Math. J. Okayama Univ. 66 (2024), 135–157

SEVERAL HOMOTOPY FIXED POINT SPECTRALSEQUENCES IN TELESCOPICALLY LOCALIZEDALGEBRAIC K-THEORY
Daniel G. Davis
Abstract. Let n ≥ 1, p a prime, and T (n) any representative of theBousﬁeld class of the telescope vn−1 F (n) of a ﬁnite type n complex.
Also, let En be the Lubin-Tate spectrum, K(En ) its algebraic K-theory
spectrum, and Gn the extended Morava stabilizer group, a proﬁnite
group. Motivated by an Ausoni-Rognes conjecture, we show that there
are two spectral sequences
E2s,t =⇒ πt−s ((LT (n+1) K(En ))hGn ) ⇐= IIE2s,tI

with common abutment π∗ (−) of the continuous homotopy ﬁxed pointsof LT (n+1) K(En ), where IE2s,t is continuous cohomology with coeﬃcientsin a certain tower of discrete Gn -modules. If the tower satisﬁes the
Mittag-Leﬄer condition, then there are isomorphisms with continuouscochain cohomology groups:∗
II ∗,∗I ∗,∗ ∼E
= Hcts (Gn , π∗ (LT (n+1) K(En ))) ∼
= E .
2

2

We isolate two hypotheses, the ﬁrst of which is true when (n, p) = (1, 2),that imply (LT (n+1) K(En ))hGn ≃ LT (n+1) K(LK(n) S 0 ). Also, we showthat there is a spectral sequences
Hcts(Gn , πt (K(En ) ⊗ T (n + 1))) =⇒ πt−s ((K(En ) ⊗ T (n + 1))hGn ).

1. Introduction
1.1. The basic characters in this work and LT (n+1) K(En ) as a Gn spectrum. Let n ≥ 1, let p be any prime, and let Sp be the symmetricmonoidal ∞-category of spectra. Let F (n) denote a ﬁnite type n complex
and let vn−1 F (n) be the telescope of a vn -self-map v on F (n). By [25, Lemma
4] and [32, page 103], vn−1 F (n) is independent of the choice of v and the
Bousﬁeld class of vn−1 F (n) is independent of the choices for F (n) and v. As
is common, we let T (n) denote a representative of this Bousﬁeld class. Also,
given a ring spectrum A (that is, an algebra in Sp; also called an E1 -ring
spectrum), K(A) denotes the algebraic K-theory spectrum of A.

Mathematics Subject Classiﬁcation. Primary 19D55; Secondary 55N20, 55P42, 55T25.
Key words and phrases. Algebraic K-theory spectrum, continuous homotopy ﬁxed
point spectrum, Lubin-Tate spectrum, Morava stabilizer group, homotopy ﬁxed point
spectral sequence, telescopic localization.
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D. G. DAVISLet K(n) be the nth Morava K-theory spectrum and let En be the LubinTate spectrum withπ∗ (En ) = W (Fpn )[[u1 , ..., un−1 ]][u±1 ],where the complete power series ring over the Witt vectors for the ﬁeld withpn elements is in degree 0 and |u| = −2. Also, Gn denotes the nth extended
Morava stabilizer group, a proﬁnite group which acts on En by maps ofcommutative algebras (for more background on this, see [19, 23, 24]). As in[19, (1.4)], let
Gn = U0 ⊋ U1 ⊋ · · · ⊋ Ui ⊋ · · ·Tbe a descending chain of open normal subgroups of Gn , with i≥0 Ui = {e}.
Then by [19, Theorem 1], [8, Section 8], [20], and [36, Theorem 1.2], there
is a diagram(1.1)

EnhGn = EnhU0 → EnhU1 → · · · → EnhUi → · · ·in the ∞-category CAlg(Sp) of commutative algebras in Sp that consists ofcontinuous homotopy ﬁxed point spectra.
As in [28, Proposition 4.22; Proposition 7.10, (d)], we let
M0 ← M1 ← · · · ← Mj ← · · ·be a tower of generalized Moore spectra: each Mj is ﬁnite of type n + 1
and an atomic µ-spectrum and the tower has the property that there is anequivalence
LF (n+1) Z ' lim(Z ⊗ Mj ),j≥0

for any spectrum Z. Let M U(p) denote the p-localization of M U . For eachj, there is a sequence a(j)0 , a(j)1 , ..., a(j)n of powers of p such that
a(j)
a(j)
(M U(p) )∗ (Mj ) ∼= (M U(p) )∗ /(v0 0 , v1 1 , ..., vna(j)n )and it is common to write
a(j)0

Mj = M (v0

a(j)1

, v1

, ..., vna(j)n ).

As recalled in Deﬁnition 4 (see [38, Deﬁnition 2.6], [32]), we use Lfn+1 todenote the Bousﬁeld localization functor that is often referred to as “ﬁnite
E(n + 1)-localization” [34]. We can now state a result that we use in Section
1.2 to deﬁne the continuous Gn -homotopy ﬁxed points of LT (n+1) K(En ).
Theorem 1.1. For every n ≥ 1 and all primes p, there is an equivalence
LT (n+1) K(En ) ' lim(colim(K(EnhUi ) ⊗ Lfn+1 Mj )).
j≥0

i≥0

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On the right-hand side of the above equivalence – and elsewhere in thispaper – we use “colim” to denote the colimit in Sp; any other colimit forspectra equipped with additional structure is marked as such. The proof of
Theorem 1.1, which is in Section 3, uses a deep result from [31] in a key way.
The Gn -action on En induces a natural Gn /Ui -action on each EnhUi , andhence, for each i and j, K(EnhUi )⊗Lfn+1 Mj has a natural Gn /Ui -action, with
Gn /Ui acting trivially on Lfn+1 Mj (and diagonally on the smash product).
It follows that in Theorem 1.1, the equivalence is Gn -equivariant.
Deﬁnition 1. Let G be any proﬁnite group. As in [11], let Sh(TG , Sp)
denote the ∞-category of Sp-valued sheaves on the Grothendieck site TG ofﬁnite continuous G-sets. Also, as in [op. cit.], PShΠ (TG , Sp) denotes the ∞category of presheaves of spectra on TG that send ﬁnite coproducts in TG toﬁnite products (see also [6, Section 2]), and PSh(OG , Sp) is the ∞-categoryof presheaves on the orbit category of G consisting of quotients of G by open
subgroups.
By applying [19, Theorem 1] and [8, Section 8], we let F ∈ PSh(OGn , Sp)
be the presheaf deﬁned by
F : Gn /U 7→ EnhU ,U an open subgroup of Gn ,where, as before, EnhU is the continuous homotopy ﬁxed point spectrum andF is actually a presheaf of commutative algebras. Thus, there is a diagram{Pj }j≥0 in PSh(OGn , Sp), with each presheaf Pj deﬁned by
Pj : Gn /U 7→ K(EnhU ) ⊗ Lfn+1 Mj ,U an open subgroup of Gn .

