TOPOLOGICAL HOCHSCHILD HOMOLOGY
OF CONNECTIVE COMPLEX K-THEORY
Christian Ausoni
27.11.2003
Abstract. Let ku be the connective complex K-theory spectrum, completed a*
*t an
odd prime p. We present a computation of the mod (p, v1) homotopy algebra*
* of the
topological Hochschild homology spectrum of ku.
Introduction
Since the discovery of categories of spectra with a symmetric monoidal smash
product, as for instance the S-modules of [EKMM], the topological Hochschild
homology spectrum T HH(A) of a structured ring spectrum A can be defined by
translating the definition of Hochschild homology of an algebra into topology, *
*using
a now standard Ä lgebra - Brave New Algebra" dictionary. The algebraic origin
of this definition sheds light on many features of topological Hochschild homo-
logy, and has also led to more conceptual proofs of results that were based on
Bökstedt's original definition [Bö1] of topological Hochschild homology for fun*
*ctors
with smash products, see for instance [SVW1]. As can be expected by analogy
with the algebraic situation, this definition also highlights the role that top*
*ological
Hochschild (co-)homology plays in the classification of S-algebra extensions. *
*See
for example [SVW2], [La] or [BJ] for applications to extensions.
The aim of this paper is to exploit the advantages of such an algebraic defi*
*nition
to compute the mod (p, v1) homotopy groups of T HH(ku) as an algebra, which we
denote by V (1)*T HH(ku). Here ku is the connective complex K-theory spectrum
completed at an odd prime p, with a suitable S-algebra structure. Let us give a
succinct description of V (1)*T HH(ku), referring to Theorem 8.15 for the compl*
*ete
structure.
0.1. Theorem. Let p be an odd prime. The graded Fp-algebra V (1)*T HH(ku)
contains a class ~2, of degree 2p2, that generates a polynomial subalgebra. The
quotient
A* = V (1)*T HH(ku) =(~2)
is finite with 4(p - 1)2 elements. Moreover A* has a top class in dimension 2p2*
* +
2p - 2, and is self-dual in the sense that An ~=A2p2+2p-2-n for all n 2 Z.
The justification for performing this computation in V (1)-homotopy is that
V (1)*T HH(ku) is a complicated but finitely presented Fp-algebra. On the other
______________
1991 Mathematics Subject Classification. 55P42 19D55.
The author was supported by the SFB 478 - Geometrische Strukturen in der Mat*
*hematik
Typeset by AM S-T*
*EX
1
2 CHRISTIAN AUSONI
hand, a presentation of the mod p homotopy algebra V (0)*T HH(ku) requires infi*
*n-
itely many generators and relations. We nevertheless evaluate the additive stru*
*cture
of V (0)*T HH(ku) in Corollary 6.8.
A first motivation for these computations is to approach the algebraic K-the*
*ory
spectrum of ku. By work of Baas, Dundas and Rognes [BDR], the spectrum K(ku)
is conjectured to represent a cohomology theory whose zeroth group classifies e*
*quiv-
alence classes of virtual two-vector bundles. It is also expected that the spec*
*trum
K(ku) is of chromatic complexity two, which essentially means that it is suitab*
*le for
studying v2-periodic and v2-torsion phenomena in stable homotopy. Thus K(ku)
should represent a form of elliptic cohomology with a genuine geometric content,
something which has long be wished for.
Topological Hochschild homology is the target of a trace map from algebraic
K-theory, which refines over the cyclotomic trace map trc : K(ku) ! T C(ku; p).
The topological cyclic homology spectrum T C(ku; p) is a very close approximati*
*on
of K(ku)p, since by Dundas [Du] and Hesselholt-Madsen [HM1] it sits in a cofibre
sequence
K(ku)p -trc!T C(ku; p) ! -1 HZp.
The spectrum T C(ku; p) is built by taking the homotopy limit of a diagram whose
vertices are the fixed point of T HH(ku) under various closed subgroups of the
circle. Thus computing T HH(ku) is a first step in the study of K(ku) by trace
maps.
A second motivation for the computations presented in this paper is to pursue
the exploration of the "brave new worldö f ring spectra and their arithmetic. *
*In
the classical case, arithmetic properties of a ring or of a ring extension are *
*to a
large extend reflected in algebraic K-theory or its approximations, as topologi*
*cal
cyclic homology, topological Hochschild homology or even Hochschild homology.
An important example is descent in its various forms. Etale descent has been co*
*n-
jectured in algebraic K-theory by Lichtenbaum and Quillen, and has been proven
for various classes of rings [RW], [HM2]. For Hochschild homology, Geller and
Weibel [GW] proved 'etale descent by showing that for an 'etale extension A ,! B
there is an isomorphism HH*(B) ~= B A HH*(A). A form of tamely ramified
descent for topological Hochschild homology, topological cyclic homology and al-
gebraic K-theory of discrete valuation rings has been proven by Hesselholt and
Madsen [HM2], see also [Ts].
Laying the foundations of a theory of extensions for S-algebras is work in p*
*rogress
(see for instance [Ro1], [Ro2]), and it is not know at this point to what exten*
*d such
descent results can be generalized. Of course this depends also on how the vari*
*ous
types of extensions are defined. In fact one might want to test a definition o*
*f an
'etale extension, or of a tamely ramified extension of S-algebras, by showing t*
*hat
it is reflected by descent in algebraic K-theory or topological Hochschild homo-
logy. Unfortunately very few computations are available. This paper provides *
*an
interesting example of what we expect to be tamely ramified descent.
The S-algebra ku has a subalgebra `, called the Adams summand. The spectrum
ku splits as an S-module into a sum of p - 1 shifted copies of `, namely
p-2`
(0.2) ku ' 2i`.
i=0
T HH OF CONNECTIVE COMPLEX K-THEORY 3
McClure and Staffeldt computed the mod p homotopy groups of T HH(`) in [MS].
This computation was was then used by Rognes and the author [AR] to further
evaluate the mod (p, v1) homotopy groups of T C(`) and K(`). In view of the abo*
*ve
splitting, one could expect that similar computations should follow quite easily
for ku. However, we found out that a computation of T HH(ku) involves some
surprising new features. For example the Bökstedt spectral sequence
E2*,*(ku) = HHFp*,*H*(ku; Fp) =) H*(T HH(ku); Fp)
has higher differentials than that for ` (Lemma 8.5), and in computing the alge*
*bra
structure of H*(T HH(ku); Fp) we have to deal with extra multiplicative extensi*
*ons
(Proposition 8.9). It turns out that the multiplicative structure of V (1)*T HH*
*(ku),
given in Theorem 8.15, is quite complicated and highly non-trivial. All this re*
*flects
the well known fact that the splitting (0.2) is not multiplicative. In fact, mu*
*ch of
the added complexity of T HH(ku), as compared to T HH(`), can be accounted
for by speculating on the extension ` ! ku. We would like to think of it as the
extension defined by the relation
(0.3) ku = `[u]=(up-1 = v1)
in commutative S-algebras. This formula holds on coefficients, since the homo-
morphism `* ! ku* is the inclusion Zp[v1] ,! Zp[u] with v1 = up-1 . First we
notice that the prime (v1) in ` ramifies, and hence the extension ` ! ku should
not qualify as 'etale. This is confirmed by the computations of V (1)*T HH(`) a*
*nd
V (1)*T HH(ku), which show that
T HH(ku) 6' ku ^` T HH(`)
(compare with the Geller-Weibel Theorem). But if we invert v1 in ` and ku we
obtain the periodic Adams summand L and the periodic K-theory spectrum KU.
Here the ramification has vanished so the extension L ! KU should be 'etale. And
indeed we have an equivalence
T HH(KU)p ' KU ^L T HH(L)p.
This is a consequence of McClure and Staffeldt's computation of T HH(L)p, which
we adapt to compute T HH(KU)p in Proposition 6.12.
Returning to ` ! ku and formula (0.3), we notice that the ramification index
is (p - 1) and that this extension ought to be tamely ramified. The behavior
of topological Hochschild homology with respect to tamely ramified extensions of
discrete valuations rings was studied by Hesselholt and Madsen in [HM2]. Let us
assume that their results hold also in the generality of commutative S-algebra.*
* The
ring ku has a maximal ideal (u), with residue ring HZp and quotient ring KU.
Following [HM2, Theorem 1.5.6] we expect to have a localization cofibre sequence
in topological Hochschild homology
! j
(0.4) T HH(HZp) -i!T HH(ku) -! T HH(ku|KU).
This requires that we can identify by d'evissage T HH(HZp) with the topological
Hochschild homology spectrum of a suitable category of finite u-torsion ku-modu*
*les.
The tame ramification of ` ! ku should be reflected by an equivalence
(0.5) ku ^` T HH(`|L) ' T HH(ku|KU).
4 CHRISTIAN AUSONI
Now T HH(`|L) can be computed using the localization cofibre sequence
! j
T HH(HZp) -i!T HH(`) -! T HH(`|L)
and McClure-Staffeldt's computation of T HH(`). Thus T HH(ku|KU) is also
known, and T HH(ku) can be evaluated from the cofibre sequence (0.4). We elabo-
rate more on this in Paragraph 9.4. At this point we do not know if this concep*
*tual
line of argument can be made rigorous. This would of course require a general-
ization of the results in [HM2] for S-algebras. But promisingly, the descripti*
*on
of V (1)*T HH(ku) it provides is perfectly compatible with our computations of *
*it
given in Theorem 8.15.
The units Zxpact as p-adic Adams operations on ku. Let be the cyclic subgr*
*oup
of order p - 1 of Zxp. Then we have a homotopy equivalence
` ' kuh
where (-)h denotes taking the homotopy fixed points. We prove the following
result as Theorem 9.2.
0.6. Theorem. Let p be an odd prime. There are homotopy equivalences of p-
completed spectra
T HH(ku)h ' T HH(`),
T C(ku; p)h ' T C(`; p), and
K(ku)h ' K(`).
We would like to interpret this Theorem as an example of tamely ramified des*
*cent
for topological Hochschild homology, topological cyclic homology and algebraic *
*K-
theory.
Let us briefly review the content of the present paper. Our strategy for com*
*put-
ing T HH(ku) can be summarized as follows. Taking Postnikov sections we obtain
a sequence of S-algebra maps ku ! M ! HZp, where M is the section ku[0, 2p-6].
Using naturality of topological Hochschild homology we construct a sequence
T HH(ku) ! T HH(ku, M) ! T HH(ku, HZp) ! T HH(HZp).
We then use this sequence to interpolate from T HH(HZp) to T HH(ku), the point
being that at each step the added complexity can be handled by essentially alge*
*braic
means. In x1 we discuss some properties of ku and compute its homology. We
present in x2 the computations in Hochschild homology that will be needed later
on as input for various Bökstedt spectral sequences. In x3 we review the defini*
*tion
of topological Hochschild homology, following [MS], [EKMM] and [SVW1], and we
set up the Bökstedt spectral sequence. We present in x4 a simplified computation
of the mod p homotopy groups of T HH(HZp), for odd primes p. We also briefly
review a computation of V (1)*T HH(`). In x5 we determine the homotopy type of
the spectrum T HH(ku, HZp). Its mod p homotopy groups V (0)*T HH(ku; HZp)
are the input for a Bockstein spectral sequence which is computed in x6. It yie*
*lds
a description of V (0)*T HH(ku) as a module over V (0)*ku, given in Corollary 6*
*.8.
