THE MOTIVIC SEGAL CONJECTURE, LECTURE 5
JOHN ROGNES
Abstract. ((TBW))
1. Some bicomodule algebras
Definition 1.1. In the dual Steenrod algebra A∗ = P (ξk | k ≥ 1) consider the ideals
e
I(n)∗ = (ξk2 | k ≥ 1, e ≥ 0, k + e ≥ n + 2)
for n ≥ −1, and
e
J(n)∗ = (ξk2 | k ≥ 2, e ≥ 0, k + e ≥ n + 2)
for n ≥ 0. Let
A(n)∗ = A∗ /I(n)∗ = P2n+1 (ξ1 ) ⊗ P2n (ξ2 ) ⊗ · · · ⊗ P22 (ξn ) ⊗ E(ξn+1 )
and
C(n)∗ = A∗ /J(n)∗ = P (ξ1 ) ⊗ P2n (ξ2 ) ⊗ · · · ⊗ P22 (ξn ) ⊗ E(ξn+1 )
be the associated quotient algebras, and let
B(n)∗ = C(n)∗ [1/ξ1 ] = P (ξ1±1 ) ⊗ P2n (ξ2 ) ⊗ · · · ⊗ P22 (ξn ) ⊗ E(ξn+1 )
be the localization obtained by inverting ξ1 .
We have inclusions
J(n + 1)∗ /
/ J(n)∗
I(n + 1)∗ /
/ I(n)∗ /
/ I(n − 1)∗
for n ≥ 0, and associated algebra homomorphisms
A∗
B(n + 1)∗
O
/ / B(n)∗
O
O
/ / C(n + 1)∗
O
/ / C(n)∗
A(n + 1)∗
/ / A(n)∗
/ / A(n − 1)∗ .
In particular, C(0)∗ = P (ξ1 ) and B(0)∗ = P (ξ1±1 ). The kernels of the algebra homomorphisms γ∗ : C(n)∗ C(0)∗ and β∗ : B(n)∗ B(0)∗ are the ideals generated by
(ξ2 , . . . , ξn+1 ), in each case.
Lemma 1.2. I(n)∗ is a Hopf ideal, for each n ≥ −1, so that there is a unique Hopf algebra
structure on A(n)∗ making A∗ A(n)∗ a Hopf algebra homomorphism. It follows that
A(n + 1)∗ A(n)∗ is also a Hopf algebra homomorphism.
Date: October 24th 2014.
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Proof. [[Calculate.]]
We use the Hopf algebra homomorphisms A∗ → A(n+1)∗ → A(n)∗ → A(n−1)∗ to view
A∗ , A(n + 1)∗ and A(n)∗ as A(n)∗ -A(n − 1)∗ -bicomodule algebras. The homomorphisms
A∗ → A(n + 1)∗ → A(n)∗ are then A(n)∗ -A(n − 1)∗ -bicomodule algebra homomorphisms.
Lemma 1.3. J(n)∗ is an A(n)∗ -A(n − 1)∗ -bicomodule ideal, for each n ≥ 0, so that there
is a unique A(n)∗ -A(n − 1)∗ -bicomodule algebra structure on C(n)∗ making A∗ C(n)∗
an A(n)∗ -A(n − 1)∗ -bicomodule algebra homomorphism.
Proof. [[Calculate.]]
If follows that C(n + 1)∗ C(n)∗ and C(n)∗ A(n)∗ are also A(n)∗ -A(n − 1)∗ bicomodule algebra homomorphisms, when C(n + 1)∗ is given the A(n)∗ -A(n − 1)∗ bicomodule algebra structure induced from the A(n+1)∗ -A(n)∗ -bicomodule algebra structure that was just defined.
