HOPF ALGEBRA STRUCTURE ON
TOPOLOGICAL HOCHSCHILD HOMOLOGY
Vigleik Angeltveit and John Rognes
July 16th 2004
Abstract. The topological Hochschild homology T HH(R) of a commutative S-
algebra (E1 ring spectrum) R naturally has the structure of a Hopf algeb*
*ra over R,
in the homotopy category. We show that under a flatness assumption this *
*makes
the B"okstedt spectral sequence converging to the mod p homology of T HH(*
*R) into
a Hopf algebra spectral sequence. We then apply this additional structure*
* to study
some interesting examples, including the commutative S-algebras ku, ko, t*
*mf, ju
and j, and to calculate the homotopy groups of T HH(ku) and T HH(ko) afte*
*r smash-
ing with suitable finite complexes. This is part of a program to make sy*
*stematic
computations of the algebraic K-theory of S-algebras, using topological c*
*yclic ho-
mology.
1. Introduction
The topological Hochschild homology T HH(R) of an S-algebra R (or an A1
ring spectrum, or a functor with smash product) was constructed in the mid-1980*
*'s
by B"okstedt [B"o1], as the natural promotion of the classical Hochschild homol*
*ogy
of an algebra in the category of vector spaces (equipped with tensor product) to
one in the category of spectra (equipped with smash product). It is the initial
ingredient in the construction by B"okstedt, Hsiang and Madsen [BHM93] of the
topological cyclic homology T C(R; p) of the S-algebra R, which in many cases
closely approximates the algebraic K-theory K(R) of R [Mc97], [Du97], [HM97].
When R is the valuation ring in a local number field, systematic computations
of the topological cyclic homology of R were made in [HM03], thereby verifying
the Lichtenbaum-Quillen conjectures for the algebraic K-theory of these fields.
Particular computations for other S-algebras, related to topological K-theory, *
*have
revealed a more general pattern of how algebraic K-theory creates a "red-shift"
in chromatic filtration [AR02], and satisfies a Galois descent property [Au]. T*
*hese
results indicate that the algebraic K-theory of a commutative S-algebra R may
be governed by some form of "S-algebraic geometry" associated to R, where the
______________
1991 Mathematics Subject Classification. 13D03, 55P43, 55S10, 55S12, 55T15, *
*55T99, 57T05.
Key words and phrases. Topological Hochschild homology, commutative S-algebr*
*a, coproduct,
Hopf algebra, topological K-theory, image of J spectrum, B"okstedt spectral seq*
*uence, Steenrod
operations, Dyer-Lashof operations.
Typeset by AM S-T*
*EX
1
2 VIGLEIK ANGELTVEIT AND JOHN ROGNES
chromatic filtration is related to a Zariski topology and Galois covers are rel*
*ated
to an 'etale topology.
Further systematic computations of the topological Hochschild homology, topo-
logical cyclic homology and algebraic K-theory of commutative S-algebras can
therefore be expected to shed light on these new geometries, and on the algebra*
*ic K-
theory functor. The present paper advances the algebraic-topological foundations
for making such systematic computations, especially by taking into account the
Hopf algebra structure present in the topological Hochschild homology of commu-
tative S-algebras. This program is continued in [BR] and [L-N], which analyze t*
*he
homological homotopy fixed point spectral sequence approximating the cyclic fix*
*ed
points of topological Hochschild homology, and the action by Steenrod operations
on the homology of the latter, respectively.
When R is a commutative S-algebra (or an E1 ring spectrum, or a com-
mutative FSP), there is an equivalence T HH(R) ' R S1 due to McClure,
Schw"anzl and Vogt [MSV97], where S1 is the topological circle and (-) S1
refers to the topologically tensored structure in the category of commutative S-
algebras. The pinch map S1 ! S1 _ S1 and reflection S1 ! S1 then induce maps
_ :T HH(R) ! T HH(R) ^R T HH(R) and O: T HH(R) ! T HH(R), which make
T HH(R) into a Hopf algebra over R in the homotopy category [EKMM97, IX.3.4].
See also theorem 3.9.
We wish to apply this added structure for computations. Such computations
are usually made by starting with the simplicial model for T HH(R), with [q] 7!
R^(q+1) = R S1q, where now [q] 7! S1qis the simplicial circle. The resulting sk*
*eleton
filtration on T HH(R) gives rise to the B"okstedt spectral sequence in homology
E2**(R) = HH*(H*(R; Fp)) =) H*(T HH(R); Fp) ,
as we explain in section 4. Compare [HM97] and [MS93]. But the coproduct
and conjugation maps are not induced by simplicial maps in this model, so some
adjustment is needed in order for these structures to carry over to the spectral
sequence. This we arrange in section 3, by using a doubly subdivided simplicial
circle to provide an alternative simplicial model for T HH(R), for which the Ho*
*pf
algebra structure maps can be simplicially defined. The verification of the Ho*
*pf
algebra relations then also involves a triply subdivided simplicial circle.
In section 4 we transport the Hopf algebra structure on T HH(R) to the B"oks*
*tedt
spectral sequence, showing in theorem 4.4 that if its initial term E2**(R) is f*
*lat as a
module over H*(R; Fp), then this term is an A*-comodule H*(R; Fp)-Hopf algebra
and the d2-differentials respect this structure. Furthermore, if every Er-term*
* is
flat over H*(R; Fp), then the B"okstedt spectral sequence is one of A*-comodule
H*(R; Fp)-Hopf algebras.
Thereafter we turn to the desired computational applications. The mod p homo*
*l-
ogy of the topological Hochschild homology of the Eilenberg-Mac Lane S-algebras
HFp and HZ was already computed by B"okstedt [B"o2]. The first non-algebraic
example, namely the topological Hochschild homology of the Adams summand
` = BP <1> of p-local connective topological K-theory, was computed for p odd by
McClure and Staffeldt in [MS93].
HOPF ALGEBRA STRUCTURE ON T HH 3
We show in section 5 how to extend these computations to include the case of
BP <1> = ku(2)at p = 2, and to more general Johnson-Wilson S-algebras BP
when p and n are such that these are commutative. In particular, we provide a p*
*roof
of B"okstedt's formula saying that the suspension map oe : R ! T HH(R) takes
the Dyer-Lashof operations Qk on the homology of the commutative S-algebra R
compatibly to the corresponding operations on the homology of the commutative
S-algebra T HH(R). See proposition 5.9.
In section 6 we do the same for the higher real commutative S-algebras ko and
tmf at p = 2. As sample results we have corollary 5.13(a) and theorem 6.2(a):
H*(T HH(ku); F2) ~=H*(ku; F2) E(oe,~21, oe,~22) P (oe,~3)
and
H*(T HH(ko); F2) ~=H*(ko; F2) E(oe,~41, oe,~22) P (oe,~3) .
Here E(-) and P (-) denote the exterior and polynomial algebras on the indicated
generators, respectively, and the classes ~,kare the conjugates of Milnor's gen*
*erators
for the dual Steenrod algebra A*.
In the more demanding section 7 we proceed to the p-local real and complex
image of J spectra j and ju, which are connective, commutative S-algebras. At
odd primes the two are homotopy equivalent. We identify the mod p homology
algebra of ju at p = 2 and of j = ju at odd primes in proposition 7.12(a) and (*
*b),
and make essential use of our results about Hopf algebra structures to show that
the corresponding B"okstedt spectral sequences for T HH collapse at the E2- and
Ep-terms, respectively, in proposition 7.13(a) and (b). Finally, in theorem 7.1*
*5 we
resolve the algebra extension questions to obtain H*(T HH(ju); Fp) as an algebr*
*a,
both for p = 2 and for p odd. This proof involves a delicate comparison with the
case of T HH(ku) for p = 2, and with T HH(`) for p odd. Again as a sample resul*
*t,
we have theorem 7.15(a):
H*(T HH(ju); F2) ~=H*(ju; F2) E(oe,~41, oe,~22) P (oe,~3) (oeb) .
Here (oeb) = E(fl2k(oeb) | k 0) is the divided power algebra on a class oeb *
*in
degree 4.
The algebra structure of H*(j; F2) is described as a split square-zero exten*
*sion
of (A==A2)* in proposition 7.12(c):
0 ! A* A2* 7K* ! H*(j; F2) ! (A==A2)* ! 0 .
Here A2 = A, and K* A2* is dual to a cyclic A2-module K
of rank 17 over F2. The A-module structure of H*(j; F2) was given in [Da75], but
this identification of the algebra structure seems to be new. The E2-term of t*
*he
B"okstedt spectral sequence for j is described in proposition 7.13(c), but it i*
*s not flat
over H*(j; F2), so the coproduct on H*(T HH(j); F2) is not conveniently describ*
*ed
by this spectral sequence. We have therefore not managed to evaluate the homolo*
*gy
of T HH(j) at p = 2 by these methods.
Next we consider the passage from the homology of T HH(R) to its homotopy,
with suitably chosen finite coefficients. This has been a necessary technical s*
*witch in
4 VIGLEIK ANGELTVEIT AND JOHN ROGNES
past computations of topological cyclic homology T C(R; p), since T C is define*
*d as
the homotopy inverse limit of a diagram of fixed-point spectra derived from T H*
*H,
and the interaction between inverse limits and homology can be difficult to con*
*trol.
The homotopy groups of an inverse limit are much better behaved. Nonetheless, it
may be that future computations of the topological cyclic homology of S-algebras
will follow a purely homological approach, see [BR] and [L-N].
In section 8 we follow the strategy of [MS93] to compute the homotopy groups*
* of
T HH(ku)^M, where M = C2 is the mod 2 Moore spectrum, and of T HH(ko)^Y ,
where Y = C2 ^ Cj is the 4-cell spectrum employed by Mahowald [M82]. The
results appear in theorems 8.13 and 8.14, respectively. In each case the method*
* is
to use the Adams spectral sequence to pass from homology to homotopy, and to
use a computation with a Morava K(1)-based B"okstedt spectral sequence to obtain
enough information about the v1-periodic towers in the abutment to completely
determine the differential structure of the Adams spectral sequence.
The present paper is based on the first author's Master's thesis [An02] at t*
*he
University of Oslo, from June 2002.
2. Hochschild and topological Hochschild homology
Let k be a graded field, i.e., a graded commutative ring such that every gra*
*ded k-
module is free, and a graded k-algebra. We recall the definition of the Hochs*
*child
homology of , e.g. from [Ma75, X.4]. The Hochschild complex C*( ) = Ck*( )
is the chain complex of graded k-modules with Cq( ) = (q+1) in degree q (all
tensor products are over k) and boundary homomorphisms @ :Cq( ) ! Cq-1 ( )
given by
@(~0 . . .~q) =
q-1X
(-1)i~0 . . .~i~i+1 . . .~q + (-1)q+ffl~q~0 . . .~q-1
i=0
where ffl = |~q|(|~0| + . .+.|~q-1 |). The Hochschild homology HH*( ) = HHk*( )
is defined to be the homology of this chain complex. It is bigraded, first by *
*the
Hochschild degree q and second by the internal grading from . When is com-
mutative the shuffle product of chains defines a product
OE: HH*( ) HH*( ) ! HH*( )
that makes HH*( ) a commutative -algebra, with unit corresponding to the in-
clusion of 0-chains ! HH*( ).
When is commutative there is also a chain level coproduct _ :C*( ) !
C*( ) C*( ) given in degree q by
Xq
(2.1) _(~0 ~1 . . .~q) = (~0 ~1 . . .~i) (1 ~i+1 . . .~q) .
i=0
HOPF ALGEBRA STRUCTURE ON T HH 5
It is essential to tensor over in the target of this chain map. When HH*( ) *
*is
flat as a -module, the chain level coproduct _ induces a coproduct
_ :HH*( ) ! HH*( ) HH*( )
on Hochschild homology. Here the right hand side is identified with the homology
of C*( ) C*( ) by the K"unneth theorem. We shall now compare this chain level
definition of the coproduct on HH*( ) with an equivalent definition given in mo*
*re
simplicial terms.
Let B*( ) = B*( , , ) be the two-sided bar construction [ML75, X.2] for the
k-algebra . It has Bq( ) = q in degree q, and is a free resolution *
*of
in the category of -bimodules. We use the bar notation ~0[~1| . .|.~q]~q+1 f*
*or
a typical generator of Bq( ). The Hochschild complex is obtained from the two-
sided bar construction by tensoring it with viewed as a -bimodule: C*( ) =
- B*( ).
When is commutative, B*( ) is the chain complex Ch ( 1) associated to
the simplicial -bimodule [q] 7! 1q. Here 1 is the simplicial 1-simplex, *
*and
1qdenotes the tensor product of one copy of for each element of 1q. See
section 3 below for more on this notation. The -bimodule structure on B*( ) is
derived from the inclusion of the two boundary points @ 1 ! 1, and C*( ) equals
the chain complex Ch ( S1) associated to the simplicial -module [q] 7! *
*S1q,
where S1 = 1=@ 1 is the simplicial circle.
There is a canonical chain level coproduct _ :B*( ) ! B*( ) B*( ) of -
bimodules, given in degree q by
Xq
_(~0[~1| . .|.~q]~q+1) = ~0[~1| . .|.~i]1 1[~i+1| . .|.~q]~q+1 .
i=0
When is commutative the chain level coproduct _ on C*( ) is derived from this,
as the obvious composite map
0
C*( ) = - B*( ) -1-_-! - (B*( ) B*( )) -_! C*( ) C*( ) .
Second, there is a shuffle equivalence sh: B*( ) B*( ) ! dB*( ) of -
bimodules, by the Eilenberg-Zilber theorem [Ma75, VIII.8.8] applied to two copi*
*es
of the simplicial -bimodule 1. Here
dB*( ) = Ch ( d 1) = Ch (( 1) ( 1))
is the chain complex associated to the simplicial tensor product of two copies *
*of
1, considered as simplicial -modules by way of the right and left actions,
respectively. This simplicial tensor product equals d 1, where the "double
1-simplex" d 1 = 1 [ 0 1 is the union of two 1-simplices that are compatibly
oriented. (So d 1 is the 2-fold edgewise subdivision of 1 [BHM93, x1].) More
explicitly, X
sh(x y) = sgn(~, )(s (x) s~(y)) ,
(~, )
6 VIGLEIK ANGELTVEIT AND JOHN ROGNES
where x 2 Bi( ), y 2 Bq-i( ), the sum is taken over all (i, q - i)-shuffles (~,*
* ),
sgn(~, ) is the sign of the associated permutation, and s (x) and s~(y) are the
appropriate iterated degeneracy operations on x and y, respectively.
Third, there is a chain equivalence ss :dB*( ) ! B*( ) of -bimodules, induc*
*ed
by the simplicial map ss :d 1 = 1 [ 0 1 ! 1 that collapses the second 1 in
d 1 to a point. It is given by
ss(x y) = x . ffl(y)
for x, y 2 Bq( ), where ffl(~0[~1| . .|.~q]~q+1) = ~0~1 . .~.q~q+1 is the augme*
*nta-
tion.
Lemma 2.2. Let be a commutative k-algebra. The maps sh O _ :B*( ) !
dB*( ) and ss :dB*( ) ! B*( ) of -bimodule complexes are mutual chain in-
verses. Hence the induced composite
dC*( ) = - dB*( ) -1-ss!C*( ) -1-(shO_)----!dC*( )
is chain homotopic to the identity.
Proof. All three chain complexes B*( ), B*( ) B*( ) and dB*( ) are free -
bimodule resolutions of , and the maps _, sh and ss are -bimodule chain maps,*
* so
it suffices to verify that the composite ss OshO_ :B*( ) ! B*( ) covers the ide*
*ntity
on . In degree zero, _(~0[]~1) = ~0[]1 1[]~1, sh(~0[]1 1[]~1) = ~0[]1 *
* 1[]~1
and ss(~0[]1 1[]~1) = ~0[]~1, as required. If desired, the explicit formula*
*s can
be composed also in higher degrees, to show that (ss O sh O _)(x) x modulo
simplicially degenerate terms, for all x 2 Bq( ). In either case, we see that s*
*sOshO_
is chain homotopic to the identity. The remaining conclusions follow by uniquen*
*ess
of inverses.
Note that dC*( ) defined above is the chain complex associated to the sim-
plicial -module d0S1, where the "double circle" d0S1 = d 1=@d 1 is the
quotient of the double 1-simplex d 1 = 1 [ 0 1 by its two end-points @d 1.
The chain equivalence 1 ss :dC*( ) = Ch ( d0S1) ! Ch ( S1) = C*( )
obtained by tensoring down the -bimodule equivalence induced from the col-
lapse map ss :d 1 ! 1, is then more directly obtained from the collapse map
ss :d0S1 ! S1 that collapses the second of the two 1-simplices in d0S1 to a poi*
*nt.
There is also a simplicial pinch map _ :d0S1 ! S1 _ S1 to the one-point union
(wedge sum) of two circles, that collapses the 0-skeleton of d0S1 to a point. *
*It
induces a map
_0: dC*( ) = Ch ( d0S1) ! Ch ( (S1 _ S1)) .
The target is the chain complex associated to the simplicial tensor product of *
*two
copies of S1, considered as a simplicial -module. It is therefore also the*
* target
of another shuffle equivalence sh: C*( ) C*( ) ! Ch ( (S1 _ S1)).
HOPF ALGEBRA STRUCTURE ON T HH 7
Proposition 2.3. Let be a commutative k-algebra. The composite map
dC*( ) -1-ss!'C*( ) -_!C*( ) C*( ) -sh!'Ch( (S1 _ S1))
of the chain level coproduct _ in C*( ) (see formula (2.1)), with the chain equ*
*iv-
alence 1 ss induced by the simplicial collapse map ss :d0S1 ! S1 and the shuf*
*fle
equivalence sh, is chain homotopic to the map
0
dC*( ) = Ch ( d0S1) -_! Ch ( (S1 _ S1))
induced by the simplicial pinch map _ :d0S1 ! S1 _ S1.