Note that x6 is not used in the later sections. In x7, we compute the mod p
homology of T HH(ku, M). The core of this paper is x8. Here we compute the mod
T HH OF CONNECTIVE COMPLEX K-THEORY 5
p homology Bökstedt spectral sequence for T HH(ku), and evaluate V (1)*T HH(ku)
as an algebra over V (1)*ku in Theorem 8.15. Finally, in x9 we compare T HH(`)
and T HH(ku) and elaborate on the properties of the extension ` ! ku mentioned
above.
Acknowledgment. It is a pleasure to acknowledge the help I recieved from John
Rognes. He suggested many improvements over an earlier draft. I especially thank
him and Vigleik Angeltveit for making a preliminary version of [AnR] available *
*to
me. I also thank Lars Hesselholt for explaining to me how [HM2] could be used to
have a guess on V (1)*T HH(ku).
Notations and conventions. Throughout the paper p will be a fixed odd prime,
and Zp will denote the p-adic integers. For an Fp-vector space V , let E(V ), P*
* (V )
and (V ) be the exterior algebra, polynomial algebra and divided power algebra
on V , respectively. If V has a basis {x1, . .,.xn}, we write V = Fp{x1, . .,.x*
*n} and
E(x1, . .,.xn), P (x1, . .,.xn) and (x1, . .,.xn) for these algebras. By defi*
*nition
(x) is the Fp-vector space Fp{flkx|k > 0} with product given by flix . fljx =
i+j h
i fli+jx, where fl0x = 1 and fl1x = x. Let Ph(x) = P (x)=(x = 0) be the
truncated polynomial algebra of height h.
In the description of spectral sequences, differentials are usually given up*
* to
multiplication by a unit.
We denote the mod p Moore spectrum by V (0). It has a periodic v1-multi-
plication 2p-2V (0) ! V (0) whose cofiber is called V (1). We define the mod p
homotopy groups of a spectrum X by V (0)*X = ß*(V (0) ^ X), and its mod (p, v1)
homotopy groups by V (1)*X = ß*(V (1) ^ X). By the symbol X 'p Y we mean
that X and Y are weakly homotopy equivalent after p-completion. We denote by fi
the primary mod p homology Bockstein, by fi0,rthe rth mod p homotopy Bockstein,
and by fi1,rthe rth mod v1 homotopy Bockstein.
For a ring R, let HR be the Eilenberg-Mac Lane spectrum associated to R. If A
is a (-1)-connected ring spectrum, we call the ring map A ! Hß0A that induces
the identity on ß0 the linearization map.
1. Connective complex K-theory
Let ku be the p-completed connective complex K-theory spectrum, having coef-
ficients ku* = Zp[u] with |u| = 2.
An E1 model. Since we would like to take ku as input for topological Hochschi*
*ld
homology, we need to specify a structured ring spectrum structure on ku. Fol-
lowing [MS, Section 9], we will take as model for ku the p-completion of the al-
gebraic K-theory spectrum of a suitable field. Let q be a topological generator
of Zxp, and let ~p1 be the set of all pth-power roots of 1 in ~Fq. We define *
*k to
be the field extensionSobtained by adjoining the elements of ~p1 to Fq. Hence
k = Fq[~p1 ] = i>0Fqpi(p-1). Quillen [Qu] proved that the Brauer lift induce*
*s a
homotopy equivalence
K(k)p -'!ku.
Notice that the inclusion k ~Fqalso induces an equivalence K(k)p ' K(~Fq)p, so
that we do not need to go all the way up to the algebraic closure to get a model
for ku. The algebraic K-theory spectrum of a commutative ring comes equipped
6 CHRISTIAN AUSONI
with a natural structure of commutative S-algebra in the sense of [EKMM], and
p-completion preserves this structure. In particular the Galois group
Gal(k=Fq) ~=Zp x Z=(p - 1)
acts on K(k)p by S-algebra maps. From now on ku will stand for K(k)p.
The Bott element. The mod p homotopy groups of ku are given by V (0)*ku =
P (u), where u is the mod p reduction of a generator of ku2. We call such a cla*
*ss u
a Bott element.
The algebraic K-theory groups of k were computed by Quillen [Qu], and are
given by 8
>< Z forn = 0,
Kn(k) = > kx forn odd > 1,
: 0 otherwise.
L
We have kx ~= lprime6=qZ=l1 . The universal coefficient formula for mod p hom*
*o-
topy implies that we have an isomorphism
~= Z x
V (0)2K(k) -! Tor 1(k , Fp) = Fp.
Identifying V (0)*ku with V (0)*K(k), a Bott element u 2 V (0)2ku corresponds
under this isomorphism to a primitive pth-root of 1 in k.
1.1. The Adams summand. Let ffi be a chosen generator of , the cyclic sub-
group of order p - 1 of Gal(k=Fq). Then ffi permutes the primitive p-th roots o*
*f 1 in
k via a cyclic permutation of order p - 1. In particular, if u 2 V (0)2ku is a *
*chosen
Bott element, then
ffi* : V (0)*ku ! V (0)*ku
maps u to ffu for some generator ff of Fxp. Let k0 be the subfield of k fixed u*
*nder
the action of . Then the homotopy fixed point spectrum kuh = K(k0)p is a
commutative S-algebra model for the p-completed Adams summand `, with coeffi-
cients `* = Zp[v1]. The spectrum ku is then a commutative `-algebra. It splits *
*as
an S-module into p-2
`
ku ' 2i`.
i=0
In V (0)*ku we have the relation up-1 = v1. We would like to think of ku as the
extension `[u]=(up-1 = v1) of ` in commutative S-algebras.
The dual Steenrod algebra. Let A* be the dual Steenrod algebra
A* = P (,1, ,2, . .). E(ø0, ø1, . .).
where ,i and øj are the generators defined by Milnor [Mi], of degree 2pi - 2 and
2pj - 1, respectively. We denote by ~,iand ~øjthe images of ,i and øj under the
canonical involution of A*. The coproduct _ on A* is given by
X pj X pj
_(~,k) = ,~j ~,i and _(~øk) = 1 ~øk+ ~øj ~,i
i+j=k i+j=k
T HH OF CONNECTIVE COMPLEX K-THEORY 7
where by convention ~,0= 1.
We view the mod p homology of a spectrum X as a left A*-comodule, i.e.
H*(X; Fp) = ß*(HFp ^ X). In particular we write
H*(HFp; Fp) = P (~,1, ~,2, . .). E(~ø0, ~ø1, . .)..
We will denote by * the coaction H*(X; Fp) ! A* H*(X; Fp). The mod p
reduction map æ : HZp ! HFp induces an injection in mod p homology, and we
identify H*(HZp; Fp) with its image in H*(HFp; Fp), namely
(1.2) H*(HZp; Fp) = P (~,1, ~,2, . .). E(~ø1, ~ø2. .)..
The homology of ku. The linearization map j : ku ! HZp has the 1-connected
cover ku[2, 1] of ku as fiber. By Bott periodicity, we can identify this cover *
*with
2ku. We assemble the iterated suspensions of the cofiber sequence
2ku -u!ku -j!HZp
into a diagram
. ._._u___// 4ku___u____// 2ku___u____//ku
(1.3) |j| |j| j||
fflffl| fflffl| fflffl|
4HZp 2HZp HZp.
Applying H*(-; Fp) we obtain an unrolled exact couple in the sense of Board-
man [Bo]. Placing 2sHZp in filtration degree -2s, it yields a spectral sequence
(1.4) E2s,t= H*(HZp; Fp) P*(x) (s,t)=) Hs+t(ku; Fp).
This is a second quadrant spectral sequence where a 2 H*(HZp; Fp) has bidegree
(0, |a|) and x has bidegree (-2, 4) and represents the image of u 2 V (0)2ku un*
*der
the Hurewicz homomorphism V (0)*ku ! H*(ku; Fp).
1.5. Theorem (Adams). There is an isomorphism of A*-comodule algebras
H*(ku; Fp) ~=H*(`; Fp) Pp-1 (x)
where H*(`; Fp) = P (~,1, ~,2, . .). E(~ø2, ~ø3, . .). A* is a sub-A*-comodul*
*e algebra
of A* and Pp-1 (x) is spherical (hence primitive).
Proof. The unrolled exact couple given by applying H*(-; Fp) to (1.3) is part o*
*f a
multiplicative Cartan-Eilenberg system. Thus the spectral sequence (1.4) is a s*
*pec-
tral sequence of differential A*-comodule algebras. It is also strongly conver*
*gent.
The E2-term of this spectral sequence is
E2*,*= P (~,1, ~,2, . .). E(~ø1, ~ø2, . .). P (x).
There is a differential d2p-2(~ø1) = xp-1 (Adams [Ad], Lemma 4), after which the
spectral sequence collapses for bidegree reasons, leaving
E1*,*= E2p-1*,*= P (~,1, ~,2, . .). E(~ø2, ~ø2, . .). Pp-1 (x).
There are no non-trivial multiplicative extensions. For instance xp-1 = 0 becau*
*se
the only other possibility would be xp-1 = ~,1, which would contradict the fact*
* that
x is primitive. For degree reasons there cannot be any non-trivial A*-comodule
extensions.
W p-2
This formula for H*(ku; Fp) reflects the splitting ku ' i=0 2i`. The cla*
*ss
v1 = up-1 2 V (0)*ku is of Adams filtration 1 and is in the kernel of the Hurew*
*icz
homomorphism, which accounts for the relation xp-1 = 0 in H*(ku; Fp).
8 CHRISTIAN AUSONI
1.6. Lemma. Let ffi : ku ! ku be the map given in 1.1. The algebra endomorphism
ffi* of H*(ku; Fp) is the identity on the tensor factor H*(`; Fp), and maps x t*
*o ffx
for some generator ff of Fxp.
Proof. By definition ` is fixed under the action of ffi, and x is the Hurewicz *
*image
of a Bott element. This implies the Lemma.
We will also need some knowledge of the integral homology of ku in low degre*
*es.
In the mod p homology of HZp, the primary Bockstein homomorphism fi is given
by
(1.7) fi : ~øi7! ~,i for alli > 1.
In particular H0(HZp; Z) = Zp and pHn(HZp; Z) = 0 for any n > 1.
1.8. Proposition. Consider the commutative graded algebra * over Zp defined
as
* = Zp[~x, ~,1]=(~xp-1= p~,1)
where ~xhas degree 2 and ~,1has degree 2p - 2. There exists a homomorphism of
Zp-algebras
~ : * ! H*(ku; Z)
such that ~ is an isomorphism in degrees 6 2p2 - 3, and such that the compositi*
*on
* -~!H*(ku; Z) -æ!H*(ku; Fp),
where æ is the mod p reduction, maps ~xto x and ~,1to ~,1.
Proof. By comparison with the case of HZp, we have primary mod p Bocksteins
fi(~øi) = ~,iin H*(ku; Fp), for all i > 2. We also have fi(x) = 0 for degree re*
*asons.
Hence the Bockstein spectral sequence
E1*= H*(ku; Fp) =) (H*(ku; Z)=torsion) Fp,
whose first differential is fi, collapses at the E2-term, leaving
E1* = E2*= Pp-1 (x) P (~,1).
Let ~x 2 H2(ku; Z) be a lift of x and ,~12 H2p-2(ku; Z) be a lift of ,~1. Since
H*(ku, Q) is polynomial over Qp on one generator in degree 2, there is a mul-
tiplicative relation ~xp-1 = a~,1in H*(ku; Z), for some a 2 pZp. The Postnikov
invariant ku[0, 2p - 4] ! 2p-1HZp of ku is of order p, so we can choose ~,1such
that a = p. We hence obtain the ring homomorphism ~. There is no torsion class
in H*(ku; Z) of degree 6 2p2 - 3, and ~ is an isomorphism in these degrees.
