , div ) op → Sp T as in Construction 3.3.5. Remark 3.3.9.
The construction M ↦ colim Φ op M Φ C (−) refines to a functor of ∞ -categoriesL ∶ TCart → CycSp gen , where we have used Proposition 3.2.8 and that the functor ( − ) Φ C k preserves colimits for every k ≥
1. Additionally, the functor L ∶ TCart → CycSp gen restricts to a functor of ∞ -categoriesL ∶ TCart ♭ → CycSp gen ♭ ≃ Ð→ CycSp ♭ as in Remark 3.3.7. Remark 3.3.10.
Let ( N , ≤ ) denote the set of natural numbers regarded as a poset with respectto the usual ordering, which means that there is a morphism m → n in ( N , ≤ ) precisely if n ≤ m .The functor ( N , ≤ ) → ( N , div ) determined by the construction n ↦ n ! is initial . Consequently,if X is a genuine cyclotomic spectum, then there is an equivalence of topological Cartier modulesTR gen ( X ) ≃ lim ←Ð ( ⋯ → X C → X C → X ) , where the maps are given by X C ( n + ) ! ≃ ( X C n + ) C n ! → ( X Φ C n + ) C n ! ≃ X C n ! . On the other hand,if M is a topological Cartier module, then there is an equivalence of genuine cyclotomic spectraL ( M ) ≃ lim Ð→ ( M → M Φ C → M Φ C → ⋯ ) , where the maps are given by M Φ C n ! ≃ ( M C n + ) Φ C n ! → ( M Φ C n + ) Φ C n ! ≃ M Φ C ( n + ) ! .Using Remark 3.3.10, we show that the functors L and TR gen form an adjunction. We followthe proof given by Antieau–Nikolaus [3, Proposition 5.2] verbatim. Proposition 3.3.11.
There is an adjunction of ∞ -categories TCart CycSp gen ← → L ←→ TR gen Proof.
Let X be a genuine cyclotomic spectrum and let M be a topological Cartier module.There is a natural transformation of functors η ∶ id → TR gen ○ L incuced by the canonical map M ≃ lim m ∈ N M C m ! → lim m ∈ N ( colim n ∈ N M Φ C n ! ) C m ! , by virtue of Remark 3.3.10, and we show that η induces an equivalence of spacesMap CycSp gen ( L ( M ) , X ) → Map
TCart ( M, TR gen ( X )) . The fully faithful functor of ∞ -categoriesCycSp gen ↪ coAlg {( − ) Φ Ck ! } ( Sp T ) admits a left adjoint given by X ↦ colim n ∈ N X Φ C n ! by [59, Section II.5], so we conclude thatMap CycSp gen ( L ( M ) , X ) ≃ Map coAlg {(−) Φ Ck ! } ( Sp T ) ( M, X ) , where M is regarded as a coalgebra using the map M ≃ M C k ! → M Φ C k ! for each k ≥
1. Similarly,we find that the fully faithful functor of ∞ -categoriesTCart ↪ Alg {( − ) Ck ! } ( Sp T ) The author learned this observation from Markus Land.
N CURVES IN K-THEORY AND TR 33 admits a right adjoint given by M ↦ lim m ∈ N M C m ! , so we conclude thatMap TCart ( M, TR gen ( X )) ≃ Map
Alg {(−) Ck ! } ( Sp T ) ( M, X ) , where X is regarded as an algebra using the map X C k ! → X Φ C k ! ≃ X for each k ≥
1. As aconsequence, it suffices to show that the identity defines a natural equivalence of spacesMap coAlg {(−) Φ Ck ! } ( Sp T ) ( M, X ) ≃ Map
Alg {(−) Ck ! } ( Sp T ) ( M, X ) but this follows by induction on k using the equalizer formula for the mapping spaces obtainedin [59, Section II.5], together with the compatible equivalences M C k ≃ M and X Φ C k ≃ X for each k ≥ (cid:3) We prove the main result of this section. As mentioned previously, this result is inspired bythe result of Blumberg–Mandell [18] which asserts that TR gen is corepresentable by the reducedtopological Hochschild homology ̃ THH ( S [ t ]) as a functor defined on the homotopy category ofthe ∞ -category of genuine cyclotomic spectra with values in the homotopy category of spectra(see [18, Theorem 6.12]). Theorem 3.3.12.
There is a natural equivalence of cyclotomic spectra with Frobenius lifts TR gen ( X ) ≃ map CycSp ( ̃
THH ( S [ t ]) , X ) for every cyclotomic spectrum X whose underlying spectrum is bounded below. The proof of Theorem 3.3.12 relies on a genuine version of the Tate orbit lemma [59, LemmaI.2.1] obtained by Antieau–Nikolaus [3].
Lemma 3.3.13. If Y is a genuine T -spectrum, then the canonical map ( Y C p ) Φ C p → Y Φ C p is an equivalence in the ∞ -category Sp T for every prime p .Proof. See Antieau–Nikolaus [3, Lemma 5.3]. (cid:3)
Proof of Theorem 3.3.12.
