THE ADAMS SPECTRAL SEQUENCE FOR T HH(M U )tS
1
HÅKON SCHAD BERGSAKER
1. A Bockstein spectral sequence
Write T for the circle group of complex numbers of modulus 1, and let Cm ⊂ T be the cyclic
group of order m. We also write C = Cp . Let X be a naive T -spectrum. We have maps
F
XT
/ X Cpn
V
/ Σ−1 X T
where F is the inclusion of fixed points and V comes from a transfer map. Using the identifications
T
X T = F (Σ∞
+ T /T, X)
T
X Cpn = F (Σ∞
+ T /Cpn , X)
they can be described as follows. Consider the projection T /Cpn → T /T . It induces a stable map
∞
∞
∞
π n : Σ∞
+ T /Cpn → Σ+ T /T , and there is also a T -transfer map tn : ΣΣ+ T /T → Σ+ T /Cpn , where
the extra suspension is suspension by the (trivial) adjoint representation of T . Applying F (−, X)T
e ∧ F (ET+ , X) we get the maps
to πn and tn gives us F and V , respectively. Replacing X by ET
F
X tT
/ X tCpn
V
/ Σ−1 X tT ,
and by taking continuous homology with respect to the corresponding Tate towers we get a sequence
of completed A∗ -comodules
(1)
0
/ H∗c (X tT )
F∗
/ H∗c (X tCpn )
/ Σ−1 H∗c (X tT )
V∗
/ 0.
Let α : T+ ∧ X → X be the T -action map, and let ρ : T → T /C be the p’th root isomorphism.
The composition
ΣX
ρt1 ∧1
/ T+ ∧ X
α
/X
defines a map σ : ΣX → X. We can pull back the T /C-action on X tC to get a T -spectrum ρ∗ X tC .
As above we get a map
σ̄ : Σρ∗ X tC → ρ∗ X tC .
(2)
To ease the notation we will often not write the ρ∗ ’s.
Lemma 1.3. The composition
∞
πn ◦ tn : ΣΣ∞
+ T /T → Σ+ T /T
is null-homotopic. The composition
F V : X tC → Σ−1 X tC
is the desuspension of σ̄.
Date: March 31, 2016.
1
2
HÅKON SCHAD BERGSAKER
∞
Proof. We will use an explicit geometric model for the transfer map tn : ΣΣ∞
+ T /T → Σ+ T /Cpn .
n
1
p
Write S (n) ' T /Cpn for the circle with T -action given by t·s = t s. Take a tubular neighborhood
of S 1 (n) ⊂ C, which is equivalent to the normal bundle η of S 1 (n) in C. Dividing out by the
complement of this neighborhood and one-point compactifying the source results in a map
p : S 2 → T h(η) ∼
= S 2 /{N, S} ,
where N and S denotes the north and south pole of S 2 , respectively. The map p just pinches N
and S together. The point N is taken as the basepoint. T -equivariantly, each circle of latitude is
a copy of S 1 (n), and this specifies the T -action in both the source and the target. Now tn is the
(desuspended) stabilization of p.
Let q : T h(η) → S 1 be the map that wraps each fiber around the target circle, with the point
at infinity mapped to the basepoint in S 1 . Here T acts trivially on the target. This map stabilizes
to πn , and since the composition q ◦ p : S 2 → S 1 is T -equivariantly contractible (slide the image of
the south pole one round back to the basepoint), this proves the first result.
For the second, consider the adjoints ...
Lemma 1.4. The sequence (1) is short exact, and is split (as A∗ -comodules) when n ≥ 2.
Proof. The induced maps of homological Tate spectral sequences form a short exact sequence of
E 2 -terms,
P (t±1 ) ⊗ H∗ (X)
f ⊗1
/ E(u) ⊗ P (t±1 ) ⊗ H∗ (X)
/ Σ−1 P (t±1 ) ⊗ H∗ (X) ,
v⊗1
where f is the inclusion, and g maps ti to zero and uti to Σ−1 ti . In other words, F∗ includes the
even coloumns and V∗ projects onto the odd coloumns. Since the first and last spectral sequence
can only have differentials in the even pages, it follows that the same is true for the middle spectral
sequence, and that the sequence of maps is short exact for every E r -term. Combined with the fact
that the composition V∗ F∗ is zero on the abutments, since π ◦ t ' ∗, this implies the first result.
