THE MOTIVIC SEGAL CONJECTURE, LECTURE 4
JOHN ROGNES
Abstract. Jeg vil snakke om de endelige underalgebraene A(n) i Steenrod-algebraen,
analysere A(n)-modulstrukturen til den kontinuerlige kohomologien ΣP (x±1 ) til Tatekonstruksjonen, og skissere Lin-Davis-Mahowald-Adams’ bevis av Lins teorem.
1. The Steenrod operations
Let p be a prime, let H = HFp be the Eilenberg–Mac Lane ring spectrum representing mod p cohomology, and let A = H ∗ (H) be the Steenrod algebra of stable mod p
cohomology operations. For p = 2 the Steenrod reduced squares
Sq i : H ∗ (X) → H ∗+i (X)
defined for i ≥ 1 (Steenrod, 1947) are known to generate A as an algebra (Serre, 1953).
For p odd the Bockstein operator
β : H ∗ (X) → H ∗+1 (X)
(satisfying β 2 = 0), together with the Steenrod reduced powers
P i : H ∗ (X) → H ∗+2i(p−1) (X)
defined for i ≥ 1 (Steenrod, 1953) are known to generate A as an algebra (Cartan, 1954,
1955). These operations are all natural in the space X. Setting Sq 0 and P 0 equal to the
identities, these operations satisfy the Cartan formula
X
Sq i (x) ∪ Sq j (y)
Sq k (x ∪ y) =
i+j=k
for p = 2, and
β(x ∪ y) = β(x) ∪ y + (−1)|x| x ∪ β(y)
X
P i (x) ∪ y + x ∪ P j (y)
P k (x ∪ y) =
i+j=k
∗
P
for p odd, with x, y ∈ H (X). Introducing the formal sum Sq(x) = i≥0 Sq i (x), we can
write the first and last of these as Sq(x ∪ y) = Sq(x) ∪ Sq(y) and P (x ∪ y) = P (x) ∪ P (y).
We have Sq i (x) = x ∪ x = x2 for |x| = i and P i (x) = x ∪ · · · ∪ x = xp for |x| = 2i. Finally,
Sq i (x) = 0 if |x| < i and P i (x) = 0 if |x| < 2i.
For p = 2 and X = RP ∞ = S ∞ /C2 we have H ∗ (X) = P (x) = F2 [x] with |x| = 1, so
Sq 0 (x) = x, Sq 1 (x) = x2 and Sq i (x) = 0 for i ≥ 2. Hence Sq(x)
= x + x2 = x(1 + x). By
P
j
j
the Cartan formula Sq(xj ) = Sq(x)j = xj (1 + x)j = i=0 i xi+j . Hence
j i+j
i j
Sq (x ) =
x
i
for i, j ≥ 0 determines the action of A on H ∗ (RP ∞ ). Only the residue class mod 2 of
the binomial coefficient matters.
Date: October 10th 2014.
1
For p odd and X = L∞ = S ∞ /Cp we have H ∗ (X) = E(x) ⊗ P (y) = Fp [x, y]/(x2 ) with
|x| = 1 and |y| = 2, where β(x) = y and β(y) = 0. Then P 0 (x) = x and P i (x) = 0 for
i ≥ 1. Furthermore, P 0 (y) = y, P 1 (y) = y p and P i (y) = 0 for i ≥ 2, so P (y) = y + y p =
P
y(1 + y p−1 ). Hence P (y j ) = P (y)j = y j (1 + y p−1 )j = ji=0 ji y j+i(p−1) , so that
j j+i(p−1)
i j
P (y ) =
y
i
for i, j ≥ 0, which determines the action of A on H ∗ (L∞ ). Only the residue class mod p
of the binomial coefficient matters.
2. The Steenrod algebra
The (non-commutative) algebra structure φ : A ⊗ A → A on A is given by composition of operations. The Steenrod squares satisfy the Adem relations (Adem, 1952)
X b − i − 1
a
b
Sq Sq =
Sq a+b−i Sq i
a
−
2i
i
for a < 2b, while the Steenrod powers satisfy
X
a b
a+i (p − 1)(b − i) − 1
P P =
(−1)
P a+b−i P i
a
−
pi
i
for a < pb and
X
X
a
b
a+i (p − 1)(b − i)
a+b−i i
a+i+1 (p − 1)(b − i) − 1
P βP =
(−1)
βP
P + (−1)
P a+b−i βP i
a
−
pi
a
−
pi
−
1
i
i
for a ≤ pb. For instance, Sq 1 Sq 1 = 0, Sq 1 Sq 2 = Sq 3 , Sq 2 Sq 2 = Sq 3 Sq 1 and Sq 3 Sq 2 = 0.
These can be used to simplify composites of the form
Sq I := Sq i1 Sq i2 . . . Sq ir
for i1 , i2 , . . . , ir ≥ 1, unless is ≥ 2is+1 for all 1 ≤ s < r. In the latter case we say
that I = (i1 , i2 , . . . , ir ) is an admissible sequence of length r, and Sq I is an admissible
monomial. For p odd the Adem relations can simplify composites of the form
β P I := β 0 P i1 β 1 . . . P ir β r
with 0 , 1 , . . . , r ∈ {0, 1} and i1 , i2 , . . . , ir ≥ 1, unless is ≥ s + pis+1 for all 1 ≤ s < r. In
the latter case we say that I = (0 , i1 , 1 , . . . , ir , r ) is an admissible sequence, and that
β P I is an admissible monomial. In fact the admissible monomials of all lengths r ≥ 0
form a basis for A ,
A = F2 {Sq I | I admissible}
for p = 2 and
A = Fp {β P I | (, I) admissible}
for p odd, so the Adem relations generate all other relations among the Sq i for p = 2,
and among β and the P i for p odd, in the Steenrod algebra A .
The Adem relations imply that the Steenrod operations Sq i are decomposable except
when i = 2j for some j ≥ 0, and that the P i are decomposable except when i = pj for
some j ≥ 0. Letting I(A ) = ker(A → Fp ) be the augmentation ideal, the quotient
Q(A ) = I(A )/I(A )2 = cok(φ : I(A ) ⊗ I(A ) → I(A ))
is the vector space of algebra indecomposables. We have
j
Q(A ) = F2 {Sq 2 | j ≥ 0}
2
for p = 2, and
j
Q(A ) = Fp {β, P p | j ≥ 0}
for p odd.
