Cyclotomic spectra and TC

In this section we give our new definition of cyclotomic spectra. The following definition gives the objects of an∞-category that will be defined in DefinitionII.1.6.

In the following, we will use that, for a spectrumX with T-action, the spectrum X^tCp can be equipped with the residualT/Cp-action which is identified with aT-action using thepth power mapT/Cp∼=T. Moreover,Cp^∞⊆Tdenotes the subgroup of elements ofp-power torsion which is isomorphic to the Pr¨ufer groupQp/Zp.(¹⁴) Again, we have a canonical identificationCp^∞/Cp∼=Cp^∞ given by thepth power map.

DefinitionII.1.1. (i) Acyclotomic spectrumis a spectrumXwithT-action together withT-equivariant mapsϕ_p:X!X^tCp for every primep.

(ii) For a fixed primep, ap-cyclotomic spectrum is a spectrumX with C_p^∞-action and aC_p^∞-equivariant mapϕ_p:X!X^tCp.

Example II.1.2. We give some examples of cyclotomic spectra.

(i) For every associative and unital ring spectrum R∈Alg_E1(Sp), the topological Hochschild homology THH(R) is a cyclotomic spectrum; cf.§III.2below.

(ii) Consider the sphere spectrumS equipped with the trivialT-action. There are canonical maps ϕp:S!S^tCp given as the composition S!S^hCp!S^tCp. These maps are T∼=T/Cp-equivariant: To make them equivariant, we need to lift the mapS!S^tCp to a mapS!(S^tCp)^h(T/Cp), as theT-action onSis trivial. But we have a natural map

S−!S^hT'(S^hCp)^h(T/Cp)−!(S^tCp)^h(T/Cp).

This defines a cyclotomic spectrum which is in fact equivalent to THH(S). Note (cf. Re-mark III.1.6 below) that it is a consequence of the Segal conjecture that the maps ϕp

are p-completions. We refer to this cyclotomic spectrum as the cyclotomic sphere and denote it also byS.

(iii) For every cyclotomic spectrum, we get ap-cyclotomic spectrum by restriction.

In particular we can consider THH(R) andSasp-cyclotomic spectra and do not distin-guish these notationally.

Remark II.1.3. One can define a slightly different notion ofp-cyclotomic spectrum, as a spectrum with aT-action and a T∼=T/Cp-equivariant map ϕp:X!X^tCp. This is what has been used (in a different language) in the literature before [19]. We however prefer to restrict the action to aCp^∞-action, since this is sufficient for the definition of TC(−, p) below and makes the construction more canonical; in particular, it is necessary for the interpretation of TC(−, p), as the mapping spectrum from Sin the ∞-category ofp-cyclotomic spectra. The functor from the∞-category of p-cyclotomic spectra with T-action to the∞-category ofp-cyclotomic spectra as defined above is fully faithful when

(¹⁴) One hasCp^∞∼=Qp/Zp∼=Q/Z(p)∼=Z[1/p]/Z, but the second author is very confused by the no-tationZ/p^∞that is sometimes used, and suggests to change it to any of the previous alternatives, or to p^−∞Z/Z.

restricted to the subcategory of p-complete and bounded below objects (so that X^tCp is also p-complete by Lemma I.2.9). However, not every Cp^∞-action on a p-complete spectrum extends to aT-action, since aCp^∞-action can act non-trivially on homotopy groups.

Let us now make the definition of cyclotomic spectra more precise, by defining the relevant∞-categories.

As in Definition I.1.2, we denote the ∞-category of spectra equipped with a T -action by Sp^BT=Fun(BT,Sp). Note that, here,Tis regarded as a topological group, and BT'CP^∞ denotes the corresponding topological classifying space. We warn the reader again that this notion of G-equivariant spectrum is different from the notions usually considered in equivariant stable homotopy theory, and we discuss their relation in§II.2 below.

Note that the ∞-category of cyclotomic spectra is the ∞-category ofX∈Sp^BT to-gether with mapsX!X^tCp. This is a special case of the following general definition.

Definition II.1.4. LetC andD be ∞-categories, and F, G:C!D be functors. The lax equalizer(¹⁵)ofF andGis the∞-category

LEq(F, G) = LEq

C ^F //

G //D defined as the pullback

LEq(F, G) //

D^∆1

(ev₀,ev₁)

C ^(F,G) //D×D

of simplicial sets. In particular, objects of LEq(F, G) are given by pairs (c, f) of an object c∈Cand a mapf:F(c)!G(c) inD.

Proposition II.1.5. Consider a cartesian diagam LEq(F, G) //

D^∆1

(ev₀,ev₁)

C ^(F,G) //D×D

(¹⁵) In accordance to classical 2-category theory it would be more precise to call this construction an inserter. But, since we want to emphasize the relation to the equalizer, we go with the current terminology. We thank Emily Riehl for a discussion of this point.

as in Definition II.1.4.

