TRANSACTIONS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 00, Number 0, 1997
THE COHOMOLOGY ALGEBRA OF A
SUBALGEBRA OF THE STEENROD ALGEBRA
GREGORY D. HENDERSON
Abstract. We compute the cohomology algebra of P (1), the subalgebra of
the Steenrod algebra generated by P 1and P p. This completes a partial *
*result
given by Arunas Liulevicius in 1962 and provides explicit representativ*
*es in
the cobar construction for all but one of the algebra generators.
Introduction.
In 1962, Arunas Liulevicius[4] used a twisted tensor product of two minimal
resolutions to compute the cohomology of a certain Hopf algebra with two algebra
generators. Up to a regrading, this Hopf algebra is the sub-Hopf algebra of the
mod-p Steenrod algebra generated by P 1and P pand is usually denoted P (1). Its
cohomology is of interest in relation to the cohomology of the Steenrod algebra,
which is the E2 term of the Adams spectral sequence. Liulevicius' results are n*
*ot
complete, however, since he only gives algebra generators and relations in the *
*case
p = 2. For p = 3, he gives algebra generators and an incomplete list of relati*
*ons,
and for p > 3 he only gives module generators over a free subalgebra [4, theore*
*ms
2, 3, and 4].
In this paper we use a different approach - the spectral sequence associated*
* to
an extension of Hopf algebras and explicit computation in the cobar constructio*
*n -
to give algebra generators and a complete list of relations in all cases. If th*
*e obvious
changes are made to bring the statement of Liulevicius's theorem 2 into agreeme*
*nt
with its proof (drop the h02s's and add h1), then our results are consistent wi*
*th
his, except for two signs in the p = 3 case. Our notation for the generators ha*
*s been
chosen to match that of [4] as much as possible, in order to make a comparison *
*as
straightforward as possible.
x1 Definitions and Background.
Fix p a prime and let Fp denote the field with p elements. We wish to compute
the cohomology of the following Fp-Hopf algebra :
Definition 1.1 ([3,Theorem 3.9] or [4,x2]). For q a positive integer which is *
*even
if p > 2, W will denote the unique noncommutative nonprimitively generated co-
commutative connected Fp-Hopf algebra of dimension p3 having algebra generators
in degrees q and pq. If we denote the algebra generator in degree q by y, the a*
*lgebra
generator in degree pq by z, and their commutator [y; z] = yz -zy in degree (p+*
*1)q
by x, then the set {xiyjzk | 0 i; j; k < p} is a vector space basis for W , an*
*d the
______________
1991 Mathematics Subject Classification. Primary 57T05 ; Secondary 57T30, 16*
*W30.
cO1997 American Mathematica*
*l Society
1
2 GREGORY D. HENDERSON
following relations define the Hopf algebra structure :
[x; y]= 0; [x; z]= 0; [y; z]= x;
xp = 0; yp = 0; zp = xp-1 y;
(x) = 0; (y) = 0; (z) = X __1__yi yj:
i+j=p i ! j !
i;j>0
If q = 1 for p = 2 or if q = 2p - 2 for p > 2, then y ! P 1and z ! P pgives an
isomorphism between W and the sub-Hopf algebra of the mod-p Steenrod algebra
generated by P 1and P p.
We will use the spectral sequence associated to an extension of Hopf algebras
as described by Adams [1,Theorem 2.3.1] to compute the cohomology of W . It is
relatively straightforward to find the E1 term of the spectral sequence - the *
*work
lies in solving the extension problem. Our method is to use explicit computati*
*on
with the cobar construction, which is something that most people avoid, but in
this case is fairly manageable. Before proceeding, then, we need to briefly re*
*call
the important facts about the cobar construction. The papers of J.F. Adams [1,
pp 33-42] and J.P. May [7, x11] and the book by J. McCleary [6, x9.2] are useful
references for this material.
The Cobar Construction.
For a graded connected R-coalgebra C, we define IC to be the elements of
positive degree (more generally the kernel of the counit ffl : C ! R). The co-
bar construction on C, Cobar (C), is the tensor algebra generated by IC with
a differential induced by the reduced coproduct on IC, : IC ! IC IC by
(c) = (c) - c 1 - 1 c. The elements are written as sums of terms of the
form [c1 | . .|.cn]; and given an external degree counting the number of slots *
*(n in
this example) and an internal degree which is the sum of the degrees of the ci *
*in
C. This gives a graded R-algebra under the total degree (the sum of the internal
and external degrees), with the product usually called the cup, or juxtapositio*
*n,
product. The differential on the cobar construction is the reduced coproduct on*
* IC
extended to give a differential graded algebra over R and has degree 1 - an exp*
*licit
formula can be found in Adams [1, p. 33].
The homology of Cobar (C) is the bigraded module H *;*(C), which is a graded
R-algebra under the total degree. If A is a graded R-algebra of finite type (fi*
*nitely
generated in each degree), then H Cobar (A*) is the cohomology of A, defined as
H *;*(A) = Ext*;*A(R; R).
[1 products.
The product in the cobar construction is induced by juxtaposition :
[a1 | . .|.an] [ [b1 | . .|.bm ] = [a1 | . .|.an | b1 | . .|.bm ];
and is clearly not commutative. When C = A* for A a cocommutative Hopf
algebra of finite type, however, the product in the cobar construction is homot*
*opy
commutative, and so the cohomology of A is commutative (in the graded sense).
This homotopy commutativity also induces a cohomology operation [1 : H n;m(A)
H r;s(A) ! H n+r-1;m+s (A) which is represented by a map on Cobar (A*) satisfyi*
*ng
(1) ffi(x [1 y) = x [ y - (-1)|x||y|y [ x - (ffix) [1 y - (-1)|x|x [1 (ffi*
*y);
THE COHOMOLOGY ALGEBRA OF P(1) 3
where |x|is the total degree of x. An explicit formula for a representation of *
*[1 can
be found in Adams [1, p.36], although Adams' formula is given for R = F2, so an
appropriate sign must be provided. If the coproduct iterated t - 1 times is wri*
*tten
X
a ! a(1) . . .a(t);
then
[a1 | . .|.as] [1 [b1 | . .|.bt] =
Xs
[a1 | . .|.ar-1 | ar(1)b1 | . .|.ar(t)bt | ar+1 | . .*
*|.as]:
r=1
We will need the signs in only one special case :
Proposition 1.2. If [a] is a cocycle in external degree 1 and even internal de*
*gree
and bi has even degree, then
Xn
[a] [1 [b1 | . .|.bn] = [b1 | . .|.abr | . .|.bn]
r=1
satisfies (1).