3. The Hopf algebra structure
The Steenrod algebra admits (Milnor, 1958) the structure of a connected, cocommutative Hopf algebra. The coproduct ψ : A → A ⊗ A is defined on the algebra generators
by
X
ψ(Sq k ) =
Sq i ⊗ Sq j
i+j=k
for p = 2, and by
ψ(β) = β ⊗ 1 + 1 ⊗ β
X
ψ(P k ) =
Pi ⊗ Pj
i+j=k
for p odd. The conjugation χ : A → A is the involutive anti-homomorphism recursively
determined by the relations
X
Sq i χ(Sq j ) = 0
i+j=k
for k ≥ 1, χ(β) = −β, and
X
1
P i χ(P j ) = 0
i+j=k
1
for k ≥ 1. For instance, χ(Sq ) = Sq , χ(Sq 2 ) = Sq 2 and χ(Sq 3 ) = Sq 2 Sq 1 .
By Milnor–Moore (1965, Section 8), any connected Hopf algebra admits a unique
conjugation χ such that φ(1 ⊗ χ)ψ = η = φ(χ ⊗ 1)ψ, and such that χ(xy) = χ(y)χ(x).
If φ is commutative, or ψ is cocommutative, then χ2 = 1.
Letting J(A ) = cok(Fp → A ) be the unit/coaugmentation coideal, the kernel
P (A ) = ker(ψ : J(A ) → J(A ) ⊗ J(A ))
consists of the coalgebra primitives in A , i.e., the elements x ∈ A satisfying ψ(x) =
j
x ⊗ 1 + 1 ⊗ x. For p = 2 let Q0 = Sq 1 and recursively define Qj = [Sq 2 , Qj−1 ] =
j
j
Sq 2 Qj−1 + Qj−1 Sq 2 for j ≥ 1. For instance, Q1 = [Sq 2 , Sq 1 ] = Sq 3 + Sq 2 Sq 1 , and
j
|Qj | = 2j − 1. For p odd let Q0 = β and define Qj+1 = [P p , Qj ] for j ≥ 0. When p = 2,
these Milnor primitives are precisely the coalgebra primitives of A , so that
P (A ) = F2 {Qj | j ≥ 0} .
[[Also discuss p odd.]]
4. Finite sub Hopf algebras
First consider p = 2. For n ≥ −1 let A(n) ⊂ A be the subalgebra generated by the
Steenrod operations Sq i for 1 ≤ i < 2n+1 . Equivalently, A(n) is the subalgebra generated
j
by the Sq i for 1 ≤ i ≤ 2n , or by the Sq 2 for 0 ≤ j ≤ n. Clearly A(n−1) is a subalgebra
of A(n) for each n ≥ 0. For instance, A(−1) = F2 ,
A(0) = F2 {1, Sq 1 }
is the exterior algebra on one generator,
•
Sq 1
3
/•
and
A(1) = F2 {1, Sq 1 , Sq 2 , Sq 1 Sq 2 , Sq 2 Sq 1 , Sq 2 Sq 2 , Sq 2 Sq 1 Sq 2 , Sq 2 Sq 2 Sq 2 }
= F2 {1, Sq 1 , Sq 2 , Sq 3 , Sq 2 Sq 1 , Sq 3 Sq 1 , Sq 5 + Sq 4 Sq 1 , Sq 5 Sq 1 }
is a finite algebra of dimension eight.
?•
Sq 2
•
Sq 1
"
/•
/•
•
?
•
/< •
.•
The next subalgebra, A(2), has dimension 64 and is more difficult to visualize. In general,
n+2
A(n) is a finite algebra of dimension 2·22 ·. . .·2n+1 = 2( 2 ) . Since each algebra generator
of A lies in some A(n), we have
[
A =
A(n) .
n
Hence each positive degree element in A is nilpotent. Clearly the coproduct ψ on A
restricts to a coproduct on A(n), and likewise for the conjugation χ, so each A(n) is a
connective, cocommutative sub Hopf algebra of A .
For n ≥ −1 let E(n) ⊂ A(n) be the subalgebra generated by the Milnor primitives Qj
for 0 ≤ j ≤ n. It turns out that these are commuting exterior generators with no further
relations, so that
E(n) = E(Q0 , Q1 , . . . , Qn )
is an exterior algebra on n + 1 generators. For instance, E(0) = A(0), while
E(1) = F2 {1, Q0 , Q1 , Q0 Q1 }
= F2 {1, Sq 1 , Sq 3 + Sq 2 Sq 1 , Sq 3 Sq 1 }
is of dimension four.
Q1
•
#
/•
Q0
•
/; •
Since each Qj is coalgebra primitive, this is a primitively generated sub Hopf algebra of
A(n). We write
[
E=
E(n) = E(Qj | j ≥ 0)
n
for the resulting primitively generated, connected, bicommutative sub Hopf algebra of
A.
[[Also discuss p odd.]]
4
5. Lin and Gunawardena’s theorems
∗
tG
In order to make sense of Ext∗,∗
A (Hc (S ; Fp ), Fp ) for G = Cp , p = 2 or p odd, we first
need to understand the A -module structure of
Hc∗ (S tG ; Fp ) = colim ΣH ∗ (BG−kV ; Fp ) .
k
We previously used to Thom isomorphisms
∼
=
Φ−kξ : H ∗+kd (BG; Fp ) −→ H ∗ (BG−kV ; Fp )
to get the additive description
Hc∗ (S tG ; Fp ) = ΣH ∗ (BG; Fp )[e−1
V ],
but the Thom isomorphisms and the homomorphisms given by multiplication by eV are
not in general A -linear, so we need to be more careful to get a description of the left hand
side as an A -module. It turns out to be convenient to specify the A -module
S structure
as a sequence of compatible A(n)-module structures, for all n, since A = n A(n).
The Thom isomorphism Φ−kξ is A(n)-linear if A(n) acts trivially on the orientation
class U−kξ , and for a fixed n, this is true whenever k is divisible by a sufficiently high
power pN of p. (When p = 2 it suffices that 2n+1 | k, because then Sq(U−kξ ) = Sq(Uξ )−k
is an integral power of
X
X
n+1
n+1
n+1
n+1
Sq(Uξ )2
=
Sq j (Uξ )2
=
Sq j·2 (Uξ2 ) ,
j≥0
j≥0
so Sq i (U−kξ ) = 0 for 0 < i < 2n+1 . These are the Sq i that generate A(n).) Likewise,
multiplication by ekV is A(n)-linear when k is divisible by the same sufficiently high power
of p. Thus the description of Hc∗ (S tG ; Fp ) as
colim ΣH ∗+kd (BG)
pN |k
where k only ranges over the multiples of pN , and the homomorphisms in the sequence are
N
given by multiplication by epV , is indeed a description given entirely within the category
of A(n)-modules and A(n)-linear homomorphisms.