(i) The pullback

LEq(F, G) //

D^∆1

(ev₀,ev₁)

C ^(F,G) //D×D is a homotopy cartesian diagram of ∞-categories.

(ii) Let X, Y∈LEq(F, G)be two objects,given by pairs(cX, fX)and (cY, fY),where cX, cY∈C, and fX:F(cX)!G(cX) and fY:F(cY)!G(cY) are maps in D. Then, the space Map_LEq(F,G)(X, Y)is given by the equalizer

Map_LEq(F,G)(X, Y)'Eq

Map_C(c_X, c_Y)

(fX)^∗G //

(f_Y)∗F //Map_D(F(c_X), G(c_Y)) .

Moreover, a map f:X!Y in LEq(F, G)is an equivalence if and only if its image in C is an equivalence.

(iii) If C and D are stable ∞-categories, and F and G are exact, then LEq(F, G) is a stable ∞-category,and the functor LEq(F, G)!C is exact.

(iv) If Cis presentable,Dis accessible,F is colimit-preserving and Gis accessible, then LEq(F, G)is presentable,and the functor LEq(F, G)!C is colimit-preserving.

(v) Assume that p:K!LEq(F, G) is a diagram such that the composite diagram K!LEq(F, G)!C admits a limit, and this limit is preserved by the functor G:C!D.

Then, padmits a limit, and the functor LEq(F, G)!C preserves this limit.

In part (ii), we denote for any∞-categoryC by Map_C:C^op×C −!S

the functor defined in [69, §5.1.3]. It is functorial in both variables, but its definition is rather involved. There are more explicit models for the mapping spaces with less functoriality, for example the Kan complex Hom^R_C(X, Y) withn-simplices given by the set of (n+1)-simplices ∆ⁿ⁺¹!C whose restriction to ∆^{0,...,n} is constant at X, and which send the vertex ∆^{n+1} toY; cf. [69,§1.2.2].

In part (iv), recall that a functor between accessible∞-categories is accessible if it commutes with small-filtered colimits for some (large enough). For the definition of accessible and presentable∞-categories, we refer to [69, Definitions 5.4.2.1 and 5.5.0.1].

Proof. For part (i), it is enough to show that the functor (ev₀,ev₁):D^∆1!D×D of ∞-categories is a categorical fibration, which follows from [69, Corollaries 2.3.2.5 and 2.4.6.5].

For part (ii), note that it follows from the definition of Hom^Rthat there is a pullback diagram of simplicial sets

Hom^R_LEq(F,G)(X, Y) //

Hom^R_D∆1(fX, fY)

Hom^R_C(c_X, c_Y) //Hom^R_D(F(c_X), F(c_Y))×Hom^R_D(G(c_X), G(c_Y)).

This is a homotopy pullback in the Quillen model structure, as the map Hom^R_D∆1(fX, fY)!Hom^R_D(F(cX), F(cY))×Hom^R_D(G(cX), G(cY))

is a Kan fibration by [69, Corollary 2.3.2.5 and Lemma 2.4.4.1]. Therefore, the diagram

Map_LEq(F,G)(X, Y) //

Map_D∆1(fX, fY)

Map_C(cX, cY) //Map_D(F(cX), F(cY))×Map_D(G(cX), G(cY))

is a pullback inS. To finish the identification of mapping spaces in part (ii), it suffices to observe that the right vertical map is the equalizer of

Map_D(F(cX), F(cY))×Map_D(G(cX), G(cY)) ////Map_D(F(cX), G(cY)), which follows by unraveling the definitions. For the final sentence of (ii), note that, if f:X!Y is a map in LEq(F, G) which becomes an equivalence inC, then, by the formula for the mapping spaces, one sees that

f^∗: Map_LEq(F,G)(Y, Z)−!Map_LEq(F,G)(X, Z)

is an equivalence for allZ∈LEq(F, G); thus, by the Yoneda lemma,f is an equivalence.

For part (iii), note first that D×D and D^∆1 are again stable by [71, Proposi-tion 1.1.3.1]. Now, the pullback of a diagram of stable∞-categories along exact functors is again stable by [71, Proposition 1.1.4.2].

For part (iv), note first that by [69, Proposition 5.4.4.3], D×Dand D^∆1 are again accessible. By [69, Proposition 5.4.6.6], LEq(F, G) is accessible. It remains to see that LEq(F, G) admits all small colimits, and that LEq(F, G)!Cpreserves all small colimits.

LetKbe some small simplicial set with a mapp:K!LEq(F, G). First, we check that an extensionp^.:K^.!LEq(F, G) is a colimit ofpif the compositionK^.!LEq(F, G)!C is a colimit ofK!LEq(F, G)!C. Indeed, this follows from the Yoneda characterization

of colimits [69, Lemma 4.2.4.3], the description of mapping spaces in part (ii), and the assumption that F preserves all small colimits. Thus, it remains to see that we can always extendpto a mapp^.:K^.!LEq(F, G) whose image inC is a colimit diagram.