Proof. It can be checked directly that ffi([a] [1 [b1 | . .|.bn]) has the corre*
*ct form.
Alternatively, let x = [b1 | . .|.bn] and y = [b2 | . .|.bn] and note that the *
*proposed
formula implies that
[a] [1 x = ([a] [1 [b1]) [ y + [b1] [ ([a] [1 y):
ffi([a] [1 [b1 | . .|.bn]) has the correct form by induction on n.
Corollary 1.3. If oe is a cocycle in external degree 1 and even internal degre*
*e,
x = [b1 | . .|.bn] and y = [c1 | . .|.cm ] for bi and cj with even degree, then
oe [1 (x [ y) = (oe [1 x) [ y + x [ (oe [1 y)
satisfies (1).
Steenrod Operations.
In [7, x11], May shows that the cohomology of a cocommutative Fp-Hopf algebra
of finite type also has cohomology operations
S"qi : H s;t(A) ! H s+i;2t(A) if p = 2,
P"i : H s;2t(A) ! H s+2i(p-1);2pt(A) if p > 2 and
fiP"i: H s;2t(A) ! H 2+2i(p-1)+1;2pt(A) if p > 2.
These operations satisfy the usual Adem and Cartan relations, as well a modified
unstable axiom :
if p = 2, then "Sqix = 0 if i < 0 or i > s and "Sqsx = x2 ;
if p > 2, then P"ix = 0 if i < 0 or 2i > s and "Pxi= xp if 2i = s ;
if p > 2, then fiP"ix = 0 if i < 0 or 2i s.
4 GREGORY D. HENDERSON
The difference between these operations and the usual Steenrod operations is
that S"q0 and P"0 do not act as the identity, and that fiP"i is not necessarily*
* the
composition of P"iwith a Bockstein. May [7,propositions 11.9 and 11.10] gives an
explicit description of "Sq0 and "P 0:
S"q0[a1 | . .|.an][=a21| . .|.a2n] if p = 2,
P"0[a1 | . .|.an] =[ap1| . .|.apn] if p > 2.
One result of this difference is that "P 0appears explicitly in the Adem and Ca*
*rtan
formulae. For example, the Cartan formula for fiP"0 on a product is
fiP"0(x [ y) = fiP"0x [ "Py0+ (-1)|x|"Px0[ fiP"0y:
An explicit formula can be found for fiP"0[a] when a is a primitive in A* (see *
*for
instance [5] 2.5.13) :
X 1
fiP"0[a] = - _____[ai | aj]:
i+j=p i ! j !
i;j>0
The Filtration Giving the Spectral Sequence.
Our main tool for computing the cohomology of a Hopf algebra is the spectral
sequence constructed by Adams [1, x2.3], which, given a central extension of Ho*
*pf
algebras A ! C ! B, computes the cohomology of C in terms of the cohomology
of A and B. The relevant facts which we will need are : the spectral sequence *
*is
trigraded, with the differentials preserving the third degree ;
Es;t;r2~= H s;i(A) Ht;j(B) ) H s+t;r(C);
i+j=r
and the spectral sequence arises from the decreasing filtration on Cobar (C*) g*
*iven
by
F nCobar (C*) =
ae fifi * oe
[c1 | . .|.cs] fifici 2 C annihilates all elements of positive degree:*
*in A
for at least n values of i
Thus the filtration which H *;*(C) inherits has the following form :
H n;*(C) = F 0Hn;*(C) F 1Hn;*(C) . . .F nHn;*(C) 0:
Finally, we will need to find representatives for elements of Er in the cobar
construction, and so will need a concrete description of Er. From the proof of
proposition 2.6 in [6], we have :
s+t;u
Zs;t;ur=F s+1Cobar (W *)
Es;t;ur= ______________________________s;t;us+t;u;
Br =F s+1Cobar (W *)
where
s s+t;u * fi s+r s+t+1;u *
Zs;t;ur= x 2 F Cobar (W ) fiffix 2 F Cobar (W )
and
s s+t;u * fi s-r s+t-1;u *
Bs;t;ur= ffiy 2 F Cobar (W ) fiy 2 F Cobar (W ):
The differential dr is induced by the map ffi.
THE COHOMOLOGY ALGEBRA OF P(1) 5
The Extension.
The extension of Hopf algebras which we will use is defined by the sub-Hopf
algebra of W generated by x. Since x is the commutator of the algebra generators
of W , the quotient Hopf algebra is commutative. Fp[x1; . .;.xn] denotes the po*
*ly-
nomial algebra on the given generators of even degree, and (x1; . .;.xn) denotes
the exterior algebra on the given generators of odd degree.
Proposition 1.4. Let A be the sub-Hopf algebra of W generated by the central
element x and let B be the quotient Hopf algebra W==A. Then A = Fp[x]=(xp),
B = Fp[y; z]=(yp; zp) with coproducts as in definition 1.1, and A ! W ! B is a
central extension of graded connected Fp-Hopf algebras.
Since the Hopf algebras we will use are finite dimensional, we may consider
the cohomology of a Hopf algebra as the homology of the cobar construction on
the dual. We will solve the extension problem by finding explicit representati*
*ves
in Cobar (W *) for those low dimensional algebra generators of H *;*W which are
involved in the relations which need to be lifted from E1 . To this end we need
descriptions of the duals of A, B, and W and the cohomology of A and B.
Proposition 1.5.
A* = Fp["x]=("xp); where "x= x* is primitive.
2 * p *
B* = Fp["y]=("yp); where "y= y and "y = z are primitive.