First note that the compositeCycSp Fr ♭ Free
ÐÐ→
TCart ♭ L Ð→ CycSp gen ♭ ≃ Ð→ CycSp ♭ is equivalent to the canonical functor CycSp Fr ♭ → CycSp ♭ as a consequence of Corollary 3.2.14combined with the formula for the functor L. Indeed, we have thatL Free ( X ) ≃ lim Ð→ ( ⊕ k ≥ X hC k pr Ð→ ⊕ k ≥ ∤ k X hC k pr Ð→ ⊕ k ≥ , ∤ k X hC k pr Ð→ ⋯ ) ≃ X, by Lemma 3.3.13, thus it follows that L Free ( X ) ≃ X as objects of the ∞ -category CycSp gen .Now, since every functor in the composite above is a left adjoint, we conclude that the compositeCycSp ♭ ≃ Ð→ CycSp gen ♭ TR gen ÐÐÐ→
TCart ♭ → CycSp Fr ♭ determines a right adjoint of the canonical functor CycSp Fr ♭ → CycSp ♭ , which shows the wantedby virtue of Proposition 2.4.3. (cid:3) Applications to curves on K-theory
We discuss an application of Theorem 3.3.12. Specifically, we prove that TR evaluated on aconnective E -ring R admits a description in terms of the spectrum of curves on the algebraicK-theory of R , extending work of Hesselholt [39] and Betley–Schlichtkrull [12].4.1. Topological Hochschild homology of truncated polynomial algebras.
In this sec-tion, we obtain a convenient description of the cyclotomic structure on lim ←Ð ̃ THH ( S [ t ]/ t n ) whichwill be instrumental in § Z [ t ]/ t n as acommutative differential graded algebra, and deduce a connectivity estimate for THH ( S [ t ]/ t n ) .Similar computations have previously been obtained by Hesselholt–Madsen [41, 40] building onthe work of [37, 51]. We refer the reader to Speirs [62] for a summary of these computations.We will begin by reviewing the notion of a graded object in a symmetric monoidal ∞ -category. Notation 4.1.1.
Let Z ds ≥ denote the set of non-negative integers regarded as a discrete category,and let C denote a presentable symmetric monoidal ∞ -category.(1) The ∞ -category of graded objects of C is defined byGr ( C ) = Fun ( Z ds ≥ , C ) , which we regard as a symmetric monoidal ∞ -category using the Day convolution symmetricmonoidal structure coming from the symmetric monoidal structure on Z ds ≥ given by multi-plication. We will denote an object X of Gr ( C ) by { X i } i ≥ . For every integer n ≥
0, theinclusion { n } ↪ Z ds ≥ induces an evaluation functorev n ∶ Gr ( C ) → C which is determined by the construction { X i } i ≥ ↦ X n .(2) The projection Z ds ≥ → { } induces a functor C → Gr ( C ) which admits a left adjointund ∶ Gr ( C ) → C determined by forming the left Kan extension along the projection Z ds ≥ → { } . Concretely,the functor und ∶ Gr ( C ) → C is determined by the construction { X i } i ≥ ↦ ⊕ i ≥ X i . Thefunctor C → Gr ( C ) canonically admits a symmetric monoidal structure since the projection Z ds ≥ → { } is a map of commutative monoids, so the left adjoint und ∶ Gr ( C ) → C canonicallyrefines to a symmetric monoidal functor of ∞ -categories (see [58, Corollary 3.8]). Remark 4.1.2. If { X i } i ≥ is a graded spectrum, then the underlying spectrum X = und ({ X i } i ≥ ) of { X i } i ≥ carries an additional grading on its homotopy groups using the formula π ∗ ( X ) ≃ ⊕ i ≥ π ∗ ( X i ) . If x ∈ π ∗ ( X i ) , then we will write w ( x ) = i , and think of this as the horizontal grading direction.See Example 4.1.8 and the discussion following the proof of Proposition 4.1.10.A pointed monoid is a monoid object in the category of pointed sets equipped with the smashproduct symmetric monoidal structure. For every n ∈ N ∪ { ∞ } , we will let Π n = { , , t, . . . , t n − } denote the pointed monoid with 0 as basepoint and whose multiplication is determined by t n = N CURVES IN K-THEORY AND TR 35
In the following, we will regard the pointed monoid Π n as an object of the ∞ -category CAlg ( S ∗ ) of pointed E ∞ -monoids, where the ∞ -category S ∗ of pointed spaces is equipped with the smashproduct symmetric monoidal structure. The ∞ -category CAlg ( S ∗ ) inherits the structure of asymmetric monoidal ∞ -category whose unit is given by the pointed monoid { , } . Example 4.1.3.
In this example, we endow the underlying pointed space of Π ∞ with two distinctgraded pointed E ∞ -monoid structures:(1) The functor Z ds ≥ → S ∗ defined by i ↦ { , t i } endows the underlying pointed space of Π ∞ with the structure of an object of the ∞ -category Gr ( S ∗ ) of graded pointed spaces with t ingrading degree 1, and we will denote this object by Π w ( t ) = ∞ . We have that Π w ( t ) = ∞ canonicallyrefines to an object of the ∞ -category CAlg ( Gr S ∗ ) since the grading is compatible with themonoid structure on Π ∞ .(2) The functor Z ds ≥ → S ∗ defined by i ↦ { , t j } if i = jn and by i ↦ { } if n does not divide i ,endows the underlying pointed space of Π ∞ with the structure of an object of the ∞ -categoryGr ( S ∗ ) with t in grading degree n , and we will similarly denote this object by Π w ( t ) = n ∞ . Asbefore, we have that Π w ( t ) = n ∞ canonically refines to an object of CAlg ( Gr S ∗ ) .For every integer n ≥
1, the assignment t ↦ t n determines a map of pointed monoids Π ∞ → Π ∞ which canonically refines to a map of E ∞ -algebras Π w ( t ) = n ∞ → Π w ( t ) = ∞ in graded pointed spaces.We have the following result which will play an important role in the following. Lemma 4.1.4.