1.2. Continuous homotopy ﬁxed points for LT (n+1) K(En ) and two
associated homotopy ﬁxed point spectral sequences. We say thata proﬁnite group G has “ﬁnite cohomological dimension” if there is some
integer s0 such that the continuous cohomology Hcs (G, M ) = 0, whenevers > s0 and M is any discrete G-module. Recall that Gn has ﬁnite virtual
cohomological dimension: that is, Gn contains an open subgroup U of ﬁnite
cohomological dimension. Also, as recalled (with more generality) in Section
4, if
X0 → X1 → · · · → Xi → · · ·is a diagram of Gn -spectra such that for each i, the Gn -action on Xi factors
through Gn /Ui (that is, Ui acts trivially on Xi ), then there is the continuous
homotopy ﬁxed point spectrum (colimi≥0 Xi )hGn , given by the totalizationof an {i ≥ 0}-indexed colimit of certain familiar cosimplicial spectra.
It is now natural to make the following deﬁnition, which follows a familiar
template in algebraic K-theory (for example, see [22, Proposition 3.1.2, lastparagraph of 3.1, proof of Theorem 4.2.6]).

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D. G. DAVIS

Deﬁnition 2. Let n ≥ 1 and let p be any prime. There is the continuous
homotopy ﬁxed point spectrumhGn(LT (n+1) K(En ))hGn := lim colim(K(EnhUi ) ⊗ Lfn+1 Mj )
,
j≥0

i≥0

where the right-hand side is given by

hGn
hGn
lim colim(K(EnhUi )⊗Lfn+1 Mj ):= lim colim(K(EnhUi )⊗Lfn+1 Mj )
.
j≥0

i≥0

j≥0

i≥0

For each j ≥ 0, by Theorem 4.1, there is an equivalencehGnfj ∏ (∗)
colim(K(EnhUi ) ⊗ Lfn+1 Mj )= (colim Pj (Gn /Ui ))hGn ' P
i≥0

i≥0

fj ∏ of the presheafwith the global sections of the Postnikov sheaﬁﬁcation P
Pj ∏ in PSh∏ (TGn , Sp) that is induced by Pj .
To state the next result, whose proof is in Section 5, we need the following notation. Given a proﬁnite group G and a tower {Aj }j≥0 of discrete∗ (G; {A }G-modules, we let Hcont
j j≥0 ) denote continuous cohomology in the∗ (G, lim
sense of Jannsen [29] and Hctsj≥0 Aj ) is continuous cochain cohomology with coeﬃcients in the stated topological G-module. Also, if X • is a
cosimplicial spectrum, then we let IIE2s,t be the E2 -term of the associatedhomotopy spectral sequenceE2s,t := lims πt (X • ) =⇒ πt−s (Tot(X • )).

II∆

Theorem 1.2. Let n ≥ 1 and let p be any prime. There are conditionallyconvergent homotopy ﬁxed point spectral sequences−1sMj )}j≥0 ) =⇒ πt−s ((LT (n+1) K(En ))hGn )(Gn ; {πt (K(En )⊗vn+1
E2s,t = Hcont

I

andE2s,t =⇒ πt−s ((LT (n+1) K(En ))hGn ).

II−1
If the tower {πt (K(En ) ⊗ vn+1
Mj )}j≥0 satisﬁes the Mittag-Leﬄer condition
for every t ∈ Z, then for all s ≥ 0, there are isomorphisms
I s,∗ ∼∼ IIE s,∗ ,E = H s (Gn , π∗ (LT (n+1) K(En ))) =
2

cts

2

−1
where for each t, πt (LT (n+1) K(En )) ∼Mj ).
= limj≥0 πt (K(En ) ⊗ vn+1

For any spectrum X with trivial Gn -action, there are two homotopy ﬁxed
point spectral sequences for π∗ ((LK(n) (En ⊗ X))hGn ) that correspond to
the two in Theorem 1.2, but by [17, Theorem 1.2], it is the second one,with its own particular IIE2∗,∗ , that is isomorphic to the strongly convergent
K(n)-local En -Adams spectral sequence for π∗ (LK(n) X). One ingredientin the construction of this Adams-type spectral sequence is that En is acommutative algebra. Similarly, K(En ) is a commutative algebra and itSEVERAL HOMOTOPY FIXED POINT SPECTRAL SEQUENCES

139

seems plausible that, in general, the second spectral sequence in Theorem 1.2
has better properties. Also, by [8, Section 4.6], it could happen that there
are cases where IIE2s,t in Theorem 1.2 is equal to the continuous cochain
cohomology group stated in the theorem, without the underlying tower of
discrete Gn -modules satisfying the Mittag-Leﬄer condition.
Since F can be regarded as a presheaf of commutative algebras, there is
the presheaf Lfn+1 ◦ K ◦ F in PSh(OGn , Sp) given by
Gn /U 7→ Lfn+1 K(F (Gn /U )) = Lfn+1 K(EnhU ),U an open subgroup of Gn ,which induces the diagram
Lfn+1 K(EnhU0 ) → Lfn+1 K(EnhU1 ) → · · · → Lfn+1 K(EnhUi ) → · · · ,with Gn /Ui acting on Lfn+1 K(EnhUi ) for each i, and hence, there is thecontinuous homotopy ﬁxed point spectrum(colim Lfn+1 K(EnhUi ))hGn .
i≥0

The next result, whose proof is in Section 5, uses this last spectrum to showthat (LT (n+1) K(En ))hGn is T (n + 1)-local.
Theorem 1.3. For each n ≥ 1 and all primes p, there is an equivalence(LT (n+1) K(En ))hGn ' LT (n+1) ((colim Lfn+1 K(EnhUi ))hGn ).
i≥01.3. Potential relationships of (LT (n+1) K(En ))hGn with an AusoniRognes conjecture involving K(En )hGn . The spectral sequences in Theorem 1.2 remind one of an Ausoni-Rognes conjecture ([2, (0.1)], [1, page 46;
Remark 10.8]; also, see the closely related [3, Conjecture 4.2]), which statesthat (a) the K(n)-local unit map η : LK(n) S 0 → En induces a map
K(LK(n) S 0 ) → K(En )hGn ,which – in this paper – we refer to as Kτ and whose target K(En )hGn is a
homotopy ﬁxed point spectrum whose construction is compatible with the
proﬁnite topology on Gn ; and (b) the map
Kτ ⊗ T (n + 1) : K(LK(n) S 0 ) ⊗ T (n + 1) → K(En )hGn ⊗ T (n + 1)
is an equivalence, so that Kτ is a T (n + 1)-equivalence.
Remark 1. The τ in Kτ is for “transposing,” since the K(n)-local unit≃

EnhGn ←− LK(n) S 0
of commutative algebras is an equivalence, by [19, Theorem 1], giving≃