2 * p *
W * = Fp["x; "y]=("xp; "yp);where "y= y and "y = z are primitive,
and ("x) = "yp "y.
ae(x ) F [fi ] if p > 2
H *;*A ~= p x where
Fp[x ] if p = 2,
x = ["x] in degree (1; (p + 1)q) and
X 1
fix= - _____["xi| "xj] = fiP"0x in degree (2; p(p + 1)q) if p > 2.
i+j=p i ! j !
i;j>0
ae(y; z) F [fi ; fi ] if p > 2
H *;*B ~= p y z where
Fp[y; z] if p = 2,
y = ["y] in degree (1; q),
X 1
fiy= - _____["yi| "yj] = fiP"0y in degree (2; pq) if p > 2,
i+j=p i ! j !
i;j>0
z = ["yp] = "P 0y in degree (1; pq) and
X 1
fiz= - _____["yip| "yjp] = fiP"0z in degree (2; p2q) if p > 2.
i+j=p i ! j !
i;j>0
6 GREGORY D. HENDERSON
Proof. This is fairly straightforward, see for instance [5, p.31]. Note that a*
*s an
algebra, B is the tensor product of two algebras with the same form as A, one w*
*ith
generator y and the other with generator z. The Hopf algebra structure of B ent*
*ers
in the action of the Steenrod algebra on H *;*B. In particular, "P 0y= z.
Lastly, note that, for this particular extension,
["xi1"yj1| . .|."xis"yjs] 2 F nCobar (W *)
if and only if jr is nonzero for at least n values of r. In particular
Proposition 1.6. If oe 2 Cobar 1;*(B*) and w 2 F nCobar *;*(W *), then oe [1 w*
* 2
F nCobar *;*(W *).
Proof. oe has the form ["yk] with k > 0, so oe [1 ["xi1"yj1| . . .| "xis"yjs] i*
*s a sum of
elements of the form ["xi1"yj1+k1| . .|."xis"yjs+ks] with kr 0.
Proposition 1.7. if 2 Cobar 2;*(B*) and w 2 F nCobar *;*(W *), then [1 w 2
F n+1Cobar *;*(W *).
Proof. is a sum of elements of the form ["yk| "yl] for k; l > 0. Hence [1 ["x*
*i1"yj1|
. . .| "xis"yjs] is a sum of elements of the form ["xi1"yj1+k1| . .|."xis"yjs+k*
*s| "yl] with
kr 0 and ["yk| "xi1"yj1+l1| . .|."xis"yjs+ls] with lr 0.
x2 The E1 Term.
The indirect argument for computing the higher differentials in the following
theorem was brought to the author's attention in [8].
Theorem 2.1. In the change of rings spectral sequence [1,theorem 2.3.1] for the
extension given in proposition 1.4,
Ep;q;r2~= Hq;iA Hp;jB ) H p+q;rW:
i+j=r
When p > 2
E*;*;*2~=(x ; y; z) Fp[fix; fiy; fiz]
and E*;*;*1is
____(h0;_O;_{oe2r+1_|_0__r_<_p_-_1})__Fp[0;_1;_0;_{2s_|_1__s_<_p_-_1};_!]_____*
*01;
h00; h0O; h0oe2r+1; h0p-11; h02s - 0oe2s-1; p1;
B@ 0O; 02s; Ooe2r+1; Op-1 2 C
1 ; O2s - 0oe2s+1; 0; A
oe2r+1oe2r0+1;oe2r+12s; 2s2s0; 1oe2r+1; 212s
where
Generator Degree Representative
in E2
h0 (1; 0; q) y
0 (2; 0; pq) fiy
1 (2; 0; p2q) fiz
0 (1; 1; (p + 2)q) xy
O (1; 2; (p2 + p + 1)q) fixy
oe2r+1; 0 r < p - 1 (1; 2r; (rp2 + (r + 1)p)q) firxz
2s ; 1 s < p - 1 (1; 2s - 1; ((s - 1)p2 + (s + 1)p + 1)q) fis-1xzx
! (0; 2p; p2(p + 1)q) fipx:
THE COHOMOLOGY ALGEBRA OF P(1) 7
When p = 2
E*;*;*2~=Fp[x ; y; z]
and
E*;*;*1= _____F2[h0;_h1;_u;_!]___(h; 3 2 2
0h1; h1; h1u; u + h0*
*!)
where
Generator Degree Representative in E2
h0 (1; 0; q) y
h1 (1; 0; 2q) z
u (2; 1; 7q) x2 y
! (0; 4; 12q) x4 :
Proof. We begin with the case p > 2. The calculation up to the E4 term is a
straightforward consequence of d2x = zy and the transgression theorem (see, for
instance, the first comment after the proof of theorem 11.8 in [7]) : since fi*
*x is
fiP"0x, we see that d2fix = 0 and d3fix = -fizz. Thus E*;*;*4is
____({O2r+1;_oe2r+1_|_0__r__p_-_1})__Fp[0;_1;_0;_{2s_|_1__s__p};_!]__________*
*01;
oe2r+1oe2r0+1; 0O2r+1; 2sO2r+1 - 0oe2r+2s-1 if r + s - 1 <;p
B@ 1oe2r+1; 02s; 2sO2r+1 - 0!oe2r+2s-2p-1 if r + s - 1 p; CA
O2r+1O2r0+1; oe2r+1O2r0+1; 2s2s0; oe2r+12s; 20
Generator Degree Representative
in E2
0 (2; 0; pq) fiy
1 (2; 0; p2q) fiz
0 (1; 1; (p + 2)q) xy
O2r+1; 0 r p - 1 (1; 2r; (rp2 + rp + 1)q) firxy
oe2r+1; 0 r p - 1 (1; 2r; (rp2 + (r + 1)p)q) firxz
2s ; 1 s p (1; 2s - 1; ((s - 1)p2 + (s + 1)p + 1)q) fis-1xzx
! (0; 2p; p2(p + 1)q) fipx:
Since ! is P"1fix, the transgression theorem implies that ! is a permanent c*
*y-
cle. Consideration of the dimensions of the generators then shows that the only
potentially nontrivial differentials on the generators of E4 are
d4O2r+1 = ff1;r212r-2 for 2 r p - 1,
d2p-22p-2 = ff2p-11O1;
d2p-22p = ff3p-11O3 and
d2p-1oe2p-1= ff4p1:
The relations in E4 imply that no new generators arise at any stage, even if the
above differentials are nonzero, although new relations will arise in those cas*
*es. This
allows us to argue indirectly that all of these differentials are nonzero. The *
*source of
the contradiction to having a zero differential comes from applying the second *
*form
8 GREGORY D. HENDERSON
of the transgression theorem to d3fix = -fizz to obtain d2p-11oe2p-1 = -p+11.