It follows that Sq i (Σxj · xk ) = Sq i (Σxj ) · xk whenever Sq i ∈ A(n) and 2n+1 | k. If we
define the binomial coefficient by the formula
j
j(j − 1) · · · (j − i + 1)
=
i(i − 1) . . . 1
i
for i ≥ 0 and j ∈ Z, then
j+k
j
≡
mod 2
i
i
whenever Sq i ∈ A(n) and 2n+1 | k. Hence the formula
j
i
j
Sq (Σx ) =
Σxi+j
i
for the A(n)-module action on H ∗ (ΣRP ∞ ; F2 ) remains valid in the localization Hc∗ (S tC2 ; F2 ).
And, since with these conventions the formula does not depend on n, it remains valid for
the entire A -module action.
5
Proposition 5.1. Hc∗ (S tC2 ; F2 ) = ΣP (x±1 ) with
j
i
j
Sq (Σx ) =
Σxi+j
i
for all i ≥ 0 and j ∈ Z.
Hc∗ (S tCp ; Fp ) = ΣE(x) ⊗ P (y ±1 ) with β(x) = y, P i (x) = 0 for i > 0, β(y) = 0 and
j
i
j
P (Σy ) =
Σy i(p−1)+j
i
for all i ≥ 0 and j ∈ Z.
In particular,
Sq i (Σx−1 ) = Σxi−1
P i (Σxy −1 ) = (−1)i Σxy (p−1)i−1
βP i (Σxy −1 ) = (−1)i+1 Σy (p−1)i
[[check the last sign]] for all i ≥ 0 because
−1
(−1)(−2) · · · (−i)
=
= (−1)i .
i
i(i − 1) · · · 1
The homomorphism : ΣP (x±1 ) → F2 given by (Σx−1 ) = 1 is A -linear. To see this,
one can check that Σx−1 is not A -module decomposable, i.e., of the form Sq i (Σxj ) for
any i≥ 1. For degree reasons we would have to have i + j = −1, but Sq i (Σx−i−1 ) =
−i−1
Σx−1 and
i
−i − 1
(−i − 1)(−i − 2) · · · (−i − i)
2i
=
≡
mod 2 ,
i
i
i(i − 1) · · · 1
which is always zero.
Theorem 5.2 (Lin). : ΣP (x±1 ) → F2 induces isomorphisms
∼
=
A
±1
# : TorA
s,t (F2 , ΣP (x )) −→ Tors,t (F2 , F2 )
and
∼
=
s,t
±1
# : Exts,t
A (F2 , F2 ) −→ ExtA (ΣP (x ), F2 ) .
Theorem 5.3 (Gunawardena). : ΣE(x) ⊗ P (y ±1 ) → Fp induces isomorphisms
∼
=
A
±1
# : TorA
s,t (Fp , ΣE(x) ⊗ P (y )) −→ Tors,t (Fp , Fp )
and
∼
=
s,t
±1
# : Exts,t
A (Fp , Fp ) −→ ExtA (ΣE(x) ⊗ P (y ), Fp ) .
In each case the Ext-isomorphism follows from the Tor-isomorphism and the natural
isomorphism
A
∗
∼
Exts,t
A (L, Fp ) = (Tors,t (Fp , L)) ,
which is valid for each left A -module L [?LDMA80, Lemma 4.3], [?AGM85, Proposition 1.2].
Lin-Davis-Mahowald-Adams and Gunawardena prove the Tor-isomorphisms by first
analyzing
A(n)
Tors,t (F2 , ΣP (x±1 ))
and
A(n)
Tors,t (Fp , ΣE(x) ⊗ P (y ±1 ))
6
for each n ≥ 0, and thereafter passing to the colimit over n. Adams-GunawardenaMiller give a more conceptual proof by first recognizing ΣP (x±1 ) = R+ (F2 ) and ΣE(x) ⊗
P (y ±1 ) = R+ (Fp ) as special cases of the Singer construction on (left) A -modules M , and
then proving more generally that : R+ (M ) → M is a Tor-equivalence.
6. The dual Steenrod algebra
Following (Milnor, 1958) let A∗ = H∗ (H) be the linear dual of the Steenrod algebra.
It inherits a product φ : A∗ ⊗ A∗ → A∗ from the ring spectrum structure on H, which is
dual to the coproduct ψ on A . It also inherits a coproduct ψ : A∗ → A∗ ⊗ A∗ , dual to
the product φ on A , which agrees with the composite
η∗
H∗ (H) ∼
= π∗ (H ∧ S ∧ H) −→ π∗ (H ∧ H ∧ H) ∼
= H∗ (H) ⊗ H∗ (H)
induced by the unit map η : S → H and the Künneth isomorphism on the right hand
side. The conjugation χ : A∗ → A∗ is dual to the conjugation χ : A → A , and agrees
with the homomorphism
γ∗
H∗ (H) = π∗ (H ∧ H) −→ π∗ (H ∧ H) = H∗ (H)
induced by the twist equivalence γ : H ∧H → H ∧H. In other words, A∗ is the connected,
commutative Hopf algebra dual to the Steenrod algebra.
The n-th space in the spectrum H is the Eilenberg–Mac Lane space K(Fp , n), and there
is a canonical homomorphism
x∗ : H̃∗+1 (K(Fp , 1)) −→ colim H̃∗+n (K(Fp , n)) = H∗ (H) .
n
∞
It is induced by the map Σ K(Fp , 1) → ΣH representing the generator x ∈ H 1 (K(Fp , 1)).
Here K(F2 , 1) ' RP ∞ and K(Fp , 1) ' L∞ for p odd, so
H∗ (RP ∞ ) = F2 {αi | i ≥ 0}
with αi dual to xi in H ∗ (RP ∞ ) = P (x), and
H∗ (L∞ ) = E(α1 ) ⊗ Fp {α2i | i ≥ 0}
with α1 dual to x and α2i dual to y i in H ∗ (L∞ ) = E(x) ⊗ P (y).
