UEHARA, HIROSHI

Osaka City University Repository

Uehara, H.
Osaka J. Math
9 (1972), 131-141
ALGEBRAIC STEENROD OPERATIONS IN THE
SPECTRAL SEQUENCE ASSOCIATED W I T H
A PAIR OF HOPF ALGEBRAS
HIROSHI UEHARA*
(Received July 10, 1971)
Araki [3], [4] and Vazquez [10] investigated behaviors of Steenrod reduced
powers in the spectral sequence associated with a fibre space in the sense of
Serre. The main purpose of this paper is to establish an algebraic analogy to
their works. For example, works of Adams [1], [2] and others, [6], [11], [12],
implicitly contain a useful, direct application of our results.
1. Steenrod operations in the spectral sequence associated with
an algebraic system @
DEFINITION 1. By a graded differential algebra @={C, δ, F, U} with a
decreasing filtration F and with cup-i-products U, we mean
1) a graded cochain complex C over the field Z2:
δ*
C: CQ-+C1-»—>Cn >Cn+1->. , where δn : Cn->Cn+1
is a morphism of graded vector spaces over Z2,
2) for each integer p, FPC is a subcomplex of C such that
i) FP+1C is a subcomplex of FPC (in notation: FpCθ>Fp+1C)
ϋ) FPC=C if p<0y and iii) FpCn=0 if p>n,
3) for each integer i there exists a Z2-linear map U : C®C->C such that if
x^FpCm's and jyei^C 1 ' ' , then ΛiU^eί1 ΛC7 W + w- ί s + ί for a=Max{p+q-i,
i
p> q], where x \Jy= U(#®.y), xUy=xUy in notations, and syt stand for
i » 0
gradings. U satisfies the following conditions:
i) U is trivial if ί<0, ii) For x^FqCm and y^FqCn, x U y=0 if i>m
i i
or ny
* This work is supported in part by the National Science Foundation research grant GP-
9585, and in part by the Deutsche Forschungsgemeinschaft while the author was a visiting
professor at Universitat des Saarlandes during the winter semester, 1970.
132 H. UEHARA
iii) xU(yUz)=(xUy)Uz, iv), l U * = # U l = * for some 1 G C M , and
v) δ(*U y)=xΌ y+y\Jx+8x\J y+x[Jδy.
i i - l i - l i i
Associating @ with an exact couple <Z>, E, i> j , k} by defining
Dp'q=Hp+g(FpC), Ep1'q=Hp+g(FpCIFp+1C), and ί, /, β as usual, we have a
spectral sequence {Ey, rfγ|7>l}. Let us define Steenrod operations in the
spectral sequence as Araki [4] and Vazquez [10] did. Define a map 0, : C^>C
by θi(x)=x[Jx+xU Sx, then we have
i ί + 1
Proposition 1. 0t- induces Steenrod operations BStiy FSt{ in the spectral
sequence associated with the algebraic system © such that
BSt{: Er-*E%rJϊ
2
* for co >
Ύ
> 2 ,
and
FStii E*'
Q
-+Epy'
2Q+p
-
i
 for O O > 7 > 1 .
They are all Z2-homomorphisms.
Proof. It is straightforward by definition that if we denote, as usual,
Zpy'
q
={xeίFpCp+q\8x<=ΞFp+yCp+g+1},B»'q={x(ΞFpCp+9\3y^
ZZq={xeiFpCp+«\8x=0}, and BZq={xGFpCp+*\syt=Cp+g-\ 8y=x}y then
θtW) czz22y--{'2Q n Zy 2***-* (zzftzϊ2* n z? 2 +*-4, Θ^B^CZBI^2' n B?L2J+*-',
θilzzηczZ!*-*-2* fΊZ%2q+p-\ and θiiBzηaBi*-*'2* Π B*-2*^-'. Note that the
restriction on γ > 2 comes from the following observation. If #e-B?'Λ, then
0,(*) = Sy U δ j = 8(y U x+y\Jy),
where δj;=^ withy^F p ^ + 1 C p + g -\ Since y
if 7 > 2 , ^, (Λ;)eβ|?I^29 for γ > 2 . Hence, θ{ induces BSti and FSίt. as stated
in Proposition 1. For xiy ^ G Z ?