Lemma 1.4. There is a short exact sequence
n+1
0 → Σ2
ξ∗
C(n)∗ −→ C(n)∗ → A(n)∗ → 0
n+1
of A(n)∗ -A(n − 1)∗ -bicomodules, where ξ∗ is given by multiplication by ξ12
n+1
B(n)∗ = colim Σ−j·2 C(n)∗
. Hence
j
has a unique A(n)∗ -A(n−1)∗ -bicomodule algebra structure making the inclusion C(n)∗ B(n)∗ an A(n)∗ -A(n − 1)∗ -bicomodule algebra homomorphism.
Proof. [[Calculate.]]
It follows that B(n + 1)∗ B(n)∗ and C(n)∗ B(n)∗ are also A(n)∗ -A(n − 1)∗ bicomodule algebra homomorphisms, when B(n+1)∗ has the A(n)∗ -A(n−1)∗ -bicomodule
algebra structure induced from the A(n + 1)∗ -A(n)∗ -bicomodule algebra structure that
was just defined.
Proposition 1.5. There is an isomorphism
C(n)∗ ∼
= A(n)∗ ⊗ (F2 A(n)∗ C(n)∗ )
of left A(n)∗ -comodules and right F2 A(n)∗ C(n)∗ -modules. There is also an isomorphism
C(n)∗ ∼
= (C(n)∗ A(n−1)∗ F2 ) ⊗ A(n − 1)∗
of right A(n − 1)∗ -comodules and left C(n)∗ A(n−1)∗ F2 -modules.
Proof. A(n)∗ , A(n − 1)∗ and C(n)∗ are connected, and C(n)∗ A(n)∗ and C(n)∗ A(n − 1)∗ are surjective, so this follows from [MM65, Theorem 4.7].
It follows that C(n)∗ is [[cofree, hence]] injective as a left A(n)∗ -comodule and as a
right A(n − 1)∗ -comodule, so the short exact sequence in Lemma 1.4 admits a splitting
in either one of these categories.
n+1
Lemma 1.6. F2 A(n)∗ C(n)∗ = P (ξ12
) and
n
n−1
C(n)∗ A(n−1)∗ F2 = P (ξ12 ) ⊗ E(ξ¯22 , . . . , ξ¯n+1 )
as subalgebras of C(n)∗ .
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Proof. We prove this inductively, starting from F2 A(n)∗ A(n)∗ = F2 and
n
A(n)∗ A(n−1)∗ F2 = E(ξ12 , ξ¯22
n−1
, . . . , ξ¯n+1 ) .
Applying F2 A(n)∗ (−) to the (split) short exact sequence in Lemma 1.4, we get an exact
sequence
0 → Σ2
n+1
ξ∗
X∗ −→ X∗ −→ F2 → 0
n+1
where X∗ = F2 A(n)∗ C(n)∗ . We have already seen that ξ12
∈ C(n)∗ lifts to X∗ . It
n+1
follows by an induction on degrees that the inclusion P (ξ12 ) ⊆ X∗ is an equality of
subalgebras of C(n)∗ .
On the other hand, applying (−) A(n−1)∗ F2 to the same (split) short exact sequence,
we get a exact sequence
n+1
0 → Σ2
ξ∗
n
n−1
Y∗ −→ Y∗ −→ E(ξ12 , ξ¯22
, . . . , ξ¯n+1 ) → 0
n
n−1
where Y∗ = C(n)∗ A(n−1)∗ F2 . We can check directly that ξ12 , ξ¯22 , . . . , ξ¯n+1 ∈ C(n)∗
lift to Y∗ . It follows by another induction on degrees that the inclusion
n−1
n
P (ξ12 ) ⊗ E(ξ¯22 , . . . , ξ¯n+1 ) ⊆ Y∗
is in fact an equality of subalgebras of C(n)∗ .
n+1
Definition 1.7. Let γ∗0 : C(n)∗ P (ξ12 ) be the composite of γ∗ : C(n)∗ C(0)∗ and
n+1
n+1
the (additive, homogeneous) retraction P (ξ1 ) P (ξ12 ), and let β∗0 : B(n)∗ P (ξ1±2 )
n+1
be the composite of β∗ : B(n)∗ B(0)∗ and the retraction P (ξ1±1 ) P (ξ1±2 ).