The following square is a pushout of E ∞ -algebras in graded pointed spaces Π w ( t ) = n ∞ Π w ( t ) = ∞ { , } Π w ( t ) = n ← → t ↦ t n ←→ t ↦ ←→ ← → Proof.
We first prove the following general assertion: Assume that M is a Π ∞ -module in S ∗ ,and let M ′ denote the cofiber of the endomorphism of M given by multiplication by t . Then thecanonical map M → M ′ induces an equivalence of Π ∞ -modules M ⊗ Π ∞ { , } → M ′ . We may assume that M = Π ∞ since the unit Π ∞ generates the ∞ -category Mod Π ∞ ( S ∗ ) undercolimits, and in this case the assertion is true by inspection. To prove the assertion of the lemma,we note that it suffices to prove that the diagram above is a pushout of E ∞ -algebras in S ∗ afterapplying the forgetful functor CAlg ( Gr S ∗ ) → CAlg ( S ∗ ) . In this case, the pushout is given byΠ ∞ ⊗ Π ∞ { , } , where the Π ∞ -module structure on Π ∞ is obtained by restriction of scalars alongthe map Π ∞ → Π ∞ in CAlg ( S ∗ ) determined by t ↦ t n . Consequently, we have that Π ∞ ⊗ Π ∞ { , } is equivalent to the cofiber of the map Π ∞ → Π ∞ given by multiplication by t n , which shows thewanted. (cid:3) There is a functor of ∞ -categories Σ ∞ ∶ Gr ( S ∗ ) → GrSp obtained by composition with the re-duced suspension spectrum functor. This functor canonically refines to a functor of ∞ -categoriesΣ ∞ ∶ CAlg ( Gr S ∗ ) → CAlg ( GrSp ) since Σ ∞ ∶ S ∗ → Sp and thus Σ ∞ ∶ Gr S ∗ → GrSp admits the structure of a symmetric monoidalfunctor. It is a consequence of Lemma 4.1.4 that the following square is a pushout of E ∞ -algebrasin the ∞ -category of graded spectra S [ s ] S [ t ] S S [ t ]/ t n ← → s ↦ t n ←→ s ↦ ←→ ← → where s is in grading degree n and t is in grading degree 1. The E ∞ -algebra S [ t ]/ t n in GrSp isdefined by S [ t ]/ t n = Σ ∞ Π w ( t ) = n . We conclude that the diagram obtained by applying the functorund ∶ GrSp → Sp to the diagram above is a pushout of E ∞ -rings. Example 4.1.5.
By the discussion above, there is an equivalence of E ∞ -rings S [ t ]/ t n = S ⊗ S [ s ] S [ t ] , for every integer n ≥
1, where S [ t ] is a module over S [ s ] by restriction of scalars along the mapof E ∞ -rings S [ s ] → S [ t ] given by s ↦ t n . There is an isomorphism of commutative rings π ( S [ t ]/ t n ) ≃ π ( S ) ⊗ π ( S [ s ]) π ( S [ t ]) ≃ Z ⊗ Z [ s ] Z [ t ] ≃ Z [ t ]/ t n by virtue of [55, Corollary 7.2.1.23] since both S and S [ t ] are connective.Presently, we discuss a graded refinement of cyclotomic spectra following [2]. The construction { X i } i ↦ { X tC p pi } i determines an endofunctor F p of the ∞ -category GrSp B T of graded spectrawith T -action for every prime p , where X tC p pi is equipped with the residual T / C p ≃ T -action. Definition 4.1.6.
The ∞ -category of graded cyclotomic spectra is defined as the pullbackGrCycSp ∏ p ( GrSp B T ) ∆ GrSp B T ∏ p ( GrSp B T × GrSp B T ) ← → ←→ ←→ ( ev , ev ) ← → ( id ,F p ) It follows from Definition 4.1.6 that a graded cyclotomic spectrum is given by a graded spec-trum with T -action { X i } ≥ together with a T -equivariant map ϕ p,i ∶ X i → X tC p pi for every prime number p and integer i ≥
0, where the target carries the residual T / C p ≃ T -action.Informally, the cyclotomic Frobenius multiplies the grading degree by p . If R is an E -algebrain the ∞ -category of graded spectra, then there is a functor of ∞ -categoriesTHH gr ∶ Alg ( GrSp ) → GrCycSpobtained by applying the cyclic bar construction in the ∞ -category GrSp of graded spectra. Wewill refer to the functor THH gr as graded topological Hochschild homology, and refer the readerto [2, Appendix A] for the details of this construction. As in Notation 4.1.1, there is a functorGrCycSp → CycSp
N CURVES IN K-THEORY AND TR 37 determined by the construction { X i } i ≥ ↦ ⊕ i ≥ X i , and this functor preserves colimits. We havethat the following diagram of ∞ -categories commutesAlg ( GrSp ) GrCycSpAlg CycSp ←→ und ← → THH gr ←→ und ← → THH where we have used that the underlying functor GrSp → Sp of Notation 4.1.1 admits a canonicalsymmetric monoidal structure.