To establish the splitting, consider the commutative diagram
H∗c (X tCpn )
8
F̄∗
0
/ H∗c (X tT )
F∗
/ H∗c (X tC )
V̄∗
V∗
H∗c (X tCpn )
7
/ Σ−1 H∗c (X tT )
/ 0.
The composition V̄∗ F̄∗ is multiplication by pn−1 integrally, hence is zero since we use mod p
coefficients. Now V∗ F̄∗ = 0, so a splitting for the unnamed map is given by F∗−1 F̄∗ .
We will use the short exact sequence (1) as a tool to compute Ext-groups. If we can compute
ExtAb∗ (Fp , H∗c (X tT )), then the splitting immediately gives us ExtAb∗ (Fp , H∗c (X tCn )) for n ≥ 2. For
n = 1 the sequence (1) is typically not split, but we will construct a spectral sequence to pass from
X tC1 to X tT . In that case the sequence represents a non-trivial element
c
tT
c
tT
δ ∈ Ext1,−1
b (H∗ (X ), H∗ (X )) .
A∗
Assume now that X is a ring spectrum and that T acts by ring maps. Then H∗c (X tT ) is a completed
A∗ -comodule algebra, and we can pull back the element δ along the unit map η : Fp → H∗c (X tT )
to an element e ∈ Ext1,−1
(Fp , H∗c (X tT )). Explicitly e is represented by the top sequence in the
Ab∗
diagram
(5)
0
/ H∗c (X tT )
/E
/ Σ−1 Fp
/ H∗c (X tT )
/ H∗c (X tC )
/0
η
0
F∗
V∗
/ Σ−1 H∗c (X tT )
/ 0,
THE ADAMS SPECTRAL SEQUENCE FOR T HH(M U )tS
1
3
where the right-hand square is a pullback.
Theorem 1.6. Let X be a naive commutative T -ring spectrum that is bounded below and of
finite type over Fp , and assume the top short exact sequence in (5) is non-split. Then there is a
(strongly?) convergent spectral sequence of algebras
c
tC
c
tT
E1∗,∗ = P (e) ⊗ Ext∗,∗
)) =⇒ Ext∗,∗
b (Fp , H∗ (X
b (Fp , H∗ (X )) ,
A∗
A∗
where e in degree (1, −1) is a permanent cycle, as are all its powers. The differentials dr have
degree () and are of the form dr (ek ⊗ x) = ek+r ⊗ dr (x). The first differential d1 is given by
d1 (ek ⊗ x) = ek+1 ⊗ σ∗ (x), where
c
tC
σ̄∗ : Ext∗,∗
)) → Ext∗,∗+1
(Fp , H∗c (X tC ))
b (Fp , H∗ (X
b
A∗
A∗
is induced by the map σ̄ in (2).
Proof. We use the spectral sequence terminology of Boardman. Write Ext(M ) for ExtAb∗ (Fp , M ).
The short exact sequence (1) induces a long exact sequence of Ext-groups
(7)
/ Exts (H∗c (X tT ))
/ Exts (H∗c (X tC ))
/ Exts (H∗c (X tT )) δ / Exts+1 (H∗c (X tT ))
/ ··· .
···
Since the homology groups are commutative algebras, these Ext-groups have induced algebra
structures, and the boundary map δ is just multiplication by e. We construct the unrolled exact
couple (truncated at s = 0)
··· f
/ Ext∗,∗ (H∗c (X tT ))
i
e
V∗
F∗
Ext∗,∗ (H∗c (X tC ))
e
/ Ext∗,∗ (H∗c (X tT ))
i
V∗
Ext∗,∗ (H∗c (X tC ))
F∗
e
/ Ext∗,∗ (H∗c (X tT ))
V∗
F∗
Ext∗,∗ (H∗c (X tC )) ,
where each triangle is an exact couple formed by coiling up (7). Since the inverse limit of the
horizontal sequence vanishes, this results in a spectral sequence converging to the (attained) colimit
Ext∗,∗ (H∗c (X tT )) with differential d1 = F∗ ◦ V∗ . The identification of the first differential follows
immediately from Lemma 1.3. Note that e has degree (1, −1), F∗ has degree (0, 0), and V∗ has
degree (0, 1).
This spectral sequence has a natural tri-grading, but we want to make one of the gradings (the
“Bockstein degree”) implicit, by instead recording the power of the element e.