K(EnhGn ) ←− K(LK(n) S 0 ),

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D. G. DAVIS

which implies that Kτ switches K(−) and (−)hGn in the case of En .
Remark 2. Currently, for every n and p, there is not a published construction of K(En )hGn or the map Kτ . In [14, Remark 1.5], it is noted that
according to Jacob Lurie, the condensed mathematics of Dustin Clausen
and Peter Scholze can be used to deﬁne K(En ) as a condensed spectrum,and then building on this, there is a candidate deﬁnition of K(En )hGn in
the condensed setting. Similarly (see [op. cit.]), by viewing K(En ) as apyknotic spectrum [5, Section 3.1], there should be a pyknotic version of the“condensed candidate” for K(En )hGn .
Remark 3. Strictly speaking, the Ausoni-Rognes conjecture referred to above−1
predicts that Kτ ⊗ vn+1
F (n + 1) is an equivalence, but this is logically thesame as Kτ ⊗ T (n + 1) being an equivalence.
From the commutative squareLT (n+1) K(LK(n)/ K(En )hGn

KτK(LK(n) S 0 )S0)/ LT (n+1) (K(En )hGn ),LT (n+1) Kτwhose vertical maps, as the usual localizations, are T (n + 1)-equivalences,
we see that if part (a) of the above Ausoni-Rognes conjecture holds, thenpart (b) holds if and only if LT (n+1) (Kτ ) is an equivalence. This leads one
to wonder about the relationship between
LT (n+1) (K(En )hGn ) and (LT (n+1) K(En ))hGn
and if there is an equivalence between(LT (n+1) K(En ))hGn and LT (n+1) K(LK(n) S 0 ).
The following result, whose proof is in Section 6, gives two hypotheses thatwhen jointly satisﬁed imply that the last two spectra
Q above are equivalent.
If a ﬁnite group H acts on a spectrum Z, we let H • Z denote the induced
cosimplicial spectrum; see Section 4 for more detail. Also, for each i ≥ 0,
we letK(η ↗ i) : LT (n+1) K(EnhGn ) → (LT (n+1) K(EnhUi ))hGn /Uibe the natural map.
Theorem 1.4. Let n ≥ 1 and let p be a prime. If(H1) the map
colim K(η ↗ i) : LT (n+1) K(EnhGn ) → colim(LT (n+1) K(EnhUi ))hGn /Ui
i≥0

i≥0

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141

is a T (n + 1)-equivalence, and(H2) for each j ≥ 0, the canonical map
colim Tot(C(n, j, i)• ) → Tot(colim C(n, j, i)• )
i≥0

i≥0

is an equivalence, where for each i,Q
C(n, j, i)• := (Gn /Ui )• (K(EnhUi ) ⊗ Lfn+1 Mj ),then there is an equivalence(LT (n+1) K(En ))hGn ' LT (n+1) K(LK(n) S 0 ).
When n = 1 and p = 2, Remark 4 shows that (H1) is true, but the validityof (H2) is still open. For pairs (n, p) 6= (1, 2), neither (H1) nor (H2) is known
to be true and below we give some considerations related to this.
Remark 4. The map K(η ↗ 0) is an equivalence, and to show that (H1)holds, it suﬃces to show that for a coﬁnal subsequence i0 , i1 , ..., il , ... of{i ≥ 0}, each map K(η ↗ il ) is an equivalence. For each i, the canonical map
EnhGn → EnhUi is a K(n)-local Gn /Ui -Galois extension [39, Theorem 5.4.4,(c)], so that by, for example, (2.1), this map is also a T (n)-local Gn /Ui Galois extension, as noted in [13, Section 4.3]. By [op. cit., Corollary 4.16],
if Gn /Ui is a p-group, then K(η ↗ i) is an equivalence. When (n, p) = (1, 2),
∼
G1 ∼
= Z×
2 = Z2 ×Z/(2) is a pro-2-group, with Z2 equal to the 2-adic integers,and hence, (H1) holds.
It is a special case of [3, Conjecture 4.2], due to Ausoni and Rognes, thatfor all n, p, and i, the canonical map
LT (n+1) K(EnhGn ) → LT (n+1) (K(EnhUi )hGn /Ui )is an equivalence. Though “Conjecture 4.2” is in general still open, thisconjecture, results in [13, Sections 1, 4], especially [op. cit., Corollary 4.16]– which was used in Remark 4, and [12, Theorems 1.3, 1.8, 1.10, 5.1, 5.6],
which include verifying special cases of “Conjecture 4.2,” give momentum
for perhaps validating (H1) in every case.
To underline the plausibility of (H1) in general, we brieﬂy highlight [13,Example 4.17] from the progress cited above. Let E be a Lubin-Tate theory∼bwith extended Morava stabilizer group Gex
n = Sn ⋊ Z, where Sn is the Morava
b is the proﬁnite completion of the integers, and forstabilizer group and Z
hK denote the continuous homotopy ﬁxed
K a closed subgroup of Gex
n , let E
points. As explained in [7, Sections 5.1, 5.2], the construction of E hK uses[19]. Now let U be an open subgroup of Gex
n such that U ∩ Sn is pro-p. By[13, Corollary 4.16] and [12], given any normal inclusion V ′ ◁ V ⊂ U of

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D. G. DAVIS

open subgroups, the canonical map≃′

LT (n+1) K(E hV ) −→ (LT (n+1) K(E hV ))hV /V′is an equivalence, and this yields a sheaf of T (n + 1)-local spectra on thesite TU .
As explained in Deﬁnition 5, (H2) holds if for every j ≥ 0, the presheaf Pj
of Section 1.1 satisﬁes “condition (iv)” with N = {Ui | i ≥ 0} (“condition(iv)” is based on [11]). Related to this is the familiar problem of showingthat a ﬁltered colimit of homotopy spectral sequences has abutment equal
to the colimit of the abutments of those spectral sequences (for example,see [35, Section 3.1.3]). Given any j, for each i there is the homotopy ﬁxed
point spectral sequence { jiEr∗,∗ }r≥1 that has the formfhGn /UihUi
i s,t,jE2 =⇒ πt−s (K(En ) ⊗ Ln+1 Mj )
where
f
i s,t
s
hUi
jE2 = H (Gn /Ui , πt (K(En ) ⊗ Ln+1 Mj )).

Then (H2) is valid if for each j, there is some r ≥ 2 and some integer s′ such
that
i s,t
′
jEr = 0, for all i ≥ 0, s > s , t ∈ Z.
1.4. Possible connections with the Ausoni-Rognes conjecture without using towers. The next result is an immediate consequence of thefollowing deﬁnition (and, for example, [8, Theorem 3.2.1] and [15, Theorem7.9]).
Deﬁnition 3. Let n ≥ 1 and set p equal to any prime. Recall that T (n + 1)
−1
F (n + 1),
is any choice of a representative from the Bousﬁeld class of vn+1
where F (n + 1) is any ﬁnite type n + 1 complex. Since
K(En ) ⊗ T (n + 1) ' colim(K(EnhUi ) ⊗ T (n + 1)),i≥0where for each i, the copy of T (n + 1) is equipped with the trivial Gn /Ui action, it is natural to deﬁne the continuous homotopy ﬁxed point spectrum
hGn(K(En ) ⊗ T (n + 1))hGn := colim(K(EnhUi ) ⊗ T (n + 1)),i≥0which, as in Deﬁnition 2, is the global sections of a Postnikov sheaﬁﬁcation.
Theorem 1.5. For each n ≥ 1 and a prime p, there is a conditionallyconvergent homotopy ﬁxed point spectral sequenceE2s,t = Hcs (Gn , πt (K(En ) ⊗ T (n + 1))) =⇒ πt−s ((K(En ) ⊗ T (n + 1))hGn ),where πt (K(En ) ⊗ T (n + 1)) is a discrete Gn -module, for each t ∈ Z.