Hence, when HH*( ) is flat over , the coproduct _ on Hochschild homology
agrees, via the identifications induced by 1 ss and sh, with the map _0 induc*
*ed by
the simplicial pinch map.
Proof. Consider the following diagram.
0
C*( ) _1___//_iiRR- (B*( ) B*( )) ______//_C*( ) C*( )
RRRR
RRRRRR |1|sh sh||
1 ss RRR fflffl| _0 fflffl|
dC*( ) _______________//Ch( (S1 _ S1))
The composite along the upper row is the coproduct _, the composite around the
triangle is chain homotopic to the identity, by lemma 2.2, and the square commu*
*tes
by naturality of the shuffle map with respect to the pinch map _0. A diagram ch*
*ase
provides the claimed chain homotopy.
We shall make use of the following standard calculations of Hochschild homol*
*ogy.
The formulas for the coproduct _ follow directly from the chain level formula (*
*2.1)
above. Let P (x) = k[x] and E(x) = k[x]=(x2) be the polynomial and exterior
algebras over k in one variable x, and let (x) = k{fli(x) | i 0} be the divi*
*ded
power algebra with multiplication
fli(x) . flj(x) = (i, j) fli+j(x) ,
where (i, j) = (i + j)!=i!j! is the binomial coefficient.
Proposition 2.4. For x 2 let oex 2 HH1( ) be the homology class of the cycle
1 x 2 C1( ) in the Hochschild complex. For = P (x) there is a P (x)-algebra
isomorphism
HH*(P (x)) = P (x) E(oex) .
The class oex is coalgebra primitive, i.e., _(oex) = oex 1 + 1 oex. For =*
* E(x)
there is an E(x)-algebra isomorphism
HH*(E(x)) = E(x) (oex) .
8 VIGLEIK ANGELTVEIT AND JOHN ROGNES
The ith divided power fli(oex) is the homology class of the cycle 1 x . . .x 2 *
*Ci( ).
The coproduct is given by
X
_(flk(oex)) = fli(oex) flj(oex) .
i+j=k
There is a K"unneth formula
HH*( 1 2) ~=HH*( 1) HH*( 2) .
Let Ph(x) = k[x]=(xh) be the truncated polynomial algebra of height h. We
write P (xi | i 0) = P (x0, x1, . .).= P (x0) P (x1) . .,.and so on. When*
* k is
of prime characteristic p it is a standard calculation with binomial coefficien*
*ts that
(2.5) (x) = Pp(flpi(x) | i 0)
as a k-algebra.
We have already noted that the Hochschild complex C*( ) is the chain com-
plex associated to a simplicial graded k-module [q] 7! Cq( ) = (q+1), with
face maps di corresponding to the individual terms in the alternating sum defin-
ing the Hochschild boundary @. In fact, this is a cyclic graded k-module, in
the sense of Connes, with cyclic structure maps tq that cyclically permute the
(q + 1) tensor factors in Cq( ), up to sign. It follows that the geometric real*
*ization
HH( ) = |[q] 7! Cq( )| admits a natural S1-action ff :HH( ) ^ S1+ ! HH( ),
and that the Hochschild homology groups are the homotopy groups of this space:
HH*( ) = ss*HH( ).
The basic idea in the definition of topological Hochschild homology is to re*
*place
the ground ring k by the sphere spectrum S, and the symmetric monoidal category
of graded k-modules under the tensor product = k by the symmetric monoidal
category of spectra, interpreted as S-modules, under the smash product ^ = ^S .
A monoid in the first category is a graded k-algebra , which is then replaced *
*by a
monoid in the second category, i.e., an S-algebra R. To make sense of this we w*
*ill
work in the framework of [EKMM97], but we could also use [HSS00] or any other
reasonable setting that gives a symmetric monoidal category of spectra.
The original definition of topological Hochschild homology was given by B"ok*
*stedt
in the mid 1980's [B"o1], inspired by work and conjectures of Goodwillie and Wa*
*ld-
hausen. The following definition is not the one originally used by B"okstedt, s*
*ince he
did not have the symmetric monoidal smash product from [EKMM97] or [HSS00]
available, but it agrees with the heuristic definition that his more complicate*
*d defi-
nition managed to make sense of, with the more elementary technology that he had
at hand.
Definition 2.6. Let R be an S-algebra, with multiplication ~: R^R ! R and unit
j :S ! R. The topological Hochschild homology of R is (the geometric realization
of) a simplicial S-module T HH(R) with
T HHq(R) = R^(q+1)
HOPF ALGEBRA STRUCTURE ON T HH 9
in simplicial degree q. The simplicial structure is like that on the simplicia*
*l k-
module underlying the Hochschild complex. More precisely, the i-th face map
di: R^(q+1) ! R^q equals idiR^ ~ ^ idq-i-1Rfor 0 i < q, while dq = (~ ^ idq-1*
*R)tq,
where tq cyclically permutes the (q+1) smash factors R by moving the last facto*
*r to
the front. The j-th degeneracy map sj: R^(q+1) ! R^(q+2) equals idj+1R^j ^idq-j*
*R.
Furthermore, the cyclic operators tq: T HHq(R) ! T HHq(R) make T HH(R) a
cyclic S-module, so that its geometric realization has a natural S1-action
ff :T HH(R) ^ S1+! T HH(R) .
The topological Hochschild homology groups of R are defined to be the homotopy
groups ss*T HH(R).
We note that when R is a commutative S-algebra there is a product that makes
T HH(R) a commutative R-algebra, with unit corresponding to the inclusion of 0-
simplices R ! T HH(R). When R is cofibrant as an S-module, we can also describe
the homotopy type of T HH(R) as the smash product R ^R^Rop R. See [EKMM97,
IX] for further discussion on these definitions of T HH(R).
3. Hopf algebra structure on T HH
Already B"okstedt noted that the simplicial structure on T HH(R), as defined
above, is derived from the simplicial structure on the standard simplicial circ*
*le
S1 = 1=@ 1. This can be made most precise in the case when R is a commuta-
tive S-algebra, in which case there is a formula T HH(R) ~=R S1 in terms of t*
*he
simplicial tensor structure on the category of commutative S-algebras. A corre-
sponding formula T HH(R) ' R |S1| in terms of the topological tensor structure
was discussed by McClure, Schw"anzl and Vogt in [MSV97]. We shall keep to the
simplicial context, since we will make use of the resulting skeletal filtration*
*s to form
spectral sequences.
We now make this "tensored structure" explicit. Let R be a commutative S-
algebra and X a finite set, and let
^
R X = R
x2X
be the smash product of one copy of R for each element of X. It is again a
commutative S-algebra. Now let f :X ! Y be a function between finite sets, and
let R f :R X ! R Y be the smash product over all y 2 Y of the maps
^
R f-1 (y) = R ! R = R {y}
x2f-1(y)
that are given by the iterated multiplication from the #f-1 (y) copies of R on *
*the
left to the single copy of R on the right. If f-1 (y) is empty, this is by defi*
*nition the
unit map j :S ! R. Since R is commutative, there is no ambiguity in how these
iterated multiplications are to be formed.
10 VIGLEIK ANGELTVEIT AND JOHN ROGNES
Note that the construction R X is functorial in both R and X. Given an
injection X ! Y and any function X ! Z, there is a natural isomorphism
R (Y [X Z) ~=(R Y ) ^(R X) (R Z) .
There is also a natural map
` ^
R ^ X+ = R ! R = R X
x2X x2X
whose restriction to the wedge summand indexed by x 2 X is R ix, where
ix :{x} ! X is the inclusion.
By naturality, these constructions all extend degreewise to simplicial finit*
*e sets
X :[q] 7! Xq, simplicial maps f :X ! Y , etc. In particular, we can define the
simplicial commutative S-algebra
i j
R X = [q] 7! R Xq
with structure maps R di, R sj, etc. There is then a useful natural map
(3.1) ! :R ^ X+ ! R X .
Consider the special case X = S1 = 1=@ 1. Here 1qhas (q + 2) elements
{x0, . .,.xq+1} where xt: [q] ! [1] has #x-1t(0) = t. The quotient S1qhas (q + *
*1)
elements, obtained by identifying x0 ~ xq+1. Then di(xt) = xt for t i and
di(xt) = xt-1 for t > i, while sj(xt) = xt for t j and sj(xt) = xt+1 for t > *
*j. So
a direct check shows that there is a natural isomorphism
(3.2) T HH(R) ~=R S1
of (simplicial) commutative S-algebras. In degree q it is the obvious identific*
*ation
R^(q+1) ~=R S1q.
There are natural maps j :* ! S1, ffl: S1 ! * and OE: S1 _ S1 ! S1 that map
to the base point of S1, retract to *, and fold two copies of S1 to one, respec*
*tively.
By naturality, these induce the following maps of commutative S-algebras:
j :R ! T HH(R)
(3.3) ffl:T HH(R) ! R
OE:T HH(R) ^R T HH(R) ! T HH(R) .
In the last case, the product map involves the identification T HH(R)^R T HH(R)*
* ~=
R (S1 [* S1), where S1 [* S1 = S1 _ S1. Taken together, these maps naturally
make T HH(R) an augmented commutative R-algebra.
There is also a natural map
(3.4) ! :R ^ S1+! T HH(R)
HOPF ALGEBRA STRUCTURE ON T HH 11
derived from (3.1), which captures part of the circle action upon T HH(R). More
precisely, the map ! admits the following factorization:
(3.5) ! = ff O (j ^ id): R ^ S1+! T HH(R) ^ S1+! T HH(R) .
This is clear by inspection of the definition of the circle action ff on the 0-*
*simplices
of T HH(R).
We would like to have a coproduct on T HH(R), coming from a pinch map
S1 ! S1 _ S1, but there is no such simplicial map with our basic model for S1. *
*To
fix this we again consider a "double model" for S1, denoted dS1, with
dS1 = ( 1 t 1) [(@ 1t@ 1) @ 1 .
Here @ 1 t @ 1 ! @ 1 is the identity map on each summand, so the two non-
degenerate 1-simplices of dS1 have opposing orientations in the geometrically r*
*eal-
ized circle. It is the quotient of the barycentric subdivision of 1 by its bou*
*ndary.
Then we have a simplicial pinch map _ :dS1 ! S1_S1 that collapses @ 1 dS1
to *, as well as a simplicial flip map O: dS1 ! dS1 that interchanges the two c*
*opies
of 1.
Remark 3.6. The simplicial set dS1 introduced here differs from the double circ*
*le
d0S1 considered in section 2, in that the orientation of the second 1-simplex h*
*as
been reversed. The switch is necessary here to make the flip map O simplicial. *
*In
principle, we could have used the same dS1 in section 2 as here, but this would*
* have
entailed the cost of discussing the anti-simplicial involution ~0[~1| . .|.~q]~*
*q+1 7!
~q+1[~q| . .|.~1]~0 of B*( ), and complicating the formula (2.1) for the chain*
* level
coproduct _. We choose instead to suppress this point.
We define a corresponding "double model" for T HH(R), denoted dT HH(R), by
dT HH(R) = R dS1 .
The pinch and flip maps now induce the following natural maps of commutative
S-algebras:
_0:dT HH(R) ! T HH(R) ^R T HH(R)
(3.7) 0
O :dT HH(R) ! dT HH(R) .
Lemma 3.8. Let R be cofibrant as an S-module. Then the collapse map ss :dS1 !
S1 that takes the second 1 to * induces a weak equivalence
ss :dT HH(R) -'!T HH(R) .
Proof. Consider the commutative diagram
B(R) oo____R ^ R _____//_B(R)
|| || |
|| || |
|| || fflffl|
B(R) oo____R ^ R _______//_R
12 VIGLEIK ANGELTVEIT AND JOHN ROGNES
of commutative S-algebras. Here B(R) = B(R, R, R) = R 1 is the two-sided bar
construction, its augmentation B(R) ! R is a weak equivalence, and the inclusion
R ^ R ! B(R) is a cofibration of S-modules. From [EKMM97, III.3.8] we know
that the categorical pushout (balanced smash product) in this case preserves we*
*ak
equivalences. Pushout along the upper row gives dT HH(R) and pushout along the
lower row gives T HH(R), so the induced map ss :dT HH(R) ! T HH(R) is indeed
a weak equivalence.
Theorem 3.9. Let R be a commutative S-algebra. Its topological Hochschild ho-
mology T HH(R) is naturally an augmented commutative R-algebra, with unit,
counit and product maps j, ffl and OE defined as in (3.3) above. In the stable*
* ho-
motopy category, these maps, the coproduct map
_ = _0O ss-1 :T HH(R) ! T HH(R) ^R T HH(R)
and the conjugation map
O = ss O O0O ss-1 :T HH(R) ! T HH(R)
naturally make T HH(R) an R-Hopf algebra.
Proof. To check that T HH(R) is indeed a Hopf algebra over R in the stable ho-
motopy category, we must verify that a number of diagrams commute. We will do
one case that illustrates the technique, and leave the rest to the reader.
Let T = T HH(R). In order to show that the diagram
(3.10) T ________//_KKKTK^R T
_| KKKfflK KKKid^OKK
| KKK KKKK
fflffl| KK%% %%
T ^R TK R KK T ^R T
KK KKKj
KKK KKK |OE
O^idKKKK%% KKK |
T ^ K%%fflffl|//
R T ____OE___T
commutes in the stable homotopy category it suffices to check that the diagram *
*of
simplicial sets
_1
tS1 ________//MMS1 _MdS1M
_2 | MMMfflMMM MMid_O0MMM
fflffl|| MMM MMM&&
1 1 MM&& 1 1
dS _ SM * MMM S _ dS
MM MMM j
MMM MMM |OE(id_ss)
O0_idMMMM&& MMM&&Mfflffl||
dS1 _ S1 _OE(ss_id)//_S1
homotopy commutes (simplicially).
HOPF ALGEBRA STRUCTURE ON T HH 13
Here tS1 = @ 2 is a "triple model" for S1, with three non-degenerate 1-simpl*
*ices.
The pinch map _1 identifies the vertices 0 and 1 in @ 2 and takes the face ffi0*
* to
the first 1 in dS1. Then the composite OE(id _ ss)(id _ O0)_1 factors as
@ 2 2 -s1! 1 ! S1 ,
and 1 is simplicially contractible. Similarly, _2 identifies the vertices 1 an*
*d 2 and
takes ffi2 to the first 1 in dS1. Then OE(ss _ id)(O0_ id)_2 factors as
@ 2 2 -s0! 1 ! S1 ,
and again this map is simplicially contractible.
Finally we use a weak equivalence tT HH(R) = R tS1 ! T HH(R), as in
lemma 3.8, to deduce that the diagram (3.10) indeed homotopy commutes.
Remark 3.11. As noted above, T HH(R) is commutative as an R-algebra. The
pinch map _ :dS1 ! S1 _ S1 is not homotopy cocommutative, so we do not expect
that the coproduct _ :T HH(R) ! T HH(R) ^R T HH(R) will be cocommutative
in any great generality.
The inclusion of the base point j :* ! S1 induces a cofiber sequence of R-
modules 1^j
R = R ^ *+ - --+! R ^ S1+-j!R ^ S1 = R
which is canonically split by the retraction map
R ^ S1+-1^ffl+--!R ^ *+ = R .
Hence in the stable homotopy category there is a canonical section ~: R ! R^S1+
to the map labeled j above. We let
(3.12) oe = ! O ~: R ! R ^ S1+! T HH(R)
be the composite map. It induces an operator oe :H*( R; Fp) ! H*(T HH(R); Fp),
which we in proposition 4.8 shall see is compatible with that of proposition 2.*
*4.
4. The B"okstedt spectral sequence
Let R be an S-algebra. To calculate the mod p homology H*(T HH(R); Fp) of
its topological Hochschild homology, B"okstedt constructed a strongly convergent
spectral sequence
(4.1) E2s,*(R) = HHs(H*(R; Fp)) =) Hs+*(T HH(R); Fp) ,
using the skeleton filtration on T HH(R). In fact,
E1s,*(R) = H*(R^(s+1); Fp) ~=H*(R; Fp) (s+1) = Cs(H*(R; Fp))
equals the Hochschild s-chains of the algebra = H*(R; Fp) over k = Fp, and the
d1-differential can as usual be identified with the Hochschild boundary operato*
*r @.
(To be quite precise, the E1-term is really the associated normalized complex
~ s, with ~ = =k, but this change does not affect the E2-term.)
This is naturally a spectral sequence of A*-comodules, where A* = H*(HFp; Fp)
is the dual of the mod p Steenrod algebra, since it is obtained by applying mod*
* p
homology to a filtered spectrum. If R is a commutative S-algebra, the spectral
sequence admits more structure.
14 VIGLEIK ANGELTVEIT AND JOHN ROGNES
Proposition 4.2. Let R be a commutative S-algebra. Then:
(a) H*(T HH(R); Fp) is an augmented commutative A*-comodule H*(R; Fp)-
algebra.
(b) The B"okstedt spectral sequence Er**(R) is an augmented commutative A*-
comodule H*(R; Fp)-algebra spectral sequence, converging to H*(T HH(R); Fp).
Proof. Assume that R is a commutative S-algebra. Then T HH(R) is an augmented
commutative R-algebra by theorem 3.9, and the relevant structure maps (3.3) are*
* all
maps of simplicial spectra. Hence they respect the skeleton filtration on T HH(*
*R),
and we have in particular a composite map of spectral sequences
Er**(R) Er**(R) ! 0Er**-OE!Er**(R)
with 0Er**the spectral sequence associated to the skeleton filtration on T HH(R*
*)^R
T HH(R). The left hand map is induced by the usual homology cross product from
the E1-term and onwards. This defines the algebra structure on Er**(R), and the
remainder is straightforward.