By assumption, the composition K!LEq(F, G)!C can be extended to a colimit diagrama:K^.!C. Moreover,F(a):K^.!Dis still a colimit diagram, whileG(a):K^.!D may not be a colimit diagram. However, we have the mapK!LEq(F, G)!D^∆1, which is given by a mapK×∆¹!Dwith restriction to K×{0}equal toF(a)|K, and restriction to K×{1} equal to G(a)|K. By the universal property of the colimit F(a), this can be extended to a map D:K^.×∆¹!D with restriction toK^.×{0} given byF(a), and restriction toK^.×{1}given byG(a). Thus, (a, D) defines a map

p^.:K^.−!C ×D×DD^∆1= LEq(F, G), whose image inCis a colimit diagram, as desired.

For part (v), use the dual argument to the one given in part (iv).

Now, we can define the∞-categories of (p-)cyclotomic spectra. Recall from Defini-tionI.1.13the functors

−^tCp: Sp^BT−!Sp^B(T/Cp)'Sp^BT and

−^tCp: Sp^BCp∞−!Sp^B(Cp∞/Cp)'Sp^BCp∞.

Definition II.1.6. (i) The∞-category of cyclotomic spectra is the lax equalizer Cyc Sp := LEq

Sp^BT ////Q

p∈PSp^BT ,

where the two functors havepth components given by the functors id: Sp^BT!Sp^BT and

−^tCp: Sp^BT!Sp^B(T/Cp)'Sp^BT.

(ii) The∞-category ofp-cyclotomic spectra is the lax equalizer Cyc Sp_p:= LEq

Sp^BCp∞ ////Sp^BCp∞

of the functors id: Sp^BCp∞!Sp^BCp∞ and−^tCp: Sp^BCp∞!Sp^B(Cp∞/Cp)'Sp^BCp∞. Corollary II.1.7. The ∞-categories Cyc Sp and Cyc Sp_p are presentable stable

∞-categories. The forgetful functors Cyc Sp!Spand Cyc Sp_p!Spreflect equivalences, are exact,and preserve all small colimits.

Proof. The Tate spectrum functorX7!X^tCpis accessible, as it is the cofiber of func-tors which admit adjoints; cf. [69, Proposition 5.4.7.7]. Moreover, the forgetful functor Sp^BG!Sp preserves all small colimits and reflects equivalences by [69, Corollary 5.1.2.3].

In particular, it is exact. Now, all statements follow from PropositionII.1.5.

For every pair of objects in a stable ∞-category C, the mapping space refines to a mapping spectrum. In fact, by [71, Corollary 1.4.2.23], for every X∈C^op the left-exact functor

Map(X,−):C −!S

from the stable∞-categoryC lifts uniquely to an exact functor map(X,−):C −!Sp.

VaryingX, i.e. by looking at the functor

C^op−!Fun^Lex(C,S)'Fun^Ex(C,Sp), we get a functor map:C^op×C!Sp.

Definition II.1.8. (i) Let X,(ϕ_p)_p∈P

be a cyclotomic spectrum. The integral topo-logical cyclic homology TC(X) ofX is the mapping spectrum map_{Cyc Sp}(S, X)∈Sp.

(ii) Let (X, ϕ_p) be a p-cyclotomic spectrum. The p-typical topological cyclic ho-mology TC(X, p) is the mapping spectrum map_{Cyc Sp}

p(S, X).

(iii) LetR∈Alg_E

1(Sp) be an associative ring spectrum. Then, TC(R):=TC(THH(R)) and TC(R, p):=TC(THH(R), p).

In fact, by PropositionII.1.5(ii), it is easy to compute TC(X) and TC(X, p).

Proposition II.1.9. (i) Let (X,(ϕp)p∈P) be a cyclotomic spectrum. There is a functorial fiber sequence

TC(X)−!X^{hT (ϕ}

hT p −can)p∈P

−−−−−−−−−−!Y

p∈P

(X^tCp)^hT, where the maps are given by

ϕ^h_p^T:X^hT−!(X^tCp)^hT and

can:X^hT'(X^hCp)^h(T/Cp)'(X^hCp)^hT−!(X^tCp)^hT, where the middle equivalence comes from the p-th power map T/C_p∼=T.

(ii) Let (X, ϕ_p)be a p-cyclotomic spectrum. There is a functorial fiber sequence TC(X, p)−!X^{hCp∞ ϕ}

hCp∞ p −can

−−−−−−−−−!(X^tCp)^hCp∞, with notation as in part (i).

Proof. Note that the fiber of a difference map is equivalent to the equalizer of the two maps. By the equivalence Fun^Ex(Cyc Sp,Sp)'Fun^Lex(Cyc Sp,S) (resp. for Cyc Spp) via composition with Ω^∞, it suffices to check the formulas for the mapping space. Thus, the result follows from PropositionII.1.5(ii).

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