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The homotopy of the K(2)-local Moore spectrum at the
prime 3 revisited
Hans-Werner Henn, Nasko Karamanov, Mark Mahowald
To cite this version:
Hans-Werner Henn, Nasko Karamanov, Mark Mahowald. The homotopy of the K(2)-local Moore
spectrum at the prime 3 revisited. Mathematische Zeitschrift, Springer, 2013, �10.1007/s00209-013-
1167-4�. �hal-00331831�
THE HOMOTOPY OF THE K(2)-LOCAL MOORE SPECTRUM AT THE
PRIME 3 REVISITED
HANS-WERNER HENN, NASKO KARAMANOV AND MARK MAHOWALD
Abstract. In this pap er we use the approach intr oduced in [5] in order to analyze the
homotopy groups of L
K(2)
V (0), the mod-3 Moore spectrum V (0) localized with respect to
Morava K-theory K(2). These homotopy groups have already been calculated by Shimomura
[12]. The results are very compli cated so that an independent verification via an alternative
approach is of interest. In fact, we end up with a result which is more precise and also differs in
some of its details from that of [12]. An additional bonus of our approach is that it breaks up
the result into smaller and more digestible chunks whi ch are related to the K(2)-locali zation
of the spectrum T MF of topological modular forms and related spectra. Even more, the
Adams-Novikov differentials for L
K(2)
V (0) can be read off from those for T MF .
1. Introduction
Let K(2) be the second Morava K-theory for the prime 3. For suitable spectra F , e.g. if F is
a finite spec trum, the homotopy groups of the B ousfield localization L
K(2)
F can be calculated
via the Adams-Novikov spectral sequence. B y [3] this spectral sequence can be identified with
the descent spectral sequence
E
s,t
2
= H
s
(G
2
, (E
2
)
t
F ) =⇒ π
t−s
(L
K(2)
F )
for the action of the (extended) Morava stabilizer g roup G
2
on E
2
∧ F where the action is via
the Goerss-Hopkins-Miller action on the Lubin-Tate spectrum E
2
(see [5] for a summary of
the necessary background material). Here we just recall that the homotopy groups of E
2
are
non-canonically isomorphic to W
F
9
[[u
1
]][u
±1
] where W
F
9
denotes the ring of Witt vectors o f F
9
,
where u
1
is of degree 0 and u is of degr e e −2. We a lso recall that G
2
is a profinite group and
its action on the profinite mo dule (E
2
)
∗
F is continuous; group cohomolo gy is, throughout this
paper, taken in the continuous sense.
The cohomological dimension of G
2
is well-known to be infinite and therefore a finite pro-
jective resolution of the trivial profinite G
2
-module Z
3
cannot exist. However, in [5] a finite
resolution of the trivial module Z
3
was constructed in terms of permutation modules. More
precisely, the gr oup G
2
is isomorphic to the product G
1
2
× Z
3
of a central subgroup (isomorphic
to) Z
3
and a group G
1
2
which is the kernel of a homomorphism G
1
2
→ Z
3
, also called the reduced
norm. One of the main technical achievements of [5] was the construction of a permutation
resolution of the trivial module Z
3
for the group G
1
2
. This resolution is self-dual in a suitable
sense (cf. section 3.4) and has the form
0 → C
3
→ C
2
→ C
1
→ C
0
→ Z
3
→ 0(1)
with C
0
= C
3
= Z
3
[[G
1
2
/G
24
]] and C
1
= C
2
= Z
3
[[G
1
2
]] ⊗
Z
3
[SD
16
]
χ. Here G
24
is a certain
subgroup of G
1
2
of order 24, isomorphic to the se midir e c t product Z/3 ⋊ Q
8
of the cyclic group
of order 3 with a no n-trivial action of the quaternion group Q
8
, and SD
16
is another subgroup,
isomorphic to the semidihedral group of order 16 (see sectio n 2.2). Furthermore, χ is a suitable
one-dimensional representation of SD
16
, defined over Z
3
, and if S is a profinite G
1
2
-set we denote
the c orrespo nding profinite permutation module by Z
3
[[S]].
Date: November 1, 2008.