Corollary 4.3. If H*(R; Fp) is a polynomial algebra over Fp, then Er**(R) col-
lapses at the E2-term, so E2**(R) = E1**(R). Furthermore, there are no nontrivi*
*al
H*(R; Fp)-module extensions.
Proof. If H*(R; Fp) = P (xi) is a polynomial algebra, where the index i ranges
through some set I, then E2**(R) = HH*(P (xi)) = P (xi) E(oexi) by proposi-
tion 2.4 (and passage to colimits). All the H*(R; Fp)-algebra generators are in*
* filtra-
tion n = 1, so all differentials on these classes are zero, since the B"okstedt*
* spectral
sequence is a right half plane homological spectral sequence. Hence the E1 -term
is a free H*(R; Fp)-module, and so also H*(T HH(R); Fp) ~=H*(R; Fp) E(oexi) is
a free H*(R; Fp)-module.
There may, as we shall see in section 5, be multiplicative extensions between
E1**(R) and H*(T HH(R); Fp), as well as A*-comodule extensions.
A flatness hypothesis is required for the spectral sequence to carry the cop*
*roduct
and full Hopf algebra structure. Our sections 5, 6 and 7 will show many examples
of B"okstedt spectral sequences with this structure.
Theorem 4.4. Let R be a commutative S-algebra and write = H*(R; Fp).
(a) If H*(T HH(R); Fp) is flat over , then there is a coproduct
_ :H*(T HH(R); Fp) ! H*(T HH(R); Fp) H*(T HH(R); Fp)
and H*(T HH(R); Fp) is an A*-comodule -Hopf algebra.
(b) If each term Er**(R) for r 2 is flat over , then there is a coproduct
_ :Er**(R) ! Er**(R) Er**(R)
and Er**(R) is an A*-comodule -Hopf algebra spectral sequence. In particular, *
*the
differentials dr respect the coproduct _.
HOPF ALGEBRA STRUCTURE ON T HH 15
Proof. Write T = T HH(R). There is a K"unneth spectral sequence with
E2**= Tor**(H*(T ; Fp), H*(T ; Fp)) =) H*(T ^R T ; Fp) .
When H*(T ; Fp) is flat over the higher Tor-groups vanish, the spectral seque*
*nce
collapses, and the map
_ :H*(T ; Fp) ! H*(T ^R T ; Fp) ~=H*(T ; Fp) H*(T ; Fp)
induces the coproduct in part (a).
For part (b) let dEr**be the spectral sequence associated to the skeleton fi*
*ltration
on dT HH(R) = R dS1, and let 0Er**be the spectral sequence associated to
T ^R T = R (S1 _ S1), as in the proof of proposition 4.2. Then there are natu*
*ral
maps of spectral sequences
0 sh
(4.5) Er**(R)- ssdEr**-_! 0Er**- Er**(R) Er**(R) .
Here ss :dEr**! Er**(R) is an isomorphism for r 2, by the algebraic analogue *
*of
lemma 3.8. The map _0: dEr**! 0Er**is induced by the simplicial pinch map _0
from (3.7). As regards the final (shuffle) map sh, we have
0E1s,*~= (s+1) (s+1) ~=E1s,*(R) E1s,*(R)
by the collapsing K"unneth spectral sequence for H*(T HHs(R) ^R T HHs(R); Fp),
and the map sh for r = 1 is the shuffle equivalence from the bigraded tensor pr*
*oduct
[E1**(R) E1**(R)]s,*. By assumption E2**(R) = HH*( ) is flat over , so by t*
*he
algebraic K"unneth spectral sequence and the Eilenberg-Zilber theorem
~= 0 1 02
sh: E2**(R) E2**(R) ~=H*(E1**(R) E1**(R)) -! H*( E**) = E**
is an isomorphism. Inductively, suppose that sh is an isomorphism for a fixed r*
* 2,
and that Er+1**(R) is flat over . Then by the algebraic K"unneth spectral sequ*
*ence
again
~= 0 r 0r+1
sh: Er+1**(R) Er+1**(R) ~=H*(Er**(R) Er**(R)) -! H*( E**) = E**
is also an isomorphism, as desired. The coproduct _ on Er**(R) is then the comp*
*os-
ite map (sh)-1 O _0O ss-1 , for r 2. The conjugation O on Er**(R) is more sim*
*ply
defined, as the composite map ss O O0O ss-1 .
Remark 4.6. The same proof shows that if only an initial sequence of terms
E2**(R), . .,.Er0**(R)
of the B"okstedt spectral sequence are flat over , then these are all A*-comod*
*ule
-Hopf algebras and the differentials d2, . .,.dr0 respect that structure.
By proposition 2.3, the coproduct (and Hopf algebra structure) on the E2-term
E2**(R) = HH*(H*(R; Fp)) that is derived from the R-Hopf algebra structure on
16 VIGLEIK ANGELTVEIT AND JOHN ROGNES
T HH(R) agrees with the algebraically defined structure on the Hochschild homol-
ogy HH*( ) of the commutative algebra = H*(R; Fp).
There are natural examples that show that the flatness hypothesis is not alw*
*ays
realistic. For example, Ausoni [Au] has studied the case of R = ku at an odd pr*
*ime
p, where H*(ku; Fp) = H*(`; Fp) Pp-1 (x) for a class x in degree 2, and alrea*
*dy
the E2-term E2**(ku) = HH*(H*(ku; Fp)) is not flat over H*(ku; Fp).
We shall also see in proposition 7.13(c) that for R = j, the connective, real
image of J-spectrum at p = 2, the E2-term E2**(j) = HH*(H*(j; F2)) is not flat
over H*(j; F2).
We say that the graded k-algebra is connected when it is trivial in negati*
*ve
degrees and the unit map j :k ! is an isomorphism in degree 0.
Proposition 4.7. Let R be a commutative S-algebra with = H*(R; Fp) connected
and such that HH*( ) is flat over . Then the E2-term of the first quadrant
B"okstedt spectral sequence
E2**(R) = HH*( )
is an A*-comodule -Hopf algebra, and a shortest non-zero differential drs,tin *
*lowest
total degree s + t, if there exists any, must map from an algebra indecomposabl*
*e to
a coalgebra primitive and A*-comodule primitive, in HH*( ).
Proof. If d2, . .,.dr-1 are all zero, then E2**(R) = Er**(R) is still an A*-com*
*odule
-Hopf algebra. If dr(xy) 6= 0, where xy is decomposable (a product), then the
Leibniz formula
dr(xy) = dr(x)y xdr(y)
implies that dr(x) 6= 0 or dr(y) 6= 0, so xy cannot be in the lowest possible t*
*otal
degree for the source of a differential.PDually, if dr(z) is not (coalgebra) pr*
*imitive,
with _(z) = z 1 + 1 z + iz0i z00i, then the co-Leibniz formula
_ O dr = (dr 1 1 dr)_
(tensor products over ) implies that some term dr(z0i) 6= 0 or dr(z00i) 6= 0, *
*so z
cannot be in the lowest possiblePtotal degree. Finally, if dr(z) is not A*-como*
*dule
primitive, with (z) = 1 z + iai zi, then the co-linearity condition
O dr = (1 dr)
implies that some term dr(zi) 6= 0, so z cannot be in the lowest possible total
degree. (The last two arguments are perhaps easier to visualize in the Fp-vect*
*or
space dual spectral sequence.)
Proposition 4.8. Let R be a commutative S-algebra. For each element x 2
Ht(R; Fp) the image oex 2 Ht+1(T HH(R); Fp) is a coalgebra primitive
_(oex) = oex 1 + 1 oex
that is represented in Er**(R) by the class oex = [1 x] 2 E21,t(R) = HH1( )t.
Proof. Note that the coproduct on T HH(R) is compatible under ! :R ^ S1+ !
T HH(R) with the pinch map R ^ dS1+! R ^ (S1 _ S1)+ , and thus under oe : R !
T HH(R) with the pinch map R ! R_ R. The claims then follow by inspection
of the definitions in section 3.
HOPF ALGEBRA STRUCTURE ON T HH 17
5. Differentials and algebra extensions
We now apply the B"okstedt spectral sequence (4.1) to compute the mod p ho-
mology of T HH(R) for the S-algebras R = BP for -1 n 1, in the cases
when these are commutative. In each case we can replace R by its p-localization
R(p)or p-completion Rp without changing the mod p homology of T HH(R), so we
will sometimes do so without further comment. The cases R = HFp and R = HZ
were first treated by B"okstedt [B"o2], and the case R = `p kup (the Adams su*
*m-
mand of p-complete connective K-theory) is due to McClure and Staffeldt [MS93,
4.2].
5.1. The dual Steenrod algebra. Let A = H*(HFp; Fp) be the Steenrod alge-
bra, with generators Sqi for p = 2 and fi and P ifor p odd. We recall the struc*
*ture
of its dual A* = H*(HFp; Fp) from [Mi60, Thm. 2]. When p = 2 we have
A* = P (,k | k 1) = P (~,k| k 1)
where ,k has degree 2k - 1 and ~,k= O(,k) is the conjugate class. Most of the t*
*ime
it will be more convenient for us to use the conjugate classes. The coproduct *
*is
given by X i
_(~,k) = ,~i ~,2j,
i+j=k
where as usual we read ~,0to mean 1. When p is odd we have
A* = P (~,k| k 1) E(~ok| k 0)
with ~,k= O(,k) in degree 2(pk -1) and ~ok= O(ok) in degree 2pk -1. The coprodu*
*ct
is given by
X pi X pi
_(~,k) = ,~i ~,j and _(~ok) = 1 ~ok+ ~oi ~,j.
i+j=k i+j=k
The mod p homology Bockstein satisfies fi(~ok) = ~,k.
Any commutative S-algebra R has a canonical structure as an E1 ring spectrum
[EKMM97, II.3.4]. In particular, its mod p homology H*(R; Fp) admits natural
Dyer-Lashof operations
Qk :H*(R; F2) ! H*+k (R; F2)
for p = 2 and
Qk :H*(R; Fp) ! H*+2k(p-1)(R; Fp)
for p odd. Their formal properties are summarized in [BMMS86, III.1.1], and
include Cartan formulas, Adem relations and Nishida relations. For p = 2, Qk(x)*
* =
0 when k < |x| and Qk(x) = x2 when k = |x|. For p odd, Qk(x) = 0 when k < 2|x|
and Qk(x) = xp when k = 2|x|. In the special case of R = HFp, the Dyer-Lashof
operations in A* = H*(HFp; Fp) satisfy
k
Qp (~,k) = ~,k+1
for all primes p, and k
Qp (~ok) = ~ok+1
for p odd. These formulas were first obtained by Leif Kristensen (unpublished),
and appeared in print in [BMMS86, III.2.2 and III.2.3].
18 VIGLEIK ANGELTVEIT AND JOHN ROGNES
5.2. The Johnson-Wilson spectra BP . For any prime p let R = BP for
-1 n 1 be the spectrum introduced by Johnson and Wilson in [JW73], with
mod p cohomology H*(R; Fp) ~=A==En = A En Fp. Here En A is the subalgebra
generated by the Milnorkprimitives Q0, . .,.Qn, which are inductively definedkby
Q0 = Sq1 and Qk = [Sq2 , Qk-1 ] for p = 2, and by Q0 = fi and Qk+1 = [P p , Qk]
for p odd, see [Mi60, x1].
This spectrum has homotopy groups
ss*BP = Z(p)[v1, . .,.vn] = ss*BP=(vk | k > n) ,
where ss*BP = Z(p)[vk | k 1] with v0 = p. The class vk is detected in the
Adams spectral sequence E**2= Ext **A*(Fp, H*(BP ; Fp)) for ss*BP by the normal-
P i
ized cobar cocycle i+j=k+1[~,i]~,2jfor p = 2 (the term for i = 0 is zero) and
P pi
- i+j=k[~oi]~,j for p odd [Ra04, p. 63]. Under the change-of-rings isomorphis*
*m to
E**2~= Ext**E*(Fp, Fp), where E* = E(~,k| k 1) for p = 2 and E(~ok| k 0) for
p odd, these cobar cocycles correspond to [~,k+1] and -[~ok], respectively. Mo*
*dulo
decomposables, we have ~,k+1 ,k+1 for p = 2 and -~ok ok for p odd.
The special cases BP <-1> = HFp, BP <0> = HZ(p), BP <1> = ` ku(p) (the
Adams summand in p-local connective topological K-theory) and BP <1> = BP
MU(p) (the p-local Brown-Peterson spectrum) have been previously studied. For
p = 2 we emphasize that ` = ku(2).
In each case -1 n 1, the spectrum BP admits the structure of an
S-algebra, see e.g. [BJ02, 3.5]. It is well known that HFp and HZ(p)admit unique
structures as commutative S-algebras, and that the p-complete Adams summand
`p kup admits at least one such structure [MS93, x9]. It remains a well-known
open problem whether BP is a commutative S-algebra for 2 n 1. For our
purposes it will suffice if the p-completion BP p admits such a structure.
Proposition 5.3. For -1 n 1 the unique map of S-algebras R = BP !
HFp identifies = H*(R; Fp) with the following sub Hopf algebra of A*:
H*(BP ; F2) = P (~,21, . .,.~,2n+1, ~,k| k n + 2)
when p = 2, and
H*(BP ; Fp) = P (~,k| k 1) E(~ok| k n + 1)
when p is odd.
Proof. See [Wi75, 1.7] and dualize.
In particular, H*(HZ; F2) = P (~,21, ~,k| k 2), H*(ku; F2) = P (~,21, ~,22*
*, ~,k| k
3), H*(BP ; F2) = P (~,2k| k 1) for p = 2, and H*(HZ; Fp) = P (~,k| k 1) E(*
*~ok|
k 1), H*(`; Fp) = P (~,k| k 1) E(~ok| k 2), H*(BP ; Fp) = P (~,k| k 1*
*) for
p odd.
The E2-term of the B"okstedt spectral sequence for BP can now be computed
from proposition 2.4. It is
E2**(BP ) = H*(BP ; F2) E(oe,~21, . .,.oe,~2n+1, oe,~k| k n +*
* 2)
HOPF ALGEBRA STRUCTURE ON T HH 19
when p = 2, and
E2**(BP ) = H*(BP ; Fp) E(oe,~k| k 1) (oe~ok| k n + 1)
when p is odd.
In all the cases for p = 2, as well as the case R = BP for p odd, the E2-term
is generated as an algebra by classes in filtration 1. Hence if furthermore n i*
*s such
that R = BP p is a commutative S-algebra (which includes the cases n 1) the
B"okstedt spectral sequence collapses at the E2-term by corollary 4.3, so E2 = *
*E1 .
The added hypothesis that BP is commutative is unrealistic for p = 2 and
n 2, by [St99, 6.5], but there is a replacement spectrum BP 0 that may ser*
*ve
as a substitute, see [St99, 2.10].
5.4. Odd-primary differentials. In the remaining cases, with p odd and n <
1, there are non-trivial dp-1 -differentials in the B"okstedt spectral sequence*
* for
T HH(BP ). These are all determined by naturality with respect to the map of
S-algebras BP ! HFp and the case R = HFp, since the map of E2-terms
E2**(BP ) ! E2**(HFp) = A* E(oe,~k| k 1) (oe~ok| k 0)
is injective. Now HFp is a connective, commutative S-algebra, and the E2-term is
free, hence flat, over A* = H*(HFp; Fp), so we are in the situation of proposit*
*ion 4.7.
The algebra generators (indecomposables) of the E2-term are in filtrations pi f*
*or
i 0, by formula (2.5), and the coalgebra primitives are all in filtration 1, *
*by
proposition 2.4, so the shortest non-zero differential in lowest total degree, *
*if any,
must go from some filtration pi to filtration 1.
Indeed, B"okstedt [B"o2, 1.3] found that in the case R = HFp there are non-
trivial dp-1 -differentials in his spectral sequence computing H*(T HH(R); Fp).*
* In
our notation, they are given by the formula
(5.5) dp-1 (flj(oe~ok)) = oe,~k+1. flj-p(oe~ok)
for j p, up to units in Fp. This way of writing B"okstedt's formula first app*
*ears
in [MS93, p. 21]. A proof of a more general result that implies (5.5) was publi*
*shed
by Hunter [Hu96, Thm. 1], as recalled below. A short, direct proof of B"oksted*
*t's
formula, in some particular cases, was later found by Ausoni [Au].
Proposition 5.6. Let R be a commutative S-algebra, and x 2 H2i-1(R; Fp) with
i 1. Then in the B"okstedt spectral sequence Er**(R) the differentials dr van*
*ish for
2 r p - 2, and there is a differential
dp-1 (flp(oex)) = oe(fiQi(x))
up to a unit in Fp.
In the case R = HFp with x = ~ok, we get i = pk and fiQi(x) = ~,k+1. This im*
*plies
B"okstedt's formula (5.5) for j = p. The general case follows from this by indu*
*ction
on j p and the coalgebra structure on the B"okstedt spectral sequence. (In mo*
*re
detail, a comparison of _dp-1 (flj(oe~ok)) = (dp-1 1 + 1 dp-1 )(_flj(oe~ok)*
*) and
20 VIGLEIK ANGELTVEIT AND JOHN ROGNES
_(oe,~k+1. flj-p(oe~ok)) shows that the difference dp-1 (flj(oe~ok)) - oe,~k+1.*
* flj-p(oe~ok)
must be a coalgebra primitive, and there are none such other than zero in its
bidegree when j > p.)
By naturality, the formula (5.5) for dp-1 holds also in the B"okstedt spect*
*ral
sequence for BP at odd primes p (whether this S-algebra is commutative or n*
*ot).
Consider its E2 = Ep-1 -term as the tensor product of the complexes E(oe,~k+1)
(oe~ok) for k n + 1 and the remaining terms H*(BP ; Fp) E(oe,~1, . .,.o*
*e,~n+1).