1
'
θi(x1+x2) = θi(x1)+θi(x2)+S(x1\Jx2)+x2 U 8x1+8x1Όx2
ί +1 i + l i + l
from the bilinearity of U. Since
SfaUxJtΞB?-*-2* ΠBϊ 2Q+||-'cJBi?zi-2Q
i + l
and
*
a
 u δ^+δ^ u x
i+l i+l
BStj and FSt{ are Z2-homomorρhisms.
For completeness sake let us show some properties of Steenrod operations
ALGEBRAIC STEENROD OPERATIONS 133
which are useful for their applications. (For example, for computation of
cohomology of the Steenrod algebra.) Let Ey * be the subvector space of E"'b
spanned by (d
s> •••, rfv+1, rfy)-cocycles and let κys+1: E^-^Ef^ be the natural
epimorphism. An element in E%\1 will be said to be ^-transgressive.
Proposition 2.
 FSί,.: E^-^E*-
2
***-* is trivial if i<p or i>p+q,
βStii Epyq-^E\^2q is trivial if i>p or i<0,
and
B&tp z== #2y—2 F^tp
Proof. If ρ>i and x^Zvy ", then θi{x)^F2p'iC2p+2g-i(zFp+1C2i'+2q-i and
Siθ^x^SxljSxGF^C2^21-^1. Hence, 0I.(x)eZ?iV2«+i)-i-1) so that by
i + 1
definition the triviality of
 F*Sίt is proved if p>i. The rest of the proof is
immediate, and hence, is omitted.
Proposition 3. If aϊΞE*% then pSt^^E^^-^ where
d = Max{p+2c—i, c}
and nSt^^E2^'2^
Proof. Recall that
If x is a representative of α, then Θ^^F^2^29^ΓίF^^C2^2^ and
8(θi(x))^F2p+2c-i+1C2p+2q-i+1f]F^+c+1C2p+2"-i+1. If i>p, thεnθi(x)£ΞZ%>2rp-i
where d=Mzx{p+2c—i, c}, while if p>iy then (9,(^)eZ|?+i'2Q. Hence, the
proof is completed.
Proposition 4. If a^Eξ'Q isg-transgressίve, then
 FSti(a)<=E%2q+p~i is also
g-transgressive. Moreover we have
( 1 ) *2Q
 BSti+1 dg+1 fc2+1(a) =
and κ^Q=κ
2
λ
q
 if\>2q.
Proof. It is obvious from Proposition 3 that
 FStj(a) is £-transgressive.
If x is a representative of a, then both sides of ( 1 ) is represented by Sx (J δx.
Hence, the proof is completed.
2. Comparison theorem in homological algebra
To prepare for later sections the algebraic Steenrod operations are
introduced by the iterated use of a comparison theorem in relative homological
134 H. UEHARA
algebra [5] (For the theorem in a more general and rigorous setting, see [8]), and
the explicit formulas of chain homotopies [1], [11] involved in the theorem are
presented in this section.
Let a: A-+B be a morphism of graded augmented algebras A and B over
a commutative ring R with unity, and let M and N be left graded modules over
algebras A and B respectively. A morphism of graded /^-modules/: M->N is
called a α-homomorphism iff f(ax)=a(a)f(x) for a^A and x e M .