The loop space structure on K(Fp , 1) ' ΩK(Fp , 2) makes H∗ (K(Fp , 1)) an algebra, and
x∗ takes all decomposables (for this so-called Pontryagin product) to zero. However, the
algebra indecomposables all turn out to map non-trivially to A∗ . These are dual to the
j
coalgebra primitives in H ∗ (K(Fp , 1)), which are the classes x2 for j ≥ 0 when p = 2, and
j
the classes x and y p for j ≥ 0 when p is odd. Thus the algebra indecomposables are α2j
for j ≥ 0 when p = 2, and α1 and α2pj for j ≥ 0 when p is odd.
For p = 2 let ξj = x∗ (α2j ) ∈ A∗ for each j ≥ 0, with |ξj | = 2j − 1 and ξ0 = 1.
For p odd let τj = x∗ (α2pj ) ∈ A∗ for each j ≥ 0, with |τj | = 2pj − 1. [[The class
x∗ (α1 ) = 1.]] Furthermore, let ξj = β∗ (τj ) = y∗ (α2pj ) ∈ A∗ , with |ξj | = 2p2 − 2, where
β∗ : H∗ (H) → H∗−1 (H) is induced by the Bockstein map β : H → ΣH, and y∗ = β∗ x∗
is induced by the map β ◦ x : K(Fp , 1) → Σ2 H representing y = β(x) ∈ H 2 (K(Fp , 1)).
Again ξ0 = 1.
Theorem 6.1 (Milnor). There are algebra isomorphisms
A∗ = P (ξk | k ≥ 1)
for p = 2, and
A∗ = E(τk | k ≥ 0) ⊗ P (ξk | k ≥ 1)
7
for p odd. The coproduct on A∗ satisfies
j
X
ψ(ξk ) =
ξi2 ⊗ ξj
i+j=k
for p = 2, and
X
ψ(τk ) = τk ⊗ 1 +
j
ξip ⊗ τj
i+j=k
ψ(ξk ) =
j
ξip
X
⊗ ξj
i+j=k
for p odd.
[[Deduce these formulas from the coaction on H∗ (K(Fp , 1))?]]
For instance,
ψ(ξ1 ) = ξ1 ⊗ 1 + 1 ⊗ ξ1
ψ(ξ2 ) = ξ2 ⊗ 1 + ξ1p ⊗ ξ1 + 1 ⊗ ξ2
for p = 2 and for p odd, and
ψ(τ0 ) = τ0 ⊗ 1 + 1 ⊗ τ0
ψ(τ1 ) = τ1 ⊗ 1 + ξ1p ⊗ τ0 + 1 ⊗ τ1
for p odd. These formulas for the coproduct in A∗ are often more convenient for calculations than the Adem relations for the product in A .
We write ξ¯j = χ(ξj ) and τ̄j = χ(τj ) for the images of the Milnor generators under the
conjugation. These are recursively determined by
X pj
ξi χ(ξj ) = 0
i+j=k
for any p and k ≥ 1, and
τk +
j
X
ξip χ(τj ) = 0
i+j=k
for p odd and k ≥ 0. Hence ξ¯1 = −ξ1 , ξ¯2 = −ξ2 + ξ1p+1 , τ̄0 = −τ0 and τ̄1 = −τ1 + τ0 ξ1p .
We shall write ξ1 in place of ξ¯1 for p = 2.
Corollary 6.2. We have algebra isomorphisms
A∗ = P (ξ¯k | k ≥ 1)
for p = 2 and
A∗ = E(τ̄k | k ≥ 1) ⊗ P (ξ¯k | k ≥ 1)
for p odd, and the coproducts satisfy
X
i
ξ¯i ⊗ ξ¯j2
ψ(ξ¯k ) =
i+j=k
for p = 2 and
ψ(τ̄k ) = 1 ⊗ τ̄k +
X
i+j=k
ψ(ξ¯k ) =
X
i
ξ¯i ⊗ ξ¯jp
i+j=k
for p odd.
8
i
τ̄i ⊗ ξ¯jp
Proof. These formulas follow from the fact that χ : A → A is an anti-homomorphism,
meaning that χ(xy) = χ(y)χ(x) for all x, y ∈ A. In terms of homomorphisms, this
asserts that χφ = φ(χ ⊗ χ)γ : A ⊗ A → A. Dually, ψχ = γ(χ ⊗ χ)ψ : A∗ → A∗ ⊗ A∗ ,
which gives the claimed formulas.
7. Comparison of bases
In order to be able to translate facts about A∗ back to A , we need to understand the
perfect pairing A ⊗ A∗ → Fp in terms of the given bases. This is the special case X = H
of the Kronecker pairing h , i : H ∗ (X) ⊗ H∗ (X) → Fp . It turns out that Sq i is dual to ξ1i
in the basis for A that is dual to the monomial basis
{ξ1e1 ξ2e2 · · · ξrer | e1 , . . . , er ≥ 0}
for A∗ . This basis for A is known as the Milnor basis, and is different from the basis
{Sq I } consisting of the admissible monomials.
Lemma 7.1. For p = 2 let j ≥ 1 and i = 2j − 1. Then
(
1 if j = 1,
hSq i , ξj i =
0 otherwise.
More generally, let e1 , e2 , . . . , er ≥ 0 and i = e1 + (22 − 1)e2 + · · · + (2r − 1)er . Then
(
1 if e2 = · · · = er = 0,
hSq i , ξ1e1 ξ2e2 · · · ξrer i =
0 otherwise.
Proof. hSq i , ξj i = hSq i , x∗ (α2j )i = hx∗ (Sq i ), α2j i = hSq i (x), α2j i. Here Sq 1 (x) = x2 and
Sq i (x) = 0 for i ≥ 2, so hSq 1 , ξ1 i = hx2 , α2 i = 1, while hSq i , ξj i = h0, α2j i = 0 for j ≥ 2.