Applying the K"unneth formula, we compute its homology to be
Ep**(BP ) = H*(BP ; Fp) E(oe,~1, . .,.oe,~n+1) Pp(oe~ok| k n + *
*1) .
At this stage the map Ep**(BP ) ! Ep**(HFp) is no longer injective, so we
seem obliged to assume that BP p is commutative. Then the Ep-term above
is generated as an algebra over H*(BP ; Fp) by classes in filtration 1, and *
*the
B"okstedt spectral sequence therefore collapses at Ep = E1 .
Proposition 5.7. Let p be any prime and -1 n 1 be such that R = BP p
admits the structure of a commutative S-algebra (e.g., n 1). Then the B"okste*
*dt
spectral sequence for R collapses at the Ep = E1 -term, which equals
E1**(BP ) = H*(BP ; F2) E(oe,~21, . .,.oe,~2n+1, oe,~k| k n +*
* 2)
when p = 2, and
E1**(BP ) = H*(BP ; Fp) E(oe,~k| 1 k n + 1) Pp(oe~ok| k n + 1)
when p is odd.
5.8. Algebra extensions. The E1 -terms above compute H*(T HH(BP ; Fp))
as a (free) = H*(BP ; Fp)-module. To determine the rest of the -Hopf alge*
*bra
structure, we need to resolve the possible algebra extensions. For this we use
the Dyer-Lashof operations that stem from the (assumed) commutative S-algebra
structure on T HH(BP ).
The map oe : R ! T HH(R) relates the Dyer-Lashof operations on R to those
of T HH(R), by the following formula of B"okstedt [B"o2, 2.9]:
Qk(oex) = oeQk(x) .
See also [MS93, p. 22]. There appears to be no published proof of this key rela*
*tion,
so we offer the following, slightly more general result. Let ff :T HH(R) ^ S1+*
* !
T HH(R) be the S1-action map, inducing the homomorphism
ff :H*(T HH(R); Fp) H*(S1+; Fp)
~=H*(T HH(R) ^ S1+; Fp) ! H*(T HH(R); Fp)
in homology. Let s1 2 H1(S1+; Fp) be the canonical generator.
HOPF ALGEBRA STRUCTURE ON T HH 21
Proposition 5.9. Let R be a commutative S-algebra. Then we have
Qk(ff(x s1)) = ff(Qk(x) s1)
for all integers k and classes x 2 H*(T HH(R); Fp). In particular, we have
Qk(oex) = oeQk(x)
for all integers k and classes x 2 H*(R; Fp).
Proof. The circle acts on T HH(R) by commutative S-algebra maps, so the right
adjoint map
eff:T HH(R) ! F (S1+, T HH(R))
is a map of commutative S-algebras, where the product structure on the right is
given by pointwise multiplication, using the diagonal map S1+! S1+^ S1+.
Let DS1+ = F (S1+, S) be the functional dual of S1+, also with the pointwise
multiplication. It has mod p homology H*(DS1+; Fp) ~= H-* (S1+; Fp) = E('1) for
a canonical class '1 2 H1(S1+; Fp) dual to the class s1 2 H1(S1+; Fp). The Dye*
*r-
Lashof operations Qk on H*(DS1+; Fp) correspond [BMMS86, III.1.2 and VIII.3] to
the Steenrod operations P -k on H-* (S1+; Fp), hence are trivial for k 6= 0. (*
*And
similarly, with different notation, when p = 2.)
There is a canonical map of commutative S-algebras
:T HH(R) ^ DS1+! F (S1+, T HH(R)) ,
given by the composition of functions
F (S, T HH(R)) ^ F (S1+, S) ! F (S1+, T HH(R)) .
Compare [LMS86, III.1]. The map is an equivalence since S1+ is a finite CW
complex (i.e., by Spanier-Whitehead duality). Hence there are homomorphisms
H*(T HH(R); Fp) -eff!H*(F (S1+, T HH(R)); Fp)
- 1
~= H*(T HH(R); Fp) H*(DS+ ; Fp)
that take x 2 H*(T HH(R); Fp) to
-1 eff(x) = x 1 + ff(x s1) '1 .
Since effand are maps of commutative S-algebras, we have
Qk( -1 eff(x)) = -1 eff(Qk(x)) .
The Cartan formula for Dyer-Lashof operations [BMMS86, III.1.1(6)] then gives
us
Qk(x 1 + ff(x s1) '1) = Qk(x) 1 + Qk(ff(x s1)) '1
since Qi(1) = 0 and Qi('1) = 0 for i 6= 0. Hence we have
Qk(x) 1 + Qk(ff(x s1)) '1 = Qk(x) 1 + ff(Qk(x) s1) '1
and can read off Qk(ff(x s1)) = ff(Qk(x) s1), as desired.
Specializing to classes x 2 H*(R; Fp), which map under j :R ! T HH(R) to
j(x) 2 H*(T HH(R); Fp), we have oex = ff(j(x) s1), and thus obtain B"okstedt's
formula Qk(oex) = oeQk(x).
The same ideas can be used to prove that oe :H*(R; Fp) ! H*+1 (T HH(R); Fp)
is a graded derivation.
22 VIGLEIK ANGELTVEIT AND JOHN ROGNES
Proposition 5.10. Let R be a commutative S-algebra. Then we have
ff(xy s1) = x . ff(y s1) + (-1)|y|ff(x s1) . y
for x, y 2 H*(T HH(R); Fp). In particular, we have the Leibniz rule
oe(x . y) = x . oe(y) + (-1)|y|oe(x) . y
for x, y 2 H*(R; Fp).
Proof. We keep the notation of the proof of proposition 5.9. Since "ffand are*
* maps
of (commutative) S-algebras, -1 "ff:H*(T HH(R); Fp) ! H*(T HH(R); Fp) E('1)
is an algebra homomorphism. Thus
xy 1 + ff(xy s1) '1 = (x 1 + ff(x s1) '1) . (y 1 + ff(y s1) *
* '1)
for x, y 2 H*(T HH(R); Fp). Multiplying out and comparing '1-coefficients gives
the claimed formulas.
We also wish to describe the A*-comodule structure on H*(T HH(R); Fp). In the
cases R = BP the following observations will suffice. The A*-comodule coact*
*ion
map
:H*(R; Fp) ! A* H*(R; Fp)
is in each case given by restricting the coproduct
_ :A* ! A* A*
given in subsection 5.1, to the subalgebra H*(R; Fp) A*, since the latter inc*
*lusion
is induced by a spectrum map R ! HFp. The operator
oe :H*( R; Fp) ! H*(T HH(R); Fp)
is induced by a spectrum map, hence is an A*-comodule homomorphism. Hence
the coaction map
:H*(T HH(R); Fp) ! A* H*(T HH(R); Fp)
satisfies
(5.11) O oe = (1 oe) .
A class x is called an A*-comodule primitive if (x) = 1 x.
Theorem 5.12. Let p be any prime and -1 n 1 be such that BP p admits
the structure of a commutative S-algebra (e.g., n 1). Then for n < 1 we have
H*(T HH(BP ); F2) = H*(BP ; F2) E(oe,~21, . .,.oe,~2n+1) P (oe,~n*
*+2)
HOPF ALGEBRA STRUCTURE ON T HH 23
when p = 2, and
H*(T HH(BP ); Fp) = H*(BP ; Fp) E(oe,~1, . .,.oe,~n+1) P (oe~on+1)
when p is odd, as primitively generated H*(BP ; Fp)-Hopf algebras.
For n = 1 we have
H*(T HH(BP ); F2) = H*(BP ; F2) E(oe,~2k| k 1)
when p = 2, and
H*(T HH(BP ); Fp) = H*(BP ; Fp) E(oe,~k| k 1)
when p is odd, as primitively generated H*(BP ; Fp)-Hopf algebras.
The A*-comodule coaction on H*(T HH(BP ); Fp) is given on H*(BP ; Fp)
by restricting the coproduct on A*. For p = 2 the algebra generators oe,~2kfor*
* 1
k n + 1 are A*-comodule primitives, while
(oe,~n+2) = 1 oe,~n+2+ ~,1 oe,~2n+1.
For p odd the algebra generators oe,~kfor 1 k n + 1 are A*-comodule primiti*
*ves,
while
(oe~on+1) = 1 oe~on+1+ ~o0 oe,~n+1.
(As usual, ~,0is read as 1 in such formulas. Thus for n = -1, oe,~1and oe~o0*
*are
also primitive.)
Proof. We first resolve the algebra extensions. For p = 2 we find that the squa*
*res
in H*(T HH(R); F2) of the algebra generators in E1**(R) are
k+1-1 2 2k+1-1 2
(oe,~2k)2 = Q2 (oe,~k) = oeQ (~,k) = 0
for k = 1, . .,.n + 1 and
k 2k
(oe,~k)2 = Q2 (oe,~k) = oeQ (~,k) = oe,~k+1
for k n + 2. In the first case we have used the easy consequence Qk(y2) = 0 of
the Cartan formula, for p = 2 and k odd.
For p odd the classes oe,~kremain exterior, since they are of odd degree in *
*the
graded commutative algebra H*(T HH(R); Fp). The p-th powers of the truncated
polynomial generators in E1**(R) are
k pk
(oe~ok)p = Qp (oe~ok) = oeQ (~ok) = oe~ok+1
for k n+1. Hence these assemble to a polynomial algebra on the single generat*
*or
oe~on+1.
All the algebra generators are of the form oex, in Hochschild filtration 1, *
*and are
primitive by proposition 4.8.
24 VIGLEIK ANGELTVEIT AND JOHN ROGNES
To compute the A*-comodule structure we use that oe is a graded derivation, *
*see
proposition 5.10, so oe(yp) = 0 and oe(1) = 0. Then for p = 2 and 1 k n + 1*
* we
have X i+1
(oe,~2k) = ,~2i oe,~2j = 1 oe,~2k
i+j=k
while X i
(oe,~n+2) = ~,i oe,~2j= 1 oe,~n+2+ ~,1 oe,~2n+1.
i+j=n+2
For p odd and 1 k n + 1 we get
X pi
(oe,~k) = ,~i oe,~j = 1 oe,~k
i+j=k
while
X pi
(oe~on+1) = 1 oe~on+1+ ~oi oe,~j = 1 oe~on+1+ ~o0 oe,~n+1.
i+j=n+1
For later reference we extract the following special cases, which correspond*
* to
n = 1. Recall that H*(ku; F2) = (A==E1)* A* and H*(`; Fp) = (A==E1)* A*.
Corollary 5.13. (a) There is an isomorphism
H*(T HH(ku); F2) ~=H*(ku; F2) E(oe,~21, oe,~22) P (oe,~3)
of primitively generated H*(ku; F2)-Hopf algebras.
The A*-comodule coaction :H*(T HH(ku); F2) ! A* H*(T HH(ku); F2) is
given on H*(ku; F2) by restricting the coproduct _ :A* ! A* A*, and on the
algebra generators by (oe,~21) = 1 oe,~21, (oe,~22) = 1 oe,~22and
(oe,~3) = 1 oe,~3+ ~,1 oe,~22.
(b) There is an isomorphism
H*(T HH(`); Fp) ~=H*(`; Fp) E(oe,~1, oe,~2) P (oe~o2)
of primitively generated H*(`; Fp)-Hopf algebras.
The A*-comodule coaction :H*(T HH(`); Fp) ! A* H*(T HH(`); Fp) is given
on H*(`; Fp) by restricting the coproduct _ :A* ! A* A*, and on the algebra
generators by (oe,~1) = 1 oe,~1, (oe,~2) = 1 oe,~22and
(oe~o2) = 1 oe~o2+ ~o0 oe,~2.
HOPF ALGEBRA STRUCTURE ON T HH 25
6. The higher real cases
For p = 2 there are a few more known examples of commutative S-algebras
such that H*(R; Fp) is a cyclic A-module. Let ko be the connective real K-theory
spectrum, with H*(ko; F2) ~= A==A1 = A A1 F2, and let tmf be the Hopkins-
Mahowald topological modular forms spectrum, with H*(tmf; F2) ~= A==A2 =
A A2 F2. Seene.g. [Re01, 21.5]. Here An A is the subalgebra generated by
Sq1, . .,.Sq2 , so A1 has rank 8 and A2 has rank 64. It is well known that ko
is a commutative S-algebra, and in the case of tmf this is a consequence of the
Hopkins-Miller theory, being presented by Goerss and Hopkins.
Proposition 6.1. There are maps of S-algebras tmf ! ko ! HF2 that induce
the following identifications:
H*(ko; F2) = P (~,41, ~,22, ~,k| k 3)
and
H*(tmf; F2) = P (~,81, ~,42, ~,23, ~,k| k 4) .
Proof. This is immediate by dualization from H*(ko; F2) ~=A==A1, cf. [St63], and
H*(tmf; F2) ~=A==A2.
We now follow the outline of section 5. By proposition 2.4 the E2-terms of t*
*he
respective B"okstedt spectral sequences are
E2**(ko) = H*(ko; F2) E(oe,~41, oe,~22, oe,~k| k 3)
and
E2**(tmf) = H*(tmf; F2) E(oe,~81, oe,~42, oe,~23, oe,~k| k 4) .
By corollary 4.3 both spectral sequences collapse at the E2-term, so E2**(R) =
E1**(R). To resolve the algebra extensions we use the Dyer-Lashof operations and
proposition 5.9. The squares in H*(T HH(ko); F2) of the algebra generators in
E1**(ko) are
(oe,~41)2= Q5(oe,~41) = oeQ5(~,41) = 0
(oe,~22)2= Q7(oe,~22) = oeQ7(~,22) = 0
by the formula Qk(y2) = 0 for p = 2 and k odd, and
k 2k
(oe,~k)2 = Q2 (oe,~k) = oeQ (~,k) = oe,~k+1
for all k 3. Similar calculations show that (oe,~81)2 = 0, (oe,~42)2 = 0 and *
*(oe,~23)2 = 0
in H*(T HH(tmf); F2), while (oe,~k)2 = oe,~k+1 for all k 4. The A*-comodule
coaction map on the resulting algebra generators is obtained from the coprodu*
*ct
on A* and formula (5.11), as in the proof of theorem 5.12. The result is as fol*
*lows:
26 VIGLEIK ANGELTVEIT AND JOHN ROGNES
Theorem 6.2. (a) There is an isomorphism
H*(T HH(ko); F2) ~=H*(ko; F2) E(oe,~41, oe,~22) P (oe,~3)
of primitively generated H*(ko; F2)-Hopf algebras. The A*-comodule structure is
given on H*(ko; F2) by restricting the coproduct on A*, and on the algebra gene*
*ra-
tors by (oe,~41) = 1 oe,~41and
(oe,~22)= 1 oe,~22+ ~,21 oe,~41
(oe,~3)= 1 oe,~3+ ~,1 oe,~22+ ~,2 oe,~41.
(b) There is an isomorphism
H*(T HH(tmf); F2) ~=H*(tmf; F2) E(oe,~81, oe,~42, oe,~23) P (oe,~4)
of primitively generated H*(tmf; F2)-Hopf algebras. The A*-comodule structure
is given on H*(tmf; F2) by restricting the coproduct on A*, and on the algebra
generators by (oe,~81) = 1 oe,~81and
(oe,~42)= 1 oe,~42+ ~,41 oe,~81
(oe,~23)= 1 oe,~23+ ~,21 oe,~42+ ~,22 oe,~81
(oe,~4)= 1 oe,~4+ ~,1 oe,~23+ ~,2 oe,~42+ ~,3 oe,~81.
Proof. Assemble the computations above.
7. The real and complex image of J
We now turn to the various image of J spectra that are commutative S-algebra*
*s.
Their mod p cohomology is no longer cyclic as a module over the Steenrod algebr*
*a,
but of rank 2, so extra work is needed to describe their homology as an A*-como*
*dule
algebra.
7.1. The image of J spectra. For any prime p let the p-local, connective compl*
*ex
image of J spectrum be ju = K(Fr)(p), where r = 3 for p = 2 and r is a prime
power that topologically generates the p-adic units for p odd. Being the locali*
*zed
algebraic K-theory of a field, ju is a commutative S-algebra [Ma77, VIII.3.1]. *
*For
p = 2 there is a cofiber sequence of spectra
3-1
(7.2) ju -~!ku(2)-_--! bu(2),
where bu ' 2ku is the 1-connected cover of ku and _3 is the Adams operation.
For odd p the cofiber sequence appears as
r-1
(7.3) ju -~!` -_--! q`
with q = 2p - 2, where ` ku(p)is the p-local, connective Adams summand and
_r is the r-th Adams operation [Ma77, V.5.16].
HOPF ALGEBRA STRUCTURE ON T HH 27
Let the 2-local, connective real image of J spectrum be j = KN (F3)(2), as in
[Ma77, VIII.3.1]. Being the localized algebraic K-theory of a symmetric bimonoi*
*dal
category, j is a commutative S-algebra. There is a cofiber sequence of spectra
3-1
(7.4) j -~!ko(2)-_--! bspin(2),
where bspin ' 4ksp is the 3-connected cover of ko [Ma77, V.5.16].
The fiber map ~: ju ! ` is a map of commutative S-algebras, at least after p-
adic completion, because there is a (discrete) model K(k0)p for `p with k0 a su*
*itable
subfield of the algebraic_closure of Fr, and applying the functor K(-)p to the *
*field
inclusions Fr k0 Fr produces the commutative S-algebra maps
jup -~!`p kup .
See [Ma77, VIII.3.2] and [MS93, x9]. Similarly, ~: j ! ko becomes a map of comm*
*u-_
tative S-algebras after 2-adic completion, since there is a (discrete)_model KO*
*(F 3)2
for ko2, and ~ can be identified with the natural map KN (F3)2 ! KO(F 3)2. See
[Ma77, VIII.2.6 and 3.2].