Definition 4.1.7.
Let { R i } i ≥ denote an E -algebra in GrSp with R ≃ und ({ R i } i ≥ ) , and defineTHH ( R ) i = ev i THH gr ({ R i } i ≥ ) for every integer i ≥
0. By definition, there is an equivalence of cyclotomic spectraTHH ( R ) ≃ und THH gr ({ R i } i ≥ ) ≃ ⊕ i ≥ THH ( R ) i . Example 4.1.8.
There is an isomorphism of graded ringsTHH ∗ ( S [ t ]) ≃ S ∗ [ t, dt ]/( dt = ηtdt ) , where η ∈ π S denotes the Hopf element and ∣ dt ∣ =
1. We may regard THH ∗ ( S [ t ]) as a gradedabelian group with w ( t ) = w ( dt ) = E ∞ -ring of THH gr ( Σ ∞ Π w ( t ) = ∞ ) isequivalent to THH ( S [ t ]) . We have the following picture:1 t t t . . . dt tdt t dt . . . . . . w degreeAlternatively, we may regard THH ∗ ( S [ t ]) as a graded abelian group with w ( t ) = w ( dt ) = n , andin this case we have the following picture:1 0 . . . t . . . t . . . . . . dt . . . tdt . . . . . . n − n n + . . . n − n n + . . . w degreeAs in Definition 4.1.7 above, there is an equivalence of cyclotomic spectraTHH ( S [ t ]/ t n ) ≃ und THH gr ( Σ ∞ Π w ( t ) = n ) ≃ ⊕ i ≥ THH ( S [ t ]/ t n ) i . Hesselholt–Madsen [40] determine the T -equivariant homotopy type of THH ( S [ t ]/ t n ) i for every i ≥ Proposition 4.1.9.
For every integer n ≥ , there is an equivalence of spectra with T -action THH ( S [ t ]/ t n ) ≃ ⊕ i ≥ THH ( S [ t ]/ t n ) i , where THH ( S [ t ]/ t n ) ≃ S and THH ( S [ t ]/ t n ) i ≃ Σ ∞+ ( S / C i ) for ≤ i ≤ n − .Proof. It remains to show that THH ( S [ t ]) ≃ S and THH ( S [ t ]/ t n ) i ≃ Σ ∞+ ( S / C i ) for 1 ≤ i ≤ n − ( S [ t ]) ≃ und THH gr ( Σ ∞ Π w ( t ) = ∞ ) ≃ ⊕ i ≥ THH ( S [ t ]) i , where THH ( S [ t ]) ≃ S and THH ( S [ t ]) i ≃ Σ ∞+ ( S / C i ) for every i ≥
1, thus we want to prove thatthere is an equivalence of spectra with T -actionTHH ( S [ t ]/ t n ) i ≃ THH ( S [ t ]) i for every 0 ≤ i ≤ n −
1. Note that the squareTHH gr ( S [ s ]) THH gr ( S [ t ]) S THH gr ( S [ t ]/ t n ) ← → s ↦ t n ←→ s ↦ ←→ ← → is a pushout in the ∞ -category of graded cyclotomic spectra, where w ( s ) = n and w ( t ) =
1. In-deed, it suffices to prove that the square is a pushout in the ∞ -category of cyclotomic spectra afterapplying the underlying functor GrCycSp → CycSp. This is a consequence of Lemma 4.1.4 sinceboth und ∶ Alg ( GrSp ) → Alg and THH ∶ Alg → CycSp preserve pushouts. Using Example 4.1.8,we conclude that there is an equivalence of spectra with T -actionTHH ( S [ t ]/ t n ) i ≃ THH ( S [ t ]) i for every 0 ≤ i ≤ n − (cid:3) We will not need to determine the T -equivariant homotopy type of THH ( S [ t ]/ t n ) i for i ≥ n as in Hesselholt–Madsen [40] (see Remark 4.1.12). Instead, we will only need a connectivityestimate for THH ( S [ t ]/ t n ) i for i ≥ n , which we deduce from a calculation of the Hochschild ho-mology groups of truncated polynomial rings over the integers. This computation was previouslyobtained by Guccione–Guccione–Redondo–Solotar–Villamayor [37]. We will let Z ⟨ y ⟩ denote thefree divided power algebra which has generators y [ ] , y [ ] , . . . with y = y [ ] and the relations that y [ i ] y [ j ] = ( i + ji ) y [ i + j ] for every pair of positive integers i and j . We have the following result: Proposition 4.1.10.
For every integer n ≥ , the Hochschild homology HH ( Z [ t ]/ t n ) is equivalentto the E -algebra given by the following differential graded algebra ( Z [ t ]/ t n ⊗ Λ ( dt ) ⊗ Z ⟨ y ⟩ , ∂ ) , with ∣ dt ∣ = and ∣ y ∣ = , whose differential is determined by ∂ ( y [ i ] ) = nt n − y [ i − ] dt and ∂ ( dt ) = . N CURVES IN K-THEORY AND TR 39
Proof.