Another examination of the dimensions of elements in E4 shows that p+11is nonze*
*ro
in E2p-1. Hence d2p-1oe2p-1 6= 0 (i.e. ff4 6= 0). In fact, d2p-1oe2p-1 = -p1: f*
*or a
direct computation, see proposition 3.4.
Next note that oe2p-1O2r+1 = 0 by the relations in E4, but if d4O2r+1 = 0 fo*
*r some
2 r p - 1, then O2r+1 survives to E2p-1 and d2p-1(oe2p-1O2r+1) = ff4p1O2r+1.
This element is nonzero in E4 and, by dimensions, is not hit by any of the poss*
*ible
differentials before d2p-1, so is nonzero in E2p-1. Hence d4O2r+1 must be nonze*
*ro.
Similarly, oe2p-12s = 0 in E4, but if d2p-22s = 0 for s = p or p - 1, then 2s
survives to E2p-1 and d2p-1(oe2p-12s) = ff4p12s. This latter element is nonzero
in E4 and, by dimensions, is not hit by any differentials before d2p-1, so is n*
*onzero
in E2p-1. Hence d2p-22p and d2p-22p-2 are nonzero. Note that 212s is_hit by
d4 of a multiple of O2s+3 if 1 s p - 2. When we set h0 = O1 and O = O3, the
proof is complete in the case p > 2.
When p = 2, the calculation begins as before, but the spectral sequence coll*
*apses
at E4.
Corollary 2.2. If p > 2, then H *;*W is a Fp[0; !] algebra with algebra genera*
*tors
h0; O; 1; 0; {oe2r+1 | 0 r < p - 1} and {2s | 1 s < p - 1}. It contains the
subalgebra
_(h0;_oe1)__Fp[0;_1]___ :
(h0oe1; oe11; h0p-11; p1)
H *;*W is also a free Fp[0; !] module with the following module generators :
Generator Degree
oe2r+1 (2r + 1; (rp2 + (r + 1)p)q) 0 r < p - 1
0oe2r+1 (2r + 3; (rp2 + (r + 2)p + 2)q) 0 r < p - 1
2s (2k + 2s; ((s - 1)p2 + (s + 1)p + 1)q)1 s < p - 1
2s1 (2s + 2; (sp2 + (s + 1)p + 1)q) 1 s < p - 1
k1 (2k; kp2q) 0 k < p
h0k1 (2k + 1; (kp2 + 1)q) 0 k < p - 1
0k1 (2k + 2; (kp2 + p + 2)q) 0 k < p
Ok1 (2k + 3; ((k + 1)p2 + p + 1)q) 0 k < p - 1:
Proof. The statement about algebras is a direct generalization of [2], III.2, p*
*ropo-
sition 11 and the definition of the filtration used to construct the spectral s*
*equence.
The statement about modules follows from the form of the relations and module
generators in E1 .
x3 Representatives for the Algebra Generators for p > 2.
In order to complete the computation of H*;*W , we must lift the relations i*
*n E1 .
We will identify the relations which must lift trivially by reason of the dimen*
*sions in
which they occur and lift the remaining relations by direct computation in the *
*cobar
construction. This will require representatives in the cobar construction, for *
*some of
the algebra generators listed in corollary 2.2. In this section we give represe*
*ntatives
for all the algebra generators except ! and verify directly the differentials w*
*hich
were determined by indirect argument in theorem 2.1.
Note that the actual computation of H *;*W does not require representatives *
*for
all of the algebra generators, and that the differentials in the spectral seque*
*nce have
THE COHOMOLOGY ALGEBRA OF P(1) 9
already been determined. We include all of the representatives and the computat*
*ion
of the differentials in this section only for completeness sake and to support *
*our
contention that these computations can be done in the cobar construction.
Since h0, oe1, 0 and 1 are in the highest filtration, their representatives *
*come
directly from H *;*B in proposition 1.5, where they correspond to y, z, fiy and*
* fiz
respectively. 0 is easily seen to be
0 = ["x| "y] + 1_2["yp| "y2]:
The remaining generators - O, oe2r+1 and 2s - correspond to multiples of powers*
* of
fix in E2, so to proceed further we need a representative for fix in E3.
Proposition 3.1. Consider the element in Cobar (W *) given by
X 1
b = - _____ "xi "xj+ ("x 1 + 1 "x)i("yp "y)j :
i+j=p i ! j !
i;j>0
Then i*b = fix where i : A ! W , and ffib = -1oe1. Furthermore, b 2 F 0, b is f*
*ix in
F 0=F 1and ffib 2 F 3. Hence b represents fix in E3.
Proof. This can, of course, be checked directly by a straightforward but tedious
computation. It is, however, true on general principles. Note first that the *
*fix
in proposition 1.5 can be described as the element ffi"xp=p in the integral cob*
*ar
construction on A*, reduced to the mod-p cobar construction. In fact, the eleme*
*nt
fix is fiP"0x, where the operation fiP"0 is essentially the pth power operation*
* (P"0)
composed with the mod-p Bockstein (fi). Our element b is constructed in the same
way in the cobar construction on W *. In the integral cobar construction on W *
**,
we have
"xp= ( "x)p = ("x 1 + "yp "y+ 1 "x)p = "xp 1 + 1 "xp+ ("yp "y)p - p ! b;
and so ffi"xp= -p ! b (recall that "yp2= 0). Upon division by p and reduction m*
*od-p,
we have b.