((more details about multiplicative structure etc))
2. The Adams E2 -term
The starting point of our computation is the homology of T HH(M U ) and T HH(BP ). First
we recall the homology of M U and BP as comodule algebras over the dual Steenrod algebra A∗ .
As an algebra, A∗ = E(τ0 , τ1 , . . . ) ⊗ P (ξ1 , ξ2 , . . . ) with τk in degree 2pk − 1 and ξk in degree
2pk − 2, assuming p is odd. When p = 2, A∗ = P (ξ1 , ξ2 , . . . ) with ξk in degree 2k − 1. Here
we are actually using the conjugates of the usual generators without changing the notation. Let
H∗ (−) = H∗ (−; Fp ).
Proposition 2.1. We have the following identifications of A∗ -comodule algebras. For p = 2,
H∗ (M U ) = P (ξk2 | k ≥ 1) ⊗ P (bi | i ≥ 2, i 6= 2k − 1)
H∗ (BP ) = P (ξk2 | k ≥ 1) ,
where P (ξk2 ) ⊂ A∗ and the bi are comodule primitives in degree 2i. For p odd,
H∗ (M U ) = P (ξk | k ≥ 1) ⊗ P (bi | i ≥ 1, i 6= pk − 1)
H∗ (BP ) = P (ξk | k ≥ 1) ,
4
HÅKON SCHAD BERGSAKER
where P (ξk ) ⊂ A∗ and the bi are comodule primitives in degree 2i.
Proposition 2.1 serves as input to a collapsing Bökstedt spectral sequence, with output the
following result. For convenience, let bpk −1 = ξk for odd primes p and b2k −1 = ξk2 for p = 2.
Proposition 2.2. There are isomorphisms of A∗ -comodule algebras
H∗ (T HH(M U )) ∼
= H∗ (M U ) ⊗ E(σbi | i ≥ 1)
H∗ (T HH(BP )) ∼
= H∗ (BP ) ⊗ E(σbpk −1 | k ≥ 1) ,
where the classes σbi are A∗ -comodule primitives.
To describe the comodule H∗c (T HH(M U )tC ), we first recall the (homological) Singer construcb , where
tion. Let M be an A∗ -comodule, and define the completed A∗ -comodule R+ (M ) = L⊗M
(
P (u±1 )
if p = 2
L=
±1
E(u) ⊗ P (t ) if p 6= 2
with u in degree −1 and t in degree −2. The A∗ -comodule structure on R+ (M ) is given by
X −r − s − 1
Sq∗s (ur ⊗ x) =
ur+s−j ⊗ Sq∗j (x)
s
−
2j
j
for p = 2, and there are corresponding but more complicated formulas for odd primes. In particular
there is always a non-trivial Bockstein going from u ⊗ x to u2 ⊗ x. The map : M → R+ (M ) given
by
∞
X
(x) =
u−r ⊗ Sq∗r (x)
r=0
for p = 2, and similarly for odd p, is a comodule map (its image is the largest actual sub-comodule
inside R+ (M )). The map induces an isomorphism on Ext-groups.
Let γ : T HH(M U ) → T HH(M U )tC be the right hand map in the Tate diagram for T HH(M U ),
and similarly for T HH(BP ). Lunøe-Nielsen and Rognes prove the following.
Theorem 2.3. There is a commutative diagram
H∗ (T HH(M U ))
R+ (H∗ (T HH(M U )))
γ∗
/ H∗c (T HH(M U )tC )
4
Φ
∼
=
of complete A∗ -comodules, and the map Φ is an isomorphism. In particular the map γ∗ is an
Ext-equivalence. An identical statement holds for BP instead of M U .
From this we can easily describe the Ext-groups of H∗c (T HH(M U )tC ) and H∗c (T HH(BP )tC ).
Corollary 2.4. There are algebra isomorphisms
c
tC
k
Ext∗,∗
b (Fp , H∗ (T HH(M U ) )) = P (ak | k ≥ 0) ⊗ P (bi | i ≥ 1, i 6= p − 1) ⊗ E(σbi | i ≥ 1)
A∗
c
tC
Ext∗,∗
b (Fp , H∗ (T HH(BP ) )) = P (ak | k ≥ 0) ⊗ E(σbpk −1 | k ≥ 1) ,
A∗
with |ak | = (1, 2pk − 1), |bi | = (0, 2i) and |σbi | = (0, 2i + 1).