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143

By [1, page 46; Remark 10.8], the Ausoni-Rognes conjecture predicts that
there is a homotopy ﬁxed point spectral sequence−1−1
Hcs (Gn , πt (K(En ) ⊗ vn+1
F (n + 1))) =⇒ πt−s (K(LK(n) S 0 ) ⊗ vn+1
F (n + 1)),and thus, it is natural to ask the following questions:−1• When T (n + 1) := vn+1
F (n + 1), the spectral sequence in Theorem
1.5 and the above conjectural spectral sequence have identical E2 terms. Is the former spectral sequence a realization of the latterone?
• In general, is there an equivalence between
K(LK(n) S 0 ) ⊗ T (n + 1) and (K(En ) ⊗ T (n + 1))hGn ?
• What is the relationship between(K(En ) ⊗ T (n + 1))hGn and K(En )hGn ⊗ T (n + 1)?
The following result gives two conditions that imply cases in which theanswers to the ﬁrst and second questions above are “yes.” The ﬁrst condition
below, (H1′ ), is logically equivalent to (H1) in Theorem 1.4.
Theorem 1.6. Let n ≥ 1, let p be a prime, and set T (n + 1) equal to−1
F (n + 1), where F (n + 1) is an atomic µ-spectrum. If
vn+1(H1′ ) the map colim K(η ↗ i) is a T (n + 1)-equivalence, andi≥0(H2′ ) the canonical map colim Tot(C(n, i)• ) → Tot(colim C(n, i)• ) is an
i≥0

i≥0

equivalence, where for each i,Q
C(n, i)• := (Gn /Ui )• (K(EnhUi ) ⊗ T (n + 1)),then there is an equivalence(K(En ) ⊗ T (n + 1))hGn ' K(LK(n) S 0 ) ⊗ T (n + 1).
The proof of Theorem 1.6 is in Section 6. As with the two conditions in
Theorem 1.4, when (n, p) = (1, 2), assumption (H1′ ) is true, by Remark 4,
but the status of (H2′ ) is unknown, and for all other pairs (n, p), neither(H1′ ) nor (H2′ ) is known to hold.
The next result, whose proof is in Section 5, has as a consequence that if
F (n + 1) is chosen to be an atomic µ-spectrum, then (by Remark 6) with−1
F (n + 1), (K(En ) ⊗ T (n + 1))hGn is T (n + 1)-local.
T (n + 1) set equal to vn+1
−1
Theorem 1.7. When n ≥ 1, p is a prime, and T (n + 1) := vn+1
F (n + 1),
where F (n + 1) is any ﬁnite type n + 1 complex, there are equivalences(K(En ) ⊗ T (n + 1))hGn ' (colim Lfn+1 K(EnhUi ))hGn ⊗ F (n + 1)
i≥0

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D. G. DAVIS' (colim Lfn+1 K(EnhUi ))hGn ⊗ T (n + 1)i≥0

' (LT (n+1) K(En ))hGn ⊗ T (n + 1).
Let n = 1, p ≥ 5, and let V (1) denote the type 2 Smith-Toda complex
S 0 /(p, v1 ). In [14], we constructed in the setting of symmetric spectra of×
simplicial sets the continuous homotopy ﬁxed points (K(E1 ) ∧ v2−1 V (1))hZp .
Here, we are using “∧” for the smash product in symmetric spectra. In
symmetric spectra, there should be a zigzag of weak equivalences between
hG1
×
– the
this (K(E1 ) ∧ v2−1 V (1))hZp and colimi≥0 (K(E1hUi ) ∧ v2−1 V (1))−1model for the continuous homotopy ﬁxed points of K(E1 ) ∧ v2 V (1) constructed in Deﬁnition 3, but we have not completed our work on this zigzag.
In [14], we also obtained a homotopy ﬁxed point spectral sequence for×(K(E1 ) ∧ v2−1 V (1))hZp with E2 -term equal to the E2 -term of the spectral
sequence given by Theorem 1.5 (with T (2) there set equal to v2−1 V (1)) andthese two spectral sequences should be isomorphic, but our work on this isincomplete, since it is closely related to the aforementioned zigzag.
2. Some basic facts about T (n)-localization
As in the introduction, n ≥ 1, p is any prime, and F (n) is a ﬁnite type ncomplex. Also, following [28, Proposition 4.22], we let
M0 ← M1 ← · · · ← M j ← · · ·be a tower of generalized Moore spectra, with each Mj ﬁnite of type n (here,
we have type n, not type n + 1, as in the introduction) and an atomic µspectrum. As recalled in Section 1.1, one feature of this tower is that for
any Z ∈ Sp,
LF (n) Z ' lim(Z ⊗ Mj ).
j≥0Given a ﬁnite type 0 complex F (0), we let T (0) := v0−1 F (0) denote thetelescope of a v0 -self-map on F (0). Thus, T (0) and HQ have the same
Bousﬁeld class.
Remark 5. Suppose that F (n) is an atomic µ-spectrum. Then each of F (n)and, by [32, proof of Lemma 2.2], the telescope vn−1 F (n) is a “ring spectrum,”in the sense of [32] (see also [18]). Here, by “ring spectrum,” we mean aleft-unital magma in the homotopy category of spectra (that need not be
associative or right-unital). It follows that T (n) can be taken to be a “ringspectrum” in the above sense, and thus, it is worth highlighting the factthat [31, proof of Lemma 2.3] proves the much stronger result that T (n) canbe set equal to an algebra in Sp.