The following square is a pushout of E ∞ -algebras over Z HH ( Z [ s ]) HH ( Z [ t ]) Z HH ( Z [ t ]/ t n ) ← → s ↦ t n ←→ s ↦ ←→ ← → since the functor − ⊗ Z ∶ CAlg → CAlg Z preserves colimits and THH ( S [ t ]/ t n ) ⊗ Z ≃ HH ( Z [ t ]/ t n ) .Recall that the pushout of E ∞ -algebras over Z above is calculated by the relative tensor product.There is an equivalence of commutative differential graded algebras HH ( Z [ s ]) ≃ Z [ s ] ⊗ Λ ( ds ) with ∣ ds ∣ = ∂ ( ds ) = Z [ s ] ⊗ Λ ( ds ) → Z [ t ] ⊗ Λ ( dt ) is determined by s ↦ t n and ds ↦ nt n − dt . Note that the Z appearing in the lower left corner ofthe pushout square above is equivalent to the commutative differential graded algebra given by Z [ s ] ⊗ Λ ( ds ) ⊗ Λ ( ε ) ⊗ Z ⟨ y ⟩ with ∣ ds ∣ = ∣ ε ∣ = ∣ y ∣ =
2, whose differential is determined by ∂ ( ε ) = s and ∂ ( y [ i ] ) = y [ i − ] ds .Since the ring homomorphism Z [ s ] → Z [ t ] given by s ↦ t n is flat, we conclude that HH ( Z [ t ]/ t n ) is given by the pushout of the following diagram of commutative differential graded algebras Z [ s ] ⊗ Λ ( ds ) Z [ t ] ⊗ Λ ( dt ) Z [ s ] ⊗ Λ ( ds ) ⊗ Λ ( ε ) ⊗ Z ⟨ y ⟩ ← → s ↦ t n , ds ↦ nt n − dt ←→ which is equivalent to the commutative differential graded algebra given by Z [ t ]/ t n ⊗ Λ ( dt ) ⊗ Z ⟨ y ⟩ with ∣ dt ∣ = ∣ y ∣ =
2, whose differential is determined by ∂ ( dt ) = ∂ ( y [ i ] ) = nt n − y [ i − ] dt ,where we have used that ε does not contribute to the pushout since Z [ s ] → Z [ t ] given by s ↦ t n is injective and ∂ ( ε ) = s . (cid:3) There is an equivalence of spectra with T -actionHH ( Z [ t ]/ t n ) ≃ ⊕ i ≥ HH ( Z [ t ]/ t n ) i , where HH ( Z [ t ]/ t n ) i ≃ THH ( S [ t ]/ t n ) i ⊗ Z . Using Proposition 4.1.10, we may regard HH ∗ ( Z [ t ]/ t n ) as a graded abelian group with w ( t ) = w ( dt ) = w ( y ) = n . As in Example 4.1.8 above, we have the following picture:1 t . . . t n − . . . . . . dt . . . t n − dt t n − dt . . . . . . . . . y yt . . . yt n − . . . . . . ydt . . . yt n − dt yt n − dt . . . . . . . . . y [ ] y [ ] t . . . . . . . . . y [ ] dt . . . ⋮ ⋮ . . . ⋮ ⋮ ⋮ . . . ⋮ ⋮ ⋮ . . . . . . n − n n + . . . n − n n + . . . ⋮ w degAs a consequence, we obtain the following result from the diagram above, which was previouslyobtained by Hesselholt–Madsen [41, § Corollary 4.1.11.
For every n ≥ and every i ≥ , the spectrum THH ( S [ t ]/ t n ) i is ℓ -connective,where ℓ = ⌊ i − n ⌋ denotes the largest integer less than i − n .Proof. It suffices to prove the assertion for HH ( Z [ t ]/ t n ) i . Note that if n divides i , then HH ( Z [ t ]/ t n ) i is concentrated in degree 2 ℓ + ℓ +
2. If n does not divide i , then HH ( Z [ t ]/ t n ) i is concentratedin degree 2 ℓ and 2 ℓ +
1. In particular, we conclude that HH ( Z [ t ]/ t n ) i is 2 ℓ -connective for everyinteger i ≥
1. This shows the wanted by Hurewicz since THH ( S [ t ]/ t n ) i ⊗ Z ≃ HH ( Z [ t ]/ t n ) i . (cid:3) Remark 4.1.12.
In this remark, we will identify the underlying spectrum of THH ( S [ t ]/ t n ) usingProposition 4.1.10. The homology of HH ( Z [ t ]/ t n ) kn is given by the homology of the complex ⋯ → → Z [ k ] n Ð→ Z [ k − ] → → ⋯ for every integer k ≥
1. It follows that there is an equivalence of spectraTHH ( S [ t ]/ t n ) kn ≃ S / n [ k − ] since the Moore spectrum S / n is uniquely characterized by π ∗ ( S / n ⊗ Z ) ≃ Z / n [ ] . Additionally,we recall that Σ ∞+ S ≃ S ⊕ S [ ] . Using the picture following the proof of Proposition 4.1.10, weconclude that there is an equivalence of spectraTHH ( S [ t ]/ t n ) ≃ S ⊕ ( ⊕ i ≥ ( n − ⊕ j = ( S ⊕ S [ ])[ i ]) ⊕ S / n [ i + ]) . In fact, it is possible to determine the underlying spectrum with T -action of THH ( S [ t ]/ t n ) . Suchan identification will be useful for computing the topological Hochschild homology of truncatedpolynomial rings over the integers or a perfectoid base ring. N CURVES IN K-THEORY AND TR 41
Example 4.1.13.