Lemma 1.8. The horizontal composites in the commutative diagram
B(n)∗
O
/ A(n)∗ ⊗ B(n)∗
O
O
C(n)∗
O
/ A(n)∗ ⊗ C(n)∗
1⊗β∗0
/ A(n)∗ ⊗ P (ξ1±2n+1 )
O
O
1⊗γ∗0
/ A(n)∗ ⊗ P (ξ12n+1 )
are isomorphisms of left A(n)∗ -comodules.
Proof. The lower horizontal composite
n+1
α∗0 : C(n)∗ −→ A(n)∗ ⊗ P (ξ12
)
induces an isomorphism
n+1
1 α∗0 : F2 A(n)∗ C(n)∗ −→ P (ξ12
),
because we can identify it with the composite of F2 A(n)∗ C(n)∗ ⊂ C(n)∗ and γ∗0 : C(n)∗ n+1
P (ξ12 ). Since A(n)∗ is connected and C(n)∗ is left A(n)∗ -comodule injective, it follows
that α∗0 is also an isomorphism. [[More details?]] The upper horizontal composite
B(n)∗ −→ A(n)∗ ⊗ P (ξ1±2
2n+1
is obtained from α∗0 by inverting multiplication by ξ1
n+1
)
, hence is also an isomorphism. Lemma 1.9. The composite of C(n)∗ A(n−1)∗ F2 ⊂ C(n)∗ and γ∗ : C(n)∗ C(0)∗ maps
n+1
e
e
2e (2k −1)
ξ¯k2 to ξ1
= ξ12 −2 for 1 ≤ k ≤ n + 1 and k + e = n + 1. Hence the composite is
n+1
e
injective, with image M (n)∗ ⊂ P (ξ1 ) the subalgebra generated by the classes ξ12 −2 for
0 ≤ e ≤ n. This subalgebra contains all classes ξ1m with m > (n − 1)2n+1 + 1.
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Proof. It follows from the recursive definition of ξ¯k = χ(ξk ) that
k
ξ¯k ≡ ξ12 −1
mod (ξ2 , . . . , ξk ) .
Hence a monomial generator
ξ1j·2
n+1
er
e
2
2 1
· · · ξn+1−e
· ξn+1−e
r
1
for C(n)∗ A(n−1)∗ F2 , where j ≥ 0, 0 ≤ r ≤ n + 1 and 0 ≤ e1 < · · · < er ≤ n, is mapped
to ξ1m with
m = j · 2n+1 + (2n+1 − 2e1 ) + · · · + (2n+1 − 2er ) = (j + r)2n+1 − (2e1 + · · · + 2er ) .
It follows that m uniquely determines j, r and e1 , . . . , er , so the composite homomorphism
is injective. It is elementary to check that the integers m that occur are those that can be
written as sums of non-negative multiples of 2n+1 − 2e for 0 ≤ e ≤ n. It is also elementary
to check that these sums include all integers greater than (n − 1)2n+1 + 1.
Definition 1.10. Let γ∗00 : C(n)∗ M (n)∗ be the composite of γ∗ : C(n)∗ C(0)∗ and
the retraction P (ξ1 ) M (n)∗ .
Lemma 1.11. The composite
γ 00 ⊗1
∗
M (n)∗ ⊗ A(n − 1)∗
C(n)∗ −→ C(n)∗ ⊗ A(n − 1)∗ −→
is an isomorphism of right A(n − 1)∗ -comodules.