7.5. Cohomology modules. Recall that
H*(`; Fp) = A==E1 ~=Fp{1, P 1, . .,.P p, . .}.,
where E1 = E(Q0, Q1) A is the exterior algebra generated by Q0 = fi and
Q1 = [P 1, fi], and that
H*(ko; F2) = A==A1 ~=F2{1, Sq4, Sq2Sq4 Sq6, Sq1Sq2Sq4 Sq7, . .}.,
where A1 = A is the subalgebra generated by Sq1 and Sq2. There are
also A-module isomorphisms
H*(bo; F2) ~= A=ASq2
H*(bso; F2) ~= 2A=ASq3
H*(bspin; F2) ~= 4A=A{Sq1, Sq2Sq3} .
See [AP76, 2.5, 2.4 and p. 501]. Here Sq2Sq3 = Sq5 + Sq4Sq1 in admissible form,
but the shorter expression is perhaps more memorable.nFor-p1odd we let An A be
the subalgebra generated by fi, P 1, . .,.P p . In particular, A1 = *
* contains
E1, and
A==A1 ~=Fp{1, P p, P 1P p -P p+1, fiP 1P p Q2, . .}..
Lemma 7.6. (a) For p = 2 the map _3 - 1 :ku(2)! 2ku(2)induces right multi-
plication by Sq2 on mod 2 cohomology:
(_3 - 1)* = Sq2 : 2A==E1 ! A==E1 .
28 VIGLEIK ANGELTVEIT AND JOHN ROGNES
(b) For p odd the map _r - 1 :` ! q` induces right multiplication by P 1on
mod p cohomology:
(_r - 1)* = P 1: qA==E1 ! A==E1 .
(c) For p = 2 the map _3 - 1 :ko(2)! bspin(2)induces right multiplication by
Sq4 on mod 2 cohomology:
(_3 - 1)* = Sq4 : 4A=A{Sq1, Sq2Sq3} ! A==A1 .
Case (c) is due to Mahowald and Milgram [MaMi76, 3.4].
Proof. The S-algebra unit map e: S(p)! ju is well-known to be 2-connected for
p = 2 and (pq-2)-connected for p odd, since this is the degree of the first ele*
*ment fi1
in the p-primary cokernel of J [Ra04, 1.1.14]. Hence e* :H*(ju; Fp) ! H*(S; Fp)*
* =
Fp is 2-coconnected for p = 2 and (pq - 2)-coconnected for p odd, meaning that
the homomorphism is injective in the stated degree, and an isomorphism in lower
degrees. In particular, H2(ju; F2) = 0 for p = 2 and Hq(ju; Fp) = 0 for p odd.
So in the long exact cohomology sequence associated to the cofiber sequence *
*(7.2)
3-1)* ~*
(7.7) 2A==E1 -(_----! A==E1 -! H*(ju; F2)
the non-zero class of Sq2 in A==E1 maps to zero under ~*, hence is in the image
of (_3 - 1)*. The latter is a (left) A-module homomorphism, and can only take
2(1) to Sq2, hence is given by (right) multiplication by Sq2. This proves (a).*
* For
part (b) we use the same argument for the exact sequence
r-1)* ~*
(7.8) qA==E1 -(_----! A==E1 -! H*(ju; Fp)
associated to (7.3), in cohomological degree q. The non-zero class of P 1in A==*
*E1
maps to zero under ~*, hence must equal the image of q(1) under (_r - 1)*. This
proves (b).
The unit map e: S(2)! j is likewise well-known to be 6-connected, since this*
* is
the degree of the first element 2 in the 2-primary cokernel of J, so e* :H*(j;*
* F2) !
H*(S; F2) = F2 is 6-coconnected. In particular, H4(j; F2) = 0. So in the long e*
*xact
cohomology sequence associated to the cofiber sequence (7.4)
3-1)* ~*
(7.9) 4A=A{Sq1, Sq2Sq3} -(_----! A==A1 -! H*(j; F2)
the non-zero class of Sq4 in A==A1 maps to zero under ~*, hence is in the image*
* of
(_3 - 1)*. The only class that can hit it is 4(1), which proves (c).
Lemma 7.10. (a) For p = 2 there is a uniquely split extension of A-modules
0 ! A==A1 ! H*(ju; F2) ! 3A==A1 ! 0 .
Hence there is a canonical A-module isomorphism H*(ju; F2) ~=A==A1{1, x}, with
x a class in degree 3.
HOPF ALGEBRA STRUCTURE ON T HH 29
(b) For p odd there is a non-split extension of A-modules
0 ! A==A1 ! H*(ju; Fp) ! pq-1A==A1 ! 0 .
As an A-module, H*(ju; Fp) is generated by two classes 1 and x in degrees 0 and
(pq - 1), respectively, with fi(x) = P p(1).
(c) There is a (unique) non-split extension of A-modules
0 ! A==A2 ! H*(j; F2) ! A A2 7K ! 0 .
The cyclic A2-module K = A2=A2{Sq1, Sq7, Sq4Sq6+Sq6Sq4} has rank 17 over F2.
As an A-module, H*(j; F2) is generated by two classes 1 and x in degrees 0 and *
*7,
respectively, with Sq1(x) = Sq8(1).
For case (b), see also [Ro03, 5.1(b)]. Case (c) is due to Davis [Da75, Thm.*
* 1],
who also shows that H*(j; F2) is a free A==A3-module.
Proof. (a) In the long exact sequence (7.7) the A-module homomorphism (_3 - 1)*
is induced up from the A1-module homomorphism
Sq2 : 2A1==E1 ! A1==E1 = F2{1, Sq2}
with kernel 2F2{Sq2} and cokernel F2{1}. Since A is flat (in fact free) over A*
*1,
it follows that ker(_3 - 1)* ~= 4A==A1 and cok(_3 - 1)* ~=A==A1. Hence there is
an extension of A-modules
0 ! A==A1 ! H*(ju; F2) ! 3A==A1 ! 0 ,
as asserted. The group of such extensions is trivial, by the change of rings is*
*omor-
phism
Ext 1A( 3A==A1, A==A1) ~=Ext 1A1( 3F2, A==A1) .
For in any A1-module extension
0 ! A==A1 ! E ! 3F2 ! 0
let x 2 E be the unique class in degree 3 that maps to 3(1). Then Sq2x = 0 sin*
*ce
A==A1 = F2{1, Sq4, Sq6, Sq7, . .}.is trivial in degree 5. Furthermore Sq1x = Sq*
*4(1)
would contradict the Adem relation Sq2Sq2 = Sq3Sq1, since Sq3Sq4 Sq7 in
A==A1. So Sq1x = 0 and the extension E is trivial.
Two choices of splitting maps for the trivial extension describing H*(ju; F2)
differ by an A-module homomorphism 3A==A1 ! A==A1, which must be zero
since A==A1 is trivial in degree 3. Therefore the splitting is unique, as claim*
*ed.
(b) Similarly, in (7.8) the A-module homomorphism (_r -1)* is induced up from
the A1-module homomorphism
P 1: qA1==E1 ! A1==E1 = Fp{1, P 1, . .,.P p-1}
30 VIGLEIK ANGELTVEIT AND JOHN ROGNES
with kernel qFp{P p-1} and cokernel Fp{1}. As above it follows that ker(_r-1)**
* ~=
pqA==A1 and cok(_r - 1)* ~=A==A1. Hence there is an extension of A-modules
0 ! A==A1 ! H*(ju; Fp) ! pq-1A==A1 ! 0 ,
as asserted. This time the group of extensions is non-trivial; in fact it is is*
*omorphic
to Z=p and generated by the extension above. To see this, we again use the chan*
*ge
of rings isomorphism
Ext1A( pq-1A==A1, A==A1) ~=Ext 1A1( pq-1Fp, A==A1)
and consider A1-module extensions
0 ! A==A1 ! E ! pq-1Fp ! 0 .
Let x 2 E be the unique class that maps to pq-1(1). Then P 1x = 0 since
A==A1 = Fp{1, P p, P 1P p, . .}.is trivial in degree (p + 1)q - 1. But fix is a*
* multiple
of P p, and this multiple classifies the extension.
To see that fix is non-zero in the case of H*(ju; Fp), recall again that the*
* first
class fi1 in the cokernel of J is in degree (pq -2) and has order p. Let c ! S(*
*p)! ju
be the usual cofiber sequence. Then by the Hurewicz and universal coefficient
theorems, the lowest class x in H*( c; Fp) sits in degree (pq - 1) and supports*
* a
non-trivial mod p Bockstein fix 6= 0. Furthermore, H*( c; Fp) ~= H*(ju; Fp) in
positive degrees * > 0, so also in H*(ju; Fp) we have fix 6= 0.
(c) In (7.9), the A-module homomorphism (_3 - 1)* is induced up from the
A2-module homomorphism
Sq4 : 4A2=A2{Sq1, Sq2Sq3} ! A2==A1 .
A direct calculation shows that A2=A2{Sq1, Sq2Sq3} has rank 24 and A2==A1 has
rank 8, as F2-vector spaces. The homomorphism Sq4 has cokernel A2==A2 = F2{1}
of rank 1, so its kernel 8K has rank 17. Here
4K = F2{Sq4, Sq6, Sq7, Sq6Sq2, Sq9, Sq10 + Sq8Sq2, Sq7Sq3,
Sq11 + Sq9Sq2, Sq10Sq2, Sq13 + Sq10Sq3, Sq11Sq2, Sq11Sq3,
Sq13Sq2 + Sq12Sq3, Sq13Sq3, Sq17 + Sq15Sq2, Sq17Sq2 + Sq16Sq3, Sq17Sq3}
as a submodule of A2=A2{Sq1, Sq2Sq3}. By another direct calculation, 4K is in
fact the cyclic A2-submodule generated by 4Sq4. The annihilator ideal turns out
to be generated by Sq1, Sq7 and Sq4Sq6 + Sq6Sq4 = Sq10 + Sq8Sq2 + Sq7Sq3 (in
admissible form), so
K ~=A2=A2{Sq1, Sq7, Sq4Sq6 + Sq6Sq4} .
Hence there is an extension of A-modules
0 ! A==A2 ! H*(j; F2) ! A A2 7K ! 0
HOPF ALGEBRA STRUCTURE ON T HH 31
with A A2 7K ~= 7A=A{Sq1, Sq7, Sq4Sq6 + Sq6Sq4}. The group of such A-
module extensions is
Ext1A(A A2 7K, A==A2) ~=Z=2 ,
and the extension is determined by the action of Sq1 on the generator x in degr*
*ee 7
that maps to 7(1). (Sq7x = 0 by the Adem relation Sq1Sq7 = 0 and the fact that
Sq1 acts injectively from degree 14 of A==A2. (Sq4Sq6 + Sq6Sq4)x = 0 since A==A2
is trivial in degree 17.)
To see that Sq1(x) = Sq8(1) 6= 0 in H*(j; Z=2), we once again use the cofiber
sequence c ! S(2)! j and the fact that c is 5-connected with ss6(c) = Z=2{ 2}.
Hence the lowest class x in H*( c; F2) sits in degree 7 and supports a non-triv*
*ial
Sq1x 6= 0. Again, H*( c; F2) ~=H*(j; F2) in positive degrees, so also in H*(j; *
*F2)
we have Sq1x 6= 0. The only possible value is Sq8 from A==A2.
We display the A2-module 4K 4A2=A2{Sq1, Sq2Sq3} below. Here (i) or
(i, j) denotes an admissible class with lexicographically leading term Sqi or S*
*qiSqj,
respectively. The arrows indicate the Sq1- and Sq2-operations. The Sq4-operatio*
*ns
can be deduced from the relations Sq4(6) = (10) and Sq4(13) = (17).
_______________________________________________________________________*
*_____________________________________________________________________________*
*_____________________________________________________________________________*
*_____________________________________________________________________________*
*__________________________________________@
__________________$$______________________________________________________*
*_____________________________________%%______________________________________*
*____________________________________
(4) (6) ________//_(7) (6, 2)_______//___________88_(9*
*)(7, 3)
_________________________*
*_____________________________________________________________________________*
*____________________________
_____________________*
*_____________________________________________________________________________*
*_____________________________________________________________________________*
*___________________________________________________________________________
(11)____________________//(13)I<<
zzz III
zzz III
zz I$$I
(10) ________________//(10,I2)________________//_(11, 3)________________//_(13*
*,:3):
III uuuu
III uuu
II$$ uu
(11, 2)__________________//_(13, 2)
______________________________________________________________________*
*_____________________________________________________________________________*
*_____________________________________________________________________________*
*_____________________________________________________
__________________%%______________________________________________________*
*__________________________________________
(17) (17, 2)_____//_(17, 3)
7.11. Homology algebras. Let us write (A==A1)* A* for the A*-comodule
subalgebra dual to the quotient A-module coalgebra A==A1 of A. For p = 2 we
recall from proposition 6.1 that:
(A==A1)* = P (~,41, ~,22, ~,k| k 3) ~=H*(ko; F2) .
For p odd
(A==A1)* = P (~,p1, ~,k| k 2) E(~ok| k 2) ,
but there is no spectrum with mod p homology realizing (A==A1)*. Both for p = 2
and for p odd there is an extension of A*-comodules
0 ! pq-1(A==A1)* ! H*(ju; Fp) -~!(A==A1)* ! 0
32 VIGLEIK ANGELTVEIT AND JOHN ROGNES
where ~ is an A*-comodule algebra homomorphism.
Likewise, we write (A==A2)* A* for the A*-comodule subalgebra dual to the
quotient A-module coalgebra A==A2 of A. For p = 2 we recall from proposition 6.1
that
(A==A2)* = P (~,81, ~,42, ~,23, ~,k| k 4) ~=H*(tmf; F2) .
There is an extension of A*-comodules
0 ! A* A2* 7K* ! H*(j; F2) -~!(A==A2)* ! 0
where ~ is an A*-comodule algebra homomorphism. Here denotes the cotensor
product [MiMo65, 2.2], and K* A2* is the A2*-comodule dual to the cyclic A2-
module K = A2=A2{Sq1, Sq7, Sq4Sq6 + Sq6Sq4}.
Proposition 7.12. (a) For p = 2, let b 2 H3(ju; F2) be the image of 3(1) in
3(A==A1)*. Then there is an A*-comodule algebra isomorphism
H*(ju; F2) ~=(A==A1)* E(b) ,
where (A==A1)* has the subalgebra structure from A*, and b is A*-comodule primi-
tive.
The Dyer-Lashof operations in H*(ju; F2) satisfy Q4(b) = 0, Q5(~,41) = *
*0,
Q7(~,22) = 0 and Q2k(~,k) = ,~k+1 for all k 3, so H*(ju; F2) is generated by
~,41, ~,22, ~,3and b as an algebra over the Dyer-Lashof algebra.
(b) For p odd, let b 2 Hpq-1(ju; Fp) be the image of pq-1(1) in pq-1(A==A1*
*)*.
There is an algebra isomorphism
(A==A1)* E(b) ~=H*(ju; Fp)
that takes the algebra generators ~,p1, ,~k, ~ok(for k 2) and b to classes ",*
*p1, ,"k,
"okand b in H*(ju; Fp), respectively. These map under ~ to ,~p1, ,~k, o~k and *
*0,
respectively. k
The Dyer-Lashof operations on H*(ju; Fp) satisfy Qpq=2(b) = 0 and Qp ("ok) =
"ok+1for all k 2, and fi("ok) = ",kfor all k 2. Thus H*(ju; Fp) is generate*
*d as
an algebra over the Dyer-Lashof algebra by ",p1, ",2, "o2and b.
The A*-comodule structure is determined by
(b) = 1 b
(",p1)= 1 ",p1- o0 b + ~,p1 1
(",2)= 1 ",2+ ~,1 ",p1+ o1 b + ~,2 1
("o2)= 1 "o2+ ~o0 ",2+ ~o1 ",p1- o0o1 b + ~o2 1 .
(The class b 2 Hpq-1(ju; Fp) maps under the connecting map for the cofiber se-
quence c ! S(p)! ju to the mod p Hurewicz image of fi1 2 sspq-2(c), so the lett*
*er
b is chosen to correspond to fi.)
HOPF ALGEBRA STRUCTURE ON T HH 33
(c) There is a square-zero extension of A*-comodule algebras
0 ! A* A2* 7K* ! H*(j; F2) -~!(A==A2)* ! 0 ,
where ~ is split as an algebra homomorphism. As an (A==A2)*-module,
ker(~) = A* A2* 7K* ~=(A==A2)* 7K*
is free of rank 17. There is an algebra isomorphism
H*(j; F2) ~=(A==A2)* (F2 7K*)
where F2 7K* is the split square-zero extension of F2 with kernel 7K* of rank*
* 17.
Proof. (a) Let x 2 H3(ju; F2) be the class that maps to 3(1) in the uniquely
split A-module extension of lemma 7.10(a). Then _(x) = x 1 + 1 x since
H*(ju; F2) = 0 for 0 < * < 3, so E(x) = F2{1, x} (no algebra structure is impli*
*ed) is
a sub-coalgebra of H*(ju; F2) and there a surjective composite A-module coalgeb*
*ra
homomorphism
A E(x) ! A H*(ju; F2) ! H*(ju; F2) .
Since the A-module extension is split, the generators Sq1 and Sq2 of A1 act tri*
*vially
on 1 and x in H*(ju; F2), so the surjection factors through a surjection
A==A1 E(x) ! H*(ju; F2) ,
which by a dimension count must be an A-module coalgebra isomorphism.
Dually, let b 2 H3(ju; F2) be the image of 3(1) in the split A*-comodule ex*
*ten-
sion
0 ! 3(A==A1)* ! H*(ju; F2) ! (A==A1)* ! 0 .