2. Hochschild homology
In this section we recall some properties of Hochschild homology and present
elementary computations that will be needed in the later sections.
Suppose R is a graded commutative and unital ring, A a graded unital R-algeb*
*ra,
and M a graded A-bimodule. Let us simply write for R . The enveloping algebra
T HH OF CONNECTIVE COMPLEX K-THEORY 9
Ae of A is the graded and unital R-algebra A Aop, and it acts on A on the le*
*ft
and on M on the right in the usual way.
If A is flat over R, the Hochschild homology of A with coefficients in M is *
*defined
as the bigraded R-module
e
HHRs,t(A, M) = TorAs,t(M, A).
Here s is the homological degree and t is the internal degree. The (two-sided) *
*bar
complex Cbar*(A), with Cbarn(A) = A (n+2), is a standard resolution of A as left
Ae -module, having the product ~ : Cbar0(A) = A A ! A of A as augmentation.
The complex M Ae Cbar*(A) is isomorphic to the Hochschild complex C*(A, M),
with Cn(A, M) = M A n , see [Lo, Chapter 1].
Suppose now that A is graded-commutative and that M is a commutative and
unital A-algebra with A-bimodule structure given by forgetting part of the A-
algebra structure. The standard product and coproduct on the bar resolution
make the Hochschild complex into a graded differential M-bialgebra with unit and
augmentation. In particular HHR*,*(A, M) is a bigraded unital M-algebra, and if
HHR*,*(A, M) is flat over M, then HHR*,*(A, M) is a bigraded unital and augment*
*ed
M-bialgebra. The unit ~
' : M -=! HHR0,*(A, M)
is given by the inclusion of the 0-cycles M = C0(A, M), and the augmentation
is the projection HHR*,*(A, M) ! HHR0,*(A, M) ~= M. There is also an R-linear
homomorphism
(2.1) oe : A ! HHR1,*(A, M)
induced by A ! C1(A, M) = M A, a 7! 1 a. It satisfies the derivation rule
oe(ab) = e(a)oe(b) + (-1)|a||b|e(b)oe(a),
where e : A ! M is the unit of M.
As usual, we write HHR*,*(A) for HHR*,*(A, A) and C*(A) for C*(A, A).
2.2. Proposition. Suppose that R consists of Fp concentrated in degree 0.
(a) Let P (x) be the polynomial Fp-algebra generated by x of even degree d. *
*Then
there is an isomorphism of P (x)-bialgebras
HHFp*,*P (x) ~=P (x) E(oex)
with oex primitive of bidegree (1, d).
(b) Let E(x) be the exterior Fp-algebra generated by x of odd degree d. Then
there is an isomorphism of E(x)-bialgebras
HHFp*,*E(x) ~=E(x) (oex)
with oex of bidegree (1, d) and with coproduct given by
X
(flkoex) = flioex E(x) fljoex.
i+j=k
Proof. This is standard, see for instance [MS, Proposition 2.1].
10 CHRISTIAN AUSONI
2.3. Proposition. Suppose that R consists of Fp concentrated in degree 0. Let
Ph(x) be the truncated polynomial Fp-algebra of height h generated by x of even
degree d, with (p, h) = 1. Then there is an isomorphism of Ph(x)-algebras
HHFp*,*Ph(x) ~=Ph(x)[zi, yj | i > 0, j > 1]= ~
where zi has bidegree (2i + 1, ihd + d) and yj has bidegree (2j, jhd + d). The *
*relation
~ is generated by 8
>< xh-1 zi = xh-1 yj = zizk = 0
ziyj = i+jixzi+j
>:
yjyg = j+gjxyj+g
for all i, k > 0 and all j, g > 1. Moreover z0 = oex, the generator zi is repre*
*sented
in the Hochschild complex C* Ph(x) by
X
(xk1 x xk2 x xk3 x . . .x xki+1 x)
k1,...,ki+1>0
k1+...+ki+1=i(h-1)
for all i > 1, and the generator yj is represented by
X
(xk1+1 xk2 x xk3 x . . .x xkj+1 x)
k1,...,kj+1>0
k1+...+kj+1=j(h-1)
for all j > 1.
Proof. Let A = Ph(x). This proposition is proven by choosing a small differenti*
*al
bigraded algebra over Ae that is a projective resolution of A as left Ae-module*
*. For
example, one can take
X*,*= Ae E(oex) (ø ),
where a 2 Ae has bidegree (0, |a|), oex has bidegree (1, d) and ø has bidegree *
*(2, dh).
The differential d of X, of bidegree (-1, 0), is given on the generators by
d(oex) = T and d(ø ) = Noex,
where T = x 1 - 1 x 2 Ae and N = (xh 1 - 1 xh)=(x 1 - 1 x) 2 Ae .
The product of A gives an augmentation X0 = Ae ! A. Now HHFp*,*Ph(x) is
isomorphic to the homology of the differential graded algebra
A Ae X*,*~= A E(oex) (ø )
with differential given by d(oex) = 0 and d(ø ) = hxh-1 oex. The class zi is r*
*epre-
sented by the cycle oex . fliø and the class yj is represented by the cycle x .*
* fljø , for
any i > 0 and j > 1. Representatives for the generators are obtained by choosing
a homotopy equivalence X* ! Cbar*(A), see for instance [BAC, page 55].
2.4. Remark. Notice that HHFp*,*Ph(x) is not flat over Ph(x), so that there is
no coproduct in this case. Moreover there are infinitely many algebra generator*
*s.
However, the set {1, zi, yj | i > 0, j > 1} of given algebra generators is also*
* a set of
T HH OF CONNECTIVE COMPLEX K-THEORY 11
Fp
Ph(x)-module generators for HHFp*,*Ph(x) . More precisely, HH*,* Ph(x) has one
Ph(x)-module generator in each non-negative homological degree, and is given by
8
Ph(x) n = 0,
><
HHFpn,*Ph(x) = > Ph-1 (x){z n-1_2}n > 1 odd,
: P
h-1 (x){y n_2} n > 2 even.
Fp
Let Bn : HHFpn,*Ph(x) ! HHn+1,* Ph(x) be Connes' operator (B0 coincides
with the operator oe given above). Then we have
B2n(yn) = zn
up to a unit in Fp, for all i > 0, where by convention we set y0 = x (see [BAC,*
* Propo-
sition 2.1]).
The next proposition shows that if we take Hochschild homology of Ph(x) with
coefficients having a lower truncation, we have both flatness and finite genera*
*tion
as an algebra.
2.5. Proposition. Suppose that R consists of Fp concentrated in degree 0. Let
Ph(x) be as above, and for 1 6 g < h, let Ph(x) ! Pg(x) be the quotient by
(xg). We view Pg(x) as a Ph(x)-algebra. Then there is an isomorphism of Pg(x)-
bialgebras
HHFp*,*Ph(x), Pg(x) ~=Pg(x) E(oex) (y)
where oex has bidegree (1, d) and y has bidegree (2, dh) and is represented in *
*the
Hochschild complex C* Ph(x), Pg(x) by
g-1X
xi xh-i-1 x.
i=0
The class oex is primitive and
X
(flky) = fliy Pg(x)fljy.
i+j=k
Proof. The proof is similar to that of Proposition 2.3, using the same resoluti*
*on
X*,*of Ph(x). The differential algebra X*,*admits a coproduct, defined as follo*
*ws.
The class oex is primitive and the coproduct on (ø ) is given by
X
(flkø ) = fliø Ae fljø.
i+j=k
By inspection this coproduct on X*,*is compatible with that on Cbar*Ph(x) under
a suitable homotopy equivalence.
12 CHRISTIAN AUSONI
2.6. Proposition. Consider P (u, u-1 ) as an algebra over P (v, v-1 ) with v =*
* uh
for some h prime to p. Then the unit
-1) -1
P (u, u-1 ) ! HHP(v,v*,* P (u, u )
is an isomorphism.
Proof. Let A = P (u, u-1 ). We have that A = P (v, v-1 )[u]=(uh = v) is flat as
P (v, v-1 )-module. The enveloping algebra is
h h
Ae = P (v, v-1 )[1 u, u 1]= (1 u) = (u 1) = v .
There is a two-periodic resolution of A as Ae -module
. .-.T!Ae-N! Ae -T! Ae,
with augmentation Ae ! A given by the product of A. Here T is multiplication
by 1 u - u 1 and N is multiplication by (1 u)h - (u 1)h =(1 u - u *
*1).
Applying A Ae - we obtain a two-periodic chain complex
h-1 0
. .-.0!A -hu---!A -! A
quasi-isomorphic to the Hochschild complex. Since huh-1 is invertible in A the
proposition follows.
2.7. Remark. Notice that the requirement that hv h-1_hbe invertible in P (v, v-*
*1 )[v 1_h]
is equivalent to the requirement that the extension P (v, v-1 ) ,! P (v, v-1 )[*
*v 1_h] be
'etale.
3. Topological Hochschild Homology
The category of S-modules (in the sense of [EKMM]) has a symmetric monoidal
smash product. Suppose that R is a unital and commutative S-algebra, that A is a
unital R-algebra, and that M is an A-bimodule. We denote the symmetric monoidal
smash-product in the category of R-modules by ^R , or simply by ^ if R = S.
We implicitly assume in the sequel that the necessary cofibrancy conditions are
satisfied. Following [MS], [EKMM], [SVW1] we define the topological Hochschild
homology spectrum of the R-algebra A with coefficients in M as the realization *
*of
the simplicial spectrum T HHRo(A, M) whose spectrum of q-simplices is
T HHRq(A, M) = M ^R A^q,
with the usual Hochschild-type face and degeneracy maps. If R = S we just write
T HH for T HHS.
From now on, we assume furthermore that A is commutative, that M is a com-
mutative and unital A-algebra, whose A-bimodule structure is induced by the unit
e : A ! M. Then T HHR (A, M) is a unital and commutative M-algebra. The unit
' : M ! T HHR (A, M)
T HH OF CONNECTIVE COMPLEX K-THEORY 13
is given by inclusion of the 0-simplices M = T HHR0(A, M). The level-wise produ*
*cts
T HHRq(A, M) ! M assemble into an augmentation
T HHR (A, M) ! M.
In particular M splits off from T HHR (A, M).
This construction of T HHR (A, M) is functorial in M. Moreover, if L ! M ! N
is a cofibration of A-bimodules, then there is a cofiber sequence of R-modules
T HHR (A, L) ! T HHR (A, M) ! T HHR (A, N).
We also have functoriality in A : if A ! B ! M are maps of commutative R-
algebras inducing the A- and B-bimodule structures on M, then there is a map of
M-algebras
T HHR (A, M) ! T HHR (B, M).
The M-algebra T HHR (A, M) has, in the derived category, the structure of an
M-bialgebra. To construct the coproduct
T HHR (A, M) ! T HHR (A, M) ^M T HHR (A, M)
in the derived category, one can use the weak equivalence T HHR (A, M) ' M ^Ae
B(A), where B(A) is the two-sided bar construction, and take advantage of the
coproduct B(A) ! B(A) ^Ae B(A) defined in the homotopy category (see [MSV],
[AnR]).
The Bökstedt spectral sequence. Let E be commutative S-algebra, for instance
E = HFp or HZp. Alternatively, one can take E such that E*A is flat over E*R.