The authors would like to thank the Mittag-Leffler Institute, Northwestern University, Universit´e Louis Pas-
teur at Strasbourg and the R uhr -Universit¨at Bochum for providing them with the opportunity to work together.
1
2 Hans-Werner Henn, Nasko Karamanov and Mark Mahowald
For any Z
3
[[G
1
2
]]-module M the resolution (1) gives rise to a first quadrant cohomological
sp e c tral sequence
E
s,t
1
= Ext
t
Z
3
[[G
1
2
]]
(C
s
, M) =⇒ H
s+t
(G
1
2
, M)(2)
refered to in the sequel a s the algebraic spectral sequence. By Shapiro’s Lemma we have
(3) E
0,t
1
= E
3,t
1
∼
=
H
t
(G
24
, M), E
1,t
1
= E
2,t
1
∼
=
(
Hom
Z
3
[SD
16
]
(χ, M) t = 0
0 t > 0 .
The bulk of our work is the calculation of this spectral sequence if M = (E
2
)
∗
(V (0)). In this case
the E
1
-term is well understood and can be interpreted in terms of modular forms in characteristic
3. In fact, it is determined by the following result which we include for the convenience of the
reader and in w hich v
1
denotes the well-known G
2
-invariant class u
1
u
−2
∈ M
4
. For the definition
of the other c lasses figuring in this result the reader is re ferred to section 5.1.
Theorem 1.1. Let M = (E
2
)
∗
(V (0)).
a) There are elements β ∈ H
2
(G
24
, M
12
), α ∈ H
1
(G
24
, M
4
) and eα ∈ H
1
(G
24
, M
12
), an
invertible G
24
-invariant element ∆ ∈ M
24
, and an isomorphism of graded algebras
H
∗
(G
24
, M)
∼
=
F
3
[[v
6
1
∆
−1
]][∆
±1
, v
1
, β, α, eα]/(α
2
, eα
2
, v
1
α, v
1
eα, αeα + v
1
β) .
b) The ring of SD
16
-invariants of M is given by t he subalgebra M
SD
16
= F
3
[[u
4
1
]][v
1
, u
±8
]
and Hom
Z
3
[SD
16
]
(χ, M) is a free M
SD
16
-module of rank 1 with generator ω
2
u
4
, i.e.
Hom
Z
3
[SD
16
]
(χ, M)
∼
=
ω
2
u
4
F
3
[[u
4
1
]][v
1
, u
±8
] .
Remark
We note that v
6
1
∆
−1
is a G
24
-invariant class in the maximal ideal of M
0
and hence
a formal power ser ies in v
6
1
∆
−1
converges in M and is also invariant. Similarly w ith u
4
1
. Of
course, the na me for ∆ is chosen to emphasize the close relation with the theory of modular
forms. For example we note that M
G
24
is isomorphic to the completion of M
3
:= F
3
[∆
±1
, v
1
]
with respect to the ideal generated by v
6
1
∆
−1
, and M
3
is isomorphic to the ring of modular
forms in characteristic 3 (cf. [2] and [1]). Similarly, M
SD
16
is isomorphic to the completion of
F
3
[v
1
, u
±8
] with respect to the ideal generated by u
4
1
= v
4
1
u
8
. The lar ger algebra F
3
[v
1
, u
±4
] is
isomorphic to the ring M
3
(2) of modular forms of level 2 (in characteristic 3) (cf. [1]). The
relation with modular forms could be made tight if in [5] we had worked with a version of E
2
which uses a deformation of the formal group of a supe rsingular curve rather than that of the
Honda formal group.
As (E
2
)
∗
(V (0)) is a graded module, the spectral sequence is trigraded. The differentials in
this spectral sequence are v
1
-linear and continuous. Therefore d
1
is completely described by
continuity and the following formulae in which we identify the E
1
-term via Theorem 1.1.