Then b is dual to x, and the dual of the above isomorphism is an A*-comodule
algebra isomorphism
H*(ju; F2) ~=(A==A1)* E(b) .
In particular, the unique A*-comodule splitting (A==A1)* ! H*(ju; F2) is an alg*
*e-
bra map, b is an A*-comodule primitive, and b2 = 0.
To determine some Dyer-Lashof operations in H*(ju; F2) we shall use the Nish*
*ida
relations and the known A*-comodule structure. Some of the Nishida relations th*
*at
we shall use are ae
Qs-1 for s even
Sq1*Qs =
0 for s odd,
and ae
Qs-2 + Qs-1Sq1* for s 0, 1 mod 4
Sq2*Qs =
Qs-1Sq1* for s 2, 3 mod 4.
See [BMMS86, III.1.1(8)].
34 VIGLEIK ANGELTVEIT AND JOHN ROGNES
The Dyer-Lashof operation Q4(b) lands in H7(ju; F2) ~=F2{~,3, ~,41b}. From t*
*he
A*-comodule structure we can read off the dual Steenrod operations Sq1*(~,3) = *
*~,22
and Sq4*(~,41b) = b, since Sqi is dual to ,i1. These are linearly independent, *
*so Q4(b)
is determined by its images under Sq1*and Sq4*. By a Nishida relation we get th*
*at
Sq1*Q4(b) = Q3(b), and Q3(b) = b2 = 0 since |b| = 3 and b is an exterior class.
By another Nishida relation Sq4*Q4(b) = Q2Sq2*(b), and Sq2*(b) = 0 since b is A*
**-
comodule primitive. Thus Sq1*Q4(b) = 0 and Sq4*Q4(b) = 0, and the only possibil*
*ity
is that Q4(b) = 0.
The operation Q5(~,41) lands in H9(ju; F2) ~= F2{~,22b}. By a Nishida relat*
*ion
Sq2*Q5(~,41) = (Q3 + Q4Sq1*)(~,41) = 0, while Sq2*(~,22b) = ~,41b 6= 0. So Q5(~*
*,41) must be
zero.
The operation Q7(~,22) lands in H13(ju; F2) ~= F2{~,22~,3, ~,41~,22b}. Its *
*image under
~ in H*(ku; F2) A* must vanish, by the Cartan formula, so in fact Q7(~,22) 2
F2{~,41~,22b}. By a Nishida relation Sq2*Q7(~,22) = Q6Sq1*(~,22) = 0, while Sq2*
**(~,41~,22b) =
(~,41)2b 6= 0, so Q7(~,22) must be zero.
The claim that Q2k(~,k) = ~,k+1follows in the same way as in A* = H*(HF2; F2*
*),
see [BMMS86, xIII.6]. It suffices to show that Sq2m*Q2k(~,k) = Sq2m*(~,k+1) for
0 m k, because the only A*-comodule primitives in H*(ju; F2) are 1 and b.
The right hand side equals ~,2kfor m = 0, and is zero otherwise, by the formula
for _(~,k+1). The Nishida relations imply that Sq1*Q2k(~,k) = Q2k-1(~,k) = ,~2*
*k,
while Sq2*Q2k(~,k) = Q2k-1Sq1*(~,k) = Q2k-1(~,2k-1) = 0 by the Cartan formula. *
*For
2 m k we have Sq2m*Q2k(~,k) = Q2k-2m-1 Sq2m-1*(~,k) = Q2k-2m-1 (0) = 0
by the formula for _(~,k) in A*. Hence Q2k(~,k) = ~,k+1in H*(ju; F2) is the on*
*ly
possibility.
(b) Let x 2 Hpq-1 (ju; Fp) be the class that maps to pq-1(1) in the A-module
extension of lemma 7.10(b). Then _(x) = x 1 + 1 x since H*(ju; Fp) = 0 for
0 < * < pq - 1, so E(x) = Fp{1, x} is a sub-coalgebra of H*(ju; Fp) and there a
surjective composite A-module coalgebra homomorphism
A E(x) ! A H*(ju; Fp) ! H*(ju; Fp) .
Dually, let b 2 Hpq-1(ju; Fp) be the image of pq-1(1) in the A*-comodule exten-
sion
0 ! pq-1(A==A1)* ! H*(ju; Fp) ! (A==A1)* ! 0 .
Then E(b) is a quotient algebra of H*(ju; Fp) and the dual of the surjection ab*
*ove
is an injective A*-comodule algebra homomorphism
H*(ju; Fp) ! A* E(b) .
We shall describe H*(ju; Fp) in terms of its image under this injection.
Since (A==A1)* is a free graded commutative algebra, we can choose an algebra
section s: (A==A1)* ! H*(ju; Fp) to the surjection ~: H*(ju; Fp) ! (A==A1)*. We
write ",p1= s(~,p1), ",k= s(~,k) and "ok= s(~ok) for the lifted classes in H*(j*
*u; Fp).
Since pq-1(A==A1) vanishes in the degrees of ~,p1and ~o2, the respective lifts*
* ",p1
and "o2are unique, and we can use the commutative S-algebra structure on ju and
HOPF ALGEBRA STRUCTURE ON T HH 35
the resulting Dyer-Lashof operations on H*(ju; Fp) to fix the other lifts by the
formulas ",k= fi("ok) and "ok+1 = Qpk("ok) for k 2. This specifies the algeb*
*ra
section s uniquely. (But s is not an A*-comodule homomorphism.)
Since |b| = pq - 1 is odd we have b2 = 0 and the exterior algebra E(b) is
a subalgebra of H*(ju; Fp). Writing i: E(b) ! H*(ju; Fp) for the inclusion, we
obtain an algebra map
s i: (A==A1)* E(b) ! H*(ju; Fp) ,
which we claim is an isomorphism. By a dimension count it suffices to show that
its composite with the injection H*(ju; Fp) ! A* E(b) is injective. We have a
diagram of algebra maps
(A==A1)* E(b) _s_i__//H*(ju; Fp)____//_A* E(b)
| | |
| |~ |
|fflffl = fflffl| fflffl|
(A==A1)* _________//_(A==A1)*_________//A*
where the vertical maps take b to zero. The lower map takes , 2 (A==A1)* to
, 2 A*, so the upper composite takes , 1 to , 1 (mod b). Hence the latter
takes , b to , b (mod b2 = 0), and it follows that the upper composite ind*
*eed
is injective.
In low degrees, H*(ju; Fp) ~=Fp{1, x, P p(1), P 1P p(1), fiP 1P p(1), . .}.i*
*s dual to
H*(ju; Fp) ~=Fp{1, b, -",p1, ",2, "o2, . .}.. By lemma 7.10(b) we have fi(x) = *
*P p(1), so
dually the A*-comodule coactions are given by
(b) = 1 b
(",p1)= 1 ",p1- o0 b + ~,p1 1
(",2)= 1 ",2+ ~,1 ",p1+ o1 b + ~,2 1
("o2)= 1 "o2+ ~o0 ",2+ ~o1 ",p1- o0o1 b + ~o2 1 .
In particular, the images in A* E(b) of these classes are 1 b, -o0 b + ~,*
*p1 1,
o1 b + ~,2 1 and -o0o1 b + ~o2 1, respectively.
The Dyer-Lashof operation Qpq=2(b) lands in Hp2q-1(ju; Fp) = Fp{",pq=21b}. H*
*ere
P*pq=2(",pq=21b) = b is non-zero, in view of the formula above for (",p1). By *
*a Nishida
relation P*pq=2Qpq=2(b) = Qq=2P*q=2(b) = 0, so Qpq=2(b) = 0.
(c) Let x 2 H7(j; F2) be the class that maps to 7(1) in the A-module extens*
*ion
of lemma 7.10(c). Then _(x) = x 1 + 1 x since H*(j; F2) = 0 for 0 < * < 7, *
*so
E(x) = F2{1, x} is a sub-coalgebra of H*(j; F2) and there is a surjective compo*
*site
A-module coalgebra homomorphism
A E(x) ! A H*(j; F2) ! H*(j; F2) .
Dually, let b 2 H7(j; F2) be the image of 7(1) in the A*-comodule extension
0 ! A* A2* 7K* ! H*(j; F2) -~!(A==A2)* ! 0 .
36 VIGLEIK ANGELTVEIT AND JOHN ROGNES
Here K* A2* has rank 17, and contains 1 2 K* A2* in degree 0. The dual of
the surjection above is an injective A*-comodule algebra homomorphism
H*(j; F2) ! A* E(b)
that may otherwise be described as the composite of the A*-comodule coaction
H*(j; F2) ! A* H*(j; F2) and the algebra surjection H*(j; F2) ! E(b).
We obtain a vertical map of A*-comodule extensions
A* A2* 7K* ______//H*(j; F2)_~___//(A==A2)*
| | |
| | |
fflffl| fflffl| fflffl|
A*{b} _________//_A* E(b)________//A*
where the right-hand square consists of A*-comodule algebra homomorphisms, and
the vertical maps are injective. At the left hand side we find the composite map
A* A2* 7K* ! A* A2* 7A2* ~= 7A* ~=A*{b}
that is dual to the surjection A{x} ~= 7A ! A A2 7K.
In the lower row the ideal A*{b} has square zero, since b2 = 0, so also the
ideal ker(~) = A* A2* 7K* is a square-zero ideal [CE56, XIV.2] in H*(j; F2).
Its module action by H*(j; F2) therefore descends to one by (A==A2)*. It can be
described in terms of the algebra product OE on A* by the following commutative
diagram with injective vertical maps:
(A==A2)* (A* A2* 7K*) ______//(A* A2* 7K*)
| |
| |
fflffl| 7OE fflffl|
A* 7A* ___________________// 7A*
The proof is easy given the algebra embedding of H*(j; F2) into A* E(b).
In fact, the square-zero ideal A* A2* 7K* is a free (A==A2)*-module of rank*
* 17.
For A2 A is a direct summand as an A2-module, so dually A* ! A2* admits an
A2*-comodule section s: A2* ! A*. For example, the image of s may be F2{~,i1~,j*
*2~,k3|
i < 8, j < 4, k < 2} A*. We then have a map
id s: 7K* ~=A2* A2* 7K* ! A* A2* 7K* .
Its composite with the inclusion A* A2* 7K* ! A* A2* 7A2* ~= 7A* factors
as the two inclusions 7K* ! 7A2* ! 7A*.
Combining id s with the (A==A2)*-module action on A* A2* 7K* we obtain
the left hand map f in a commuting diagram
(A==A2)* 7K* ______//(A==A2)* 7A2*
|f| |~=|
fflffl| fflffl|
A* A2* 7K* ________//A* A2* 7A2* .
HOPF ALGEBRA STRUCTURE ON T HH 37
Here A* A2* 7A2* ~= 7A* and the right hand isomorphism exhibits 7A* as a
free (A==A2)*-module on the generators given by the section s: 7A2* ! 7A*.
(It is a case of the Milnor-Moore comodule algebra theorem [MiMo65, 4.7].) The
upper map is injective, hence so is
f :(A==A2)* 7K* ! A* A2* 7K* .
But both sides have the same, finite dimension over F2 in each degree, so in fa*
*ct f
is an isomorphism of (A==A2)*-modules.
The fact that ~ splits as an algebra homomorphism is clear since (A==A2)* is
a free graded commutative algebra over F2. However, the splitting is not an A*-
comodule homomorphism, and the (A==A2)*-module isomorphism f is not an A*-
comodule isomorphism. The A*-comodule algebra structure on H*(j; F2) may, if
desired, be obtained by describing the image of the algebra generators under the
A*-comodule algebra embedding H*(j; F2) ! A* E(b).
Proposition 7.13. (a) For p = 2 the B"okstedt spectral sequence Er**(ju) colla*
*pses
at the E2-term, with
E1**(ju) ~=H*(ju; F2) E(oe,~41, oe,~22, oe,~k| k 3) (oeb) .
(b) For p odd the B"okstedt spectral sequence Er**(ju) collapses at the Ep-t*
*erm,
with
E1**(ju) ~=H*(ju; Fp) E(oe,"p1, oe,"2, oe"ok| k 2) (oeb) .
(c) The B"okstedt spectral sequence for j at p = 2 has E2-term
E2**(j) ~=HH*((A==A2)*) HH*(F2 7K*)
where
HH*((A==A2)*) ~=(A==A2)* E(oe,~81, oe,~42, oe,~23, oe,~k| k 4)
and
HHq(F2 7K*) ~=[( 7K*) q ]Cq [( 7K*) q+1 ]Cq+1 .
In particular, E2**(j) is not flat as a module over H*(j; F2).
Proof. (a) The B"okstedt spectral sequence for ju at p = 2 begins
E2**(ju) = H*(ju; F2) E(oe,~41, oe,~22, oe,~k| k 3) (oeb)
Proposition 4.7 applies, so a shortest non-zero differential must map from an a*
*lgebra
indecomposable to a coalgebra primitive and A*-comodule primitive. Here we are
referring to the A*-comodule H*(ju; F2)-Hopf algebra structure on E2**(ju). The
only possible algebra indecomposables are the fl2k(oeb) in degrees 2k+2 , for k*
* 2.
The coalgebra primitives are H*(ju; F2){oe,~41, oe,~22, oe,~k| k 3, oeb}, all*
* in filtration
s = 1, and contain the A*-comodule primitives E(b) F2{oe,~41, oe,~k| k 4, o*
*eb}.
These are in degrees 4, 5, 7, 8, 2k and 2k + 3, for k 4. The image of a diffe*
*rential
on fl2k(oeb) must be in total degree 2k+2 - 1, for k 2, but these degrees do *
*not
38 VIGLEIK ANGELTVEIT AND JOHN ROGNES
contain any simultaneous coalgebra- and comodule primitives. Therefore there are
no nonzero differentials, and the spectral sequence collapses at the E2-term.
(b) For p odd the spectral sequence begins
E2**(ju) = H*(ju; Fp) E(oe,"p1, oe,"k| k 2) (oeb, oe"ok| k 2)*
* .
By proposition 5.6 we have E2 = Ep-1 and there are differentials
dp-1 (flp(oe"ok)) = oe,"k+1
for k 2. This uses the relation fiQpk("ok) = fi("ok+1) = ",k+1. There is al*
*so a
potential differential
dp-1 (flp(oeb)) = oe(fiQpq=2(b)) ,
but fiQpq=2(b) is in degree p2q - 2 of H*(ju; Fp), which is a trivial group, so*
* this
differential is zero. Hence
Ep**(ju) = H*(ju; Fp) E(oe,"p1, oe,"2) Pp(oe"ok| k 2) (oeb) .
Proposition 4.7 applies again, so a shortest differential must map from one of *
*the
algebra indecomposables flpk(oeb) in degrees pk+1 q, for k 2. Its target must*
* be lie
among the coalgebra primitives, which are H*(ju; Fp){oe,"p1, oe,"2, oe"ok| k *
*2, oeb}, all
in filtration s = 1. The target must also be A*-comodule primitive. The formulas
for the A*-comodule structure on H*(ju; Fp) imply the following formulas:
(oeb)= 1 oeb
(oe,"p1)= 1 oe,"p1- o0 oeb
(oe,"2)= 1 oe,"2+ ~,1 oe,"p1+ o1 oeb
(oe"o2)= 1 oe"o2+ ~o0 oe,"2+ ~o1 oe,"p1- o0o1 oeb .
The oe"okare A*-comodule primitives for k 3, in view of the relations (oe"ok)*
*p =
oe"ok+1for k 2, the formula for (oe"o2), and the fact that ~o0p= ~o1p= (o0o1*
*)p = 0.
Thus the simultaneous coalgebra- and comodule primitives are contained in
E(b) Fp{oe,"p1, oe"ok| k 3, oeb}. These are in degrees pq, pq + 1, 2pq - 1*
*, 2pq,
2pk and 2pk + pq - 1 for k 3. The image of a differential dr(flpk(oeb)) is in*
* degree
pk+1 q - 1 for k 2, which contains none of the possible target classes. Hence*
* there
are no further differentials, and the spectral sequence collapses at the Ep-ter*
*m.
(c) By the K"unneth formula
E2**(j) = HH*(H*(j; F2)) ~=HH*((A==A2)*) HH*(F2 7K*) .
Here the first tensor factor was identified in the discussion of tmf in section*
* 6. By
the following lemma 7.14,
HHq(F2 7K*) ~=[ 7K*q ]Cq [ 7K*(q+1)]Cq+1 ,
and e.g. HH1(F2 7K*) is not flat as an F2 7K*-module.
HOPF ALGEBRA STRUCTURE ON T HH 39
Lemma 7.14. Let k be a field and V a graded k-vector space. The Hochschild
homology of the split square zero extension k V , with unit (1, 0) and multipl*
*ication
(k1, v1) . (k2, v2) = (k1k2, k1v2 + k2v1), is
HHq(k V ) ~=[V q]Cq [V (q+1)]Cq+1
where [V q]Cq V q denotes the invariants of the cyclic group Cq of order q *
*acting
by cyclic permutations on V q, and [V q]Cq denotes the coinvariants of this a*
*ction.
When dim kV 2 the Hochschild homology HH*(k V ) is not flat as a module
over k V .
Proof. We compute HH*(k V ) as the homology of the normalized Hochschild
complex NC*(k V ) with
NCq(k V ) = (k V q) (V V q) ~=V q V (q+1).
Since V is a square zero ideal, the Hochschild boundary @ is the direct sum ov*
*er
q 1 of the operators
1 + (-1)qtq: V q ! V q,
i.e., 1 + (-1)qtq on the V q-summand and zero on the V (q+1)-summand, where
tq(v1 . . .vq) = (-1)fflvq v1 . . .vq-1 with ffl = |vq|(|v1| + . .+.|vq-1*
* |). Let the
generator T 2 Cq act on V as (-1)q+1tq, so @ is the sum of the operators 1 - T
(and T qacts as the identity). Clearly, then, the Hochschild homology is the di*
*rect
sum over q 1 of the kernels
ker(1 - T ) = [V q]Cq
in degree q, and the cokernels
cok(1 - T ) = [V q]Cq
in degree (q - 1), plus the term k = V 0 in degree 0.