To find ffib, we apply ffi to ffi"xp, divide by p and reduce mod-p. Since ff*
*i2 = 0, we
obtain
0 = fiP"0["yp| "y] + ffib;
which should be regarded as the Kudo transgression theorem (ffifiP"0["x] = -fiP*
*"0ffi["x]
where b = fiP"0["x]). The Cartan formula for fiP"0(oe1h0) using "P 0h0 = oe1, "*
*P 0oe1 = 0
and 1 = fiP"0oe1 completes the proof.
The representative for O can now easily be seen to be
O = b["y] + 1["x]:
Unfortunately, the representatives for oe2r+1 and 2s are not quite as straightf*
*or-
ward. Once the answers are known, however, they are not too difficult to check.
Note that b 2 F 0Cobar (W *) and 1 2 F 2Cobar (W *).
10 GREGORY D. HENDERSON
Proposition 3.2. Let U(0)0= 1 and U(0)i= 0 for i 6= 0. We define
U(r)i= bU(r-1)i+ 1(oe1 [1 U(r-1)i) + 1U(r-1)i-1
for r 1 and 0 i r, with the convention that U(r)i= 0 for i outside this rang*
*e.
Then r
X k (r)
ffiU(r)i= - Uk ["y(k-i)p]
k=i+1 k - i
and
U(r)i ribr-ii1 in F 2i=F 2i+2.
Proof. This is a straightforward induction. Note that if w 2 Cobar (W *) is in *
*even
external degree, then ffi(oe1 [1 w) = oe1w - woe1 + oe1 [1 (ffiw), and that U(r*
*)iis in
even external degree since b and 1 are. Also, if w 2 F s, then oe1 [1 w 2 F sby
proposition 1.5.
Proposition 3.3. If 0 r < p - 1, then oe2r+1 is represented by
Xr 1
______U(r)k["y(k+1)p];
k=0 k + 1
and oe2r+1 broe1 mod F 3.
Proof. A simple calculation shows that the given element has zero differential,*
* lies
in F 1and is congruent to br["yp] modulo F 2(also modulo F 3).
Proposition 3.4.
p-2 !
X 1 (p-1) p
ffi ______Uk ["y(k+1)p] = -1;
k=0 k + 1
so d2p-1oe2p-1 = -p1.
Proof. The formula is easily checked, the element lies in F 1and is congruent to
bp-1 ["yp] modulo F 2. Note that U(r)r= r1.
Proposition 3.5. If 1 s < p - 1, then 2s is represented by
s-1X 1 s-1X
______U(s-1)k["y(k+1)p| "x] + ______1_______U(s-1)k["y(k+2)p| "*
*y];
k=0 k + 1 k=0 (k + 1)(k + 2)
and 2s bs-12 mod F 3.
Proof. A short computation shows that the given element has trivial differentia*
*l,
lies in F 1and is congruent to bs-1["yp| "x] modulo F 2(also modulo F 3).
THE COHOMOLOGY ALGEBRA OF P(1) 11
Proposition 3.6. If 0 r p - 1, then O2r+1 is represented by
U(r)0["y] + U(r)1["x]
in E4.
Proof. The given element lies in F 1, is congruent to br["y] modulo F 2and its *
*dif-
ferential lies in F 5, so the element is in Z*;*;*4.
Proposition 3.7.
p-2 !
X 1 (p-2) p-3X 1 (p-2)
ffi ______Uk ["y(k+1)p| "x] + ______________Uk ["y(k+2)p| "y]=
k=0 k + 1 k=0 (k + 1)(k + 2)
p-11["y];
so d2p-22p-2 = p-11O1.
Proof. Direct computation shows that the formula for the differenetial holds. T*
*he
element is in F 1and is congruent to bp-2 ["yp| "y] modulo F 2. ["y] is a repre*
*sentative
for O1 by proposition 3.6.
Proposition 3.8.
p-2 !
X 1 (p-1) p-3X 1 (p-1)
ffi ______Uk ["y(k+1)p| "x] + ______________Uk ["y(k+2)p| "y]
k=0 k + 1 k=0 (k + 1)(k + 2)
p-2X 1
= -p1["x] + U(p-1)p-21["y] + p-11 _______________["y(p-l)p| "y(l+1)*
*p| "y];
l=1 (p - l) ! (l + 1) !
so d2p-22p = -p-11O3.
Proof. The formula for the differential holds by direct computation. The element
lies in F 1 and is congruent to bp-1 ["yp | "x] modulo F 2, so represents 2p. *
*The
differential lies in F 2p-1 and is congruent to -bp-11["y] modulo F 2p. But bp-*
*1 is
p-11b in F 2p-2=F 2p-1modulo the image of ffi (use 1[1b repeatedly and proposit*
*ion
1.6), and b["y] = U(1)0["y], which is O3 in F 1=F 2.
Proposition 3.9. If 2 r p - 1, then
i j r-2X 1
ffi U(r)0["y] + U(r)1["x]= - (k + 2)U(r)k+2["y(k+1)p| "x] + ______["y(k+*
*2)p|;"y]
k=0 k + 2
so d4O2r+1 = r(r - 1)212r-2.
Proof. The formula for the differential holds by a short computation. The eleme*
*nt
represents O2r+1 by proposition 3.6, the differential lies in F 5and is congrue*
*nt to
r(r - 1)br-2 21["yp| "x] modulo F 6. Finally, in F 4=F 5, br-2 21is 21br-2 modu*
*lo the
image of ffi (use 1 [1 br-1 twice and proposition 1.6).
x4 The Extension Problem.