Proof. Let α∗00 denote the composite right A(n − 1)∗ -comodule homomorphism. The induced homomorphism
α∗00 1 : C(n)∗ A(n−1)∗ F2 −→ M (n)∗
is an isomorphism, because we can identify it with the composite of C(n)∗ A(n−1)∗ F2 ⊂
C(n)∗ and γ∗00 : C(n)∗ M (n)∗ . Since A(n−1)∗ is connected and C(n)∗ is right A(n−1)∗ comodule injective, it follows that α∗00 is also an isomorphism. [[More details?]]
n+1
on C(n)∗ is not directly compatible with multiplication with
Multiplication with ξ12
on M (n)∗ , because the retraction P (ξ1 ) M (n)∗ does not commute with such mulξ1
tiplication. This is why the following result does not immediately follow by localization
from the previous lemma.
2n+1
Lemma 1.12. The composite
β∗ ⊗1
B(n)∗ −→ B(n)∗ ⊗ A(n − 1)∗ −→ B(0)∗ ⊗ A(n − 1)∗
is an isomorphism of right A(n − 1)∗ -comodules.
Proof. Consider the commutative diagram of right A(n − 1)∗ -comodules
Σ−j·2
n+1
C(n)∗
B(n)∗
/ Σ−j·2n+1 C(0)∗ ⊗ A(n − 1)∗
/ / Σ−j·2n+1 M (n)∗ ⊗ A(n − 1)∗
/ B(0)∗ ⊗ A(n − 1)∗
where the left hand vertical map is a canonical map from the definition of B(n)∗ as a
colimit, and the upper horizontal composite is the isomorphism α∗00 of Lemma 1.11. For
each fixed degree, there exists a sufficiently large j such that the two vertical maps and
the upper right hand horizontal map are all isomorphism in that degree. This uses that
A(n)∗ , A(n − 1)∗ and the kernel of C(0)∗ M (n)∗ are bounded above and below. Hence
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the two remaining horizontal maps are also isomorphisms in the fixed degree. Since the
lower horizontal map does not depend on j, it is an isomorphism in every degree.
2. The dual bimodule coalgebras
We now dualize the definitions and results of the previous section.
Definition 2.1. Let A = (A∗ )∗ be the Steenrod algebra, let
A(n) = (A(n)∗ )∗
be the Hopf algebra dual to A(n)∗ , let
C(n) = (C(n)∗ )∗
be the A(n)-A(n − 1)-bimodule coalgebra dual to C(n)∗ , and let
n+1
B(n) = (B(n)∗ )∗ = lim Σ−j·2 C(n)
j
be the A(n)-A(n − 1)-bimodule coalgebra dual to B(n)∗ .
Then A(n) A(n + 1) A are Hopf algebra homomorphisms, there is a short exact
sequence
n+1
ξ
0 → A(n) → C(n) −→ Σ2 C(n) → 0
of A(n)-A(n − 1)-bimodules, where ξ is dual to ξ∗ , and the diagram
A o
B(n + 1) o
o B(n)
o C(n + 1) o
O
o C(n)
O
O
O
o A(n)
A(n + 1) o
consists of A(n)-A(n − 1)-bimodule coalgebra homomorphisms. There is an isomorphism
C(n) ∼
= A(n) ⊗ (F2 ⊗A(n) C(n))
of left A(n)-modules and right F2 ⊗A(n) C(n)-comodules, and an isomorphism
C(n) ∼
= (C(n) ⊗A(n−1) F2 ) ⊗ A(n − 1)
of right A(n − 1)-modules and left C(n) ⊗A(n−1) F2 -comodules. In particular, C(n) is free
as a left A(n)-module and also as a right A(n − 1)-module (but not as an A(n)-A(n − 1)bimodule).
Definition 2.2. Let C(0) = F2 {Sq k | k ≥ 0} and B(0) = F2 {Sq k | k ∈ Z}, with Sq k
dual to ξ1k . The surjection B(0) C(0) maps Sq k to 0 for k < 0. Let β : B(0) B(n)
n+1
be dual to β∗ , and let β 0 : F2 {Sq j·2
| j ∈ Z} B(n) be the restriction of β to the
subspace spanned by the Sq k with 2n+1 |k.