In this example, we describe a cyclotomic structure on the product ∏ n ≥ X ⊗ Σ ∞+ ( S / C n ) , where X denotes a cyclotomic spectrum whose underlying spectrum is bounded below as follows:Using [3, Lemma 2.11], we conclude that the canonical map of spectra with T -action ( ∏ n ≥ X ⊗ Σ ∞+ ( S / C n )) tC p → ∏ n ≥ ( X ⊗ Σ ∞+ ( S / C n )) tC p is an equivalence for every prime number p . As a consequence, we may regard the product aboveas a cyclotomic spectrum whose cyclotomic Frobenius is induced by the T -equivariant map X ⊗ Σ ∞+ ( S / C n ) id ⊗ ψ p ÐÐÐ→ X ⊗ Σ ∞+ ( S / C pn ) tC p ϕ p ⊗ id ÐÐÐ→ X tC p ⊗ Σ ∞+ ( S / C pn ) tC p ℓ Ð→ ( X ⊗ Σ ∞+ ( S / C pn )) tC p where ℓ denotes the map induced by the canonical lax symmetric monoidal structure of the Tateconstruction (see [59, Theorem I.3.1]).Finally, we obtain a convenient description of the cyclotomic structure of lim ←Ð ( X ⊗ ̃ THH ( S [ t ]/ t n )) for every cyclotomic spectrum X whose underlying spectrum is bounded below using the con-nectivity result obtained in Corollary 4.1.11. Proposition 4.1.14.
There is an equivalence of cyclotomic spectra lim ←Ð ( X ⊗ ̃ THH ( S [ t ]/ t n )) ≃ ∏ n ≥ X ⊗ Σ ∞+ ( S / C n ) for every cyclotomic spectrum X whose underlying spectrum is bounded below, where the cyclo-tomic structure on the product is described in Example 4.1.13.Proof. For every integer n ≥
2, we let ̃ THH ( S [ t ]) < n denote the cyclotomic spectrum defined by ̃ THH ( S [ t ]) < n = n − ⊕ i = Σ ∞+ ( S / C i ) , whose cyclotomic Frobenius is induced by Σ ∞+ ( S / C i ) → Σ ∞+ ( S / C pi ) hC p , where Σ ∞+ ( S / C pi ) hC p = pi ≥ n . By construction, the canonical projection ̃ THH ( S [ t ]/ t n ) → ̃ THH ( S [ t ]) < n determines a map of cyclotomic spectra such that the following diagram commutes ̃ THH ( S [ t ]/ t n + ) ̃ THH ( S [ t ]) < n + ̃ THH ( S [ t ]/ t n ) ̃ THH ( S [ t ]) < n ← → ←→ ←→ ← → where the left vertical map is induced by the canonical map of pointed monoids Π n + → Π n , andthe right vertical map is given by the projection. We obtain a map of cyclotomic spectra(3) lim ←Ð ( X ⊗ ̃ THH ( S [ t ]/ t n )) → lim ←Ð ( X ⊗ ̃ THH ( S [ t ]) < n ) for every cyclotomic spectrum X whose underlying spectrum is bounded below, and we showthat this map is an equivalence of cyclotomic spectra. First note that the forgetful functorCycSp → Sp B T preserves both of the limits appearing in (3) by virtue of Corollary 4.1.11 and the assumption that the underlying spectrum of X is bounded below. Consequently, it sufficesto show that the map of spectra with T -actionlim ←Ð ( X ⊗ ̃ THH ( S [ t ]) < n ) → lim ←Ð ( X ⊗ ̃ THH ( S [ t ]/ t n )) induced by the inclusion ̃ THH ( S [ t ]) < n ↪ ̃ THH ( S [ t ]/ t n ) is an equivalence. There is a commuta-tive diagram of cofiber sequences of spectra with T -action X ⊗ ̃ THH ( S [ t ]) < n + X ⊗ ̃ THH ( S [ t ]/ t n + ) X ⊗ ⊕ i ≥ n + ̃ THH ( S [ t ]/ t n + ) i X ⊗ ̃ THH ( S [ t ]) < n X ⊗ ̃ THH ( S [ t ]/ t n ) X ⊗ ⊕ i ≥ n ̃ THH ( S [ t ]/ t n ) i ← → ←→ ← → ←→ ←→ ← → ← → where the right vertical map is induced by the composite ̃ THH ( S [ t ]/ t n + ) i → ̃ THH ( S [ t ]/ t n ) i ↪ ̃ THH ( S [ t ]/ t n ) . Thus, it suffices to show that lim ←Ð n ⊕ i ≥ n ( X ⊗ ̃ THH ( S [ t ]/ t n ) i ) ≃ , which now follows from Corollary 4.1.11. Indeed, we have thatlim ←Ð n ⊕ i ≥ n ( X ⊗ ̃ THH ( S [ t ]/ t n ) i ) ≃ lim ←Ð n ( lim ←Ð m ⊕ i ≥ m ( X ⊗ ̃ THH ( S [ t ]/ t n ) i )) since the diagonal N → N × N is an initial functor, and using Corollary 4.1.11, we conclude thatthe limit appearing in the parenthesis on the right hand side of the equivalence above vanishes.In conclusion, we have proved that the map appearing in (3) is an equivalence. To finish theproof, we show that the map of cyclotomic spectra ∏ n ≥ X ⊗ Σ ∞+ ( S / C n ) → lim ←Ð ( X ⊗ ̃ THH ( S [ t ]) < n ) . induced by the projection maps is an equivalence, where the cyclotomic structure on the productis defined in Example 4.1.13. Using the Milnor lim -sequence, this follows from the fact that if ⋯ proj ÐÐ→ A ⊕ A ⊕ A ÐÐ→ A ⊕ A ÐÐ→ A is a tower of abelian groups, then the limit is given by the product ∏ n ≥ A n . (cid:3) Curves on K-theory.