Proof. A standard calculation gives
k
Ext∗,∗
A∗ (Fp , H∗ (M U )) = P (ak | k ≥ 0) ⊗ P (bi | i ≥ 1, i 6= p − 1)
with ak in bi-degree (1, 2pk − 1) and bi in bi-degree (0, 2i). Proposition 2.2 and Theorem 2.3 then
combine to prove the corollary for M U . The case BP is similar.
THE ADAMS SPECTRAL SEQUENCE FOR T HH(M U )tS
1
5
We now want to use the spectral sequence of the previous section to pass from Ext of the cyclic
Tate construction to Ext of the circle Tate construction. First we have to show that the extension
(5) is non-trivial.
Lemma 2.5. The short exact sequence
/ H∗c (T HH(M U )tT )
0
/E
/ Σ−1 Fp
/0
c
tT
representing e ∈ Ext1,−1
b (Fp , H∗ (T HH(M U ) ) does not split as A∗ -comodules. Similarly for BP .
A∗
Proof. First consider the corresponding extension for the sphere spectrum
0 → P (t±1 ) → ES → Σ−1 Fp → 0
(6)
pulled back from
0 → H∗c (S tT ) → H∗c (S tC ) → Σ−1 H∗c (S tT ) → 0 .
(7)
Identifying H∗c (S tC ) with R+ (Fp ) = E(u) ⊗ P (t±1 ), the right hand map in (7) sends u ⊗ 1 to Σ−1 1.
This can be seen by looking at the induced map on homological Tate spectral sequences. There is
a Bockstein β(u ⊗ 1) = t ⊗ 1 in R+ (Fp ), so the extension (7) is non-trivial. Since both u ⊗ 1 and
t ⊗ 1 are in ES , the extension (6) is also non-trivial.
After identifying H∗c (T HH(M U )tC ) with R+ (H∗ (T HH(M U ))), the map induced from S tC →
T HH(M U )tC sends u ⊗ 1 to u ⊗ 1. We have a map of extensions
0
/ H∗c (S tT )
/ ES
0
/ H∗c (T HH(M U )tT )
/E
v
/ Σ−1 Fp
/0
/ Σ−1 Fp
/ 0,
where the middle map sends u ⊗ 1 to u ⊗ 1. Hence v(u ⊗ 1) = Σ−1 1 and the extension E is
non-trivial as well. The same argument works for BP .
We will start with the calculation for X = T HH(BP ). Using Corollary 2.4, the spectral
sequence in Theorem 1.6 is
c
tT
P (e) ⊗ P (ak | k ≥ 0) ⊗ E(σbpk −1 | k ≥ 1) =⇒ Ext∗,∗
b (Fp , H∗ (T HH(BP ) )) .
A∗
The map γ : T HH(BP ) → ρ∗ T HH(BP )tC is T -equivariant, which implies that the d1 -differential
is induced from σ : ΣT HH(BP ) → T HH(BP ) via the Ext-isomorphism γ∗ .
We have the following picture of a part of the E1 -page, where we use the Adams grading
convention with t − s as the horizontal axis and s as the vertical axis. For this picture we choose
p = 3. We have only drawn the algebra generators, in addition to the target of the d1 -differential.
5
4
3
ea0 σξ1
2
1
e
a0
a1
σξ1
0
−3 −2 −1
0
1
2
3
4
5
6
7
8
9
10
6
HÅKON SCHAD BERGSAKER
We already know that e is a permanent cycle, and so is a0 since there is nothing in (Adams)
bi-degree (−1, 2). In homology, σ vanishes on the σξk classes, hence d1 = σ∗ is zero on the σξk in
the corresponding Ext. What remains are the ak classes, and we will show that d1 (ak ) = −ea0 σξk ,
as indicated in the picture for k = 1. The element ak is represented in the (completed) cobar
complex by
X
i
b ∗c (T HH(BP )tC ) ,
−
[τi ]ξjp ∈ A∗ ⊗H
i+j=k
which maps to −[τ0 ]σξk under 1 ⊗ σ. Since [τ0 ]σξk represents a0 σξk , this shows that d1 (ak ) =
eσ∗ (ak ) = −ea0 σξk .