The skeletal filtration of T HHR (A, M) induces in E*-homology a conditionally
convergent spectral sequence [EKMM, Th. 6.2 and 6.4]
e) R
(3.1) E2p,q(A, M) = TorE*(Ap,q(E*A, E*M) =) Ep+q T HH (A, M).
We call this spectral sequence the Bökstedt spectral sequence. If E*A is flat o*
*ver
E*R, we can identify the E2-term of this spectral sequence with
E2p,q(A, M) = HHE*Rp,q(E*A, E*M).
In good cases, the rich structure of T HHR (A, M) carries over to the Bökstedt
spectral sequence. Indeed, the unit, augmentation and product of T HHR (A, M)
are compatible with the skeletal filtration. In particular, it is a spectral se*
*quence
of differential unital and augmented E*M-algebras. If moreover Er*,*(A, M) is f*
*lat
over E*M for all r, it is also a spectral sequence of differential E*M-bialgebr*
*as. On
the E2-term of the Bökstedt spectral sequence, these structures coincide with t*
*he
corresponding structures for Hochschild homology described in the previous sect*
*ion.
See [AnR] for a detailed discussion of the Hopf-algebra structure on topological
Hochschild homology and on the Bökstedt spectral sequence.
Finally, if E*E is flat over E*, then E*A and E*M are E*E-comodule algebras
and the Bökstedt spectral sequence is one of differential E*E-comodules.
14 CHRISTIAN AUSONI
The map oe. There is a map ! : S1+^ A ! T HHR (A) defined in [MS, Proposition
3.2], and which is induced by the S1-action on the 0-simplices. It splits in t*
*he
homotopy category as the sum of the unit and a map oe : A ! T HHR (A).
Composing the latter with the map T HHR (A) ! T HHR (A, M) induced by the
unit e : A ! M, we obtain a map
(3.2) oe : A ! T HHR (A, M).
Let us assume E*A is flat over E*R. The interplay between the homomorphisms
oe* : E*A ! E*+1 T HHR (A, M) and oe : E*A ! HHE*R1,*(E*A, E*M) given above
is described in the following proposition. Let us first specify a notation.
3.3. Notation. We denote by [w] the class in E*T HHR (A, M) represented by w in
the E2-term of the Bökstedt spectral sequence.
3.4. Proposition (McClure-Staffeldt). For any a 2 E*A we have oe*(a) = [oea]
in E*+1 T HHR (A, M).
Proof. This is Proposition 3.2 of [MS].
In the case R = S, the map oe has another useful feature : it commutes with *
*the
Dyer-Lashof operations. Let us denote by Qi the Dyer-Lashof operation of degree
2i(p - 1) on the mod p homology of a commutative S-algebra.
3.5. Proposition (Bökstedt). For any a 2 H*(A; Fp) we have
Qioe*(a) = oe*(Qia)
in H*+2i(p-1)+1(T HH(A); Fp).
Proof. Bökstedt gives a proof of this proposition in [Bö2, Lemma 2.9]. His appr*
*oach
is to analyze the pth-reduced power of the map S1+^A ! T HH(A). Another elegant
proof is presented by Angeltveit and Rognes in [AnR].
4. Topological Hochschild Homology of Zp
In a very influential but unpublished paper, Bökstedt [Bö2] computed the hom*
*o-
topy type of the HZ-module T HH(HZ). In this section we present a simplified
computation of V (0)*T HH(HZp) for p > 3, since we will need this result in the
sequel.
We start by computing the Bökstedt spectral sequence
(4.1) E2s,t(HZp) = HHFps,tH*(HZp; Fp) =) Hs+t(T HH(HZp); Fp).
The description of H*(HZp; Fp) given in (1.2) and Proposition 2.2 imply that the
E2-term of this spectral sequence is
H*(HZp; Fp) E(oe,~1, oe,~2, . .). (oe~ø1, oe~ø2, . .).
where a 2 H*(HZp; Fp) has bidegree (0, |a|), the class oe,~ihas bidegree (1, 2p*
*i- 2)
and oe~øjhas bidegree (1, 2pj - 1), for i, j > 1.
4.2. Lemma. There are multiplicative relations
[oe~øi]p = [oe~øi+1]
in H*(T HH(HZp); Fp), for all i > 1.
Proof. This follows from the Dyer-Lashof operations Qpi~øi= ~øi+1. By Proposi-
tions 3.4 and 3.5 we have
i pi
[oe~øi]p = oe*(~øi)p = Qp oe*(~øi) = oe*(Q ~øi) = oe*(~øi+1) = [oe~øi+*
*1].
T HH OF CONNECTIVE COMPLEX K-THEORY 15
4.3. Lemma. In the spectral sequence (4.1) we have dr = 0 for 2 6 r 6 p - 2, a*
*nd
there are differentials
dp-1 (flp+k oe~øi) = oe,~i+1. flkoe~øi
for all i > 1 and k > 0. Taking into account the algebra structure, this leaves
Ep*,*(HZp) = H*(HZp; Fp) E(oe,~1) Pp(oe~ø1, oe~ø2, . .)..
At this stage the spectral sequence collapses.
Proof. The E2-term of the spectral sequence (4.1) is flat over H*(HZp; Fp), so *
*this
is a spectral sequence of unital augmented differential H*(HZp; Fp)-bialgebras,*
* at
least until a differential puts an end to flatness. The mod p primary Bockstei*
*n fi
in the mod p homology of a ring spectrum is a derivation. From the relation
p
[oe,~i] = oe*(~,i) = oe* fi(~øi) = fi oe*(~øi) = fi([oe~øi-1] ) *
*= 0
in H*(T HH(HZp); Fp) we know that each class oe,~ifor i > 2 must be hit by a
differential. The rich algebra structure of the spectral sequence now only lea*
*ves
enough freedom for the claimed pattern of differentials. To see this, one can *
*for
example work dually. The H*(HZp; Fp)-dual of E2*,*(HZp) is given by
2 # # # # #
E*,*(HZp) = H*(HZp; Fp) E(oe,~1, oe,~2, . .). P (oe~ø1, oe~ø2, . .)*
*..
Here we used that the Fp-module generators of E(oe,~1, oe,~2, . .). (oe~ø1, o*
*e~ø2, . .).
form a basis of the free module E2*,*(HZp) over H*(HZp; Fp), and that the dual
basis is generated, under the product, by the duals oe,~#iand oe~øj#of oe,~iand*
* oe~øj,
respectively. By induction on i > 1 one checks that the vanishing of oe,~#i+1in
#
E1*,*(HZp) must be accounted for by a differential
(dp-1 )# (oe,~#i+1) = (oe~øi#)p.
In particular we also have differentials
# #k #p+k
(dp-1 )# oe,~i+1. (oe~øi) = (oe~øi)
for all k > 0. This leaves
p # # # #
E*,*(HZp) = H*(HZp; Fp) E(oe,~1) Pp(oe~ø1, oe~ø2, . .)..
At this stage the spectral sequence collapses for bidegree reasons. Dualizing a*
*gain,
we get the Lemma.
4.4. Remark. Bökstedt's original argument to prove this Lemma relies on a Kudo-
type formula for differentials in the spectral sequence
HHFps,tH*(A; Fp) =) H*(T HH(A); Fp),
namely n+1
dp-1 (flp+k oex) = oe(fiQ ___2x) . flkoex
whenever x 2 H*(A; Fp) is a class of odd degree n. See [Bö2] or [Hu].
16 CHRISTIAN AUSONI
4.5. Proposition. There is an isomorphism of A*-comodule algebras
H*(T HH(HZp); Fp) ~=H*(HZp; Fp) E([oe,~1]) P ([oe~ø1]).
The A*-coaction * is given on the tensor factor H*(HZp; Fp) by the inclusion in
the coalgebra A*. The class [oe,~1] is primitive and
*([oe~ø1]) = 1 oe~ø1+ ~ø0 [oe,~1].
Proof. By Lemma 4.3 the E1 -term of the spectral sequence (4.1) is
E1*,*(HZp) = H*(HZp; Fp) E(oe,~1) Pp(oe~ø1, oe~ø2, . .)..
Lemma 4.2 implies that the subalgebra Pp(oe~ø1, oe~ø2, . .).of E1*,*(HZp) lifts*
* as a sub-
algebra P ([oe~ø1]) of H*(T HH(HZp); Fp). There are no further possible multipl*
*ica-
tive extensions. The A*-coaction on the tensor factor H*(HZp; Fp) is determined
by naturality with respect to the unit map HZp ! T HH(HZp). The values of
*(oe,~1) and *(oe~ø1) follow by naturality with respect to oe : HZp ! T HH(H*
*Zp),
which is expressed in the formula
(4.6) *oe* = (1 oe*) *.
We use that in A* H*(HZp; Fp) we have
*(~,1) = ~,1 1 + 1 ~,1and *(~ø1) = ~ø1 1 + 1 ~ø1+ ~ø0 ~,1,
and that oe* is a derivation.
4.7. Theorem (Bökstedt). For any prime p > 3 there is an isomorphism of
Fp-algebras
V (0)*T HH(HZp) ~=E(~1) P (~1),
where |~1| = 2p - 1 and |~1| = 2p.
Proof. The proof we give here is adapted from the proof of [AR, Proposition 2.6*
*].
Since HFp ' V (0) ^ HZp, the spectrum V (0) ^ T HH(HZp) is an HFp-module. In
particular the Hurewicz homomorphism
V (0)*T HH(HZp) ! H*(V (0) ^ T HH(HZp); Fp)
is an injection with image the A*-comodule primitives. Let ~1 and ~1 be classes
that map respectively to [oe,~1] and [oe~ø1] - ~ø0[oe,~1] under this homomorphi*
*sm. By
inspection these classes generate the subalgebra of A*-comodule primitive eleme*
*nts
in H*(V (0) ^ T HH(HZp); Fp).
4.8. Remark. Bökstedt proved also that there are higher mod p homotopy Bock-
steins r r
fi0,r(~p1) = ~p1-1~1
in V (0)*T HH(HZp), for all r > 1. This implies a homotopy equivalence of HZp-
modules `
T HH(HZp) 'p HZp _ 2k-1 HZ=pvp(k),
k>1
where vp is the p-adic valuation.
Let ` be the Adams summand defined in 1.1. A very similar computation can
be performed for T HH(`), yielding a description of the Fp-algebra V (1)*T HH(`*
*).
T HH OF CONNECTIVE COMPLEX K-THEORY 17
4.9. Theorem (McClure-Staffeldt). For any prime p > 5 there are isomor-
phisms of Fp-algebras
H*(T HH(`); Fp) ~=H*(`; Fp) E(oe,~1, oe,~2) P (oe~ø2)
and
V (1)*T HH(`) ~=E(~1, ~2) P (~2),
where |~1| = 2p - 1, |~2| = 2p2 - 1, and |~2| = 2p2.
Proof. See [MS, Corollary 7.2] for a computation of V (0)*T HH(`). The descript*
*ion
of V (1)*T HH(`) given here was made explicit in [AR, Proposition 2.6]. We brie*
*fly
review this computation. The linearization map ` ! HZp is injective in mod p
homology and induces an injection on the E2-terms of the respective Bökstedt
spectral sequences. By comparison this determines both the differentials and t*
*he
multiplicative extensions in the Bökstedt spectral sequence for `, and the desc*
*ription
of H*(T HH(`); Fp) given follows. Now there is an equivalence V (1) ^ ` ' Fp, *
*so
V (1) ^ T HH(`) is an HFp-module and the Hurewicz homomorphism
V (1)*T HH(`) ! H*(V (1) ^ T HH(`); Fp)
is injective with image the A*-comodule primitives. The homotopy classes ~1,
~2 and ~2 have as image the primitive homology classes [oe,~1], [oe,~2] and [oe*
*~ø2] -
~ø0[oe,~2].