As an application of Theorem 3.3.12 and Proposition 4.1.14, weobtain the desired description of TR evaluated on a connective E -ring R in terms of the spectrumof curves on K ( R ) extending work of Hesselholt [39] and Betley–Schlichtkrull [12]. We will beginby recalling the following notation (see the discussion following Lemma 4.1.4). Notation 4.2.1.
Recall that for every integer n ≥
1, the following square is a pushout of E ∞ -rings S [ t ] S [ t ] S S [ t ]/ t n ← → t ↦ t n ←→ t ↦ ←→ ← → N CURVES IN K-THEORY AND TR 43
In particular, there is a map of E ∞ -rings S [ t ]/ t n → S determined by the assignment t ↦
0. If R is a connective E -ring, then we define the E -ring R [ t ]/ t n by R [ t ]/ t n = R ⊗ S [ t ]/ t n , and wehave that π ∗ ( R [ t ]/ t n ) ≃ ( π ∗ R )[ t ]/ t n . We obtain a map of connective E -rings R [ t ]/ t n → R suchthat the kernel of the induced ring homomorphism π ( R [ t ]/ t n ) ≃ ( π R )[ t ]/ t n → π R is given by the nilpotent ideal ( t ) . If E ∶ Alg cn → Sp is a functor, then we will let E ( R [ t ]/ t n , ( t )) denote the fiber of the induced map of spectra E ( R [ t ]/ t n ) → E ( R ) .We recall the definition of the spectrum of curves on algebraic K-theory following Hessel-holt [39], which is based on a previous variant studied by Bloch [15] in his work on the relationshipbetween algebraic K-theory and crystalline cohomology. Definition 4.2.2.
The spectrum of curves on K-theory is defined byC ( R ) = lim ←Ð Ω K ( R [ t ]/ t n , ( t )) for every connective E -ring R .A fundamental result of Hesselholt [39, Theorem 3.1.10] asserts that if R is a discrete commu-tative Z / p j -algebra for some j ≥
1, then there is a natural equivalence of spectra TR ( R ) ≃ C ( R ) .As a consequence, Hesselholt [39, Theorem C] proves that the homotopy groups of the p -typicalsummand of C ( A ) is isomorphic to the de Rham–Witt complex WΩ ∗ A in the case where A is asmooth algebra over a perfect field of characteristic p . In [12, Theorem 1.3], Betley–Schlichtkrullestablish a variant of the result of Hesselholt for topological cyclic homology on discrete associa-tive rings after profinite completion, where the inverse limit in the definition of the spectrum ofcurves on K-theory is replaced by a limit over a diagram which additionally encodes the transfermaps R [ t ]/ t m → R [ t ]/ t mn determined by t ↦ t n . Our main result is the following: Theorem 4.2.3.