Theorem 1.2. There are elements
∆
k
∈ E
0,0,24k
1
, b
2k+1
∈ E
1,0,16k+8
1
,
b
2k+1
∈ E
2,0,16k+8
1
,
∆
k
∈ E
3,0,24k
1
for each k ∈ Z satisfying
∆
k
≡ ∆
k
, b
2k+1
≡ ω
2
u
−4(2k+1)
,
b
2k+1
≡ ω
2
u
−4(2k+1)
, ∆
k
≡ ∆
k
(where the congruences are modulo the ideal (v
6
1
∆
−1
) resp. (v
4
1
u
8
) and in the case of ∆
0
we
even have equality ∆
0
= ∆
0
= 1) such that
d
1
(∆
k
) =
(−1)
m+1
b
2.(3m+1)+1
k = 2m + 1
(−1)
m+1
mv
4.3
n
−2
1
b
2.3
n
(3m−1)+1
k = 2m.3
n
, m 6≡ 0 (3)
0 k = 0
d
1
(b
2k+1
) =
(−1)
n
v
6.3
n
+2
1
b
3
n+1
(6m+1)
k = 3
n+1
(3m + 1)
(−1)
n
v
10.3
n
+2
1
b
3
n
(18m+11)
k = 3
n
(9m + 8)
0 else
The homotopy of the K(2)-local Moore spectrum at the prime 3 revisited 3
d
1
(b
2k+1
) =
(−1)
m+1
v
2
1
∆
2m
2k + 1 = 6m + 1
(−1)
m+n
v
4.3
n
1
∆
3
n
(6m+5)
2k + 1 = 3
n
(18m + 17)
(−1)
m+n+1
v
4.3
n
1
∆
3
n
(6m+1)
2k + 1 = 3
n
(18m + 5)
0 else .
It turns out that the d
2
-differential o f this spectral sequence is determined by the following
principles: it is non-triv ial if and only if v
1
-linearity and sparseness of the resulting E
2
-term
permit it, and in this case it is determined up to sign by these two pro perties. The remaining
d
3
-differential turns out to be trivial. More precisely we have the following result.
Proposition 1.3.
a) The differential d
2
: E
0,1,∗
2
→ E
2,0,∗
2
is determined by
d
2
(∆
k
α) =
(−1)
m+n+1
v
6.3
n
+1
1
b
3
n+1
(6m+1)
k = 2.3
n
(3m + 1)
(−1)
m+n
v
10.3
n+1
+1
1
b
3
n+1
(18m+11)
k = 2.3
n
(9m + 8)
0 else
d
2
(∆
k
eα) =
(
(−1)
m
v
11
1
b
18m+11
k = 6m + 5
0 else .
b) The d
3
-differential is trivial.
Remark 1 on notation
Of course, the elements ∆
k
α and ∆
k
eα are only names for elements in the
E
2
-term which ar e represented in the E
1
-term as products, but which are no longe r products in
the E
2
-term. Similar abuse of notation will be used in Theorem 1.4, Proposition 1.5, Theorem
1.6 and in section 6 a nd 8.
Next we use that the element β of Theorem 1.1 lifts to an element with the same name in
H
2
(G
1
2
, M
12
) resp. in H
2
(G
2
, M
12
). In fact this latter element detects the image o f β
1
∈ π
10
(S
0
)
in π
10
(L
K(2)
V (0)). The previous results yield the following E
∞
-term as a module over F
3
[β, v
1
].
Theorem 1.4. As an F
3
[β, v
1
]-module the E
∞
-term of the algebraic spectral sequence (2) for
M = (E
2
)
∗
/(3) is isomorphic t o a direct sum of cyclic modules generated by the following
elements and with the following annihilator ideals:
a) For E
0,∗,∗
∞
we have the following generators with respective annihilator ideals
1 = ∆
0
(βv
2
1
)
∆
m
β m 6= 0 (v
2
1
)
α (v
1
)
∆
2m+1
α (v
1
)
∆
2.3
n
(3m−1)
α m 6≡ 0 mod (3) (v
1
)
∆
2m
eα (v
1
)
∆
2m+1
eα m 6≡ 2 mod (3) (v
1
)
∆
2.3
n
(3m+1)
αβ (v
1
)
∆
2.3
n
(3m−1)
αβ m ≡ 0 mod (3) (v
1
)
∆
2m+1
eαβ m ≡ 2 mod (3) (v
1
) .
b) For E
1,∗,∗
∞
we have the following generators with respective annihilator ideals
b
1
(β)
b
2.3
n
(3m−1)+1
m 6≡ 0 mod (3) (v
4.3
n
−2
1
, β) .
c) For E
2,∗,∗
∞
we have the following generators with respective annihilator ideals
b
3
n+1
(6m+1)
(v
6.3
n
+1
1
, β)
b
3
n
(6m+5)
m ≡ 1 mod (3) (v
10.3
n
+1
1
, β) .