5. The homotopy type of T HH(ku, HZp)
The linearization map j : ku ! HZp makes HZp into a commutative and unital
ku-algebra. Our aim in this section is to determine the homotopy type of the
HZp-algebra T HH(ku, HZp).
We first compute its mod p homology, using the Bökstedt spectral sequence
(5.1)
E2s,t(ku, HZp) = HHs,tH*(ku; Fp), H*(HZp; Fp) =) Hs+t(T HH(ku, HZp); Fp).
The algebra homomorphism j* : H*(ku; Fp) ! H*(HZp; Fp) is the edge homomor-
phism
H*(ku; Fp) i E10,* E20,*= H*(HZp; Fp)
of the spectral sequence (1.4) described in the proof of Theorem 1.5. It is the*
*refore
given by
P (~,1, ~,2, . .). E(~ø2, ~ø3, . .). !Pp-1P(x)(~,1, ~,2, . .). E(~ø1, ~*
*ø2, . .).,
~,i7! ~,iif i > 1,
~øi7! ~øiif i > 2,
x 7! 0.
By Propositions 2.2 and 2.5, the E2-term of (5.1) is
H*(HZp; Fp) E(oex, oe,~1, oe,~2, . .). (y, oe~ø2, oe~ø3, . .).
where a 2 H*(HZp; Fp) has bidegree (0, |a|), a class oe! for ! 2 H*(ku; Fp) has
bidegree (1, |!|), and y has bidegree (2, 2p - 2).
Recall that a class oe! is represented in the Hochschild complex by 1 ! an*
*d y
is represented by 1 xp-2 x.
18 CHRISTIAN AUSONI
5.2. Lemma. The classes oe,~1and y in E2*,*(ku, HZp) are permanent cycles, and
in H*(T HH(ku, HZp); Fp) there is a primary mod p Bockstein
fi([y]) = [oe,~1].
Proof. To detect the mod p Bockstein claimed in this lemma we will need some
knowledge of the integral homology of T HH(ku, HZp). For integral computa-
tions it is more convenient to work with T HHSp(ku, HZp), because in this way
the ground ring for the Bökstedt spectral sequence is H*(Sp; Z) = Zp instead of
H*(S; Z) = Z (recall that H*(ku, Z) and H*(HZp, Z) are Zp-algebras). The natu-
ral map T HH(ku, HZp) ! T HHSp(ku, HZp) is an equivalence after p-completion,
and induces an isomorphism of the mod p homology Bökstedt spectral sequences. It
follows that the mod p homology Bockstein spectral sequences for T HH(ku, HZp)
and T HHSp(ku, HZp) are also isomorphic.
The class oe,~1is a permanent cycle for bidegree reasons. On the other hand*
* y
generates the component of total degree 2p in the E2-term of the Bökstedt spect*
*ral
sequence (5.1). We claim that [oe,~1] is the mod p reduction of a class of ord*
*er
p in integral homology. Then [oe,~1] must be in the image of the primary mod p
Bockstein, and this forces fi([y]) = [oe,~1]. In particular y is also a permane*
*nt cycle.
It remains to prove the claim that [oe,~1] is the mod p reduction of a class*
* of order
p in integral homology. Consider the commutative diagram
H*(ku; Z) __oe*_//_H*+1 (T HHSp(ku, HZp); Z)
|æ| æ||
fflffl| oe* fflffl|
H*(ku; Fp) _____//_H*+1 (T HHSp(ku, HZp); Fp)
where æ is the mod p reduction. If ~,12 H2p-2(ku; Z) is the class defined in 1*
*.8,
then
æoe*(~,1) = oe*æ(~,1) = oe*(~,1) = [oe,~1]
in H2p-1(T HHSp(ku, HZp); Fp). In particular [oe,~1] is the reduction of an int*
*egral
class. We now prove that pH2p-1(T HHSp(ku, HZp); Z) = 0, which implies the
claim.
The Bökstedt spectral sequence converging to H*(T HHSp(ku, HZp); Z) has an
E2-term given by
(5.3) E~2*,*(ku, HZp) = TorH*(ku^Spku;Z)*,*H*(ku; Z), H*(HZp; Z) .
Recall the graded ring homomorphism ~ : * ! H*(ku; Z) defined in Proposi-
tion 1.8. Since * is torsion free and ~ is an isomorphism in degrees 6 2p2 - 3*
*, the
map
e*= * Zp * ! H*(ku ^Spku; Z)
is also an isomorphism in this range of degrees. In particular, the Tor group o*
*f (5.3)
is isomorphic to
HHZp*,* *, H*(HZp; Z)
T HH OF CONNECTIVE COMPLEX K-THEORY 19
in total degrees 6 2p2 - 3. Here the *-bimodule structure of H*(HZp; Z) is giv*
*en
by the ring homomorphism * ! H*(HZp; Z) that sends ~xto 0 and ~,1to a lift of
~,1in H*(HZp; Z). There is a free resolution X* of * as e*-module
d2 d1 e
0 _____//_ e*{w}_____// e*{oe~x, oe,~1}//_ *
having as augmentation the product e*! *. The bidegree of the generators is
|oe~x| = (1, 2), |oe,~1| = (1, 2p - 2) and |w| = (2, 2p - 2). The differential *
*is given by
d1(oe~x)= 1 ~x- ~x 1,
d1(oe,~1)= 1 ~,1- ~,1 1, and
p-1 p-1
d2(w) = (1 ~x - ~x 1)=(1 ~x- ~x 1) oe~x- poe,~1.
In total degrees 6 2p2 - 3, the E2-term (5.3) is isomorphic to
HH*,* *, H*(HZp; Z) = H*(H*(HZp; Z) e*X*).
By inspection we have E~21,2p-2(ku, HZp) = Fp{oe,~1}, and the remaining groups *
*in
E~2*,*(ku, HZp) of total degree 2p - 1 are all trivial. For degree reasons ther*
*e are no
differentials affecting E~21,2p-2(ku, HZp). This proves that
H2p-1(T HHSp(ku, HZp); Z) ~=Fp{[oe,~1]}.
5.4. Lemma. There are multiplicative relations
[y]p = [oe~ø2] and [oe~øi]p = [oe~øi+1]
in H*(T HH(ku, HZp); Fp), for all i > 2.
Proof. The linearization j : ku ! HZp induces a map of HZp-algebras
j : T HH(ku, HZp) ! T HH(HZp).
By naturality of oe, we have j*([oe,~1]) = [oe,~1] and j*([oe~øi]) = [oe~øi] fo*
*r all i > 2.
The Bockstein fi : H2p(T HH(HZp); Fp) ! H2p-1(T HH(HZp); Fp) is injective and
maps [oe~ø1] to [oe,~1]. Thus the Bockstein fi([y]) = [oe,~1] implies that j*([*
*y]) = [oe~ø1].
Therefore the multiplicative relations for H*(T HH(ku, HZp); Fp) follow from the
ones for H*(T HH(HZp); Fp) given in Lemma 4.2.
5.5. Lemma. The Bökstedt spectral sequence (5.1) behaves as follows.
(a) For 2 6 r 6 p - 2 we have dr = 0, and there is a differential
dp-1 (flp+k oe~øi) = oe,~i+1. flkoe~øi
for all i > 2 and k > 0. Taking into account the algebra structure, this leaves
Ep*,*(ku, HZp) = H*(HZp; Fp) E(oex, oe,~1, oe,~2) Pp(oe~ø2, oe~ø3, . .)*
*. (y).
(b) For p 6 r 6 2p - 2 we have dr = 0, and there is a differential
d2p-1(flp+k y) = oe,~2. flky
for all k > 0. Taking into account the algebra structure, this leaves
E2p*,*(ku, HZp) = H*(HZp; Fp) E(oex, oe,~1) Pp(y, oe~ø2, oe~ø3, . .*
*)..
At this stage the spectral sequence collapses.
20 CHRISTIAN AUSONI
Proof. The proof is similar to that of Lemma 4.3. Here also the spectral sequen*
*ce
is one of unital and augmented differential H*(HZp; Fp)-bialgebras, at least un*
*til a
differential puts an end to flatness. Part (a) of the Lemma also follows by nat*
*urality
with respect to j : T HH(ku, HZp) ! T HH(HZp). This time the class oe,~2is a
boundary because [oe,~2] = fi([y]p) = 0. Again, an algebraic argument implies t*
*hat
the only possibility is a differential
d2p-1(flpy) = oe,~2.
The differential d2p-1(flp+k y) = oe,~2. flky are then detected using the copro*
*duct
on flp+k y. This leaves the E2p-term as claimed, where all classes lie in filt*
*ration
degrees < 2p, and the spectral sequence collapses.
5.6. Proposition. There is an isomorphism of A*-comodule algebras
H*(T HH(ku, HZp); Fp) ~=H*(HZp; Fp) E([oex], [oe,~1]) P ([y])
where [oex] and [oe,~1] are A*-comodule primitives and the coaction on [y] is
*([y]) = ~ø0 [oe,~1] + 1 [y].
Proof. The Bökstedt spectral sequence described in Lemma 5.5 is strongly conver-
gent and has an E1 -term given by
E1*,*(ku, HZp) = H*(HZp; Fp) E(oex, oe,~1) Pp(y, oe~ø2, oe~ø3, . .)*
*..
We have the multiplicative extensions [y]p = [oe~ø2] and [oe~øi]p = [oe~øi+1] e*
*stablished
in Lemma 5.4. There are no further possible multiplicative extensions. The clas*
*ses
[oex] and [oe,~1] are comodule primitives by (4.6). The homomorphism
j* : H*(T HH(ku, HZp); Fp) ! H*(T HH(HZp); Fp)
maps the class [y] to [oe~ø1], which has coaction *([oe~ø1]) = ~ø0 [oe,~1] + *
*1 [oe~ø1].
The formula for the coaction on [y] follows by naturality because j* is injecti*
*ve in
the relevant degrees.
5.7. Theorem. For any prime p > 3 there is an isomorphism of Fp-algebras
V (0)*T HH(ku, HZp) ~=E(z, ~1) P (~1)
with |z| = 3, |~1| = 2p - 1 and |~1| = 2p.
Proof. The proof is the same as for Theorem 4.7. Here z, ~1 and ~1 map respecti*
*vely
to [oex], [oe,~1] and [y] - ~ø0[oe,~1] under the Hurewicz homomorphism
V (0)*T HH(ku, HZp) ! H*(V (0) ^ T HH(ku, HZp); Fp).
Notice that as in 4.8 we have mod p Bocksteins
r pr-1
fi0,r(~p1) = ~1 ~1
in V (0)*T HH(ku, HZp).
5.8. Corollary. For any prime p > 3, there is a homotopy equivalence
T HH(ku, HZp) 'p S3+^ T HH(HZp).
T HH OF CONNECTIVE COMPLEX K-THEORY 21
6. The mod p homotopy groups of T HH(ku)
In this section we compute V (0)*T HH(ku) as a module over P (u) = V (0)*ku.