There is a natural equivalence of spectra TR ( X ) ≃ lim ←Ð Ω TC ( X ⊗ ̃ THH ( S [ t ]/ t n )) for every cyclotomic spectrum X whose underlying spectrum is bounded below.Proof. There is a natural equivalence of spectraTR ( X ) ≃ map CycSp ( ̃
THH ( S [ t ]) , X ) by virtue of Theorem 3.3.12, so we conclude thatTR ( X ) ≃ Eq ( map Sp B T ( ⊕ n ≥ Σ ∞+ ( S / C n ) , X ) ∏ p map Sp B T ( ⊕ n ≥ Σ ∞+ ( S / C n ) , X tC p ) ← →← → ) ≃ Eq ( ∏ n ≥ map Sp B T ( Σ ∞+ ( S / C n ) , X ) ∏ p ∏ n ≥ map Sp B T ( Σ ∞+ ( S / C n ) , X tC p ) ← →← → ) by [59, Proposition II.1.5], where the top map is induced by the cyclotomic structure of ̃ THH ( S [ t ]) ,and the bottom map is induced by the cyclotomic structure of X . Note that Σ ∞+ ( S / C n ) is dual-izable regarded as a spectrum with T -action for every n ≥
1. Indeed, we have that T -equivariant Atiyah duality identifies its dual with Σ ∞− + ( S / C n ) . Thus, the equalizer above is equivalent toEq ( ∏ n ≥ map Sp B T ( S , Σ ∞− + ( S / C n ) ⊗ X ) ∏ p ∏ n ≥ map Sp B T ( S , Σ ∞− + ( S / C n ) ⊗ X tC p ) ← →← → ) ≃ Eq ( map Sp B T ( S , ∏ n ≥ Σ ∞− + ( S / C n ) ⊗ X ) ∏ p map Sp B T ( S , ∏ n ≥ Σ ∞− + ( S / C n ) ⊗ X tC p ) ← →← → ) ≃ Eq ( map Sp B T ( S , ∏ n ≥ Σ ∞− + ( S / C n ) ⊗ X ) ∏ p map Sp B T ( S , ( ∏ n ≥ Σ ∞− + ( S / C n ) ⊗ X ) tC p ) ← →← → ) , where the final equivalence follows since the canonical T -equivariant composite map of spectra ( ∏ n ≥ X ⊗ Σ ∞− + ( S / C n )) tC p ≃ Ð→ ∏ n ≥ ( Σ ∞+ ( S / C n ) ⊗ X ) tC p ≃ Ð→ ∏ n ≥ Σ ∞+ ( S / C n ) ⊗ X tC p is an equivalence, where the first map is an equivalence as explained in Example 4.1.13, and thesecond map is an equivalence since Σ ∞+ ( S / C n ) is compact. Under these identifications, the topmap in the equalizer above is induced by the trivial cyclotomic structure of S , and the bottommap is induced by the following T -equivariant map of spectraΣ ∞− + ( S / C n ) ⊗ X id ⊗ ϕ p ÐÐÐ→ Σ ∞− + ( S / C n ) ⊗ X tC p . There is a commutative diagram of spectra with T -action (cf. [62, Lemma 8]) X ⊗ Σ ∞+ ( S / C n ) X ⊗ Σ ∞+ ( S / C pn ) hC p X ⊗ Σ ∞+ ( S / C pn ) tC p X tC p ⊗ Σ ∞+ ( S / C n ) X tC p ⊗ Σ ∞+ ( S / C pn ) hC p X tC p ⊗ Σ ∞+ ( S / C pn ) tC p ( X ⊗ Σ ∞+ ( S / C n )) tC p ( X ⊗ Σ ∞+ ( S / C pn ) hC p ) tC p ( X ⊗ Σ ∞+ ( S / C pn )) tC p ← → id ⊗ ψ p ←→ ϕ p ⊗ id ← → id ⊗ can ←→ ϕ p ⊗ id ←→ ϕ p ⊗ id ← → id ⊗ ψ p ←→ ≃ ← → id ⊗ can ←→ ←→ ℓ ← → ( id ⊗ ψ p ) tCp ← → ( id ⊗ ε ) tCp where ε ∶ Σ ∞+ ( S / C pn ) hC p → Σ ∞+ ( S / C pn ) denotes the canonical map, and ℓ denotes the map in-duced by the lax symmetric monoidal structure of the Tate construction ( − ) tC p (see [59, TheoremI.3.1]). The lower horizontal composite is induced by the following C p -equivariant compositeΣ ∞+ ( S / C n ) ψ p Ð→ Σ ∞+ ( S / C pn ) hC p ǫ Ð→ Σ ∞+ ( S / C pn ) where Σ ∞+ ( S / C n ) and Σ ∞+ ( S / C pn ) are equipped with the trivial C p -action. The cofiber of thecomposite above is an induced spectrum with C p -action, hence an equivalence after applyingthe Tate construction ( − ) tC p . We conclude that the lower horizontal composite in the diagramabove is an equivalence. From this, we conclude that the bottom map in the equalizer computingTR ( X ) above is induced by the following T -equivariant map of spectra X ⊗ Σ ∞+ ( S / C n ) id ⊗ ψ p ÐÐÐ→ X ⊗ Σ ∞+ ( S / C pn ) tC p ϕ p ⊗ id ÐÐÐ→ X tC p ⊗ Σ ∞+ ( S / C pn ) tC p ℓ Ð→ ( X ⊗ Σ ∞+ ( S / C pn )) tC p N CURVES IN K-THEORY AND TR 45 which precisely is the definition of the cyclotomic structure of the product of Example 4.1.13. Itfollows from Proposition 4.1.14 that TR ( X ) is naturally equivalent to the following spectrumlim ←Ð ΩEq ( map Sp B T ( S , X ⊗ ̃ THH ( S [ t ]/ t n )) ∏ p map Sp B T ( S , ( X ⊗ ̃ THH ( S [ t ]/ t n )) tC p ) ← →← → ) , where the top map is induced by the trivial cyclotomic structure of the sphere S and the bottommap is induced by the cyclotomic structure of X ⊗ ̃ THH ( S [ t ]/ t n ) . We conclude thatTR ( X ) ≃ lim ←Ð Ω map
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Fachbereich Mathematik und Informatik, Westf¨alische Wilhelms–Universit¨at M¨unster, Germany
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