SEVERAL HOMOTOPY FIXED POINT SPECTRAL SEQUENCES

145

Remark 6. Again, let F (n) be an atomic µ-spectrum, so that as in Remark
5, vn−1 F (n) is a “ring spectrum.” Then if Z is any spectrum, Z ⊗ vn−1 F (n)is T (n)-local. To verify this, it suﬃces to show that the equivalent spectrum vn−1 F (n) ⊗ Z is vn−1 F (n)-local (since vn−1 F (n) and T (n) have the same
Bousﬁeld class), and this conclusion is reached by noting that the argumentin [37, proof of Proposition 1.17, (a)] goes through here. The observationthat this argument applies in this context also occurs in the antepenultimate
paragraph of [32, proof of Lemma 2.2].
We return to letting F (n) denote a ﬁnite type n complex that is notnecessarily an atomic µ-spectrum. We recall some standard notation (forexample, see [32, Deﬁnition 3.1]).
Deﬁnition 4. For each n ≥ 1 and every prime p, Lfn denotes the Bousﬁeld
localization functor determined by the spectrum T (0) ⊕ T (1) ⊕ · · · ⊕ T (n).
By [32, Corollary 3.5], Lfn is smashing.
Remark 7. By [32, Proposition 3.2], there is an equivalence
Lfn F (n) ' vn−1 F (n).
We believe the following result is fairly well-known (for example, see [9,3.2; Theorem 3.3] and [27, Corollary 2.2]), but we do not know of a reference
to it in the literature that – relative to the setup and deﬁnitions in this paper– is straightforward to follow, and so we give a proof. We learned of thisresult from [21, Fact 2.11, 2] and in the case when Z = S 0 , the result is [32,Proposition 5.1].
Theorem 2.1. Given Z ∈ Sp, n ≥ 1, and p any prime, there is an equivalence
LT (n) Z ' LF (n) Lfn Z.
Proof. Since Lfn is smashing and, as in [32, Proposition 5.1], there is thediagram {vn−1 Mj }j≥0 of telescopes, there are equivalences
LF (n) Lfn Z ' lim(Z ⊗ Lfn Mj ) ' lim(Z ⊗ vn−1 Mj ).
j≥0

j≥0

Each Mj is an atomic µ-spectrum, so that each Z ⊗ vn−1 Mj is T (n)-local,
and hence, the three displayed expressions above are T (n)-local. Now weonly need to show that the composition
Z → Lfn Z → LF (n) Lfn Z
of canonical maps is a T (n)-equivalence.
We ﬁx a choice for T (n): let
T (n) = vn−1 M0 = colim Σ−kd M0 ,
k≥0

146

D. G. DAVIS

where d is a ﬁxed integer determined by the self-map used to form thetelescope. As in Remark 6, since M0 is an atomic µ-spectrum, the argumentin [37, proof of Proposition 1.17, (a)] shows that Lfn S 0 ⊗ Z ⊗ Σ−kd M0 is M0 local, and hence, F (n)-local, since M0 and F (n) have the same Bousﬁeld
class (by [26, page 5]), for all k ≥ 0. This justiﬁes the third step in thefollowing chain of equivalences, whose ﬁrst step applies the fact that Σ−kd M0
is a ﬁnite spectrum:(LF (n) Lfn Z) ⊗ T (n) ' colim lim(Lfn S 0 ⊗ Z ⊗ Mj ⊗ Σ−kd M0 )k≥0 j≥0' colim LF (n) (Lfn S 0 ⊗ Z ⊗ Σ−kd M0 ) ' colim(Lfn S 0 ⊗ Z ⊗ Σ−kd M0 )k≥0k≥0
' Lfn S 0 ⊗ Z ⊗ T (n) ' Z ⊗ Lfn T (n) ' Z ⊗ Lfn (Lfn M0 ) ' Z ⊗ Lfn M0' Z ⊗ T (n).
It follows that the aforementioned composition is a T (n)-equivalence.

□

Remark 8. In [13, Section 4.3], the authors work with T (n)-local and T (n)local pro-Galois extensions in the sense of [39]. Theorem 2.1 above showsthat T (n) satisﬁes [8, Assumption 1.0.3] and so [8] can be used to study“T (n)-local proﬁnite Galois extensions” (especially ones that are consistentand of ﬁnite virtual cohomological dimension), which diﬀer slightly fromT (n)-local pro-Galois extensions.
As in [34, Section 3], set
K(≤ n) := K(0) ⊕ K(1) ⊕ · · · ⊕ K(n),where K(0) = HQ, and as is standard, we let Ln denote the Bousﬁeld
localization functor LK(≤n) . Notice that given a p-local spectrum Z, thecanonical K(≤ n)-equivalence Lfn Z → Ln Z (see [32, page 113]) is an equivalence when Z is K(≤ n)-local, since there are equivalences≃

Lfn Z ' Lfn S 0 ⊗ Z ' Ln S 0 ⊗ Lfn S 0 ⊗ Z ' Ln Lnf Z −→ Ln Ln Z ' Z.
Therefore, if a spectrum Z is K(≤ n)-local, there are equivalences
(2.1)

≃

LK(n) Z ' LF (n) Ln Z ←− LF (n) Lfn Z ' LT (n) Z,

where the ﬁrst and last steps applied [28, Proposition 7.10, (e)] and Theorem
2.1, respectively. The observation in (2.1) is not original: it is stated in [4,Section 3] and [31, proof of Corollary 4.20, (iv)], and the latter reference
gives a proof.

SEVERAL HOMOTOPY FIXED POINT SPECTRAL SEQUENCES

147

3. A proof of Theorem 1.1
We continue to let n ≥ 1 and p denotes a prime. Also, we use the notation“ colimA ” for the colimit in the ∞-category A of ring spectra.
Now letA0 → A1 → · · · → Ai → Ai+1 → · · ·be a diagram in the ∞-category AT (n) of T (n)-local ring spectra. The colimitAT (n)colimi≥0
Ai of this diagram in AT (n) satisﬁes

A

T (n)
colimi≥0
Ai ' LT (n) (colimAi≥0 Ai )in AT (n) . Then a special case of [31, Corollary 4.31], which is due to Land,
Mathew, Meier, and Tamme, is the remarkable fact that there is an equivalenceA

T (n)LT (n+1) K(colimi≥0Ai ) ' LT (n+1) (colim LT (n+1) K(Ai ))i≥0

in Sp, which simpliﬁes to(3.1)

LT (n+1) K(LT (n) (colimAi≥0 Ai )) ' LT (n+1) (colim K(Ai ))i≥0

in Sp.
We recall that (1.1) is the diagram EnhGn = EnhU0 → EnhU1 → · · · in the
category CAlg(Sp) of commutative algebras. Let “ colimCAlg(Sp) ” denotethe colimit in CAlg(Sp). By [19, Deﬁnition 1.5, Lemma 6.2 and its proof,
Proposition 6.4], there is an equivalenceCAlg(Sp)En ' LK(n) (colimi≥0

EnhUi )in CAlg(Sp). Also, the sequential colimits
CAlg(Sp)

colimi≥0

EnhUi

and

hUi
colimAi≥0 Enare equivalent in Sp to colim EnhUi , and the equivalencei≥0

hUi ' colim
colimAi≥0 Eni≥0CAlg(Sp)