Proposition 2.3. (a) The composite
n+1
A(n) ⊗ F2 {Sq j·2
1⊗β 0
| j ∈ Z} −→ A(n) ⊗ B(n) −→ B(n)
is an isomorphism of left A(n)-modules. Hence B(n) is a free left A(n)-module with basis
n+1
{Sq j·2
| j ∈ Z}.
(b) The composite
β⊗1
B(0) ⊗ A(n − 1) −→ B(n) ⊗ A(n − 1) −→ B(n)
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is an isomorphism of right A(n − 1)-modules. Hence B(n) is a free right A(n − 1)-module
with basis {Sq k | k ∈ Z}.
3. The algebraic Singer construction
Recall that B(n) is an A(n)-A(n − 1)-bimodule. For any left A(n − 1)-module M , the
tensor product B(n) ⊗A(n−1) M becomes a left A(n)-module. If M is a left A(n)-module,
the inclusion B(n) B(n + 1) of A(n)-A(n − 1)-bimodules induces a left A(n)-module
homomorphism
(1)
B(n) ⊗A(n−1) M −→ B(n + 1) ⊗A(n) M .
It is compatible with the isomorphisms
∼
=
B(0) ⊗ M ∼
= B(0) ⊗ A(n − 1) ⊗A(n−1) M −→ B(n) ⊗A(n−1) M
and
∼
=
B(0) ⊗ M ∼
= B(0) ⊗ A(n) ⊗A(n) M −→ B(n + 1) ⊗A(n) M
from Proposition 2.3(b), hence the left A(n)-module homomorphism (1) is in fact an
isomorphism.
Definition 3.1. For any left A -module M let the Singer construction
R+ (M ) = colim B(n) ⊗A(n−1) M
n
be the A -module whose underlying A(n)-module structure respects the isomorphism
∼
=
B(n) ⊗A(n−1) M −→ R+ (M ), for each n ≥ 0.
For any A -module M , the composite A(n)-A(n−1)-bimodule homomorphism B(n) C(n) A induces an A(n)-module homomorphism
B(n) ⊗A(n−1) M −→ A ⊗A(n−1) M .
For varying n these are compatible with the isomorphisms in the colimit system defining
R+ (M ) and the evident surjective homomorphisms
A ⊗A(n−1) M A ⊗A(n) M ,
hence induce a homomorphism
R+ (M ) −→ colim A ⊗A(n) M ∼
= A ⊗A M ∼
=M.
n
Definition 3.2. For each left A -module M let the augmentation
: R+ (M ) −→ M
be the A -module homomorphism whose underlying A(n)-module homomorphism is given
by the composite
B(n) ⊗A(n−1) M −→ A ⊗A(n−1) M −→ M .
Theorem 3.3 (W. H. Lin). The augmentation is a Tor-isomorphism, in the sense that
A
# : TorA
∗,∗ (F2 , R+ (M )) −→ Tor∗,∗ (F2 , M )
is an isomorphism in all bidegrees, for all left A -modules M .
Lemma 3.4. (a) The functor R+ from left A -modules to left A -modules is exact.
(b) If M is free then R+ (M ) is flat.
(c) If M is free then 1 ⊗ : F2 ⊗A R+ (M ) −→ F2 ⊗A M is an isomorphism.
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Proof. (a) Let 0 → M 0 → M → M 00 → 0 be a short exact sequence of left A -modules.
In particular, it is a short exact sequence of left A(n − 1)-modules. Since B(n) is free,
hence flat, as a right A(n − 1)-module, the sequence
0 → B(n) ⊗A(n−1) M 0 → B(n) ⊗A(n−1) M → B(n) ⊗A(n−1) M 00 → 0
of left A(n)-modules remains exact. Hence
0 → R+ (M 0 ) → R+ (M ) → R+ (M 00 ) → 0
is a short exact sequence of left A -modules.