For an example of the failure of flatness, let V = k{x, y} with x, y in odd *
*degrees
and q = 1. Then HH1(k V ) ~=V VC22 ~=k{oex, oey, xoex, xoey yoex, yoey} i*
*s not
flat over k V .
Theorem 7.15. (a) For p = 2 there is an isomorphism
H*(T HH(ju); F2) ~=H*(ju; F2) E(oe,~41, oe,~22) P (oe,~3) (oeb)
of H*(ju; F2)-Hopf algebras.
(b) For p odd there is an isomorphism
H*(T HH(ju); Fp) ~=H*(ju; Fp) E(oe,"p1, oe,"2) P (oe"o2) (oeb)
of H*(ju; Fp)-Hopf algebras.
40 VIGLEIK ANGELTVEIT AND JOHN ROGNES
Proof. (a) In view of proposition 7.13(a) we must identify the possible algebra
extensions between the E1 -term
E1**(ju) ~=H*(ju; F2) E(oe,~41, oe,~22, oe,~k| k 3) (oeb)
and the abutment H*(T HH(ju); F2). Here H*(ju; F2) ~=(A==A1)* E(b) by propo-
sition 7.12.
In H*(T HH(ju); F2) we have (oeb)2 = Q4(oeb) = oeQ4(b) = 0, (oe,~41)2 = 0,
(oe,~22)2 = 0 and (oe,~k)2 = oe,~k+1 for all k 3, by proposition 5.9 and the *
*state-
ment about Dyer-Lashof operations in proposition 7.12(a). It remains to prove
that we can find classes
fl2k 2 H4.2k(T HH(ju); F2)
that are represented by fl2k(oeb) in E1**(ju) and satisfy fl22k= 0, for all k *
* 0. We
have just seen that we can take fl1 = oeb. So fix a number k 1, and assume
inductively that we have chosen classes fl2m for 0 m < k that are represented*
* by
fl2m (oeb) and satisfy fl22m= 0.
We shall prove below that a class fl2k representing fl2m (oeb) can be chosen*
* so
that its square fl22kis both an H*(ju; F2)-coalgebra primitive and an A*-comodu*
*le
primitive. All such coalgebra- and comodule primitives are included among
E(b) F2{oe,~41, oe,~22, oe,~m | m 3, oeb) .
When k 1, the only such class in the degree of fl22kis oe,~k+3= (oe,~k+2)2. S*
*o either
fl22k= 0 or fl22k= (oe,~k+2)2. In the latter case we change fl2k by subtracting*
* oe,~k+2,
which does not alter the representative at the E1 -term. Thereby we have achiev*
*ed
fl22k= 0, which will complete the inductive step.
To show that fl22kcan be arranged to be a coalgebra- and comodule primitive,
we make use of the maps of E1 -terms and abutments induced by the commutative
S-algebra homomorphism ~: ju ! ku(2). The target E1 -term is
E1**(ku) ~=H*(ku; F2) E(oe,~21, oe,~22, oe,~k| k 3)
with abutment
H*(T HH(ku); F2) ~=H*(ku; F2) E(oe,~21, oe,~22) P (oe,~3) .
Here H*(ku; F2) ~= (A==E1)*. Note that in degrees less than that of fl2k, ~ ma*
*ps
H*(T HH(ju); F2) modulo classes that square to zero, which by the inductive hy-
pothesis is (A==A1)* P (oe,~3), injectively into H*(T HH(ku); F2) modulo clas*
*ses
that square to zero, which is (A==E1)* P (oe,~3). We shall refer to this prop*
*erty as
the "near-injectivity of ~".
So choose any class fl2k in H*(T HH(ju); F2) that is represented by fl2k(oeb*
*) in
E1**(ju). We shall arrange that its image ~fl2k squares to zero in H*(T HH(ku);*
* F2).
If not, we can write
~fl2k c . (oe,~3)`
HOPF ALGEBRA STRUCTURE ON T HH 41
with c 2 (A==E1)* not equal to zero, modulo classes in H*(T HH(ku); F2) that
square to zero, and modulo similar terms with c of lower degree (or equivalentl*
*y,
with higher exponent `). So c . (oe,~3)` is the "leading term" in H*(T HH(ku); *
*F2)
modulo classes that square to zero.
We divide into three cases. First, if ` = 0 and ~fl2k c with c 2 (A==E1)*,*
* we can
apply the Hopf algebra counit ffl: T HH(R) ! R to see that c = ffl(~fl2k) = ~(f*
*flfl2k)
is in the image of ~: H*(ju; F2) ! H*(ku; F2), so that in fact c 2 (A==E1)*. We
can then replace fl2k by fl2k - c without altering its representative in E1 , a*
*nd thus
eliminate the case ` = 0.
Second, if ` = 1 and ~fl2k c . oe,~3with c 2 (A==E1)*, we need to argue
that c 2 (A==A1)*. If not, we can write c = ,~21. d with d 2 (A==A1)*. Here
_(c) c 1 + d ~,21modulo terms with first tensor factor in degree less that
|d|. Note from the structure of H*(T HH(ku); F2) that ~fl2k c . oe,~3modulo
classes that square to zero, and similar terms e . (oe,~3)` with ` 2, so |e| *
*< |d|. So
~fl2k c oe,~3+ d ~,21oe,~3modulo classes that square to zero, and terms *
*with
first factor in lower degree. This must be the image of fl2k under 1 ~, so
fl2k c oe,~3+ d x
for some class x 2 H*(T HH(ju); F2) with ~(x) ~,21oe,~3. But there exists no *
*such
class x, since H*(T HH(ju); F2) modulo classes that square to zero is trivial in
degree 10. (This is clear by inspection of the B"okstedt spectral sequence, wh*
*ere
the group that could cause a problem, E22,8(ju), is in fact zero.) This contrad*
*iction
shows that c 2 (A==A1)*, and we can alter fl2k by c . oe,~3without altering the
representative at E1 , and thus eliminate the case ` = 1.
Third, for the remaining cases ` 2 we consider the coalgebra coproduct. We
find X
_(~fl2k) c . (oe,~3)i (oe,~3)j
i+j=`
in H*(T HH(ku); F2) H*(ku;F2)H*(T HH(ku); F2), modulo classes that square to
zero and similar terms with c of lower degree. Since ` 2 this sum includes so*
*me
terms c . (oe,~3)i (oe,~3)j with both i and j positive, so that c . (oe,~3)i a*
*nd (oe,~3)j are
both in degree less than that of fl2k. This is the image under ~ ~ of _(fl2k*
*) in
H*(T HH(ju); F2) H*(ju;F2)H*(T HH(ju); F2), so it follows by the near-injectivi*
*ty
of ~ that c in fact lies in (A==A1)* (A==E1)*.
Then we can change the chosen fl2k in H*(T HH(ju); F2) by subtracting c.(oe,*
*~3)`
from it, and thus remove the "leading" term c . (oe,~3)` from ~fl2k. By repeati*
*ng this
process we can arrange that fl2k has be chosen so that ~fl2k is zero modulo cla*
*sses
that square to zero, i.e., that ~fl22k= 0.
Then _(fl2k) fl2k 1+1 fl2k modulo classes that square to zero. For _(~fl22*
*k) =
0 so any other terms in _(fl2k) must map under ~ to classes that square to zero,
hence square to zero themselves by the near-injectivity of ~. Hence fl22kis a c*
*oalgebra
primitive.
Next consider the A*-comodule coaction. If (fl2k) 1 fl2k modulo classes
that square to zero, then (fl22k) = 1 fl22kand fl22kis an A*-comodule primit*
*ive, as
desired. Otherwise, we can write
(fl2k) a (oe,~3)`
42 VIGLEIK ANGELTVEIT AND JOHN ROGNES
with a 2 A* in positive degree, modulo classes in A* H*(T HH(ju); F2) that squa*
*re
to zero, and modulo similar terms with a of lower degree. Then (fl22k) a2 (o*
*e,~3)2`
modulo terms with a of lower degree. Applying ~ yields a2 (oe,~3)2` = 0, sin*
*ce
~fl22k= 0, so a2 = 0. This is impossible for a 6= 0, so we conclude that fl22ki*
*s indeed
an A*-comodule primitive.
This completes the proof.
(b) Also in the odd primary case we must identify the algebra extensions bet*
*ween
the E1 -term from proposition 7.13(b)
E1**(ju) = H*(ju; Fp) E(oe,"p1, oe,"2, oe"ok| k 2) (oeb)
and the abutment H*(T HH(ju); Fp). In the latter we have (oeb)p = Qpq=2(oeb) =
oeQpq=2(b) = 0 and (oe"ok)p = Qpk(oe"ok) = oeQpk("ok) = oe"ok+1, by proposition*
*s 5.9
and 7.12(b). The classes oe,"p1and oe,"2are in odd degree, and therefore have s*
*quare
zero, since p is odd.
It remains to prove that we can find classes flpk 2 Hpk+1q(T HH(ju); Fp) that
are represented by flpk(oeb) in E1**(ju) and satisfy flppk= 0, for all k 0. W*
*e have
just verified this for k = 0.
The remaining inductive proof follows exactly the same strategy as in the p *
*= 2
case. Instead of working modulo classes that square to zero we work modulo clas*
*ses
with p-th power equal to zero. The S-algebra map ~: ju ! ` induces an algebra
homomorphism in homology that has the required "near-injectivity" property, etc.
In the tricky case when ~flpk c . oe"o2, modulo classes in H*(T HH(`); Fp) th*
*at
have trivial p-th power, and modulo similar terms with c of lower degree, we ha*
*ve
c 2 (A==E1)* and must argue that in fact c 2 (A==A1)*. Writing c = ~,f1. d, wi*
*th
d 2 (A==A1)* and 0 f < p we assume 0 < f < p and reach a contradiction. The
coaction ~flpk contains a term f,~f-11d ~,1oe~o2, and we must check that the*
*re is no
class x 2 H*(T HH(ju); Fp) with ~x = ~,1oe~o2. This is again clear by the B"oks*
*tedt
spectral sequence. The rest of the proof can safely be omitted.
8. Topological K-theory revisited
We now wish to pass from the homology of the spectra T HH(ku) and T HH(ko)
to their homotopy. The first case is the analogue for p = 2 of the discussion *
*in
[MS93, xx5-7], where McClure and Staffeldt compute the mod p homotopy groups
ss*(T HH(`); Z=p) of the Adams summand ` ku(p)for odd primes p.
The idea is to compute homotopy groups using the Adams spectral sequence
Es,t2= Exts,tA*(Fp, H*(X; Fp)) =) sst-s(X)^p,
which is strongly convergent for bounded below spectra X of Zp-finite type. Thi*
*s is
difficult for X = T HH(ku) or T HH(ko) at p = 2, just as for T HH(`) at p odd, *
*but
becomes manageable after introducing suitable finite coefficients, e.g. after s*
*mashing
T HH(ku) with the mod 2 Moore spectrum M = C2, or smashing T HH(ko) with
the Mahowald spectrum Y = C2^Cj [Ma82]. Here Cf denotes the mapping cone of
a map f. Neither of these finite CW spectra are ring spectra, so an extra argum*
*ent
is needed to have products on T HH(ku) ^ M and T HH(ko) ^ Y . We shall manage
with the following weak version of a ring spectrum:
HOPF ALGEBRA STRUCTURE ON T HH 43
Definition 8.1. A ~-spectrum is a spectrum R with a unit j :S ! R and multi-
plication ~: R ^ R ! R that is left and right unital, but not necessarily assoc*
*iative
or commutative.
The following result is similar to [Ok79, 1.5], but slightly easier.
Lemma 8.2. Let R be a ~-spectrum and let Sk -f!S0 -i!Cf -ss!Sk+1 be a cofiber
sequence such that idR ^ idCf ^ f :R ^ kCf ! R ^ Cf is null-homotopic. Then
there exists a multiplication ~ :(R ^ Cf) ^ (R ^ Cf) ! (R ^ Cf) that makes R ^ *
*Cf
a ~-spectrum and idR ^ i :R ! R ^ Cf a map of ~-spectra.
Proof. A choice of null-homotopy provides a splitting m: R ^ Cf ^ Cf ! R ^ Cf
in the cofiber sequence
R ^ kCf -id^id^f----!R ^ Cf -id^id^i----!R ^ Cf ^ Cf -id^id^ss----!R ^ *
*k+1 Cf
which satisfies m(id ^ id ^ i) ' id (right unitality). The difference id - m(id*
* ^ i ^
id): R ^ Cf ! R ^ Cf restricts trivially to R, so extends over a map OE: R ^ Sk*
*+1 !
R ^ Cf. Using that the Spanier-Whitehead dual of Cf is -(k+1)Cf, together
with the null-homotopy hypothesis again, shows that OE extends further to a map
_ :R ^ k+1 Cf ! R ^ Cf such that we can alter m by _ O (id ^ id ^ ss) so as to
also make m(id ^ i ^ id) ' id (left unitality), without destroying the right un*
*itality.
Then the required multiplication is the composite
(R ^ Cf) ^ (R ^ Cf) -id^fl^id----!R ^ R ^ Cf ^ Cf -~^id^id----!R ^ Cf ^ Cf -m! *
*(R ^ Cf) .
In the case f = 2: S0 ! S0, the Moore spectrum M = C2 has cohomology
H*(M; F2) = E(Sq1), which equals E1==E(Q1) as an E1-module. By the Cartan
formula Sq2 acts nontrivially on H*(M ^ M; F2), and therefore M does not split
off from M ^ M. So M is not a ~-spectrum, and the map 2: M ! M is not
null-homotopic. It must therefore factor as the composite
(8.3) M -ss!S1 -j!S0 -i!M .
See e.g. [AT65, 1.1].
Similarly, in the case f = j :S1 ! S0 the mapping cone Cj has cohomology
H*(Cj; F2) = E(Sq2). By the Cartan formula Sq4 acts nontrivially on H*(Cj ^
Cj; F2), and therefore Cj does not split off from Cj ^ Cj. In particular, the m*
*ap
j : Cj ! Cj is not null-homotopic, and must factor as the composite
(8.4) Cj -ss!S3 -! S0 -i!Cj .
Here 2 ss3(S) is the Hopf invariant one class. The Mahowald spectrum Y =
C2 ^ Cj has cohomology H*(Y ; F2) = E(Sq1, Sq2), which equals A1==E(Q1) as an
A1-module.
44 VIGLEIK ANGELTVEIT AND JOHN ROGNES
Lemma 8.5. (a) T HH(ku) ^ M is a ~-spectrum.
(b) T HH(ko) ^ Cj and T HH(ko) ^ Y are ~-spectra.
Proof. (a) Let T = T HH(ku). By (8.3) the map 1 ^ 2: T ^ M ! T ^ M factors
through 1 ^ j : T ! T , which in turn factors as
T -1^j-!T ^ ku ! T ,
since T = T HH(ku) is a ku-module spectrum. But j 2 ss1(S) maps to zero in
ss1(ku), so this map is null-homotopic, and lemma 8.2 applies.
(b) Let T = T HH(ko) and R = T HH(ko)^Cj. By (8.4) the map 1^j : R ! R
factors through 1 ^ : 3T ! T , which in turn factors as
3T -1^-!T ^ ko ! T ,
since T = T HH(ko) is a ko-module spectrum. But 2 ss3(S) maps to zero in
ss3(ko), so 1 ^ j : R ! R is null-homotopic, and lemma 8.2 applies again to pro*
*ve
that R is a ~-spectrum under T .
Using once more that 1^j : R ! R is null-homotopic we get that 1^2: R^M !
R ^ M is null-homotopic by (8.3), and so R ^ M = T HH(ko) ^ Y is a ~-spectrum
under R.
Lemma 8.6. Let B A be a sub Hopf algebra and let N be an A*-comodule
algebra. Then there is an isomorphism of A*-comodule algebras
(A==B)* N ~=A* B* N .
Here B* is the quotient Hopf algebra of A* dual to B, and (A==B)* = A* B* Fp
is dual to A B Fp.
Proof. This is analogous to the usual G-homeomorphism G=H x X ~=G xH X for
a G-space X and subgroup H G. Let i: (A==B)* ! A* be the inclusion and
:N ! A* N the coaction. The composite homomorphism
(A==B)* N -i-! A* A* N -OE-1!A* N
equalizes the two maps to A* B* N, and hence factors uniquely through A* B*
N. An explicit inverse can be constructed using the Hopf algebra conjugation O
on A*.
Recall the n-th connective Morava K-theory spectrum k(n), with homotopy
ss*k(n) = Fp[vn] where |vn| = 2(pn -1), and cohomology H*(k(n); Fp) ~=A==E(Qn)
[BM72]. Dually, H*(k(n); Fp) ~=(A==E(Qn))* A*. In particular, for n = 1 and
p = 2 we have k(1) ' ku ^ M with H*(k(1); F2) ~=(A==E(Q1))* ~=P (~,1, ~,22, ~,k*
*| k
3). The n-th periodic Morava K-theorynspectrum K(n) is the telescope v-1nk(n) of
the iterated maps vn :k(n) ! 2(1-p )k(n), with homotopy ss*K(n) = Fp[vn, v-1n].
These are (non-commutative) S-algebras for all n [Ro89].
HOPF ALGEBRA STRUCTURE ON T HH 45
Proposition 8.7. (a) There are A*-comodule algebra isomorphisms
H*(T HH(ku) ^ M; F2) ~=(A==E(Q1))* E(oe,~21, oe,~22) P (oe,~3)
~=H*(k(1); F2) E(~1, ~2) P (~) .