EnhUi

is in A.
For each i, EnhUi is K(n)-local, so that EnhUi is T (n)-local and colim EnhUi
i≥0

is K(≤ n)-local. Then by (2.1),
LK(n) (colim EnhUi ) ' LT (n) (colim EnhUi ).
i≥0

i≥0

Furthermore, (1.1) is a diagram in AT (n) and the last equivalence gives
hUi
hUi
hUi
LK(n) (colimAi≥0 En ) ' LK(n) (colim En ) ' LT (n) (colim En )
i≥0

i≥0

148

D. G. DAVIS
hUi' LT (n) (colimAi≥0 En ).

hUi
Since En ' LK(n) (colimAi≥0 En ) in A, we ﬁnd that there is an equivalence
hUi
En ' LT (n) (colimAi≥0 En )in A. Therefore, equivalence (3.1) yields the equivalenceshUi
LT (n+1) K(En ) ' LT (n+1) K(LT (n) (colimAi≥0 En ))' LT (n+1) (colim K(EnhUi ))i≥0in Sp. We recall the tower
M0 ← M1 ← · · · ← M j ← · · ·from Section 1.1 of type n + 1 generalized Moore spectra. Then by Theorem
2.1, there is an equivalence
LT (n+1) K(En ) ' lim(colim(K(EnhUi ) ⊗ Lfn+1 Mj )),
j≥0

i≥0

which completes the proof of Theorem 1.1.
4. Continuous homotopy fixed points and Postnikov
sheafification
In this section, we brieﬂy recall some background material on continuous homotopy ﬁxed point spectra and we make some observations about
relationships with (pre)sheaves of spectra.
Let G be any proﬁnite group. Let N be a collection of open normal subgroups of G that is coﬁnal in the collection of all the open normal subgroupsof G. Suppose that {XN }N ∈N is a diagram of G-spectra, consisting of asingle map XN → XN ′ for each inclusion N ′ ⊂ N in N , such that for eachN ∈ N , the G-action on XN factors through G/N . Then colimN ∈N XN has
an induced G-action, there is the continuous homotopy ﬁxed point spectrum(colimN ∈N XN )hG , and if one of the conditions(i) G has ﬁnite virtual cohomological dimension;(ii) there is a ﬁxed integer m such that Hcs (N ′ , πt (colimN ∈N XN )) = 0,
for all s > m, t ∈ Z, and N ′ ∈ N ; and(iii) there is a ﬁxed integer r such that πt (colimN ∈N XN ) = 0, for allt>rholds, then by [8, Theorem 3.2.1; page 5038: 2nd paragraph] and [16, page911], there is an equivalenceQ(4.1)(colim XN )hG ' Tot(colim( (G/N )• XN )),
N ∈N

N ∈N

SEVERAL HOMOTOPY FIXED POINT SPECTRAL SEQUENCES

where for each N ,Q

149(G/N )• XN is a cosimplicial spectrum that satisﬁes

Q
Tot( (G/N )• XN ) ' (XN )hG/N ,Q
Qwith ( (G/N )• XN )([0]) = (G/N )0 XN = XN and for each n ≥ 1,
Q
Q
( (G/N )• XN )([n]) = (G/N )n XN
is the product of copies of XN indexed by the set (G/N )n , which is the nfold
Q product of copies of G/N . As is well-known, the cosimplicial spectrum(G/N )• XN that can be used in this discussion is not unique.
In general (that is, even when none of conditions (i) – (iii) are satisﬁed),associated to colimN ∈N XN is the presheaf P ∈ PSh(OG , Sp) deﬁned by
G/U 7→ P(G/U ) := (colim XN )hU ,N ∈N

U an open subgroup of G,

where (colimN ∈N XN )hU is the continuous U -homotopy ﬁxed points (forexample, see [8, Proposition 3.3.1] and [6, Section 2]). This presheaf extends
canonically to a presheaf P∏ in PSh∏ (TG , Sp), by sending ﬁnite coproducts
in TG to ﬁnite products in Sp.
In the other direction, let F be a presheaf in PSh(OG , Sp), so that foreach N ∈ N , F(G/N ) has a natural G-action that factors through the
G/N -action, and, as usual, the latter action yields the cosimplicial spectrum
Qe∏(G/N )• F(G/N ). Let F ∈ Sh(TG , Sp) denote the Postnikov sheaﬁﬁcationof the canonical presheaf F∏ in PSh∏ (TG , Sp) that is induced by F. Thenby [11, Construction 4.6, proof of Proposition 4.9],Fe∏ (∗) ' Tot(colim F∏ ((G/N )•+1 )),N ∈N

where for each N , (G/N )•+1 is the usual simplicial object in TG associatedto G/N . For each n ≥ 0, the stabilizer subgroup in G of the G-action on any
elementin (G/N )n+1 is N , so that there is an isomorphism (G/N )n+1 ∼
=
`
G/N
in
T
.
It
follows
that
there
is
an
equivalencenG(G/N )
Q
Fe∏ (∗) ' Tot(colim (G/N )• F(G/N ))N ∈N(for example, see [11, proof of Proposition 4.9, (7) in Proposition 4.11]).
Each inclusion N ′ ⊂ N in N induces the projection π : G/N ′ → G/Nin OG and the map F(π) : F(G/N ) → F (G/N ′ ), with source and target
equipped with the induced G-action, is G-equivariant, so that as at the
beginning of this section, colimN ∈N F(G/N ) has a G-action.
The following result is an immediate consequence of the above.

150

D. G. DAVISTheorem 4.1. Let G be a proﬁnite group and let F ∈ PSh(OG , Sp), with
F(G/N ) equipped with the natural G/N -action for each N ∈ N . If G andcolimN ∈N F(G/N ) satisfy one of conditions (i) – (iii), then(colim F(G/N ))hG ' Fe∏ (∗),
N ∈N

where the right-hand side of this equivalence is the global sections of the
Postnikov sheaﬁﬁcation of F∏ .
Deﬁnition 5. Given a proﬁnite group G and F ∈ PSh(OG , Sp), we deﬁne“condition (iv)” to be(iv) there is some integer d ≥ 0 such that for each N ∈ N , the G/N action on F(G/N ) is weakly d-nilpotent.
We refer the reader to [11, Deﬁnition 4.8] for the meaning of “weakly dnilpotent.” Our consideration of condition (iv) is partly motivated by [op.
cit., Propositions 4.9, 4.16], and [op. cit., Proposition 4.16, Theorem 4.26
(see its proof and the paragraph above Theorem 4.25)] give scenarios implying that condition (iv) holds. When this condition is satisﬁed, [op. cit.,Lemma 2.34] gives
Q
hG/NFe∏ (∗) ' colim Tot(.
• F(G/N )) ' colim F(G/N )(G/N )N ∈N

N ∈N

Now we put together the various strands of discussion of this section in
the following result.
Theorem 4.2. Let G be a proﬁnite group and suppose that {XN }N ∈N is
a diagram of G-spectra consisting of a unique map XN → XN ′ wheneverN ′ ⊂ N in N , such that for each N ∈ N , the G-action on XN factors
through G/N . Let Fe∏ be the Postnikov sheaﬁﬁcation of the presheaf F∏ in
PSh∏ (TG , Sp) that is determined by the presheaf
F : G/U 7→ (colim XN )hU ,N ∈N

U an open subgroup of G.