12 GREGORY D. HENDERSON
Theorem 4.1. When p > 3, H *;*W is
____(h0;_O;_{oe2r+1_|_0__r_<_p_-_1})__Fp[0;_1;_0;_{2s_|_1__s_<_p_-_1};_!]_____*
*0ae1;
h00; Oh0 - 10; Ooe2r+1 if r = p - 2,
BB 0O; h02s - 0oe2s-1; Ooe2r+1 + (r + 2)2r+21 otherwise; C
BB ae CC
BB 02s; O2s - 0oe2s+1; oe2r+1oe2r0+1+ _1__r0+1p-11if r + r0 = p -;2CC
BB 1oe2r+1; oe2r+1h0 - r2r1; oe2r+1oe2r0+1 otherwise; CC
BB ae 1 p-2 0 CC
BB 2; 2s2s0+ ___ss001 if s + s = p - 1; CC
BB 0 2s2s0 otherwise; CC
BB 8 1_p-2 CC
BB >< oe2r+12s + sO1 if r + s = p - 1; CCC
@ > oe2r+12s + _1__r+1h0p-21if r + s = p - 2; A
:
oe2r+12s otherwise
where
Generator Degree Representative
in Cobar (W *)
h0 (1; q) ["y]
0 (2; pq) fiy in proposition 1.5
1 (2; p2q) fiz in proposition 1.5
0 (2; (p + 2)q) ["x| "y] + 1_2["yp| "y*
*2]
O (3; (p2 + p + 1)q) b["y] + 1["x]
(b in proposition 3.*
*1)
oe2r+1 0 r < p - 1 (2r + 1; (rp2 + rp + p)q) proposition 3.3
2s 1 s < p - 1 (2s; (sp2 - p2 + sp + p + 1)q) proposition 3.5
! (2p; p2(p + 1)q):
When p = 3, H *;*W is
________________(h0;_O;_oe1;_oe3)__F3[0;_1;_0;_2;_!]_________________
0 0h0 + oe10; Oh0 - 10; Ooe1 - 21; 1oe1; 02 - 01; 1;
B@ 0O + 0oe3; 20+ 02; Ooe3; 1oe3; 22+ 01; C
h02 - 0oe1; oe12 + h01; oe1h0; oe1oe3 - 21; A
O2 - 0oe3 oe32 - O1; oe3h0 - 21;
where
Generator Degree Representative in Cobar (W *)
h0 (1; q) ["y]
oe1 (1; 3q) ["y3]
0 (2; 3q) ["y2| "y] + ["y| "y2]
0 (2; 5q) ["x| "y] - ["y3| "y2]
2 (2; 7q) ["y3| "x] - ["y6| "y]
1 (2; 9q) ["y6| "y3] + ["y3| "y6]
O (3; 13q) b["y] + 1["x] (b in proposition 3.1)
oe3 (3; 15q) b["y3] + 1["y6] (b in proposition 3.1)
! (6; 36q):
THE COHOMOLOGY ALGEBRA OF P(1) 13
Proof. Our aim here is to lift the relations given in theorem 2.1 from E1 to H*
* *;*W .
To reduce the work involved, we note that E1 is a free graded commutative alge-
bra over Fp[0; !], and so H *;*W is also a free graded commutative algebra over
any elements which lift 0 and ! (a direct generalization of [2], III.2, proposi*
*tion
11). We also note that the algebra generators h0; oe1; 0; and 1 are in the high*
*est
filtration, and so don't require lifting. The subalgebra these elements generat*
*e is
_(h0;_oe1)__Fp[0;_1]___ H *;*W:
(h0oe1; oe11; h0p-11; p1)
The Relations Which Must Be Lifted.
The external and internal degrees of the remaining relations and the module
generators given in corollary 2.2 shows that the relations we must lift for gen*
*eral p
are :
h0O a multiple of 01;
h0oe2r+1 a multiple of 12r for 1 r < p - 1,
Ooe2r+1 a multiple of 12r+2 for 0 r < p - 2,
oe2r+1oe2r0+1 a multiple of p-11 for 0 r; r0 < p - 1 and r + r0 = p - 2,
oe2r+12s a multiple of Op-21 for 0 r < p - 1, 1 s < p - 1
and r + s = p - 1,
oe2r+12s a multiple of h0p-21 for 0 r < p - 1, 1 s < p - 1
and r + s = p - 2,
2s2s0 a multiple of 0p-21 for 1 s; s0< p - 1
and s + s0= p - 1.
For p = 3 there are four additional relations to be lifted :
20 a multiple of 20;
h00 a multiple of oe10;
O0 a multiple of oe30;
02 a multiple of 10:
At first sight it may seem a hopeless task to lift all of these relations. F*
*ortunately,
many relations are dependant on others, and the work is much less than it first
appears. We begin with those which must be lifted by direct computation, of
which there are only five.
Lifts of the "Straightforward" Relations.
The following computations take place in Cobar (W *) using the representativ*
*es
found in section 3. The equivalence relation ~ is "differs by a coboundary", so*
* that
equivalent cocycles represent the same element in H *;*(W ).
Lemma 4.2.
Oh0 = ffi 1_2b["y2]+ 10
and, for p = 3:
2
0h0 = ffi ["x| "y]- oe10:
14 GREGORY D. HENDERSON
Lemma 4.3.
oe2r+1h0 ~ r2r1 for 1 r < p - 1.
Proof. This is proved by showing that
oe2r+1h0 r2r1 + ffi(br["x] + ar) mod F 4;
for some ar 2 F 2, by induction on r. Since oe2r+1h0 is a multiple of 2r1 in
H *;*(W ), and this element is in F 3, this will be sufficient to prove the res*
*ult.
The case r = 1 holds since
ffi(b["x] + 1 [1 2) oe3h0 - 21 mod F 4.
Note that 2 = ["yp| "x] + 1_2["y2p| "y] oe1["x] mod F 2, and that, if w 2 F l,*
* then
1 [1 w 2 F l+1by proposition 1.6.
For the general case we use 2s bs-1oe1["x] mod F 2(in E2), oe2r+1 broe1 mod
F 3(proposition 3.3) and
ffi(b(br-1 ["x] + ar-1 ) + 1(oe1 [1 br-1 )["x] + 1 [1 2r)
oe2r+1h0 - r2r1 mod F 4:
Lemma 4.4.
oe2p-52 ~ 1_2h0p-21:
Proof. We have
p-3
X 1 (p-3)
ffi ______________Uk ["y(k+2)p| "x]
k=0 (k + 1)(k + 2)
p-4X 1 !