It follows that
hSq i , ξje i = hSq i , Φ(ξj ⊗ · · · ⊗ ξj )i = hΨ(Sq i ), ξj ⊗ · · · ⊗ ξj i
X
X
=
hSq i1 ⊗ · · · ⊗ Sq ie , ξj ⊗ · · · ⊗ ξj i =
i1 +···+ie =i
hSq i1 , ξj i · · · hSq ie , ξj i ,
i1 +···+ie =i
where Φ and Ψ denote the e-fold products and coproducts, respectively. Here hSq i1 , ξj i · · · hSq ie , ξj i
equals 1 if i1 = · · · = ie = 1 and j = 1, and is 0 otherwise. Hence hSq i , ξje i is 1 if i = e
and j = 1, or if e = 0, and is 0 otherwise. In the same fashion,
hSq i , ξ1e1 · · · ξrer i = hSq i , Φ(ξ1e1 ⊗ · · · ⊗ ξrer )i = hΨ(Sq i ), ξ1e1 ⊗ · · · ⊗ ξrer i
X
X
hSq i1 , ξ1e1 i · · · hSq ir , ξrer i ,
=
hSq i1 ⊗ · · · ⊗ Sq ir , ξ1e1 ⊗ · · · ⊗ ξrer i =
i1 +···+ir =i
i1 +···+ir =i
where Φ and Ψ now denote r-fold products and coproducts, respectively. Here hSq i1 , ξ1e1 i · · · hSq ir , ξrer i
equals 1 if i1 = e1 and e2 = · · · = er = 0, and is 0 otherwise. Hence hSq i , ξ1e1 · · · ξrer i
equals 1 if i = e1 and e2 = · · · = er = 0, and is 0 otherwise.
8. Finite quotient Hopf algebras
The finite sub Hopf algebras A(n) ⊂ A have dual finite quotient Hopf algebras A(n)∗ =
A∗ /I(n)∗ .
9
Definition 8.1. For p = 2 and n ≥ −1 let
I(n)∗ = (ξ12
n+1
2
n
2
, ξk | k ≥ n + 2) ⊂ A∗
, ξ22 , . . . , ξn2 , ξn+1
e
be the ideal generated by the listed elements ξk2 with k ≥ 1, e ≥ 0 and k + e ≥ n + 2,
and let
A(n)∗ = A∗ /I(n)∗ = P2n+1 (ξ1 ) ⊗ P2n (ξ2 ) ⊗ · · · ⊗ E(ξn+1 )
be the quotient algebra. [[Also discuss p odd.]]
Here Ph denotes the truncated polynomial algebra of height h, with P2 = E. Note that
n+2
A(n)∗ is a finite algebra of dimension 2n+1 · 2n · . . . · 2 = 2( 2 ) . For instance, A(−1)∗ = F2 ,
A(0)∗ = E(ξ1 ) and A(1)∗ = P4 (ξ1 ) ⊗ E(ξ2 ). Since I(n)∗ ⊂ I(n−1)∗ we get an infinite
sequence of surjective algebra homomorphisms
A∗ → · · · → A(n)∗ → A(n−1)∗ → · · · → A(0)∗ → F2 .
Lemma 8.2. I(n)∗ is a Hopf ideal in A∗ .
This means that the coproduct ψ : A∗ → A∗ ⊗ A∗ maps I(n)∗ into the kernel I(n)∗ ⊗
A∗ + A∗ ⊗ I(n)∗ of A∗ ⊗ A∗ → A(n)∗ ⊗ A(n)∗ , hence induces a coproduct
ψ : A(n)∗ −→ A(n)∗ ⊗ A(n)∗
making the diagram of horizontal extensions
/ A∗
I(n)∗
/ A(n)∗
ψ
I(n)∗ ⊗ A∗ + A∗ ⊗ I(n)∗
/ A∗ ⊗ A∗
ψ
/ A(n)∗ ⊗ A(n)∗
commute.
Proof. It suffices to observe that
e
e
ψ(ξk2 ) = ψ(ξk )2 =
X
j
e
(ξi2 ⊗ ξj )2 =
i+j=k
X
j+e
ξi2
e
⊗ ξj2
i+j=k
j+e
maps to zero in A(n)∗ ⊗ A(n)∗ whenever k ≥ 1, e ≥ 0 and k + e ≥ n + 2. In fact ξi2
e
I(n)∗ under these conditions, unless i = 0, in which case j = k and ξj2 ∈ I(n)∗ .
∈
It follows that A(n)∗ is itself a connected, commutative Hopf algebra, and the infinite
sequence above consists of Hopf algebra homomorphisms. This also implies that I(n)∗
is closed under the conjugation, hence can equally well be generated by the conjugate
e
classes ξ¯k2 for k ≥ 1, e ≥ 0 and k + e ≥ k + 2.
Proposition 8.3. The quotient Hopf algebra A(n)∗ of A∗ is dual to the sub Hopf algebra
A(n) of A .
Proof. The dual of A(n)∗ consists of the classes in A that annihilate I(n)∗ under the
Kronecker pairing with A∗ . For 1 ≤ i < 2n+1 we have
e
hSq i , ξs2 i = 0
whenever s ≥ 1, e ≥ 0 and s + e ≥ n + 2 (which for s = 1 implies e ≥ n + 1), so these
Sq i annihilate the ideal generators of I(n)∗ . A general class in I(n)∗ is a sum of terms
e
α · ξs2 , with α ∈ A∗ , and
X
e
e
hSq k , α · ξs2 i =
hSq i , αihSq j , ξs2 i .
i+j=k
10
For 1 ≤ k < 2n+1 we have 0 ≤ j < 2n+1 , so the right hand factor vanishes unless j = 0
e
and i = k, leaving hSq k , αihSq 0 , ξs2 i, which is zero for degree reasons. Hence the algebra
generators of A(n) all lie in the dual of A(n)∗ , proving that A(n) is contained in that
dual.
j
It remains to prove that the images of the Sq 2 for 0 ≤ j ≤ n suffice to span the algebra
indecomposables of the dual of A(n)∗ . By duality, this is equivalent to asking that the
j
coalgebra primitives P (A(n)∗ ) map injectively to the span of ξ12 for 0 ≤ j ≤ n.
[[Refer to Milnor (1958) for the opposite inclusion?]]
2
[[The quotient Hopf algebra E(n)∗ = A∗ /(ξ12 , . . . , ξn+1
, ξk | k ≥ n+2) = E(ξ1 , . . . , ξn+1 )
of A∗ is dual to the sub Hopf algebra E(n) of A .]]