The strategy we use is similar to that developed by McClure and Staffeldt [MS]
for computing V (0)*T HH(`) as a P (v1)-module, except that we use the mod u
Bockstein spectral sequence
(6.1) E1*= V (0)*T HH(ku, HZp) =) V (0)*T HH(ku)=(u-torsion) P(u) Fp
instead of the Adams spectral sequence.
6.2. Proposition. Let X be a connective ku-module of finite type. There is a
one-column, strongly convergent spectral sequence
E1*= V (0)*(HZp ^ku X) =) V (0)*X=(u-torsion) P(u) Fp,
called the mod u Bockstein spectral sequence. Its rth differential is denoted f*
*iu,rand
decreases degree by 2r + 1. There is an isomorphism of P (u)-modules
M
V (0)*X ~=P (u) E1* Pr(u) im(fiu,r).
r>1
Moreover, if X is a ku-algebra, then this is a spectral sequence of differentia*
*l alge-
bras.
6.3. Remark. The grading of the target group V (0)*X=(u-torsion) P(u) Fp is
such that a class a P(u) 1 has the degree of its representative of minimal deg*
*ree in
V (0)*X=(u-torsion) x Fp
Proof. This is very similar to the mod p Bockstein spectral sequence, see for i*
*n-
stance [Mc, Theorem 10.3]. We just sketch the proof. Consider the diagram (1.3)
and prolong it to the right by desuspending. Applying V (0)*(- ^ku X) we obtain
an unrolled exact couple. Placing V (0)*( 2sHZp ^ku X) in filtration degree -2s,
it yields a spectral sequence
V (0)*(HZp ^ku X) P (u, u-1 ) =) V (0)*X P(u) P (u, u-1 ).
Here the class u represents the Bott element and has bidegree (-2, 4). Strong
convergence follows from the assumptions on X. This spectral sequence is one of
differential P (u, u-1 )-modules. In particular all columns are isomorphic at *
*each
stage. Extracting the column of filtration 0 and taking fiu,r = u-r dr, we obt*
*ain
the mod u Bockstein spectral sequence.
The ring P (u) is a graded principal ideal domain. Since X is connective of
finite type the graded module V (0)*X splits as a sum of shifted copies of P (u)
and its truncations (namely the quotients by an ideal generated by a homogeneous
element). If there is a differential fiu,r(a) = b, then by definition of fiu,r *
*the class b
is the image under
V (0)*X ! V (0)*(HZp ^ku X)
of a class ~bnot divisible by u, such that ur-1~b6= 0 and ur~b= 0. The descript*
*ion
of the P (u)-module V (0)*X given follows. Finally, if X is a ku-algebra, then *
*our
22 CHRISTIAN AUSONI
unrolled exact couple is part of a multiplicative Cartan-Eilenberg system (see *
*[CE],
XV.7) with
H(p, q) = V (0)* ( pku= qku) ^ku X) ,
and hence this spectral sequence is one of differential algebras.
Let K(1) be the Morava K-theory, with coefficients K(1)* = P (v1, v-11). We
compute K(1)*T HH(ku), which allows us to determine the E1 -term of (6.1). It
will then turn out that only one pattern of differentials is possible.
6.4. Proposition. There is an isomorphism of K(1)*-algebras
K(1)*ku ~=P (u, u-1 ) K(1)0`.
Here u is the Hurewicz image of the Bott element and on the right-hand side the
K(1)*-module structure is given by the inclusion K(1)* ! P (u, u-1 ) with v1 =
up-1 .
Proof. The isomorphism
K(1)*` ~=K(1)* K(1)0`
W p-2
is established in [MS, Proposition 5.3.(a)]. The splitting ku ' i=0 2i` imp*
*lies
that the formula claimed for K(1)*ku holds additively. The multiplication-by-u
map 2ku ! ku induces an isomorphism K(1)*-2 ku ~=K(1)*ku since for its cofiber
HZp we have K(1)*HZp = 0. Thus multiplication by u is invertible in K(1)*ku.
The relation v1 = up-1 follows from the corresponding relation in V (0)*ku.
6.5. Theorem. The unit map ku ! T HH(ku) induces isomorphisms
~=
K(1)*ku -! K(1)*T HH(ku)
and ~
v-11V (0)*ku -=!v-11V (0)*T HH(ku).
Proof. McClure and Staffeldt [MS, Th. 5.1 and Cor. 5.2] prove the corresponding
statements for `. Their argument extends to this case, and we just outline it,
referring to [MS] for further details. By Proposition 6.4 we have an isomorphism
-1 Fp ~= K(1)*
HHK(1)**,*P (u, u ) HH*,*(K(1)0`) -! HH*,* (K(1)*ku).
By Proposition 2.6 and [MS, Proposition 5.3.(c)] the unit for each of the tensor
factors on the left-hand side is an isomorphism. This implies that the unit
K(1)*ku ! HHK(1)**,*(K(1)*ku)
is an isomorphism. The isomorphism K(1)*ku ~=K(1)*T HH(ku) follows from the
collapse of the Bökstedt spectral sequence
E2s,t= HHK(1)*s,t(K(1)*ku) =) K(1)s+tT HH(ku).
Finally, by Lemma 5.4 of [MS] the first isomorphism claimed implies the second
one.
T HH OF CONNECTIVE COMPLEX K-THEORY 23
6.6. Definition. Let
8
>< p - 2 if n = 0,
a(n) => pn+1 - pn + pn-1 - . .+.p2 - p if n > 1 is odd,
: pn+1 - pn + pn-1 - . .+.p3 - p2 + p - 2 if n > 2 is even, and
8
>< 0 if n = 0, 1,
b(n) => pn-1 - pn-2 + . .+.p - 1 if n > 2 is even,
: pn-1 - pn-2 + . .+.p2 - p if n > 3 is odd.
We are now ready to describe the differentials in the spectral sequence (6.1*
*). By
Theorem 5.7 its E1-term is
E1*= V (0)*T HH(ku, HZp) ~=E(z, ~1) P (~1).
6.7. Theorem. The u-Bockstein spectral sequence (6.1) for T HH(ku) has differ-
entials (
n z~b(n)1 if n > 0 is even,
fiu,a(n)(~p1) = b(n)
~1~1 if n > 1 is odd.
Proof. Since we have the relation v1 = up-1 in V (0)*ku, we know from Theorem 6*
*.5
that u-1 V (0)*T HH(ku) = P (u, u-1 ). In particular the E1 -term consists sol*
*ely
of a copy of Fp in degree 0, and V (0)0T HH(ku, HZp) = Fp is the subgroup of
permanent cycles in E1*. The classes z and ~1 are infinite cycles. As can be ch*
*ecked
by induction on n, the pattern of differentials given is the only one that leav*
*es
E1* = Fp.
6.8. Corollary. There is an isomorphism of P (u)-modules
M
V (0)*T HH(ku) ~=P (u) Pa(n)(u) In,
n>0
where In is the graded Fp-module
( b(n+1) pn+1 b(n)+pnj
E(~1~1 ) P (~1 ) Fp{z~1 |j = 0, . .,.p - 2}, n > 0 even,
In = b(n+1) n+1 b(n)+pnj
E(z~1 ) P (~p1 ) Fp{~1~1 |j = 0, . .,.p - 2}, n > 1 odd.
Proof. The Ea(n)-term of the u-Bockstein spectral sequence is given by
( pn b(n) b(n+1)
P (~1 ) E(z~1 , ~1~1 ) for n > 0 even,
Ea(n)*= pn b(n+1) b(n)
P (~1 ) E(z~1 , ~1~1 ) for n > 1 odd.
Using the description of fiu,a(n)given above, one checks that im(fiu,a(n)) ~=In*
*. The
Corollary then follows from Proposition 6.2.
6.9. Remark. A computation of V (0)*T HH(ku) at the prime 2 has been performed
by Angeltveit and Rognes [AnR]. Their argument is similar to that of [MS] at odd
primes, and involves the Adams spectral sequence.
24 CHRISTIAN AUSONI
6.10. Corollary. In any presentation, the P (u)-algebra V (0)*T HH(ku) has in-
finitely many generators and infinitely many relations.
Proof. This follows from Corollary 6.8 by degree and u-torsion considerations.
We view this Corollary as a motivation for pursuing, in the next two section*
*s, a
description of the algebra structure of V (1)*T HH(ku). By analogy with the cas*
*e of
T HH(`) one expects it to be nicer then the structure of V (0)*T HH(ku). Indeed,
it will turn out that V (1)*T HH(ku) admits finitely many generators and relati*
*ons.
The periodic case. Let KU and L denote the periodic complex K-theory spec-
trum and the periodic Adams summand, both completed at p. They inherit a
commutative S-algebra structure as the E(1)-localizations of ku and `, respec-
tively. The homotopy type of the spectrum T HH(L)p was computed by McClure
and Staffeldt [MS, Theorem 8.1], and is given as
(6.11) T HH(L)p ' L _ LQ ,
where LQ denotes the rationalization of the spectrum L. Their argument can be
applied to compute T HH(KU)p, the only new ingredient being the computation of
K(1)*T HH(ku) given in Theorem 6.5. We therefore formulate without proof the
following proposition.
6.12. Proposition. There is an equivalence T HH(KU)p ' KU _ KUQ .
7. Coefficients in a Postnikov section
In this section we will assume that p > 5. Let M be the Postnikov section
M = ku[0, 2p - 6] of ku with coefficients
æ ku ifn 6 2p - 6,
Mn = n
0 otherwise.
It is known ([Ba, Theorem 8.1]) that the Postnikov sections of a commutative S-
algebra can be constructed within the category of commutative S-algebras. We can
therefore assume that the natural map OE : ku ! M is a map of commutative S-
algebras. In this section we compute the mod p homology groups of T HH(ku, M)
using the Bökstedt spectral sequence. This will be useful in performing the cor*
*re-
sponding computations for T HH(ku).
The mod p homology of M is given by an isomorphism of A*-comodule algebras
H*(M; Fp) ~=H*(HZp; Fp) Pp-2 (x),
where x = OE*(x) under the map OE* : H*(ku; Fp) ! H*(M; Fp). The proof of this
statement is a variation of the proof of Theorem 1.5.
By Propositions 2.2 and 2.5 the Bökstedt spectral sequence
(7.1) E2*,*(ku, M) = HHFp*,*H*(ku; Fp), H*(M; Fp) =) H*(T HH(ku, M); Fp)
has an E2-term given by
E2*,*(ku, M) ~=H*(HZp; Fp) Pp-2 (x) E(oex, oe,~1, oe,~2, . .). (y, oe~ø2*
*, oe~ø3, . .)..
T HH OF CONNECTIVE COMPLEX K-THEORY 25
Let ffi : ku ! ku be the operation corresponding to a chosen generator of ,
as described in 1.1. It restricts to a map of S-algebras ffi : M ! M by natu-
rality of the Postnikov section. Since topological Hochschild homology is func*
*to-
rial in both variables we have S-algebra maps ffi : T HH(ku) ! T HH(ku) and
ffi : T HH(ku, M) ! T HH(ku, M), inducing morphisms of spectral sequences
ffi* : E**,*(ku) ! E**,*(ku) and ffi* : E**,*(ku, M) ! E**,*(ku, M). Suppose ch*
*osen a
Bott element u 2 V (0)2ku, and let ff 2 Fxpbe such that ffi*(u) = ffu.
7.2. Definition. A class w in Er*,*(ku) or Er*,*(ku, M) has ffi-weight n 2 Z=(*
*p - 1)
if ffi*(w) = ffnw. Similarly, a class v in H*(T HH(ku); Fp), V (0)*T HH(*
*ku),
V (1)*T HH(ku), or H*(T HH(ku, M); Fp) has ffi-weight n if ffi*(v) = ffnv.
