Equivalence of TC

In this section, we prove parts (i) and (ii) of TheoremII.3.8. In other words, ifX is a genuine cyclotomic spectrum whose underlying spectrum is bounded below, then

TC^gen(X)'TC(X).

We recall the definition of TC^gen(X) in Definition II.4.4 and diagram (1) below. The proof relies on some consequences of the Tate orbit lemma (LemmaI.2.1), which we record first. Here and in the following, we identifyCpⁿ/Cp^m∼=C_pn−m, so for example if X is a spectrum withCpⁿ-action, thenX^hCpm is considered as a spectrum with C_pn−m-action via the identificationCpⁿ/Cp^m∼=C_pn−m.

LemmaII.4.1. Let X∈Sp^BCpn be a spectrum withCpⁿ-action that is bounded below.

Then, the canonical morphism X^tCpn!(X^tCp)^hCpn−1 is an equivalence.

Proof. First, by applying the Tate orbit lemma toXhC_pn−2, we see that, for n>2, the norm morphism

N:XhC_pn−!(XhC_pn−1)^hCp is an equivalence. Then, by induction, the map

N:XhC_pn−!(XhC_p)^hCpn−1 is an equivalence. This norm morphism fits into a diagram

XhC_pn

//X^hCpn //

X^tCpn

(XhC_p)^hCpn−1 //(X^hCp)^hCpn−1 //(X^tCp)^hCpn−1,

which commutes, since all maps in the left square are norm maps, and the right vertical map is defined as the cofiber. Now, since the left and middle vertical maps are equiva-lences and the rows are fiber sequences, it follows that the right vertical morphism is an equivalence as well.

In the following lemma, we use the Tate construction forTdefined in CorollaryI.4.3.

Lemma II.4.2. If X is a spectrum bounded below with T-action, then (X^tCp)^hT is p-complete and the canonical morphism X^tT!(X^tCp)^hT exhibits it as the p-completion of X^tT.

Remark II.4.3. This lemma leads to a further simplification of the fiber sequence TC(X)!X^{hT (ϕ}

as one can identify the final term with the profinite completion of X^tT, in case X is a cyclotomic spectrum bounded below.

Proof. The canonical morphism (X^tCp)^hT!(X^tCp)^hCp∞'lim

←−(X^tCp)^hCpn is an equiv-alence, since both sides arep-complete by LemmaI.2.9andCp^∞!Tis ap-adic equiva-lence. Thus, by LemmaII.4.1, in the commutative diagram

X^tT //

all corners except possibly for X^tT are equivalent. Thus, it remains to prove that the canonical mapX^tT!lim←−(X^tCpn) is ap-completion.

By Lemma I.2.6, we may assume that X is bounded, so that, by a filtration, we may assume thatX=HM is an Eilenberg–MacLane spectrum. Note that, in this case, theT-action onM is necessarily trivial. We may also assume thatM is torsion free by passing to a 2-term resolution by torsion-free groups. We find that

πi(HM^tT) =

M, ifieven,

0, ifiodd, and πi(HM^tCpn) =

M/pⁿ, ifi even, 0, ifi odd.

The maps in the limit diagram are given by the obvious projections M!M/pⁿ, by LemmaI.4.4, so the result follows.

Now, let X be a genuine p-cyclotomic spectrum in the sense of Definition II.3.1.

Let us recall the definition of TC^gen(X, p) by B¨okstedt–Hsiang–Madsen [23]. First,X has genuineC_pn-fixed points X^Cpn for alln>0, and there are maps F:X^Cpn!X^Cpn−1 forn>1 which are the inclusion of fixed points. Moreover, for alln>1, there are maps R:X^Cpn!X^Cpn−1, and R and F commute (coherently). The maps R:X^Cpn!X^Cpn−1 arise as the composition of the mapX^Cpn!(Φ^CpX)^Cpn−1 that exists for any genuineC_pn -equivariant spectrum, and the equivalence (Φ^CpX)^Cpn−1'X^Cpn−1 which comes from the genuine cyclotomic structure onX.

These structures determine TC^gen(X, p) as follows.