4 Hans-Werner Henn, Nasko Karamanov and Mark Mahowald
d) For E
3,∗,∗
∞
we have the following generators with respective annihilator ideals
∆
2m
(v
2
1
)
∆
3
n
(6m±1)
(v
4.3
n
1
, βv
2
1
)
∆
m
α (v
1
)
∆
m
eα (v
1
) .
To get at H
∗
(G
1
2
, (E
2
)
∗
/(3)) we s till need to know the extensions between the filtration
quotients. They are given by the following result.
Proposition 1.5. The F
3
[β, v
1
]-module generators of the E
∞
-term of Theorem 1.4 can be lifted
to elements (with the same name) in H
∗
(G
1
2
; (E
2
)
∗
/(3)) such that t he relations defining the
annihilator ideals of Theorem 1.4 continue to hold with the following exceptions
v
1
α = b
1
v
1
∆
2.3
n
(9m+2)
α = (−1)
m+1
b
2.3
n+1
(9m+2)+1
v
1
∆
2.3
n
(9m+5)
α = (−1)
m+1
b
2.3
n+1
(9m+5)+1
v
1
∆
6m+1
eα = (−1)
m
b
2(9m+2)+1
v
1
∆
6m+3
eα = (−1)
m+1
b
2(9m+5)+1
β
b
3
n+1
(6m+1)
= ±∆
3
n
(6m+1)
eα
βb
3
n+1
(18m+11)
= ±∆
3
n
(18m+11)
eα
β
b
18m+11
= ±∆
6m+4
α .
Apart from the last group of β-extensions (which are simple consequences of the calculation
of H
∗
(G
2
, (E
2
)
∗
/(3, u
1
)), cf. [4]) one can summarize the result by saying that nontrivial v
1
-
extensions exist only between E
0,1,∗
∞
and E
1,0,∗+4
∞
and there is such an ex tension whenever
sparseness permits it, and then the corresponding relation is unique up to s ign. Unfortunately
this is not clear a priori, but needs proof and the proof gives the exact value of the sign. In
contrast determining the sign for the β-relations would r e quire an extra effort.
The main re sults can now be stated as follows.
Theorem 1 .6. As an F
3
[β, v
1
]-module H
∗
(G
1
2
, (E
2
)
∗
/(3)) is isomorphic to the direct s um of
the cyclic modules generated by t he following elements and with the following annihilator ideals
1 = ∆
0
(βv
2
1
)
∆
m
β m 6= 0 (v
2
1
)
α (βv
1
)
∆
2m+1
α (v
1
)
∆
2.3
n
(3m−1)
α m 6≡ 0 mod (3) (v
4.3
n+1
−1
1
, βv
1
)
∆
2m
eα (v
1
)
∆
2m+1
eα m 6≡ 2 mod (3) (v
3
1
, βv
1
)
∆
2.3
n
(3m+1)
αβ (v
1
)
∆
2.3
n
(3m−1)
αβ m ≡ 0 mod (3) (v
1
)
∆
2m+1
eαβ m ≡ 2 mod (3) (v
1
)
b
3
n+1
(6m+1)
(v
6.3
n
+1
1
, βv
1
)
b
3
n
(6m+5)
m ≡ 1 mod (3) (v
10.3
n
+1
1
, βv
1
)
∆
2m
(v
2
1
)
∆
3
n
(6m±1)
(v
4.3
n
1
, βv
2
1
)
∆
2m+1
α (v
1
)
∆
2m
α m 6≡ 2 mod (3) (v
1
)
∆
2m
eα (v
1
)
∆
3
n
(6m+5)
eα m 6≡ 1 mod (3) (v
1
) .