7.3. Lemma. In E2*,*(ku, M) the classes belonging to the tensor factor
H*(HZp, Fp) E(oe,~1, oe,~2, . .). (y, oe~ø2, oe~ø3, . .).
have ffi-weight 0, while x and oex have ffi-weight 1.
Proof. This is proven by inspection of the action of ffi* on the Hochschild com-
plex.
7.4. Lemma. There is an isomorphism of spectral sequences
E**,*(ku, M) ~=Pp-2 (x) E**,*(ku, HZp).
Proof. It suffices to prove by induction that for all r > 2, the following two *
*asser-
tions hold.
(1) There is an isomorphism Er*,*(ku, M) ~=Pp-2 (x) Er*,*(ku, HZp),
(2) For each 0 6 i 6 p - 3 the dr-differential maps Fp{xi} Er*,*(ku, HZp)*
* to
itself.
For r = 2 assertion (1) holds. Each algebra generator of E2*,*(ku, M) that can
support a differential has ffi-weight 0. Since differentials preserve the ffi-w*
*eight, the
first non-trivial differential maps Fp{xi} E2*,*(ku, HZp) to itself. In parti*
*cular it
is detected by the morphism of spectral sequences
'**,*: E**,*(ku, M) ! E**,*(ku, HZp),
induced by the linearization ' : M ! HZp, and is given by dp-1 (flpoe~øi) = oe,*
*~i+1for
i > 2. As in the case of E**,*(ku, HZp), taking into account the bialgebra stru*
*cture,
this leaves
Ep*,*(ku, M) ~=Pp-2 (x) H*(HZp, Fp) E(oex, oe,~1, oe,~2) Pp(oe~ø2, oe~ø3,*
* . .). (y).
Again (1) holds and algebra generators that can support a differential are of f*
*fi-
weight 0. We can repeat the argument until we reach E2p*,*(ku, M), where the
spectral sequence collapses for bidegree reasons.
26 CHRISTIAN AUSONI
7.5. Proposition. There is an isomorphism of A*-comodule algebras
H*(T HH(ku, M); Fp) ~=H*(M; Fp) E([oex], [oe,~1]) P ([y])
where [oex] and [oe,~1] are A*-comodule primitives and the coaction on [y] is
*([y]) = ~ø0 [oe,~1] + 1 [y].
Proof. The Bökstedt spectral sequence described in Lemma 7.4 is strongly conver-
gent and has an E1 -term given by
E1*,*(ku, M) = H*(M; Fp) E(oex, oe,~1) Pp(y, oe~ø2, oe~ø3, . .)..
The map '1 is surjective and its kernel is the ideal generated by x. The mult*
*i-
plicative and comodule extensions are detected using the homomorphism
'* : H*(T HH(ku, M); Fp) ! H*(T HH(ku, HZp); Fp).
8. The V (1) homotopy groups of T HH(ku)
In this section we compute the Bökstedt spectral sequence
(8.1) E2*,*(ku) = HHFp*,*H*(ku; Fp) =) H*(T HH(ku); Fp)
and describe V (1)*T HH(ku) as an algebra over V (1)*ku, for primes p with p > *
*5.
We treat the case p = 3 separately at the end of the section. Unless otherwise
specified, we assume throughout this section that p > 5.
Recall the description of H*(ku; Fp) given in Theorem 1.5, and let Pp-1 (x) *
*be
the subalgebra of H*(ku; Fp) generated by x 2 H2(ku; Fp). Let us denote by 2*,*
the bigraded Pp-1 (x)-algebra HH*,* Pp-1 (x) . It has generators
æ z
i of bidegree (2i + 1, (2p - 2)i + 2) fori > 0,
yj of bidegree (2j, (2p - 2)j + 2) forj > 1,
subject to the relations given in Proposition 2.3. By Propositions 2.2 and 2.3,*
* the
E2-term of the Bökstedt spectral sequence (8.1) is given by
E2*,*(ku) = H*(`; Fp) E(oe,~1, oe,~2, . .). (oe~ø2, oe~ø3, . .). *
* 2*,*.
Notice that E2*,*(ku) is not flat over H*(ku; Fp), so there is no coproduct str*
*ucture
on this spectral sequence.
8.2. Lemma. Any class in the tensor factor
H*(`; Fp) E(oe,~1, oe,~2, . .). (oe~ø2, oe~ø3, . .).
of E2*,*(ku) has ffi-weight 0. The generators x, zi and yj of 2*,*, for i > 0*
* and
j > 1, have ffi-weight 1.
Proof. This is a consequence of the action of ffi* on H*(ku; Fp) which was desc*
*ribed
in Lemma 1.6. For zi and yj it follows from the fact that a representative for*
* zi
or yj in the Hochschild complex of Pp-1 (x) consists of a sum of terms having a
number of factors x that is congruent to 1 modulo (p - 1).
The S-algebra map OE : ku ! M from previous section induces a map OE :
T HH(ku) ! T HH(ku, M) and a morphism of spectral sequences OE* : E**,*(ku) !
E**,*(ku, M). The term E2*,*(ku, M) was given in 7.1.
T HH OF CONNECTIVE COMPLEX K-THEORY 27
8.3. Lemma. The homomorphism OE2 : E2*,*(ku) ! E2*,*(ku, M) is characterized
as follows. On the tensor factor H*(`; Fp) it is the inclusion into H*(HZp; Fp*
*),
on the factor E(oe,~1, oe,~2, . .). (oe~ø2, oe~ø3, . .).it is the identity, a*
*nd on 2*,*it is
given by
OE2(x)= x,
OE2(zi)= oex . fliy for alli > 0,
OE2(yj)= x . fljy for allj > 1.
Proof. The homomorphism OE* : H*(ku; Fp) ! H*(M; Fp) is given by the tensor
product of the inclusion of H*(`; Fp) into H*(HZp; Fp) and the projection of Pp*
*-1 (x)
onto Pp-2 (x). This lemma follows from a computation in Hochschild homology,
using the resolution given in the proof of Proposition 2.3.
8.4. Definition. Let 1*,*be the submodule of 2*,*generated by
æ z
i for0 6 i 6 p - 1,
yj for1 6 j 6 p - 1
over Pp-1 (x). Then 1*,*is closed under multiplication, and hence is a subalge*
*bra
of 2*,*.
8.5. Lemma. The Bökstedt spectral sequence (8.1) behaves as follows.
(a) For 2 6 r 6 p - 2 we have dr = 0, and there are differentials
dp-1 (flp+k oe~øi) = flkoe~øi. oe,~i+1
for all k > 0 and all i > 2. Taking into account the algebra structure, this le*
*aves
Ep*,*(ku) = H*(`; Fp) E(oe,~1, oe,~2) Pp(oe~ø2, oe~ø3, . .). 2**
*,*.
(b) For p 6 r 6 2p - 2 we have dr = 0, and there are differentials
d2p-1(zi) = zi-p . oe,~2 for alli > p,
d2p-1(yp) = x . oe,~2,
d2p-1(yj) = yj-p . oe,~2 for allj > p.
Taking into account the algebra structure, this leaves
*
* 1
E2p*,*(ku) = H*(`; Fp) E(oe,~1) Pp(oe~ø2, oe~ø3, . .). Fp{oe,~2} *
* *,* .
Here oe,~2. ! = 0 for any ! 2 1*,*of positive total degree. At this stage the *
*spectral
sequence collapses.
Proof. We use the morphism of spectral sequences
OEr : Er*,*(ku) ! Er*,*(ku, M)
whose description on the E2-term is given in Lemma 8.3. The kernel of OE2 :
E2*,*(ku) ! E2*,*(ku, M) is the ideal generated by xp-2 and xp-3 yj for all j >*
* 1.
In particular any w 2 ker OE2 is of ffi-weight p - 2. On the other hand, any a*
*lge-
bra generator v of E2*,*(ku) is of ffi-weight 0 or 1. Since differentials pres*
*erve the
ffi-weight, this implies that the differentials of E**,*(ku) originating on alg*
*ebra gen-
erators are detected by OE*. Thus we obtain the claimed dp-1 and d2p-1 differen*
*tial.
In E2p*,*(ku) all algebra generators lie in filtration degrees smaller than 2p,*
* so the
spectral sequence collapses for bidegree reasons.
28 CHRISTIAN AUSONI
8.6. Lemma. For 0 6 i 6 p - 2 let Wi be the Fp-vector space of elements of
ffi-weight i in H*(T HH(ku); Fp). Then multiplication by x : W1 ! W2 is an iso-
morphism.
Proof. The map ffi : T HH(ku) ! T HH(ku) is an isomorphism. Therefore the
additive isomorphism H*(T HH(ku); Fp) ~= Tot E1*,*(ku) preserves the ffi-weight,
and it suffices to check the corresponding statement for E1*,*(ku). This follo*
*ws
from Lemma 8.2 by inspection.
8.7. Lemma. Let OE* : H*(T HH(ku); Fp) ! H*(T HH(ku, M); Fp) be the algebra
homomorphism induced by OE : ku ! M. Then ker(OE*|W1[W2 ) = 0.
Proof. By Lemma 8.3, the morphism OE1 : E1*,*(ku) ! E1*,*(ku, M) is injective *
*on
the vector space of elements of ffi-weight 1 or 2. This implies the correspond*
*ing
statement for OE*.
The H*(ku, Fp)-algebra H*(T HH(ku); Fp) differs from its associated graded
E1*,*(ku) by multiplicative extensions. More precisely, the subalgebra
Pp(oe~ø2, oe~ø3, . .). (Fp{oe,~2} 1*,*)
of E1*,*(ku) lifts to the subalgebra * of H*(T HH(ku); Fp) defined as follows.
8.8. Definition. We define * to be the (graded) commutative unital Pp-1 (x)-
algebra with generators 8
>:y~j 1 6 j 6 p - 1i,
[oe~ø2],
and relations 8 p-2
>>>x ~zi= 0 0 6 i 6 p - 2,
>>>xp-2 ~yj= 0 1 6 j 6 p - 1,
>>>
>< ~yi~yj= x~yi+j i + j 6 p - 1,
>>>~zi~yj= x~zi+j i + j 6 p - 1,
>>>~yi~yj= x~yi+j-p[oe~ø2]i + j > p,
>>>
>: ~zi~yj= x~zi+j-p[oe~ø2]i + j > p,
~zi~zj= 0 0 6 i, j 6 p - 1.
Here by convention [~y0] = x, and the degree of the generators is |~zi| = 2pi +*
* 3,
|~yj| = 2pj + 2 and |[oe~ø2]| = 2p2.
Beware that in * we have xp-2 ~zp-16= 0, which accounts for an extension
xp-2 ~zp-1= [oe,~2].
8.9. Proposition. There is an isomorphism of H*(`; Fp)-algebras
H*(T HH(ku); Fp) ~=H*(`; Fp) E([oe,~1]) *.
Proof. The Bökstedt spectral sequence (8.1) converges strongly and its E1 -term
is given in Lemma 8.5.b. For 1 6 i 6 p - 1, we define ~yi2 H2pi+2(T HH(ku); Fp)
by induction on i, with the following properties :
T HH OF CONNECTIVE COMPLEX K-THEORY 29
(1) ~yihas ffi-weight 1,
(2) ~yireduces to iyi modulo lower filtration in E1*,*(ku), and
(3) OE*(~yi) = x[y]i + ic~ø1[z][y]i-1 in H2pi+2(T HH(ku, M); Fp) for some c*
* 2 Fxp
independent of i.