Proposition 5. Let £: £->M be a R-split exact resolution of M in the
category
 AWl of left A-modules and let η: %)->N be a R-split exact resolution of N
in the category βΉJl. Theny for any a-homomorphism f: M-+N there exists a
a-chain map extension F: ϊ-^2) of fin the sense that
1) for each n> 0, F
n
: X
n
-^Y
n
 is a a-homomorphism, and
2) d
n
F
n
=FH_$n for n>\ and fβ=ηFOy where
! s I s I I s I
/ "V. VJ FJ " \F
ψ V Y d
x
 ψ d
n
 ψ ψ
3) If Fy Ff are a-chain map extensions of'/, then there exists a a-chain homotopy
h: X—>2) connecting F with F\
Proof. First let us observe that the proposition is the usual comparison
theorem in case when A=B and a is the identity map. The following remarks
enable us to reduce the proposition to the classical theorem; 1) any S-module
Z can be considered as an ^4-module by definition az=a(a)z for a^A and
z^Zy 2) any morphism £: Z->Z' in
 BWl can be regarded as a morphism in Λ3Jί
by considering Z, Zr as A -modules because
g(az) = g(a(a)z) = a(a)g(z) = ag(z),
3) a i?-homomorphism k: X-> Y is a α-homomorphism iff k is a morphism in
A^SSl considering Y as an ^4-module. For k(ax)= a(a)k(x)=ak(x). From 1)
and 2), η: 2)—>N can be considered as a i?-split exact complex of N in
 A^ΰiy and
from 3) /: M—>N is a morphism in
 A2JΪ. It follows from the usual comparison
theorem that there exists a chain map extension F of/ in
 ASΰl. From 3) ί1 is a
<2-homomorphism. It is immediate to see the rest of the proof. This proves
the proposition.
Let us apply the proposition to the following case. Let A be a co-
commutative Hopf algebra over Z2 and let a: A->A®A be the cocommutative
ALGEBRAIC STEENROD OPERATIONS 135
comultiplication Δ. Since M=Z2 and N=Z2®Z2^Z2 can be considered by
augmentations as a left ^4-module and a left 4^ ®^t-module respectively, the
α-map/: Z2->Z2®Z2 defined by/(l)=l<g)l, can be extended to a Δ-chain map
h°: ϊ-^3t®X by the direct application of the proposition, where X is a Z2-split
exact resolution of Z2. If p: 96(g)9i-^ 9£®9£ is the twisting chain map, then ρh°
is again a Δ-chain map extension of /, because Δ is cocommutative. Hence,
there exists a Δ-chain homotopy h1 connecting h° with ρh°. Since ph1 is a
Δ-chain homotopy and since hι-\-phλ is a Δ-chain map extension of the trivial
Δ-homomorphism 0: Z2->Jmdiy there exists a Δ-chain homotopy h
2
 connecting
h1 and pk\ By the iterated use of the same arguments we have a sequence of
Δ-chain homotopies {A0, A1, •••, A% •••}. Hence, we have
Proposition 6. L^ A be a cocommutative Hopf algebra over Z2 and let
Δ: A ^A®A be the comultiplication. If£: £->Z2 is a Z2-split exact resolution of
the A-module Z2, then there exists a sequence of A-homomorphisms hι\ &->X®£
for i=0y 1,..., n> ••• such that 1) k° is a grade preserving A-chain map and 2) for
i>0 h* is a A-chain homotopy connecting h*~ι with ph1'1 which raises the homological
dimensions by i and preserves the grading\ where p: ϊ®ϊ->X®X is the twisting
chain map.
Consider a diagram
U
Hom
Λ
(ϊ, Z2)®HomΛ(X, Z2).-' . .>Hom A ( ϊ , Z2)
X \ / * "
Hom^0i4(X®X, Z2)
where X is the Z2-chain map defined by X(f®g)(x®y)=f(x)g(y) for /,
Hom
Λ
(ϊ, Z2) and for x>
DEFINITION 2. The cup-z-product U in the cochain complex C =
i
Hom
Λ
(X, Z2) is defined by h*-X.