If G, colimN ∈N XN , and F satisfy any one of conditions (i) – (iv), then there
are equivalencesQ(colim XN )hG ' Fe∏ (∗) ' Tot(colim• XN ).
N ∈N

N ∈N(G/N )

Proof. In general (that is, even when none of conditions (i) – (iv) hold), foreach n ≥ 0, there are equivalencesQ
Q
colim (G/N )n F(G/N ) ' colim (G/N )n colim
F(G/N ′ )
′
N ∈N
N ∈N
N ∈N
Q
′
= colim (G/N )n colim
colim
XN ′′ )hN′′′
N ∈N
N ∈N N ∈N
QQ′′ ) ' colim' colim (G/N )n (colim
X
N(G/N )n XN ,
′′
N ∈N

N ∈N

N ∈N

SEVERAL HOMOTOPY FIXED POINT SPECTRAL SEQUENCES

151

where the ﬁrst and last steps are because the diagonal of N × N is coﬁnal in
N × N , and the penultimate step uses [8, Proposition 3.3.1: (2), (3)]. Hence(again, in general), there are equivalences
Q
Q
Fe∏ (∗) ' Tot(colim (G/N )• F(G/N )) ' Tot(colim (G/N )• XN ).
N ∈N

N ∈N

These last two equivalences, together with the discussion at the beginningof this section, imply the conclusion of the theorem when (i), (ii), or (iii) is
satisﬁed.
Now suppose that condition (iv) holds: the desired conclusion comes fromour last two equivalences andhG/NFe∏ (∗) ' colim F(G/N )hG/N = colim (colim
XN ′ )hN' (colim XN )hG ,
′
N ∈N

N ∈N

N ∈N

N ∈N

where the last step follows from the fact that for each N ,hG/N(colim
XN ′ )hN' (colim
XN ′ )hG ,
′
′
N ∈N

N ∈N

□

by [op. cit., Proposition 3.3.1, (4)].
5. Proofs of Theorems 1.2, 1.3, and 1.7

We let n be any positive integer and p a prime. Also, we let {Mj }j≥0 be
the tower of ﬁnite type n + 1 spectra that is described in Section 1.1.
−1
Mj ,Now we prove Theorem 1.2. Let j ≥ 0: by Remark 7, Lfn+1 Mj ' vn+1
so that−1
Mjcolim(K(EnhUi ) ⊗ Lfn+1 Mj ) ' LT (n+1) (colim K(EnhUi )) ⊗ vn+1
i≥0

i≥0
−1
−1
Mj .
' (LT (n+1) K(En )) ⊗ vn+1 Mj ' K(En ) ⊗ vn+1

−1
Mj }j≥0
Also, by [32, Proposition 5.1] and Theorem 2.1, there is a tower {vn+1fand as a tower, it is levelwise equivalent to {Ln+1 Mj }j≥0 . With these observations, Theorem 1.2 is an immediate consequence of [22, Proposition 3.1.2],[15, Theorems 8.5, 8.8], and [29, Theorem 2.2].
To prove Theorem 1.3, we set “T ” in the notation of [8, Section 6.1] equal
to T (0) ⊕ T (1) ⊕ · · · ⊕ T (n + 1). We havehGn
(LT (n+1) K(En ))hGn ' lim colim(Lfn+1 K(EnhUi ) ⊗ Mj )
j≥0

i≥0

' lim (colim Lfn+1 K(EnhUi )) ⊗ Mj
j≥0

hGn

i≥0

' lim (colim Lfn+1 K(EnhUi ))hGn ⊗ Mj
j≥0

i≥0

' LF (n+1) ((colim Lfn+1 K(EnhUi ))hGn )
i≥0



152

D. G. DAVIS' LF (n+1) Lfn+1 ((colim Lfn+1 K(EnhUi ))hGn )i≥0' LT (n+1) ((colim Lfn+1 K(EnhUi ))hGn ),i≥0where the ﬁrst step is because Lfn+1 is smashing; the third step follows fromthe fact that each Mj is a ﬁnite spectrum and Gn has ﬁnite virtual cohomological dimension (for example, see [35, Proposition 3.10] and [8, Theofrem 3.2.1]); because colim Lfn+1 K(EnhUi ) is T -local (since Ln+1
is smashing),i≥0fhUi hGn(colim Ln+1 K(En ))is T -local, by [8, proof of Lemma 6.1.5], and thisi≥0yields the ﬁfth step; and Theorem 2.1 gives the last step. This completes
the proof of Theorem 1.3.
We continue with the above context and prove Theorem 1.7: given any−1choice of F (n + 1), with T (n + 1) now set equal to vn+1
F (n + 1),(K(En ) ⊗ T (n + 1))hGn ' Tot colim' Tot(colim
i≥0

Q

i≥0

(Gn /Ui )•

Q


f
hUi
(Gn /Ui )• (K(En ) ⊗ Ln+1 F (n + 1))


Lfn+1 K(EnhUi )) ⊗ F (n + 1)

' (colim Lfn+1 K(EnhUi ))hGn ⊗ F (n + 1)i≥0

' Lfn+1 ((colim Lfn+1 K(EnhUi ))hGn ) ⊗ F (n + 1)i≥0
' (colim Lfn+1 K(EnhUi ))hGn ⊗ T (n + 1)i≥0

' (LT (n+1) K(En ))hGn ⊗ T (n + 1),
where the second equivalence applies the fact that F (n + 1) is a ﬁnite spectrum, the fourth equivalence uses the justiﬁcation given above for the ﬁfth
step of the proof of Theorem 1.3, and the last equivalence is by Theorem
1.3. In this sequence of equivalent expressions, the ﬁrst, fourth, sixth, andseventh ones are the ones explicitly required by Theorem 1.7.

6. How the two hypotheses give(LT (n+1) K(En ))hGn ' LT (n+1) K(LK(n) S 0 )In this section we prove Theorem 1.4. Also, since the proof is helpful
for verifying the related Theorem 1.6, after giving the proof of the former
result, we prove the latter one.
Let n ≥ 1, with p equal to any prime. We let {Mj }j≥0 be the tower ofgeneralized Moore spectra from Section 1.1, with each Mj a ﬁnite spectrum

SEVERAL HOMOTOPY FIXED POINT SPECTRAL SEQUENCES

153

of type n + 1 and an atomic µ-spectrum, and T (n + 1) denotes a represen−1
tative of the Bousﬁeld class of vn+1
M0 . We assume that the following two
statements are true:(H1) The map
colim K(η ↗ i) : LT (n+1) K(EnhGn ) → colim(LT (n+1) K(EnhUi ))hGn /Ui
i≥0

i≥0

is a T (n + 1)-equivalence.
Q(H2) For each j ≥ 0, with C(n, j, i)• := (Gn /Ui )• (K(EnhUi ) ⊗ Lfn+1 Mj )for all i ≥ 0, the canonical map
colim Tot(C(n, j, i)• ) → Tot(colim C(n, j, i)• )
i≥0