(b) It suffices to prove this when M = A . For each n ≥ 0 we have that A is free as
a left A(n − 1)-module, hence R+ (A ) is a direct sum of suspensions of copies of B(n) as
a left A(n)-module. Since B(n) is free as a left A(n)-module, it follows that R+ (A ) is
free, hence flat, as a left A(n)-module. Thus
TorA
s,t (N, R+ (A )) = colim Tors,t (N, R+ (A )) = 0
A(n)
n
for each right A -module N and every positive s ≥ 1, which implies that R+ (A ) is flat
as left A -module.
(c) It suffices to prove this when M = A . When viewed as a left A(n − 1)-module,
A = colimm≥n−1 A(m), so we can realize the source F2 ⊗A R+ (A ) of 1 ⊗ as the colimit
colim F2 ⊗A(n) R+ (A ) = colim F2 ⊗A(n) B(n) ⊗A(n−1) A
n
n
= colim F2 ⊗A(n) B(n) ⊗A(n−1) A(m)
m≥n−1
= colim F2 ⊗A(n) B(n) ⊗A(n−1) A(n − 1)
n
∼
= colim F2 ⊗A(n) B(n)
n
since the pairs (m, n) with m = n − 1 are cofinal among the pairs (m, n) with m ≥ n − 1
in the product partial ordering.
By Proposition 2.3(a), we know that the left A(n)-module B(n) is free on the generators
Sq k for 2n+1 |k. The homomorphism
F2 ⊗A(n) B(n) −→ F2 ⊗A(n+1) B(n + 1)
is induced by the left A(n)-module homomorphism B(n) B(n + 1) followed by the
canonical surjection F2 ⊗A(n) B(n+1) F2 ⊗A(n+1) B(n+1). It can therefore be rewritten
as a homomorphism
F2 {Sq k | 2n+1 |k} −→ F2 {Sq k | 2n+2 |k}
sending Sq k to 0 when k is of the form j · 2n+1 with j odd, and mapping Sq 0 to Sq 0 .
Hence
colim F2 ⊗A(n) B(n) ∼
= F2 ,
n
0
sending 1 ⊗ Sq to the generator.
Clearly the target of is F2 ⊗A A ∼
= F2 . To check that 1⊗ : F2 → F2 is an isomorphism
it suffices to check that the class 1 ⊗ Sq 0 ∈ F2 ⊗A(0) B(0), which maps to the generator in
the source, also maps to the generator in the target, but this is clear since Sq 0 (1) = 1. Proof of Theorem 3.3. Let
· · · → F2 → F1 → F0 → M → 0
be a resolution of M by free left A -modules. Then by exactness and flatness
· · · → R+ (F2 ) → R+ (F1 ) → R+ (F0 ) → R+ (M ) → 0
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is a resolution of R+ (M ) by flat left A -modules. Hence TorA
∗,∗ (F2 , R+ (M )) is the homology of the complex
· · · → F2 ⊗A R+ (F2 ) → F2 ⊗A R+ (F1 ) → F2 ⊗A R+ (F0 ) → 0 .
Since each Fs is free, induces an isomorphism from the latter chain complex to the
complex
· · · → F2 ⊗A F2 → F2 ⊗A F1 → F2 ⊗A F0 → 0 ,
whose homology computes TorA
∗,∗ (F2 , M ). Hence the induced homomorphism # of homology groups is also an isomorphism.
[[Recognize R+ (F2 ) = colimn B(n) ⊗A(n−1) F2 as ΣP (x±1 ).]]
References
[AGM85] J. F. Adams, J. H. Gunawardena, and H. Miller, The Segal conjecture for elementary abelian
p-groups, Topology 24 (1985), no. 4, 435–460.
[MM65] John W. Milnor and John C. Moore, On the structure of Hopf algebras, Ann. of Math. (2) 81
(1965), 211–264.
Department of Mathematics, University of Oslo, Norway
E-mail address: rognes@math.uio.no
URL: http://folk.uio.no/rognes
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