Denoting Hom^^^,, Z2) by CptS for each homological dimension p>0 and
the grading s>0, we have the cochain complex
C*s= {C^ίorp^O, 1, ...,n, •••}
such that C={C*s\s = 0, 1, ...}. Then f\Jg= {j(f®g)^Cp+q'i's+t for
M
 and g^Cg>t. It is immediate to see by definition the coboundary
formula
136 H. UEHARA
Kf Vg) = f Vg+S U/+S/ ϋg+f U δg .
i i-1 ί-1 i >
For
= d*h»X(f®g) = X(f®g)h'd
= h<-i'X(f®g)+h<-»X(g®f)+hi*X(δ®l+l®S)(f®g)
= fUg+g Uf+Sf Ug+fl)8g.
i-1 i-1 i i
DEFINITION 3. Algebraic Steenrod operation
 ASq{: H
p
'
s(A)^H2p~it2s{A)
is defined by
 ASq^)=f U/, where f e i P f % 4 ) is represented by f(=Cp's with
δ/=0, and the bar over / (J / stands for the cohomology class.
1
Adams [1] and others (for example, see [11]) computed explicitely a
Δ-homomorphism hι in case when X is the bar resolution B(A). If
A(a) =
for flGi, then we have
for odd ί,
o
 ΣJ^I y l
n
...nnϊnn
for even ί,
K (KI -1 j) g gfj H
i α'P;_1+1-flί; ι βP/+11 - 1 « j .
Let us sketch the method of computation for completeness sake. Let S be
the contracting homotopy for B(A), then t=S® 1 + 5 ® S is a contracting
homotopy for B(A)®B(A). Define Ag = Δ, then A?50 = ί0Ag determine A? by
/*;(##) =Δ(α)Ai(#). Inductively A0 is obtained easily. Define hl = t
o
(h%+pho
o
)y
then h\ is calculated by h\S0=t1{hl+ph\)S0-^-t1hl and /*ί(α*) = Δ(ά)h\(x).
Repeat this process, we get the above formula.
ALGEBRAIC STEENROD OPERATIONS 137
3. @(Γ, Λ) associated with a pair of Hopf algebras (Γ, Λ)
Let (Γ, Λ) be a pair of connected locally finite cocommutative Hopf algebras
over Z2 such that the subhopf algebra Λ is central in Γ. Then we have a
sequence of Hopf algebras
Λ Λ Γ - ^ Ω = Γ/ /(Λ) Γ
where the inclusion i and the projection π are morphisms of Hopf algebras (see
[1]). In this setting we are going to associate with a pair of Hopf algebras
(Γ, Λ) a graded differential algebra @(Γ, Λ)= {C, δ, F, U } with a decreasing
i
filtration F and with cup-z-products U so that behaviors of algebraic Steenrod
operations can be discussed in the spectral sequence {Ey, a\} associated with
Recall the filtration in the bar construction B(Γ) which Adams introduced
in [1]. For each integer p define a subcomplex FpB(T) of B(T) such that
FpB(Γ)n is the Γ-submodule of B(Γ)n=Γ®I(Γ)n generated by elements of the
form γ[γ
x
 | ••• 17«] with the property that γ 5 e/(Λ) for at least (n—p) values of s.
Then it is immediate to see that F is the canonical increasing filtration in B(T).
Define the product filtration F in B(Γ)®B(Γ) by
Fp(B(T)®B(T)) =pU^Fp_sB(T)®FsB(Γ) .
Then (B(T)®B(T)J F) is a resolution of Γ®Γ-module Z2 with the increasing
X
filtration F. Let Δ: Γ-^Γ®Γ be the cocommutative diagonal and let p be the
twisting chain map of B(Γ)® B(Γ). Then we have
Theorem 1. There exists a sequence of A-homomorphisms
for i=0y 1, •••, n, ••• such that 1) h° is a A-chain map which preserves grading and
filtration, 2) hι is a A-chain homotopy connecting A'"1 and ph*'1 which preserves
grading, raises homological dimension by i, and satisfies the filtration condition
hi(FpB(T))czFa(B(T)®B(T))
for a=Min{2ρyρ+i}.