Definition II.4.4. LetX be a genuinep-cyclotomic spectrum. Define TR(X, p) = lim

←−RX^Cpn, which has an action ofF, and

TC^gen(X, p) := Eq

TR(X, p) ^id //

F //TR(X, p) 'lim

←−REq

X^{Cpn R} //

F //X^Cpn−1 .

To compare this with our definition, we need the following lemma, which follows directly from (the discussion following) PropositionII.2.13.

Lemma II.4.5. Let X be a genuine Cpⁿ-equivariant spectrum. There is a natural pullback diagram of spectra

X^{Cpn R} //

(Φ^CpX)^Cpn−1

X^hCpn //X^tCpn for all n>1.

Note that, while the upper spectra depend on the genuineC_pn-equivariant structure ofX, the lower spectra only depend on the underlying spectrum withC_pn-action.

Proposition II.4.6. Let X be a genuine Cpⁿ-equivariant spectrum. Assume that the underlying spectrum is bounded below. Then,for every n>1,there exists a canonical pullback square

X^Cpn //

(Φ^CpX)^Cpn−1

X^hCpn //(X^tCp)^hCpn−1. Proof. Combine LemmaII.4.1 with LemmaII.4.5.

Corollary II.4.7. Let X be a genuine Cpⁿ-equivariant spectrum. Assume that the spectra

X,Φ^CpX,Φ^Cp2X, ...,Φ^Cpn−1X

are all bounded below. Then,we have a diagram

X^Cpn //

Φ^CpnX

(Φ^Cpn−1X)^hCp //

(Φ^Cpn−1X)^tCp

(Φ^Cp2X)^hCpn−2

//...

(Φ^CpX)^hCpn−1

//((Φ^CpX)^tCp)^hCpn−2

X^hCpn //(X^tCp)^hCpn−1

which exhibits X^Cpn as a limit of the right side (i.e. an iterated pullback).

Proof. Use induction onnand PropositionII.4.6.

Remark II.4.8. One can use the last corollary to give a description of the full subcat-egory ofCpⁿSp consisting of those genuineCpⁿ-equivariant spectra all of whose geometric fixed points are bounded below. An object then explicitly consists of a sequence

X, Φ^CpX, ..., Φ^CpnX

of bounded below spectra withCpⁿ/C_pi-action together with Cpⁿ/C_pi-equivariant mor-phisms

Φ^CpiX−!(Φ^Cpi−1X)^tCp

for all 16i6n. One can prove these equivalences for example similarly to our proof of Theorem II.6.3 in §II.6 and §II.5. Vice versa, one can also deduce our TheoremII.6.3 from these equivalences.

Note that this description is quite different from the description as Mackey functors, which is in terms of the genuine fixed points; a general translation appears in work of Glasman [39]. One can also go a step further and give a description of the category of all genuineC_pn-spectra orT-spectra (without connectivity hypothesis) along these lines, but the resulting description is more complicated and less tractable as it contains a lot of coherence data. This approach has recently appeared in work of Ayala–Mazel-Gee–

Rozenblyum [8].

Let us note the following corollary. This generalizes a result of Ravenel [82], that proves the Segal conjecture forCpⁿ, by reduction toCp. This had been further general-ized to THH by Tsalidis [90], and then by B¨okstedt–Bruner–Lunøe-Nielsen–Rognes [21, Theorem 2.5], who proved the following theorem under a finite-type hypothesis.

Corollary II.4.9. Let X be a genuine C_pn-equivariant spectrum,and assume that for any Y∈{X,Φ^CpX, ...,Φ^Cpn−1X},the spectrum Y is bounded below,and the map

(Y^Cp)^∧_p−!(Y^hCp)^∧_p

induces an isomorphism onπifor all i>k. Then,the map (X^Cpn)^∧_p!(X^hCpn)^∧_p induces an isomorphism on πi for all i>k.

As in [21, Theorem 2.5], this result also holds true if one instead measures connectiv-ity after taking function spectra from someW in the localizing ideal of spectra generated byS[1/p]/S. Note that, for cyclotomic spectra, one needs to check the hypothesis only forY=X.