i≥0

is an equivalence.
To prove Theorem 1.4, we use each assumption only once and the usages
are marked with “By (H1)” and “applies (H2).”−1
By Remark 6, for every j and any spectrum Z, Z ⊗ vn+1
Mj is T (n + 1)f
local, and hence, by Remark 7, Z ⊗ Ln+1 Mj is T (n + 1)-local. Also, for all
j and Z,Z ⊗ Lfn+1 Mj ' (LT (n+1) Z) ⊗ Lfn+1 Mj ,since Z ⊗ Lfn+1 Mj ' (LLf M Z) ⊗ Lfn+1 Mj .
n+1 j
A helpful tool for our argument is the natural equivalence
Z hH ' LT (n+1) (ZhH ) = LT (n+1) (colim Z),
H

where Z is any T (n + 1)-local spectrum with an action by a ﬁnite group H([30, Theorem 1.5], partly [33]; there are helpful presentations of this resultin [10, Section 1], [30, page 350]). The action of H on Z induces a diagram
BH → SpT (n+1) , where SpT (n+1) is the ∞-category of T (n+1)-local spectra,
and in the above equivalence, the rightmost expression is the colimit of thisdiagram.
By (H1), the map (colimi≥0 K(η ↗ i)) ⊗ T (n + 1) is an equivalence, which
implies that for each j, the map colimi≥0 (K(η ↗ i) ⊗ Lfn+1 Mj ) is an equivalence: that is, the canonical maps
(LT (n+1) K(EnhGn )) ⊗ Lfn+1 Mj → colim((LT (n+1) K(EnhUi ))hGn /Ui ⊗ Lfn+1 Mj )i≥0are equivalences. This gives the last step in the equivalences
LT (n+1) K(LK(n) S 0 ) ' lim(K(EnhGn ) ⊗ Lfn+1 Mj )j≥0' lim((LT (n+1) K(EnhGn )) ⊗ Lfn+1 Mj )
j≥0

154

D. G. DAVIS' lim colim((LT (n+1) K(EnhUi ))hGn /Ui ⊗ Lfn+1 Mj ).
j≥0 i≥0

Now we let i and j be ﬁxed non-negative integers and consider(LT (n+1) K(EnhUi ))hGn /Ui ⊗ Lfn+1 Mj .
The homotopy ﬁxed points and (−) ⊗ Lfn+1 Mj commute to yield the natural
equivalence(LT (n+1) K(EnhUi ))hGn /Ui ⊗ Lfn+1 Mj ' ((LT (n+1) K(EnhUi )) ⊗ Lfn+1 Mj )hGn /Ui ,because the homotopy orbits and (−) ⊗ Lfn+1 Mj commute. In more detail,we have(LT (n+1) K(EnhUi ))hGn /Ui ⊗ Lfn+1 Mj' LT (n+1) (colim LT (n+1) K(EnhUi )) ⊗ Lfn+1 Mj
Gn /Ui' (colim LT (n+1) K(EnhUi )) ⊗ Lfn+1 Mj
Gn /Ui' LT (n+1) ((colim LT (n+1) K(EnhUi )) ⊗ Lfn+1 Mj )Gn /Ui' LT (n+1) (colim((LT (n+1) K(EnhUi )) ⊗ Lfn+1 Mj ))Gn /Ui' ((LT (n+1) K(EnhUi )) ⊗ Lfn+1 Mj )hGn /Ui .
Since ((LT (n+1) K(EnhUi )) ⊗ Lfn+1 Mj )hGn /Ui ' (K(EnhUi ) ⊗ Lfn+1 Mj )hGn /Ui ,we obtain the natural equivalence(LT (n+1) K(EnhUi ))hGn /Ui ⊗ Lfn+1 Mj ' Tot(C(n, j, i)• ).
Putting the conclusions of the last two paragraphs together and thenpushing further, we obtain
LT (n+1) K(LK(n) S 0 ) ' lim colim((LT (n+1) K(EnhUi ))hGn /Ui ⊗ Lfn+1 Mj )j≥0 i≥0' lim colim Tot(C(n, j, i)• ) ' lim Tot(colim C(n, j, i)• )j≥0 i≥0j≥0' lim colim(K(EnhUi ) ⊗ Lfn+1 Mj )
j≥0

i≥0

hGn

i≥0

= (LT (n+1) K(En ))hGn ,where the third equivalence applies (H2) and the fourth equivalence uses
(4.1).
Now we prove Theorem 1.6. We continue the conventions used above,−1
but now we deﬁne T (n + 1) := vn+1
F (n + 1), where F (n + 1) is an atomicµ-spectrum. We assume the validity of the following two statements:(H1′ ) The map colim K(η ↗ i) is a T (n + 1)-equivalence.
i≥0SEVERAL HOMOTOPY FIXED POINT SPECTRAL SEQUENCES

155(H2′ ) The canonical map colim Tot(C(n, i)• ) → Tot(colim C(n, i)• ) is an
i≥0

i≥0

equivalence, where for each i,Q
C(n, i)• := (Gn /Ui )• (K(EnhUi ) ⊗ T (n + 1)).
We use each assumption just once and we mark the occurrences with “by(H1′ )” and “by (H2′ ).”We have
Q(K(En ) ⊗ T (n + 1))hGn ' Tot(colim (Gn /Ui )• (K(EnhUi ) ⊗ T (n + 1)))i≥0Q
' colim Tot( (Gn /Ui )• (K(EnhUi ) ⊗ T (n + 1)))i≥0

' colim(K(EnhUi ) ⊗ T (n + 1))hGn /Ui ,i≥0where the ﬁrst equivalence applies (4.1) and the second equivalence is by(H2′ ). Since F (n + 1) is an atomic µ-spectrum, Z ⊗ T (n + 1) is T (n + 1)local for any spectrum Z, by Remark 6. Thus, as in the above proof of
Theorem 1.4, there is a natural equivalence(K(EnhUi ) ⊗ T (n + 1))hGn /Ui ' (LT (n+1) K(EnhUi ))hGn /Ui ⊗ T (n + 1),
for each i. We now conclude that
(K(En ) ⊗ T (n + 1))hGn ' colim(K(EnhUi ) ⊗ T (n + 1))hGn /Uii≥0' colim((LT (n+1) K(EnhUi ))hGn /Ui ⊗ T (n + 1))i≥0' (colim(LT (n+1) K(EnhUi ))hGn /Ui ) ⊗ T (n + 1)i≥0

' LT (n+1) K(LK(n) S 0 ) ⊗ T (n + 1) ' K(LK(n) S 0 ) ⊗ T (n + 1),
where the penultimate step is by (H1′ ).
Acknowledgements
I thank Martin Frankland for a discussion about LT (n) , Niko Naumann for
introducing me to a version of “condition (iv)” in Deﬁnition 5, Akhil Mathew
and John Rognes for helpful interactions, and Philip Hackney and Justin
Lynd for stimulating conversations related to working with ∞-categories.
Also, I thank the referee for helpful comments that improved the paper invarious places.
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Daniel G. Davis
Department of Mathematics
University of Louisiana at Lafayette
217 Maxim Doucet Hall, P.O. Box 43568
Lafayette, Louisiana, 70504-3568 USA
e-mail address: dgdavis@louisiana.edu(Received February 25, 2023 )(Accepted August 7, 2023 )