Proof. In virtue of Proposition 6 it remains only to prove that hι
n
 shown
in §2 satisfies the filtration condition. By denoting
Δ(7) = Σ 7'<g>7"
the three formulas hι
n
 ([γ11 ••• 17n]) show that for each j with n>j>\ exactly one
of the three elements γ y , 7j, and y'/ appears solely between bars. For example,
138 H. UEHARA
if i is odd, each of the elements γ ί , — , γP'o, y'p'0 + 19 — , 7P'ί, 7 ί 1 + i — , 7ί2> •••,
7 P ' _ 1 + I , •••, 7P'> ΎP. +U •"•> 7W appears solely in | | . I t follows that if
[Ύil ••* \Ύfϊ\^FpB(Γ)
u
, each t e r m of the s u m on the right hand sides of the
formulas contains at least (n—p) elements in /(Λ). By definition of the product
X
filtration F we obtain
h'(FpB(Γ))czFp+i(B(Γ)®B(Γ)).
If p>i, then the proof is complete, because Min {p+i, 2p}=p+i. If i>p,
it is seen that among / products
(or 7pt _1 +r 7pf if i is even) there exist at least (i—p) products contained in
/(Λ). Otherwise, at least (p+ί) products are not contained in /(Λ). Then y
s
are not in /(Λ) for at least (p+ί) values of s. This is a contradiction. It fol-
lows that each term of the sum for hi ([rγ11 ••• 17W]) has at least
(n-p)+(i-p) = n+i-2p
elements in /(Λ). Therefore,
h'(FpB(Γ))<zF2p(B(Γ)®B(T))
if i>p, where Min {p+i, 2p}=2p. This completes the proof.
Now let us dualize what we have obtained in this section. Let (C, δ) be
the cochain complex Hom
Γ
(i?(Γ), Z2) over Z2. For each integer p define a
subcomplex FP(C) by the image of
Hom
Γ
 {B{T)IFp_β{T), Z2)
under the dual of the projection
Then it is seen that (C, δ, F) is a cochain complex with a decreasing filtration.
Let us call it Adams filtered complex associated with (Γ, Λ).
Theorem 2. Let (C, δ, F) be Adams filtered complex associated with a pair
of Hopf algebras over Z2. Then there exist a Z2-lίnear map U : C®C->C such
i
that @(Γ, Λ)= {C, δ, F, U} is a graded differential algebra with a decreasing
filtration F and with cup-ί-products in the sense of Definition 1.
Proof. Let h£: B(Γ)->B(T)®B(T) be the Δ-homomorphism in Theorem 1
and define (J : C®C->C by h**X as was considered in Definition 2. Since U
ALGEBRAIC STEENROD OPERATIONS 139
is the cup-z-product in C=Hom
Γ
(Z?(Γ), Z2), it is easy to see that U satisfies all
the necessary conditions except the filtration condition. Consequently, it is
sufficient to show that if fζΞFpCm>s and g^FgCn'\ then f \Jg^FΛCm+n~isΛt
i
for a=M.3x{p-{~q—/, p, q}. Consider first the case when
a = Mzx{p+q—i, p, q} = p+q—i ,
then
Min{(α-l)+ί, 2(α-l)} = (a-l)+i = p+q-1
except the case when p=q=i. By Theorem 1
If h\x) = 2 x'®x"
for *eF,. 1β(Γ),,+,.M +,, then Λ/eFjδfΓ),,, and / e F ^ ( Γ ) , , , with the pro-
perty that ξ-\-v=pJrg—1) p+σ^wί+w, and θ~\-v=s-\-t. Then
f
because ξ<p or y<q. Therefore, / U G F ^ " 1 1 ^ " " 1 " ^ . If a=p, then
p>q and i>q. In this case also,
can be shown because
A' (^_ l JB(Γ))cJP ί_1 + ί(β(Γ)®β(Γ))ΠF,p_2(B(T)®B(T)).
Hence, the proof is completed.
From Theorem 2 and Proposition 1 we obtain
Theorem 3. Let (Γ, Λ) be a pair of connected locally finite cocommutative
Hopf algebras over Z2 such that Λ is central in Γ, and let {Ey, dy} be Adams
spectral sequence associated with the system @(Γ, Λ). Then there exist algebraic
Steenrod operations
 BSt{: E
p
y
q
-^E\^z\'2q for o o >
Ύ
> 2 and
 FSti'. E%
2q+p
-'for
4. Some properties of algebraic Steenrod operations
Theorem 4.
 flS/,- and FSt{ defined in Adams spectral sequence satisfy
Propositions 2, 3, and 4.