+ _____________________U(p-3)k["y(k+3)p| "y]
k=0(k + 1)(k + 2)(k + 3)
= oe2p-5 ["yp| "x] + 1_2["y2p|-"y]1_2p-21h0
= oe2p-52 - 1_2p-21h0:
The next lemma shows how to lift one of the relations which is not in questi*
*on.
This is necessary to prove lemma 4.6.
Lemma 4.5.
oe2r+11 - _1___rf+f1i(br+1 + a0r) mod F 4 for 0 r < p - 1
and some a0r2 F 2.
Proof. The case r = 0 is oe11 = -ffi(b - oe1 [1 1). The induction step uses
oe2r+1 boe2r-1 mod F 2and
ffi(b(br+ a0r-1) + 1(oe1[1br) + 1[1oe2r+1) -(r + 1)oe2r+11 mod F 4:
THE COHOMOLOGY ALGEBRA OF P(1) 15
Lemma 4.6.
Ooe2r+1 ~ -(r + 2)2r+21 for 0 r < p - 2.:
Proof. With the notation of lemma 4.5 and the proof of lemma 4.3, we have
ffi (1 [1 ["x])oe2r+1 + b(h0 [1 oe2r+1) - ["x](1 [1 oe2r+1) - r_+_2_r(+b1r+1*
*["x] + ar+1)
+ _1___r[+"1x](br+1 + a0r) + __1__ro+e11(h0 [1 br+1) - __1__r(+o1e1*
* [1 br+1)h0
Ooe2r+1 + (r + 2)2r+21 mod F 4
(Recall that O = b["y] + 1["x] = bh0 + 1["x]). Since Ooe2r+1 is a multiple of 2*
*r+21
in H *;*(W ) and that element is in F 3, the claim is established.
The Remaining Relations.
The relations which remain to be lifted are determined by the relations which
have already been lifted. As before, the computations here are in Cobar (W *), *
*and
the equivalence ~ is "differs by a coboundary".
Lemma 4.7. For p = 3:
20 ~ -02;
0O ~ -0oe3;
02 ~ 01:
Proof. If 20~ a02, then we note that 021 is non-zero in E1 (by corollary
2.2), hence in H *;*W , and use the results of lemmas 4.2 and 4.6 :
021 ~ 0Ooe1 ~ -oe10O ~ 0h0O ~ -0Oh0
~ -010 ~ -201 ~ -a021;
so a = -1. If 0O ~ a0oe3, we use lemma 4.3 and proceed as before :
021 ~ . .~.0h0O ~ h00O ~ ah00oe3 ~ -a0oe3h0 ~ -a021;
so a = -1. If 02 ~ a01, we note that 0h01 is non-zero in E1 (by corollary
2.2), hence in H *;*W , and use lemmas 4.2 and 4.4 :
0h01 ~ -0oe12 ~ -oe102 ~ 0h02
~ h002 ~ ah001 ~ a0h01;
so a = 1.
16 GREGORY D. HENDERSON
Lemma 4.8. In H *;*W :
if 2s2s0 = as;s00p-21 for 1 s; s0< p - 1
and s + s0= p - 1,
then oe2r+1oe2r0+1= rar;r0+1p-11; for 0 r; r0 < p - 1,
r + r0 = p - 2 and r 6= 0,
oe2r+12s = sar+1;sh0p-21 for 0 r < p - 1, 1 s < p - 1
and r + s = p - 2,
and oe2r+12s = rar;sOp-21 for 0 r < p - 1, 1 s < p - 1
and r + s = p - 1.
Proof. Note first that 0p-11is nonzero by corollary 2.2. If oe2r+1oe2r0+1 ~ cp-*
*11
for 0 r; r0 < p - 1, r + r0 = p - 2 and r 6= 0, then 1 r < p - 1 and 0 r0 < *
*p - 2.
Lemma 4.6 and the relation O2r - 0oe2r+1, which lifted trivially, imply that
c0p-11~ 0oe2r+1oe2r0+1 ~ O2roe2r0+1 ~ 2rOoe2r0+1
~ -(r0+ 2)2r2r0+21 ~ -(r0+ 2)ar;r0+10p-11:
Since -(r0+ 2) p - r0- 2 = r mod p, the first result holds.
If oe2r+12s ~ ch0p-11for 0 r < p - 1, 1 s < p - 1 and r + s = p - 2, then
0 r < p - 2, so lemmas 4.2 and 4.6 imply that
c0p-11~ c10p-21~ cOh0p-21~ Ooe2r+12s
~ -(r + 2)2r+212s ~ -(r + 2)2r+22s1 ~ -(r + 2)ar+1;s0p-11:
As before, -(r + 2) p - r - 2 = s mod p, and the second result holds..
Finally, if oe2r+12s ~ cOp-21for 0 r < p - 1, 1 s < p - 1 and r + s = p - *
*1,
then 1 r; s < p - 1 and lemmas 4.2 and 4.3 imply that
c0p-11~ c10p-21~ cOh0p-21~ -ch0Op-21~ -h0oe2r+12s
~ oe2r+1h02s ~ r2r12s ~ r2r2s1 ~ rar;s0p-11:
Note that the r = 0 case of the first result in lemma 4.8 can be computed si*
*nce
oe1oe2p-3 = -oe2p-3oe1, but doesn't fit into the pattern at this point.
Lemma 4.9. If 1 s; s0< p - 1 and s + s0= p - 1, then as;s0= - 2__ss0a1;p-2.
Proof. Induction on s. The case s = 1 is trivial and if 2 s < p - 1, then
s0as;s00p-11 ~ s02s2s01 ~ s02s02s1 ~ s0as0;s0p-11~ 0oe2s0+1oe2s-1
~ -0oe2s-1oe2s0+1 ~ -(s - 1)as-1;s0+10p-11;
and the general case follows.