Let k be a field. We sometimes use the notation A//B = A ⊗B k for the tensor product
over an augmented subalgebra B of a k-algebra A. It is the coequalizer in left A-modules
/
/ A//B
A⊗B
/A
of the two homomorphisms A ⊗ B → A taking a ⊗ b to ab and a(b), respectively,
where : B → k is the augmentation. As such, it equals the quotient A/AI(B), where
I(B) = ker() is the augmentation ideal of B, and AI(B) is the left ideal in A generated
by I(B). If B is normal in A, in the sense that AI(B) = I(B)A, then this quotient is a
quotient algebra of A, but this does not hold in general.
If A is a Hopf algebra and B a sub Hopf algebra, then for a ∈ A, b ∈ I(B) we have
ψ(ab) = ψ(a)ψ(b) and ψ(b) ∈ B ⊗ I(B) + I(B) ⊗ B, so ψ(ab) ∈ A ⊗ AI(B) + AI(B) ⊗ A.
Hence AI(B) is a Hopf ideal in A, and A//B inherits a coalgebra structure from A,
making the quotient map A → A//B a unital coalgebra homomorphism.
[[More generally so for A a left or right B-module coalgebra.]]
The following theorem is due to Milnor–Moore (1965, Theorem 4.4). It implies that a
Hopf algebra is free as a (left or right) module over any sub Hopf algebra, as long as they
are both connected.
Theorem 8.4. If B is a connected Hopf algebra and A is a connected left B-module
coalgebra such that i : B → A is injective, then
A∼
= B ⊗ (k ⊗B A)
as left B-modules and right k ⊗B A-comodules. If A is instead a connected right B-module
coalgebra, then
A∼
= (A ⊗B k) ⊗ B
as right B-modules and left A ⊗B k-comodules.
Dually, we can consider the cotensor product A∗ B∗ k over a unital quotient coalgebra
B∗ of a k-coalgebra A∗ . It is the equalizer in left A∗ -comodules
/
/ A∗
A∗ B∗ k
/ A∗ ⊗ B∗
of the two homomorphisms A∗ → A∗ ⊗ B∗ given by the coproduct on A∗ followed by
the projection to B∗ in the right hand factor, and A∗ tensored with the unit of B∗ ,
respectively. If A∗ → B∗ is dual to B ⊂ A, the cotensor product is dual to the tensor
product A ⊗B k = A/AI(B), hence is the left A∗ -subcomodule of A∗ that annihilates
AI(B) under the pairing with A. It is not in general a subcoalgebra of A∗ . If A∗ → B∗ is
a surjection of Hopf algebras, then the two homomorphisms are algebra homomorphisms,
so the equalizer is a subalgebra of A∗ , making the inclusion A∗ B∗ k → A∗ an (augmented)
algebra homomorphism.
[[More generally so for A∗ a left or right B∗ -comodule algebra.]]
11
Here is the dual Milnor–Moore theorem (1965, Theorem 4.7).
Theorem 8.5. If B∗ is a connected Hopf algebra and A∗ is a connected left B∗ -comodule
algebra such that j : A∗ → B∗ is surjective, then
A∗ ∼
= B∗ ⊗ (k B∗ A∗ )
as left B∗ -comodules and right k B∗ A∗ -modules. If A∗ is instead a connected right
B∗ -comodule algebra, then
A∗ ∼
= (A∗ B∗ k) ⊗ B∗
as right B∗ -comodules and left A∗ B∗ k-modules.
n
n−1
Proposition 8.6. The subalgebra E(ξ12 , ξ¯22 , . . . , ξ¯n+1 ) = A(n)∗ A(n−1)∗ F2 of A(n)∗ is
dual to the quotient coalgebra A(n)//A(n−1) = A(n) ⊗A(n−1) F2 of A(n).
Proof. We know that A(n) is a free right A(n−1)-module by the Milnor–Moore theorem,
n+1
n+2
so A(n)⊗A(n−1) F2 has dimension 2( 2 ) /2( 2 ) = 2n+1 , hence by duality A(n)∗ A(n−1)∗ F2
also has this dimension. It is the equalizer of two algebra homomorphisms
/
A(n)∗
/ A(n)∗ ⊗ A(n−1)∗ ,
hence is a subalgebra of A(n)∗ . It contains the n + 1 elements ξ12 , ξ¯22
X e
i+e
e
ξ¯i2 ⊗ ξ¯j2 ,
ψ(ξ¯k2 ) =
n
n−1
, . . . , ξ¯n+1 , because
i+j=k
i+e
and ξ¯j2 maps to zero in A(n−1)∗ for i + j = k ≥ 1, e ≥ 0 and k + e ≥ n + 1, unless
e
e
j = 0. Thus the image of ξ¯k2 in A(n)∗ ⊗ A(n−1)∗ is ξ¯k2 ⊗ 1 under both homomorphisms.
n
n−1
These n + 1 elements generate the exterior algebra E(ξ12 , ξ¯22 , . . . , ξ¯n+1 ) inside A(n)∗ , of
dimension 2n+1 . Hence, by a dimension count, this is the whole of A(n)∗ A(n−1)∗ F2 . 9. Some bicomodule algebras and bimodule coalgebras
Definition 9.1. For n ≥ 0 let
n
2
2
J(n)∗ = (ξ22 , . . . , ξn2 , ξn+1
, ξk | k ≥ n + 2) ⊂ A∗
e
be the ideal generated by the elements ξk2 with k ≥ 2, e ≥ 0 and k + e ≥ n + 2, and let
C(n)∗ = A∗ /J(n)∗ = P (ξ1 ) ⊗ P2n (ξ2 ) ⊗ · · · ⊗ E(ξn+1 )
be the quotient algebra. Let C(n) ⊂ A be the dual sub coalgebra.
For instance, C(0)∗ = P (ξ1 ) is dual to C(0) = F2 {Sq i | i ≥ 0}, and C(1)∗ = P (ξ1 ) ⊗
E(ξ2 ).
The ideal J(n)∗ is not a Hopf ideal, so C(n)∗ is not a Hopf algebra. However, it admits
interesting left and right coactions.
Lemma 9.2. J(n)∗ is an A(n)∗ -A(n−1)∗ bicomodule ideal in A∗ .