140 H. UEHARA
Theorem 5. Let (Γ, Λ) and (Γ", Λ') be pairs of Hopf algebras over Z2 both
of which satisfy the conditions stated before, and let Ey and E'Ί be Adams spectral
sequences associated with @(Γ, Λ) and @(Γ", Λ') respectively. Iff: (Γ, Λ)-^(Γ", Λ')
be a morphism of pairs of Hopf algebras, then f induces a sequence of homomorphisms
φ
γ
: Έ
r
Ί
->E
Ί
 for γ > l such that
φy FSti = FStj φy and φ2y_2 BSt{ = BStt φy
for
Ύ
>2.
Proof. It is obvious t h a t / induces a chain map B(f): fi(Γ)->β(Γ/)
preserving filtrations and gradings. If hι and hf* are Δ-homomorphisms in
Theorem 1, then we have h/iB(f)=(B(f)®B(f))hi. Consequently, B(f) induces
a morphism ©(/): @(Γ", Λ')-»@(Γ, Λ). By a morphism Θ(/) of the system
@ we mean that ©(/) is a chain map compatible with gradings, filtrations, and
cup-/-products. Therefore, it is straightforward to verify the theorem.
i π
Theorem 6. Let A -> Γ -> Ω be a sequence of Hopf algebras as stated
before, and let {Ey} be Adams spectral sequence associated with (Γ, Λ). Then the
natural maps B(π): B(Γ)->B(Ω) and B(i): B(A)->B(Γ) induce isomorphisms
B(π)*: Hp(Ω)-»Ei-° and B(i)*: E°2>*->Hg(A) respectively. If Ep2>° and E°2'q are
identified with Hp(Cί) and Hg(A) respectively, then BS^: E^^Eψ-*'0 coincides
with
 Ω
Sqr #*(Ω)-»# 2*-'(Ω), and
 F5ί, : E%-
q
 -> El'2q-1 coincides with
ASq{: H*(A)->H2*-'(A). Moreover, BSt£: E^-^Ei*-*'2* for i<p and
FStj: EZ'-^EZ29**-* for i>p are induced by rSqr. Hp+*(Γ)-+H2p+2*-'(Γ).
Proof. Adams has shown in [1] that 5(zr)* and β(ί)* are isomorphisms.
Hence, a morphism of pairs of Hopf algebras π: (Γ, Λ)->(f2, Z2) induces the
isomorphism φ2: Ei
p
'
0
-^Eξt0 for each/), because
E?-° = E'J'° = Hp(Ω).
Since
 BStg: E^-^Eί
2
*-*-
0
 is exactly ^Sq^ H^Ω^H^'^Ω), we obtain
BSti φ2=φ2 ΩSqi. Similarly, φ2 FSti=ASqj φ2. From the facts that HP+9(Γ) is
filtered by FpHp+g(T)=ZVlBV with the property that EV=FpHp+q{Ω)l
Fp+Ήp+«(Ω) and that
 r
Sq4 maps F
PHP+*(Γ) into F*p-iH*p+2q-*{Γ)<zIPp+**-i{Γ\
it is immediate to see that
 BSti: E^-^E
2?-*'2* is induced by
 Γ
Sq4. The rest of
the proof is obvious. Hence, the proof is complete.
In a subsequent paper the author wishes to discuss higher cohomology
operations involved in the Cartan formula and Massey-Uehara products.
OKLAHOMA STATE UNIVERSITY
ALGEBRAIC STEENROD OPERATIONS 141
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ALGEBRAIC STEENROD OPERATIONS IN THE SPECTRAL SEQUENCE ASSOCIATED WITH A PAIR OF HOPF ALGEBRAS

Uehara, Hiroshi

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