Proof. Note that, for 16i6n, one has Φ^Cp(Φ^Cpi−1X)=Φ^CpiX. Therefore, using Lemma II.4.5 for the genuine C_p-equivariant spectrum Φ^Cpi−1X, we see that the as-sumption is equivalent to the condition that

Φ^CpiX−!(Φ^Cpi−1X)^tCp

induces an isomorphism on πi for all i>k after p-completion. Now, we use that, in CorollaryII.4.7, all the short vertical maps induce an isomorphism onπ_ifor alli>kafter p-completion. It follows that the long left vertical map does the same, as desired.

Theorem II.4.10. Let X be a genuine p-cyclotomic spectrum such that the under-lying spectrum is bounded below. Then, there is a canonical fiber sequence

TC^gen(X, p)−!X^{hCp∞ ϕ}

hCp∞ p −can

−−−−−−−−−!(X^tCp)^hCp∞ In particular, we get an equivalence TC^gen(X, p)'TC(X, p).

Proof. By Proposition II.4.6 and the equivalenceX'Φ^CpX, we inductively get an equivalence

X^Cpn'X^hCpn×

(X^tCp)^hCpn−1X^hCpn−1×...×_XtCpX;

cf. also CorollaryII.4.7. Here, the projection to the right is always the tautological one, and the projection to the left isϕ_p(or more precisely the induced map on homotopy fixed points). Under this equivalence, the map R:X^Cpn!X^Cpn−1 corresponds to forgetting

the first factor andF:X^Cpn!X^Cpn−1 corresponds to forgetting the last factor followed R⁰⁰ forgets the first factor, and F⁰⁰ forgets the last factor and projects (X^tCp)^hCpk! (X^tCp)^hCpk−1. Also, ϕ_p denotes the various maps X^hCpk!(X^tCp)^hCpk, and can the various mapsX^hCpk!(X^tCp)^hCpk−1. It is easy to see that the diagram commutes (since the lower-right is a product; this can be checked for every factor separately), and by design the vertical fibers areX^Cpn andX^Cpn−1 with the induced map given byR−F.

Now, we compute the fibers horizontally. We getX^hCpn via the diagonal embedding for the upper line and analogously (X^tCp)^hCpn−1 via the diagonal embedding for the lower line. Therefore, the fiber of R−F:X^Cpn!X^Cpn−1 is equivalent to the fiber of ϕ_p−can:X^hCpn!(X^tCp)^hCpn−1. Finally, we take the limit over n to get the desired result.

Recall from [32, Lemma 6.4.3.2] Goodwillie’s definition of the integral topological cyclic homology for a genuine cyclotomic spectrumX in the sense of DefinitionII.3.3. It is defined(²²)by the pullback square

TC^gen(X) //

Note that the homotopy fixed points (X_p^∧)^hT in the lower-right corner are equivalent to (X_p^∧)^hCp∞, since the inclusion C_p^∞!Tis ap-adic equivalence. Taking homotopy fixed points commutes with p-completion, since the mod-pMoore spectrum M(Z/p)=S/p is a finite spectrum, and fixed points of ap-complete spectrum stayp-complete. It follows that the right vertical map is a profinite completion, and therefore also the left vertical map is a profinite completion.

(²²) This is not the definition initially given by Goodwillie, but it is equivalent to the corrected version of Goodwillie’s definition as we learned from B. Dundas, and we take it as a definition here.

TheoremII.4.11. LetX be a genuine cyclotomic spectrum such that the underlying spectrum is bounded below. Then,there is a canonical fiber sequence

TC^gen(X)!X^{hT (ϕ} In particular, we get an equivalenceTC^gen(X)'TC(X).

Proof. We consider the commutative diagram

Here, the lower line is the product over allpof thep-completions of the fiber sequences from Theorem II.4.10, and the square is the pullback defining TC^gen(X). Now, by LemmaI.2.9and using thatC_p^∞!Tis a p-adic equivalence, the natural maps

(X^tCp)^hT−!((X_p^∧)^tCp)^hCp∞

are equivalences for all primesp. Thus, we can fill the diagram to a commutative diagram

TC^gen(X) //

Now the result follows formally as the lower line is a fiber sequence, the left square is a pullback and the right map is an equivalence.

Dans le document On topological cyclic homology (Page 59-65)