This results in the E2 -term
E2 = P (e) ⊗ P (a0 , ap1 , ap2 , . . . ) ⊗ E(σξk )/(ea0 σξk | k ≥ 1) .
Since the ak classes with k ≥ 1 are now gone, there is nothing for dr (σξk ) to hit, so all the σξk
have to be permanent cycles. (This also follows from the fact that the image of dr is divisible by
er .) The only class dr (apk ) could hit is ep a0 σξk , but this class was already killed by d1 . Hence the
spectral sequence collapses from the E2 -term.
Since the σξk has odd total degree, they are automatically exterior classes in the abutment,
at least for odd primes. (Use Steenrod operations in Ext for p = 2?) Since V∗ (ak ) = a0 σξk in
Ext∗,∗
(Fp , H∗c (T HH(BP )tT )), which lies in the kernel of multiplication by e, the relation ea0 σξk =
Ab∗
0 still holds in the abutment.
For M U , there are extra classes bi and σbi on the s = 0 line, and differentials d1 (bi ) = eσbi .
So the bi classes die, while the σbi remain but are now e-torsion. Again the spectral sequence
collapses from the E2 -term and the multiplicative extensions are similarly resolved. Summing up,
we have the following result.
Theorem 2.8. There are algebra isomorphisms
p
p
c
tT
Ext∗,∗
b (Fp , H∗ (T HH(M U ) )) = P (e) ⊗ P (a0 , ak | k ≥ 1) ⊗ P (bi ) ⊗ E(σbi )/(ea0 σξk , eσbi )
A∗
p
c
tT
Ext∗,∗
b (Fp , H∗ (T HH(BP ) )) = P (e) ⊗ P (a0 , ak | k ≥ 1) ⊗ E(σξk )/(ea0 σξk ) .
A∗
Let
A = P (e) ⊗ P (a0 , apk | k ≥ 1) ⊗ P (bpi ) ⊗ E(σbi )/(ea0 σξk , eσbi )
B = P (e) ⊗ P (a0 , apk | k ≥ 1) ⊗ E(σξk )/(ea0 σξk ) .
From the splitting of (1) we immediately get the following consequence.
Corollary 2.9. Let n ≥ 2. There are identifications
c
tCpn
Ext∗,∗
)) = A ⊕ Σ−1 A
b (Fp , H∗ (T HH(M U )
A∗
Ext∗,∗
(Fp , H∗c (T HH(BP )tCpn ))
Ab∗
= B ⊕ Σ−1 B ,
with A and B corresponding to subalgebras.
3. Adams differentials
We focus on the case BP . The Adams spectral sequence we consider is
c
tT
tT
Ext∗,∗
b (Fp , H∗ (T HH(BP ) )) =⇒ π∗ (T HH(BP ) ) ,
A∗
with E2 -term given explicitly by Theorem 2.8. Here is a zoomed out picture of the E2 -term.
(Again we only draw the algebra generators.)
THE ADAMS SPECTRAL SEQUENCE FOR T HH(M U )tS
1
7
5
ap1
0
−5
e a0
0
σξ1
5
10
ap2
15
σξ2
20
25
30
35
40
45
The element a0 is a permanent cycle, by comparing with the corresponding spectral sequence
for S tT , and probably for other reasons as well. The map
s,t+1
V∗ : Exts,t
(Fp , H∗c (T HH(BP )tT ))
A∗ (Fp , H∗ (T HH(BP ))) → Ext b
A∗
sends ak to a0 σξk , for k ≥ 1. This is clear from the computation of d1 = F∗ V∗ above. Since the
Adams spectral sequence for T HH(BP ) collapses (McClure-Staffeldt), the image a0 σξk = V∗ (ak )
has to be a permanent cycle. What about σξk ? And especially apk ?
For e, we can say the following. Since T HH(BP )tT ' T F (BP ), which is connective, we know
that all the ek classes has to support non-trivial differentials. A differential on e will leave ep , so
2
a later differential will have to kill ep , which will leave ep , and so on. So the Adams spectral
sequence will not collapse after finitely many pages. It seems hard to analyze these differentials
directly; maybe it’s possible to map to or from this Adams spectral sequence to a modified Adams
spectral sequence of some sort, as is done by Ravenel in the case S tT . What makes this easier
in Ravenel’s case is the fact that the norm-restriction cofiber sequence splits for S, which I don’t
think is true for T HH(BP ).