This means that the left A(n)∗ -coaction A∗ → A∗ ⊗ A∗ → A(n)∗ ⊗ A∗ maps J(n)∗ into
the kernel A(n)∗ ⊗ J(n)∗ of A(n)∗ ⊗ A∗ → A(n)∗ ⊗ C(n)∗ , and that the right A(n−1)∗ coaction A∗ → A∗ ⊗ A∗ → A∗ ⊗ A(n−1)∗ maps J(n)∗ into the kernel J(n)∗ ⊗ A(n−1)∗
of A∗ ⊗ A(n−1)∗ → C(n)∗ ⊗ A(n−1)∗ .
12
e
Proof. The left and right coactions take ξk2 with k ≥ 2, e ≥ 0 and k + e ≥ n + 2 to the
images of
X j+e
e
e
ψ(ξk2 ) =
ξi2 ⊗ ξj2
i+j=k
in A(n)∗ ⊗ A∗ and A∗ ⊗ A(n−1)∗ , respectively. Regarding the left A(n)∗ -coaction, ξi2
e
e
with i + j = k maps to zero in A(n)∗ unless i = 0, so ψ(ξk2 ) maps to 1 ⊗ ξk2 , which lies
j+e
in the ideal A(n)∗ ⊗ J(n)∗ . Regarding the right A(n−1)∗ -coaction, ξi2 with i + j = k
e
lies in J(n)∗ unless i = 0 or i = 1, so ψ(ξk2 ) maps to
j+e
e
k−1+e
1 ⊗ ξk2 + ξ12
e
e
2
⊗ ξk−1
,
e
2
and ξk2 and ξk−1
both map to zero in A(n−1)∗ . Both coactions are algebra maps, so this
implies the claim.
It follows that these coactions induce a left A(n)∗ -coaction C(n)∗ → A(n)∗ ⊗ C(n)∗
and a right A(n−1)∗ -coaction C(n)∗ → C(n)∗ ⊗ A(n−1)∗ , and that these two coactions
commute. In particular the diagram
A∗
/ A(n)∗ ⊗ A∗ ⊗ A(n−1)∗
/ A∗ ⊗ A∗ ⊗ A∗
Ψ
C(n)∗
/ A(n)∗ ⊗ C(n)∗ ⊗ A(n−1)∗
commutes.
Lemma 9.3. The surjection A∗ → C(n)∗ is a homomorphism of A(n)∗ -A(n−1)∗ bicomodule algebras. Dually, the inclusion C(n) ⊂ A is a homomorphism of A(n)-A(n−1)
bimodule coalgebras.
Definition 9.4. We define
B(n)∗ = C(n)∗ [ξ1−1 ] = colim Σ−k C(n)∗ ,
k
−k
and let B(n) = limk Σ C(n) be the dual of B(n)∗ .
Lemma 9.5. Multiplication by ξ12
module homomorphism.
n+1
n+1
: Σ2
C(n)∗ → C(n)∗ is an A(n)∗ -A(n−1)∗ bico-
n+1
n+1
Proof. The left A(n)∗ -coaction takes ξ12
to 1 ⊗ ξ12 , while the right A(n−1)∗ -coaction
n+1
n+1
n+1
to ξ12 ⊗ 1, hence multiplication by ξ12
commutes with both coactions. takes ξ12
Lemma 9.6. The short exact sequence
2n+1
0→Σ
ξ12
n+1
·
C(n)∗ −→ C(n)∗ −→ A(n)∗ → 0
and the injection
n+1
C(n)∗ −→ B(n)∗ = colim Σ−j·2
j
C(n)∗
both consist of A(n)-A(n−1) bicomodules and A(n)-A(n−1) bicomodule homomorphisms.
Dually, the short exact sequence
n+1
0 → A(n) −→ C(n) −→ Σ2
C(n) → 0
and the surjection
n+1
B(n) = lim Σ−j·2
j
C(n) −→ C(n)
both consist of A(n)-A(n−1) bimodules and A(n)-A(n−1) bimodule homomorphisms.
13
n
n
Proof. Note that I(n)∗ is the ideal generated by J(n)∗ and ξ12 , and multiplication by ξ12
acts injectively on C(n)∗ .
10. The A(n)-module structure of ΣP (x±1 ) [[and ΣE(x) ⊗ P (y ±1 )]]
Consider the A(1)-module structure on ΣP (x±1 ), where A(1) ⊂ A is generated by Sq 1
and Sq 2 .
Sq 2
...
Σx
−5
%
/ Σx−4
−4
Σx
;
/ Σx−2
−3
−2
Σx
−1
$
/ Σ1
Σx
;
0
Sq 1
/ Σx2
Σx
2
3
%
/ Σx4
4
It is 4-periodic, and is obtained by non-trivial extensions from infinitely many copies of
the cyclic A(1)-module A(1)//A(0) = F2 {1, Sq 2 , Sq 1 Sq 2 , Sq 2 Sq 1 Sq 2 }, suspended by all
integer multiples of four. The quotient by the submodule A(1)F<0 generated by classes
in negative degree appears as follows
Σx
−1
0
Σx
;
/ Σx2
2
Σx
3
%
/ Σx4
Σx
;
4
5
/ Σx6
6
Σx
7
%
/ Σx8
.: . . ,
8
and there is a short exact sequence of A(1)-modules
ΣP (x±1 )
ΣP (x±1 )
−→ Σ4
→ 0.
A(1)F<0
A(1)F<0
These patterns generalize to all larger n.
0 → A(1)//A(0) −→
Definition 10.1. Let F<k = F2 {Σxi | i + 1 < k} ⊂ ΣP (x±1 ) be the subspace of classes
in degree less than k, and let A(n)F<k ⊂ ΣP (x±1 ) be the left A(n)-submodule generated
by F<k .
We get a tower of surjections
ΣP (x±1 )
ΣP (x±1 )
. . . −→
−→
−→ . . . ,
A(n)F<k
A(n)F<k+1
in the category of A(n)-modules. Since A(n) is bounded above (and below) we get an
isomorphism
ΣP (x±1 )
ΣP (x±1 ) ∼
,
= lim
k A(n)F<k
where in each degree the limit is achieved at a finite stage.
n+1
Multiplication by x2
acts A(n)-linearly on ΣP (x±1 ), hence induces isomorphisms
n+1
Σ2
ΣP (x±1 ) ∼ ΣP (x±1 )
.
=
A(n)F<k
A(n)F<k+2n+1
Restricting k = −j · 2n+1 to integer multiples of 2n+1 , we can write
±1
ΣP (x±1 ) ∼
)
n+1 ΣP (x
ΣP (x±1 ) ∼
.