Here (oe,~21) = 1 oe,~21, (oe,~22) = 1 oe,~22, and (oe,~3) = 1 oe,~3+ *
*~,1 oe,~22.
The classes ~1 = oe,~21, ~2 = oe,~22and ~ = oe,~3+ ~,1. oe,~22(in degrees 3,*
* 7 and 8,
respectively) are A*-comodule primitives.
(b) There are A*-comodule algebra isomorphisms
H*(T HH(ko) ^ Y ; F2) ~= (A==E(Q1))* E(oe,~41, oe,~22) P (oe,~3)
~= H*(k(1); F2) E(~1, ~2) P (~) .
Here (oe,~41) = 1 oe,~41, (oe,~22) = 1 oe,~22+ ~,21 oe,~41and (oe,~3) =*
* 1 oe,~3+ ~,1
oe,~22+ ~,2 oe,~41.
The exterior classes ~1 = oe,~41and ~2 = oe,~22+,~21.oe,~41are A*-comodule p*
*rimitives,
while ~ = oe,~3+ ~,1. oe,~22has
(~) = 1 ~ + ~,21 ~,1. ~1 + (~,2+ ~,31) ~1 .
The squared class ~2 = (oe,~3)2 is A*-comodule primitive.
In each case, the homology algebra on the left hand side has the unit and pr*
*od-
uct induced by the ~-spectrum structure from lemma 8.5. This product is in fact
associative (and graded commutative), in view of the exhibited additive and mul-
tiplicative isomorphism with the (associative) algebra on the right hand side.
Proof. (a) By corollary 5.13(a) there is an A*-comodule algebra isomorphism
H*(T HH(ku) ^ M; F2) ~=(A==E1)* (E1==E(Q1))* E(oe,~21, oe,~22) P (oe,~*
*3)
with the diagonal A*-comodule structure on the first two tensor factors, and the
claimed coaction on the remaining generators.
Since H*(M; F2) ~=(E1==E(Q1))* is in fact an A*-comodule algebra, there is an
A*-comodule algebra isomorphism
(A==E1)* (E1==E(Q1))* ~=A* E1* (E1==E(Q1))* ~=(A==E(Q1))*
by lemma 8.6. We are free to replace the polynomial generator oe,~3by the primi*
*tive
class ~, since their difference ~,1. oe,~22has square zero.
(b) By theorem 6.2(a) there is an A*-comodule algebra isomorphism
H*(T HH(ko) ^ Y ; F2) ~=(A==A1)* (A1==E(Q1))* E(oe,~41, oe,~22) P (oe,*
*~3) ,
with the claimed coaction on the exterior and polynomial generators. Again, by
lemma 8.6 there is an isomorphism of A*-comodule algebras
(A==A1)* (A1==E(Q1))* ~=A* A1* (A1==E(Q1))* ~=(A==E(Q1))* .
The classes ~1, ~2 and ~ are defined as in the statement of the proposition, and
their coactions are obtained by direct calculation.
46 VIGLEIK ANGELTVEIT AND JOHN ROGNES
Lemma 8.8. (a) The E2-term of the Adams spectral sequence for ss*(T HH(ku) ^
M) is
E**2~=P (v1) E(~1, ~2) P (~)
with ~1, ~2, ~ and v1 = [,2] in bidegrees (0, 5), (0, 7), (0, 8) and (1, 3), re*
*spectively.
(b) The E2-term of the Adams spectral sequence for ss*(T HH(ko) ^ Y ) is
2
E**2~= P (v1) E(~2, ~3) E(~2){~1} P (~ )
with ~1, ~2, ~3 = ~1~, ~2 and v1 = [,2] in bidegrees (0, 5), (0, 7), (0, 13), (*
*0, 16)
and (1, 3), respectively. It is the homology of the algebra
P (v1) E(~1, ~2) P (~)
with respect to the differential d(~) = v1~1, with cycles ~1, ~2 and v1.
Proof. (a) By change of rings, the E2-term of the Adams spectral sequence for
X = T HH(ku) ^ M is
E**2= Ext**A*(F2, (A==E(Q1))* E(~1, ~2) P (~))
~=Ext **E(Q1)*(F2, E(~1, ~2) P (~))
~=P (v1) E(~1, ~2) P (~)
as a graded algebra. Here E(Q1)* = E(,2) and Ext **E(,2)(F2, F2) = P (v1) with
v1 = [,2] in the cobar complex.
(b) The E2-term of the Adams spectral sequence for X = T HH(ko) ^ Y is
E**2= Ext**A*(F2, (A==E(Q1))* E(~1, ~2) P (~))
~=Ext **E(Q1)*(F2, E(~1, ~2) P (~))
~=Ext **E(Q1)*(F2, F2{1, ~1, ~2, ~, ~1~2, ~1~, ~2~, ~1~2~} P (~2))*
* .
Here
F2{1, ~1, ~2, ~, ~1~2, ~1~, ~2~, ~1~2~} ~=F2{1, ~2, ~1~, ~1~2~} E(Q1)*{~1, ~1*
*~2}
as E(Q1)*-comodules. For ~1, ~2 and ~2 are A*-comodule primitives, while (~)
maps to 1 ~ + ,2 ~1 in E(Q1)* H*(T HH(ko) ^ Y ; F2). So
2
E**2= P (v1) E(~2, ~1~) E(~2){~1} P (~ ) .
We let ~3 = ~1~ to obtain the claimed formula.
To determine the differentials in the Adams spectral sequence for ss*(T HH(k*
*u)^
M) or ss*(T HH(ko) ^ Y ) we first compute the v1-periodic homotopy. This is in
turn easy to derive from the K(1)-homology. Recall that BP*BP ~= BP*[tk | k 1]
with |tk| = 2(2k - 1) and K(1)* = F2[v1, v-11] with |v1| = 2.
HOPF ALGEBRA STRUCTURE ON T HH 47
Lemma 8.9. For p = 2 there are isomorphisms of K(1)*-algebras
k 2
K(1)*(ku) ~=K(1)*[tk | k 1]=(v1t2k= v21tk) ~=K(1)*[uk | k 1]=(uk = uk)
and
k 2
K(1)*(ko) ~=K(1)*[tk | k 2]=(v1t2k= v21tk) ~=K(1)*[uk | k 2]=(uk = uk) .
Proof. We first follow the proof of [MS93, 5.3]. We have K(1)*(BP ) ~=K(1)* BP*
BP*(BP ) ~=K(1)*[tk | k 1], where |tk| = 2(2k-1). The spectrum ku(2)= BP <1>
can be constructed from BP by Baas-Sullivan cofiber sequences killing the class*
*es
vk+1 for k 1, which map to
k
jR (vk+1 ) v1t2k+ v21tk mod (jR (v2), . .,.jR (vk))
by [Ra04, 6.1.13]. So each jR (vk+1 ) 2 K(1)*(BP ) is not a zero divisor mod
(jR (v2), . .,.jR (vk)), and
k
K(1)*(ku) ~=K(1)*[tk | k 1]=(v1t2k= v21tk) .
k
Substituting uk = v1-21 tk, the relations become u2k= uk for each k 1.
The cofiber sequence
ko -j!ko -c!ku -@! 2ko
induces a short exact sequence
0 ! K(1)*(ko) -c*!K(1)*(ku) -@*! 2K(1)*(ko) ! 0
since multiplication by j is zero in K(1)-homology. The connecting map @ right
multiplies by Sq2 in mod 2 cohomology, so right "comultiplies" with the dual cl*
*ass
~,21in mod 2 homology, which corresponds to t1 in BP*BP [Za72, p.488]. From the
coproduct formula for (tj) [Ra04, A2.1.27(e)] it follows that @*(t1) = 2(1) w*
*hile
@*(tk) = 0 for k 2. Hence we can identify K(1)*(ko) with the claimed subalgeb*
*ra
of K(1)*(ku), via the K(1)*-algebra homomorphism c*.
Lemma 8.10. The unit maps R ! T HH(R) induce isomorphisms
~=
K(1)*(ku) -! K(1)*T HH(ku)
and ~
K(1)*(ko) -=!K(1)*T HH(ko) .
Proof. The proof of [MS93, 5.3] continues as follows. There is a K(n)-based
B"okstedt spectral sequence
E2**= HHK(n)**(K(n)*(R)) =) K(n)*T HH(R)
48 VIGLEIK ANGELTVEIT AND JOHN ROGNES
for every S-algebra R, derived like the one in (4.1), but by applying K(n)-homo*
*logy
to the skeleton filtration of T HH(R). The identification of the E2-term uses *
*the
K"unneth formula for Morava K-theory. When K(n)*(R) ~= K(n)* Fp K(n)0(R)
is concentrated in degrees * 0 mod |vn|, we can rewrite the E2-term as
E2**~=K(n)* Fp HHFp*(K(n)0(R)) .
This is the case for n = 1, p = 2 and R = ku, whenQK(1)0(ku)mis the colimit ove*
*r m
of theQalgebrasmF2[ukQ| 1m k m]=(u2k= uk) ~= 2i=1F2. Then for each m the un*
*it
map 2i=1F2 ! HHF2*( 2i=1F2) is an isomorphism, so by passage to the colimit
the unit map
K(1)0(ku) ! HHF2*(K(1)0(ku))
is an isomorphism. Hence the K(1)-based B"okstedt spectral sequence collapses at
the edge s = 0, and the unit map ku ! T HH(ku) induces the asserted isomor-
phism.
Likewise, K(1)0(ko) is the colimit of the algebras F2[uk | 2 k m]=(u2k=
Q 2m-1
uk) ~= i=1 F2, and the same argument shows that the unit map ko ! T HH(ko)
induces an isomorphism in K(1)-homology.
The mod 2 Moore spectrum M admits a degree 8 self-map v41:M ! -8 M
that induces multiplication by v41in ku*(M) = k(1)* = P (v1). Smashing with Cj
yields a self-map v41:Y ! -8 Y , which admits a fourth root v1: Y ! -2 Y (up
to nilpotent maps). It induces multiplication by v1 in ko*(Y ) = k(1)* = P (v1)*
*. See
[DM82, 1.2].
Lemma 8.11. Let X be a spectrum such that K(1)*(X) = 0. Then v-11ss*(X ^
Y ) = 0 and v-41ss*(X ^ M) = 0.
Proof. Recall that j is the homotopy fiber of the map _3 - 1: ko(2) ! bspin(2).
The unit map e: S ! j induces an equivalence of mapping telescopes
v-11(S ^ Y ) -'!v-11(j ^ Y ) .
See e.g. the case n = 0 of [M82, 1.4]. Here v-11(ko(2)^ Y ) ' v-11k(1) = K(1) a*
*nd
likewise v-11(bspin(2)^ Y ) ' K(1), so there is a cofiber sequence of spectra
v-11Y ! K(1) -_!K(1) .
Furthermore, v-11Y ' v-41Y sits in a cofiber sequence
v-41 M -j!v-41M ! v-41Y
where j is nilpotent (j4 = 0).
From the first cofiber sequence it follows that if K(1)*(X) = 0, then ss*(X ^
v-11Y ) = v-11ss*(X ^ Y ) = 0. From the second cofiber sequence it then follows*
* that
multiplication by j is an isomorphism on ss*(X ^ v-41M) = v-41ss*(X ^ M). Since
j is nilpotent this implies that v-41ss*(X ^ M) = 0.
HOPF ALGEBRA STRUCTURE ON T HH 49
Corollary 8.12. The unit maps induce isomorphisms
~= -4
K(1)* = v-41ss*(ku ^ M) -! v1 ss*(T HH(ku) ^ M)
and ~
K(1)* = v-11ss*(ko ^ Y ) -=!v-11ss*(T HH(ko) ^ Y ) .
Proof. Apply lemmas 8.10 and 8.11 to the cofiber of the unit map R ! T HH(R),
for R = ku and R = ko, respectively.
Theorem 8.13. Consider the Adams spectral sequence
E**2= P (v1) E(~1, ~2) P (~)
for ss*(T HH(ku) ^ M), with ~1 = oe,~21, ~2 = oe,~22, ~ = oe,~3+ ~,1. oe,~22and*
* v1 = [,2]
in bidegrees (s, t) = (0, 3), (0, 7), (0, 8) and (1, 3), respectively. Recursiv*
*ely define
n-3
~n = ~n-2 ~2
for n 3. Likewise define r(1) = 2, r(2) = 4, r(n) = 2n + r(n - 2), s(1) = 3,
s(2) = 7 and s(n) = 2n + s(n - 2), for n 3. So ~n has bidegree (0, s(n)) and
2r(n) + s(n) = 2n+2 - 1.
Then the Adams spectral sequence has differentials generated by
n-1 r(n)
dr(n)(~2 ) = v1 ~n
for all n 1. This leaves the E1 -term
M1 n
E**1= P (v1){1} Pr(n)(v1){~n} E(~n+1 ) P (~2 ) .
n=1
Hence ss*(T HH(ku) ^ M)n= ss*(T HH(ku); Z=2) isngenerated as a P (v1)-module
by elements 1, xn,m = ~n~2 m and x0n,m= ~n~n+1 ~2 m for n 1 and m 0.
Here |xn,m | = s(n) + 2n+3 m and |x0n,m| = s(n) + s(n + 1) + 2n+3 m. The module
structure is generated by the relations vr(n)1xn,m = 0 and vr(n)1x0n,m= 0 for n*
* 1,
m 0.
Proof. The classes ~1, ~2 and ~ were introduced in proposition 8.7, and the E2-
term was found in lemma 8.8. By corollary 8.12 the abutment ss*(T HH(ku) ^ M)
is all v1-torsion, except the direct summand ss*(ku ^ M) = P (v1), which is inc*
*luded
by the unit map. Hence every class ~n is v1-torsion, so there is some integer r*
*(n)
such that vr(n)1~n is hit by a differential.
Suppose by induction that the dr(k)-differentials for 1 k < n have been fo*
*und,
leaving the term
n-1
E**r(n-1)+1= P (v1) E(~n, ~n+1 ) P (~2 )
n-1M k
Pr(k)(v1){~k} E(~k+1 ) P (~2 ) .
k=1
50 VIGLEIK ANGELTVEIT AND JOHN ROGNES
Consider the P (v1)-module generated by ~n. Let r be minimal such that vr1~n is
a boundary. The source x of such a differential cannot be divisible by v1, sin*
*ce r
is minimal, so dr(x) = vr1~n where x has Adams filtration s = 0 and even total
degree. Furthermore, vr1~n is not v1-torsion at this term, so x cannot be v1-to*
*rsion.
Likewise, ~n+1 does not annihilate vr1~n, so ~n+1 cannot annihilate x either. T*
*his
forces x 2 P (~2n-1). By lemma 8.5, T HH(ku) ^ M is a ~-spectrum, so dr is a
derivationnand-the1Leibniz rule showsnthat-x1= ~2n-1,nsince-dr1on any higher po*
*wer
of ~2 must be divisible by ~2 . Hence dr(~2 ) = vr1~n, and by a degree
count we must have r = r(n).
To complete the induction step, we must compute the Er(n)+1-term of the
Adams spectral sequence. The dr(n)-differentialkdoes not affect the summands
Pr(k)(v1){~k} E(~k+1 ) P (~2 ) for 1 k < n, and is zero on E(~n+1 )
P (~2n). It acts on P (v1){1, ~n, ~2n-1, ~n~2n-1}, leaving Pr(n)(v1){~n} P (*
*v1)
E(~n+2 ), where by definition ~n+2 = ~n~2n-1. This shows that the term P (v1)
E(~n, ~n+1 P (~2n-1) at the Er(n-1)+1-term gets replaced by the direct sum of
P (v1) E(~n+1 , ~n+2 ) P (~2n) and Pr(n)(v1){~n} E(~n+1 ) P (~2n).
Theorem 8.14. Consider the Adams spectral sequence
2
E**2= P (v1) E(~2, ~3) E(~2){~1} P (~ )
for ss*(T HH(ko) ^ Y ), with ~1 = oe,~41, ~2 = oe,~22+ ~,21. oe,~41, ~3 = oe,~4*
*1(oe,~3+ ~,1. oe,~22),
~2 = (oe,~3)2 and v1 = [,2] in bidegrees (s, t) = (0, 5), (0, 7), (0, 13), (0, *
*16) and
(1, 3), respectively. Recursively define
n-3
~n = ~n-2 ~2
for n 4. Likewise define r(1) = 1, r(2) = 4, r(n) = 2n + r(n - 2), s(1) = 5,
s(2) = 7 and s(n) = 2n + s(n - 2), for n 3. So ~n has bidegree (0, s(n)) and
2r(n) + s(n) = 2n+2 - 1.
Then the Adams spectral sequence has differentials generated by
n-1 r(n)
dr(n)(~2 ) = v1 ~n
for all n 2. This leaves the E1 -term
M1 n
E**1= P (v1){1} Pr(n)(v1){~n} E(~n+1 ) P (~2 ) .
n=1
Hence ss*(T HH(ko) ^nY ) = ss*(T HH(ko); Y )nis generated as a P (v1)-module*
* by
elements 1, xn,m = ~n~2 m and x0n,m= ~n~n+1 ~2 m for n 1 and m 0. Here
|xn,m | = s(n) + 2n+3 m and |x0n,m| = s(n) + s(n + 1) + 2n+3 m. The module stru*
*cture
is generated by the relations vr(n)1xn,m = 0 and vr(n)1x0n,m= 0 for n 1, m *
*0.
Proof. Starting with an imagined E1-term
E**1= P (v1) E(~1, ~2) P (~)
and differential d1(~) = v1~1, the proof is the same as for theorem 8.13.
HOPF ALGEBRA STRUCTURE ON T HH 51
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Department of Mathematics, Massachusetts Institute of Technology, USA
E-mail address: vigleik@math.mit.edu
Department of Mathematics, University of Oslo, Norway
E-mail address: rognes@math.uio.no