THE COHOMOLOGY ALGEBRA OF P(1) 17
Lemma 4.10. as;s0= - 1__ss0, and so
2s2s0 = - 1__ss00p-21 for 1 s; s0< p - 1 and s + s0= p - 1,
oe2r+1oe2r0+1= - __1___r0+p1-11 for 0 r; r0 < p - 1 and r + r0 = p - 2,
oe2r+12s = - _1___rh+01p-21 for 0 r < p - 1, 1 s < p - 1
and r + s = p - 2 and
oe2r+12s = - 1_sOp-21 for 1 r; s < p - 1 and r + s = p - 1
in H *;*W .
Proof. By lemma 4.4 oe2p-52 = 1_2h0p-21. Thus a1;p-2 = ap-2;1 = 1_2, and so
as;s0= - 1__ss0and the result follows. The r = 0 case for oe2r+1oe2r0+1 now fi*
*ts into
the pattern.
Elimination of Redundant Relations.
An examination of the statements of theorems 2.1 and 4.1 shows that four rel*
*a-
tions have been dropped between E1 and H *;*(W ). These four are h0p-11, Op-11,
212s and p1and can be derived from the other relations in H *;*(W ). When
r = 0 and r0 = p - 2, we have the relation oe1oe2p-3 - p-11. This implies that
p-11= oe1oe2p-3, and so h0p-11, Op-11and p1are all zero. For r = s - 1 and
1 s < p - 1, the relation Ooe2r+1 + (r + 2)2r+21 gives 12s = - _1__s+1Ooe2s-1.
This implies that 212s is zero. This completes the proof of theorem 4.1.
For completeness we include the remaining case (p = 2).
Theorem 4.11 [4, theorem 3]. When p = 2,
H *;*W = _____F2[h0;_h1;_u;_!]___(h; 3 2 2
0h1; h1; h1u; u + h0*
*!)
where
Generator Degree Representative in Cobar (W *)
h0 (1; q) ["y]
h1 (1; 2q) ["y2]
u (3; 7q) ["x| "x| "y] + ["x"y2| "y| "y] + ["y2| "x"y| "y] *
*+ ["y2| "y2| "x]
! (4; 12q):
Proof. No extension problems arise and the representatives can be checked direc*
*tly.
x5 Comparison with the Results of Liulevicius. We conclude by comparing
our results with those of Liulevicius [4]. Note that the only difference betwee*
*n our
notation and his is that we write oe1 for his h1 and 2 for his 0. We have done *
*this
to combine several of his relations.
18 GREGORY D. HENDERSON
Results for p > 3.
For p > 3, Liulevicius gives module generators for H *;*W over Fp[0; !] [4,t*
*he-
orem 2]. Our results agree with the list of generators given in the proof of h*
*is
theorem, which unfortunately does not match the list given in the statement of *
*the
theorem. For comparison, note that
h0 = y;
h1 = z;
0 = fiy;
1 = fiz;
h2;0= x; and
2;0 = fix:
Results for p = 3.
For p = 3, Liulevicius gives algebra generators and a partial list of relati*
*ons for
H *;*W [4,theorem 4]. Our algebra generators agree, but we have written oe1 for*
* his
h1 and 2 for his 0, and our O is the negative of his. Changing his notation to
match ours, we have :
Liulevicius_ Ours_
oe1h0 = 0 also holds
1oe1 = 0 also holds
21h0 = 0 redundant
0 = {h1; h0; h1} also holds
0 = {h0; h1; h0} also holds
oe3= {h1; 1; h1} also holds
O = -{h0; h1; 1} also holds
oe10 = h02 also holds
oe1O = -12 also holds
h0oe3 = -12 also holds
1oe3 = 0 also holds
oe10 = -h00 also holds
1h0 = -oe12 also holds
oe1oe3= -21 oe1oe3= 21
h0O = 10 also holds
02 = -20 also holds
1O = -oe32 1O = oe32
0O = -0oe3
O2 = 0oe3
Ooe3 = 0
02 = 01
22 = -01
THE COHOMOLOGY ALGEBRA OF P(1) 19
Nowhere in Liulevicius' proof does he show that his list of relations is com*
*plete,
nor does he explicitly claim that it is. In fact we have added five relations.*
* The
relation h021is redundant, even in Liulevicius' partial list, as was shown at t*
*he
end of the proof of theorem 4.1. It can be checked directly that our representa*
*tives
for 2, 0, oe3 and O are homologous to the Massey products given by Liulevicius
(better still O = {1; h1; h0})
There are two relations in which we disagree with Liulevicius' signs. It ca*
*n be
checked directly that oe3oe1 + 21- ffi(b["y6]) = 0, for the representatives we *
*have
given, thus oe1oe3 = -oe3oe1 = 21. It is more difficult to give a direct argume*
*nt that
1O = oe32. Alternatively, we can use the argument given in proposition 4.8 and
the relation 22= -01.
Results for p = 2.
These, of course, are the same.
References.
References
[1] J. F. Adams, On the Non-Existence of Elements of Hopf Invariant One, Annals*
* of Mathe-
matics 72 (1960), 20-104.
[2] N. Bourbaki, Commutative Algebra, Addison-Wesley Publishing Co., 1972.
[3] Gregory D. Henderson, Low Dimensional Cocommutative Connected Hopf Algebras*
*, Journal
of Pure and Applied Algebra 102 (1995), 173-193.
[4] Arunas L. Liulevicius, The Cohomology of a Subalgebra of the Steenrod Algeb*
*ra, Transactions
of the American Mathematical Society 104 (1962), 443-449.
[5] Arunas Liulevicius, The Factorization of Cyclic Reduced Powers by Secondary*
* Cohomology
Operations, Memoirs of the American Mathematical Society, No. 42, American *
*Mathematical
Society, 1962.
[6] John McCleary, User's Guide to Spectral Sequences, Mathematics Lecture Seri*
*es no. 12,
Publish or Perish, Inc., 1985.
[7] J. P. May, A General Algebraic Approach to Steenrod Operations, The Steenro*
*d Algebra and
its Applications, Lecture Notes in Mathematics, vol. 168, Springer-Verlag, *
*1970, pp. 153-229.
[8] Haynes R. Miller, Private Communication to Clarence W. Wilkerson.
Mathematics Department, Pennsylvania State University, University Park, PA
16802
E-mail address: gdh@math.psu.edu