= lim
= lim Σ−j·2
j A(n)F<−j·2n+1
j
A(n)F<0
14
.< . .
Proposition 10.2. There is an A(n)-module [[coalgebra?]] isomorphism
∼
=
C(n) ⊗A(n−1) F2 −→
ΣP (x±1 )
A(n)F<0
mapping the class of c ⊗ 1 to the class of c(Σx−1 ).
Proof. The displayed map is well defined because
j
n
j −1−2n
Sq 2 (Σx−1 ) = Sq 2 (Σx2
) ∈ A(n)F<0
j
for 0 < j < n, and these Sq 2 generate A(n−1).
By the Milnor–Moore theorem C(n) is free as a right A(n−1)-module, and c(Σx−1 )
lies in A(n)F<2n+1 when c ∈ A(n), so there is a map of short exact sequences
0
/ A(n) ⊗A(n−1) F2
/ C(n) ⊗A(n−1) F2
/ Σ2n+1 C(n) ⊗A(n−1) F2
/0
0
/ A(n)F<2n+1 /A(n)F<0
/ ΣP (x±1 )/A(n)F<0
/ ΣP (x±1 )/A(n)F<2n+1
/0
of left A(n)-modules, where the right hand vertical map is isomorphic to the 2n+1 -th
suspension of the middle vertical map. The left hand vertical map is surjective because
Sq i (Σx−1 ) = Σxi−1
for 0 ≤ i < 2n+1 generate the target as an A(n)-module, and these Sq i lie in A(n). That
target contains precisely one generator in each congruence class of degrees modulo 2n+1 ,
so it has the same dimension as A(n) ⊗A(n−1) F2 . Hence the left hand vertical map is
an isomorphism, and it follows by induction that the middle vertical map is also an
isomorphism.
Theorem 10.3. There is an A(n)-module isomorphism
∼
=
B(n) ⊗A(n−1) F2 −→ ΣP (x±1 ) .
Proof. The canonical homomorphism
n+1
n+1
B(n) ⊗A(n−1) F2 ∼
= lim Σ−j·2 C(n) ⊗A(n−1) F2 −→ lim Σ−j·2 C(n) ⊗A(n−1) F2
j
j
ΣP (x±1 ) ∼
) ∼
n+1 ΣP (x
∼
= lim
= ΣP (x±1 )
= lim Σ−j·2
j
j
A(n)F<0
A(n)F<−j·2n+1
±1
is an isomorphism, because in each degree both limits are achieved for all sufficiently
n+1
large j, so Σ−j·2 C(n) ⊗A(n−1) F2 maps isomorphically and compatibly to both sides, in
a range of degrees that grows to cover all degrees as j increases to ∞.
To proceed from here, Davis and Mahowald [?LDMA80, Lemma 1.3] obtained the following splitting, after base change along A(n) ⊂ A .
Lemma 10.4. There is an A -module isomorphism
ΣP (x±1 ) ∼ M j·2n+1
A ⊗A(n)
Σ
(A ⊗A(n−1) F2 ) .
=
A(n)F<0
j≥0
Proof. The composition
µ : A ⊗A(n) C(n) ⊂ A ⊗A(n) A → A
15
induces a splitting µ ⊗A(n−1) 1 of the short exact sequence
n+1
0 → A ⊗A(n−1) F2 −→ A ⊗A(n) C(n) ⊗A(n−1) F2 −→ A ⊗A(n) Σ2
C(n) ⊗A(n−1) F2 → 0
of left A -modules. Hence there is an A -module splitting of the short exact sequence
n+1
0 → A ⊗A(n−1) F2 −→ A ⊗A(n) ΣP (x)/A(n)F<0 −→ A ⊗A(n) Σ2
(ΣP (x)/A(n)F<0 ) → 0 .
Iterating, these combine to define the asserted isomorphism.
Corollary 10.5.
A(n)
Tors,t (F2 ,
ΣP (x±1 ) ∼ M
n+1
A(n−1)
)=
Tors,t
(F2 , Σj·2 F2 ) .
A(n)F<0
j≥0
Proof. Apply TorA
s,t (F2 , −) and change-of-rings.
Corollary 10.6.
A(n)
Tors,t (F2 , ΣP (x±1 )) ∼
=
M
A(n−1)
Tors,t
(F2 , Σj·2
n+1
F2 ) .
j∈Z
Proof. Pass to the (achieved) limit over the desuspensions ΣP (x±1 )/A(n)F<k for (negative) multiples k of 2n+1 .
Corollary 10.7.
A∗
±1
∼
TorA
s,t (F2 , ΣP (x )) = Tors,t (F2 , F2 ) .
Proof. Pass to the colimit over n, checking that only the summand with j = 0 survives.
This approach requires some careful control of the splitting maps and their behavior
under passage from A(n) to A(n+1). We shall instead give the details in the more
conceptual argument of [?AGM85].
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.
[Ade52] José Adem, The iteration of the Steenrod squares in algebraic topology, Proc. Nat. Acad. Sci. U.
S. A. 38 (1952), 720–726.
[Car54] Henri Cartan, Sur les groupes d’Eilenberg-Mac Lane. II, Proc. Nat. Acad. Sci. U. S. A. 40 (1954),
704–707 (French).
[LDMA80] W. H. Lin, D. M. Davis, M. E. Mahowald, and J. F. Adams, Calculation of Lin’s Ext groups,
Math. Proc. Cambridge Philos. Soc. 87 (1980), no. 3, 459–469.
[Mil58] John Milnor, The Steenrod algebra and its dual, Ann. of Math. (2) 67 (1958), 150–171.
[MM65] John W. Milnor and John C. Moore, On the structure of Hopf algebras, Ann. of Math. (2) 81
(1965), 211–264.
[Ser53] Jean-Pierre Serre, Cohomologie modulo 2 des complexes d’Eilenberg-MacLane, Comment. Math.
Helv. 27 (1953), 198–232 (French).
[Ste47] N. E. Steenrod, Products of cocycles and extensions of mappings, Ann. of Math. (2) 48 (1947),
290–320.
, Homology groups of symmetric groups and reduced power operations, Proc. Nat. Acad.
[Ste53]
Sci. U. S. A. 39 (1953), 213–217.
Department of Mathematics, University of Oslo, Norway
E-mail address: rognes@math.uio.no
URL: http://folk.uio.no/rognes
16