Let ~y1= [y1]. Then (1) and (2) are obviously satisfied and by Lemma 8.3 OE*(~y*
*1)
x[y] modulo filtration lower then 2. By the splitting of the 0 simplices in T H*
*H we
deduce that OE*(~y1) x[y] modulo classes of filtration 1. The filtration-1 p*
*art of
H2p+2(T HH(ku, M); Fp) is Fp{~ø1[z]}. On the other hand the map
j* : H*(T HH(ku); Fp) ! H*(T HH(ku, HZp); Fp)
satisfies j*(~y1) = c~ø1[z] for some unit c 2 Fp, because j*(~y1) 6= 0 since ~y*
*1is not
divisible by x and ~ø1[z] generates H2p+2(T HH(ku, HZp); Fp). The homomorphism
j* factorizes as '*OE*, so this proves that
(8.10) OE*(~y1) = x[y] + c~ø1[z].
Assume that ~yisatisfying conditions (1), (2) and (3) has been defined for s*
*ome
1 6 i 6 p - 2. The class ~y1~yihas ffi-weight 2 so is divisible by x in a uniqu*
*e way.
Let ~yi+1= x-1 ~y1~yi. Then ~yi+1satisfies conditions (1) and (2). By inspectio*
*n there
is no nonzero class w of ffi-weight 1 in H2p(i+1)+2(T HH(ku, M); Fp) with xw = *
*0,
so we can write
OE*(~yi+1) = x-1 OE*(~y1)OE*(~yi),
which proves that ~yi+1satisfies (3).
Next, we define ~z0= c[z0], where c 2 Fp is the same as in condition (3) abo*
*ve.
The class ~z0~yihas ffi-weight 2, and we define ~zi= x-1 ~z0~yi. Then ~zihas f*
*fi-weight
1, and if i > 1 it reduces to iczi modulo lower filtration in E1*,*(ku). Moreov*
*er we
have OE*(~zi) = c[z][y]i for 0 6 i 6 p - 1.
The relations ~yi~yj= x~yi+jand ~zi~yj= x~zi+jfor i + j 6 p - 1 are satisfie*
*d by
definition of ~ziand ~yj. It remains to check the following relations :
8 p-2
>>>x ~zi= 0 0 6 i 6 p - 2,
>>>>~zi~zj= 0 0 6 i, j 6 p - 1,
>>:~yi~yj= x~yi+j-p[oe~ø2]i + j > p.
~zi~yj= x~zi+j-p[oe~ø2]i + j > p.
The class xp-2 ~yjis of ffi-weight 0 and is in the kernel of multiplication by *
*x. It
follows that it is in the ideal generated by [oe,~2], so for degree reasons it *
*must be
zero. Similarly we have xp-2 ~zi= 0 if 0 6 i 6 p - 2. The product ~zi~zjis of f*
*fi-weight
2 and in the kernel of OE*, so must be zero. If i + j > p, we have
OE*(~yi~yj) = x2[y]i+j + (i + j)c~ø1x[z][y]p-i-j = OE*(x~yi+j-p[oe~ø2*
*]).
Since both ~yi~yjand x~yi+j-p[oe~ø2] have ffi-weight 2, they must be equal. The*
* proof
that ~zi~yj= x~zi+j-p[oe~ø2] for i + j > p is similar.
Finally, the class [oe,~2] maps to zero via
j* : H*(T HH(ku); Fp) ! H*(T HH(ku, HZp); Fp),
30 CHRISTIAN AUSONI
and hence must be divisible by x. The only possibility left is a multiplicative*
* exten-
sion xp-2 zp-1 = [oe,~2] (up to multiplication by a unit). We have now establi*
*shed
all possible multiplicative relations involving the classes x, ~zi, ~yjand [oe~*
*ø2]. The
associated graded of * is isomorphic to
Pp(oe~ø2, oe~ø3, . .). (Fp{oe,~2} 1*,*).
This proves the proposition.
8.11. Proposition. The A*-coaction on H*(T HH(ku); Fp) is as follows :
- on H*(`; Fp) it is induced by inclusion into the coalgebra A* ;
- the classes x, [oe,~1] and ~z0are primitive ;
- on the remaining algebra generators, we have
*(~zi)= 1 ~zi+ i~ø0 [oe,~1]~zi-1 fori > 1,
*(~y1)= 1 ~y1+ ~ø0 (x[oe,~1] + ~,1~z0) + ~ø1 ~z0,
*(~yj)= 1 ~yj+ i~ø0 ([oe,~1]~yj-1+ ~,1~zj-1) + i~ø1 ~zj-1+
+ i~ø0~ø1 [oe,~1]~zj-2 forj > 2,
*([oe~ø2])= 1 [oe~ø2] + ~ø0 xp-2 ~zp-1.
Proof. The class x is known to be primitive. On classes in the image of oe*, li*
*ke ~z0,
[oe,~1] and [oe~ø2], the coaction is determined by (4.6).
The class ~y1was defined such that OE*(~y1) = x[y] + c~ø1[z]. By Proposition*
* 7.5 we
have
*OE*(~y1) = 1 x[z] + ~ø0 (x[oe,~1] + c~,1[z]) + ~ø1 c[z]
in A* H*(T HH(ku, M); Fp). Since OE* is injective on classes of ffi-weight 1 *
*we have
by naturality
*(~y1) = 1 ~y1+ ~ø0 (x[oe,~1] + ~,1~z0) + ~ø1 ~z0.
The product formulas x~yj= ~y1~yj-1and x~zi= ~z0~yiallow us to compute inductiv*
*ely
the coaction on ~yjand ~zifor 2 6 j 6 p - 1 and 1 6 i 6 p - 1. Again, by Lemma *
*8.6
there is no indeterminacy upon dividing *(x~zi) and *(x~yj) by 1 x.
Recall from [Ok] that V (1) is a ring spectrum if and only if p > 5. Our ne*
*xt
aim is to describe V (1)*T HH(ku) as an algebra over V (1)*ku if p > 5, and as a
module over V (1)*ku if p = 3.
8.12. Remark. The obstruction in [Ok, Example 4.5] for V (1) to be a ring spect*
*rum
at p = 3 vanishes when V (1) is smashed with HZp or ku. In particular V (1) ^
T HH(ku) is a ring spectrum at p = 3. However this relies on the ku-algebra
structure of T HH(ku), and the product on V (1)*T HH(ku) for p = 3 is not natur*
*al.
For instance T C(ku; p) and K(ku) are not ku-algebras.
We define a Pp-1 (u) algebra *. It is the counterpart of * in V (1)-homoto*
*py
and is abstractly isomorphic to it.
T HH OF CONNECTIVE COMPLEX K-THEORY 31
8.13. Definition. Assume p > 3, and let * be the (graded) commutative unital
Pp-1 (u)-algebra with generators
8
>< ai 0 6 i 6 p - 1,
>: bj 1 6 j 6 p - 1,
~2,
and relations 8 p-2
>>>u ai = 0 0 6 i 6 p - 2,
>>>up-2 bj = 0 1 6 j 6 p - 1,
>>>
>< bibj = ubi+j i + j 6 p - 1,
>>>aibj = uai+j i + j 6 p - 1,
>>>bibj = ubi+j-p~2 i + j > p,
>>>
>: aibj = uai+j-p~2 i + j > p,
aiaj = 0 0 6 i, j 6 p - 1.
Here by convention b0 = u, and the degree of the generators is |ai| = 2pi + 3,
|bj| = 2pj + 2 and |~2| = 2p2.
8.14. Remark. Let us describe the Pp-1 (u)-algebra * more explicitly. The cla*
*ss
~2 generates a polynomial subalgebra P (~2) *. The quotient algebra Q* =
*=(~2) is a finite graded Pp-1 (u)-algebra with 2(p - 1)2 elements, and is giv*
*en
additively as
Q* = Pp-1 (u) Pp-2 (u){a0, b1, a1, b2, . .,.ap-2 , bp-1 } Pp-1 (u){ap-*
*1 }.
In particular Q* has up-2 ap-1 as top-class in dimension 2p2 - 1, and has the
remarkable property of satisfying the duality relation
Qn ~=Q2p2-1-n
for any 0 6 n 6 2p2 - 1.
8.15. Theorem. Let * be the Pp-1 (u)-algebra defined above. There is an iso-
morphism
V (1)*T HH(ku) ~=E(~1) *,
where ~1 is of degree 2p-1. If p > 5 this is an isomorphism of Pp-1 (u)-algebra*
*s. If
p = 3 it is, at least, an isomorphism of Pp-1 (u)-modules such that P (~2) incl*
*udes
as a subalgebra in V (1)*T HH(ku).
Theorem 0.1 follows from this result. The quotient A* featured there can be
identified with Q* E(~1).
Proof for p > 5. Since V (1) ^ ` ' HFp, the spectrum V (1) ^ T HH(ku) is an
HFp-module and its homology is given by
H*(V (1) ^ T HH(ku); Fp) ~=A* E([oe,~1]) *.
The Hurewicz map
V (1)*T HH(ku) ! H*(V (1) ^ T HH(ku); Fp)
32 CHRISTIAN AUSONI
is injective with image the A*-comodule primitives. We identify V (1)*T HH(ku)
with its image (in particular Pp-1 (u) is identified with Pp-1 (x)). Consider *
*the
following classes in H*(V (1) ^ T HH(ku); Fp) :
a0 = ~z0,
b1 = ~y1- ~ø0x[oe,~1] - ~ø1~z0,
~1 = [oe,~1],
~2 = [oe~ø2] - ~ø0xp-2 [zp-1 ].
By Proposition 8.11 these classes are comodule primitives. Lemma 8.6 also holds*
* for
H*(V (1) ^ T HH(ku); Fp). We define inductively bj+1 = u-1 b1bj, for 1 6 j 6 p *
*- 2,
and ai = u-1 a0bi, for 1 6 i 6 p - 1. These classes bj and ai are primitive by
construction. By inspection, the classes ai, bj and ~2 satisfy the relations o*
*ver
Pp-1 (u) given in Definition 8.13. There is an isomorphism
H*(V (1) ^ T HH(ku); Fp) ~=A* E(~1) *.
where the Pp-1 (u)-algebra E(~1) * consists of A*-comodule primitives.
Proof for p = 3. Applying V (1)*T HH(ku, -) to the diagram (1.3) we obtain an
unrolled exact couple and a strongly convergent spectral sequence of algebras
(8.16) E2*,*= V (1)*T HH(ku, HZp) P (u) =) V (1)*T HH(ku)
analogous to the mod u spectral sequence of Proposition 6.2. Here u is of bideg*
*ree
(-2, 4) and represents the mod v1 reduction of the Bott element. By Theorem 5.7
we have
E20,*= V (1)*T HH(ku, HZp) ~=E(z, ~1, ") P (~1)
where " has degree 2p - 1 with a primary v1 Bockstein fi1,1(") = 1. We deduce
from Theorem 6.7 that there is a differential
d2(~1) = uz.
The ku-module structure of T HH(ku) implies a differential
d4(") = u2.
At this point the spectral sequence collapses and this leaves
2
E1*,*= E5*,*= E(~1) P (~31) E(") Fp{z, z~1} E(z~1) P2(u) .
Defining ~2 = ~31, ai = z~i1and bj = z"~j-11we obtain the P2(u)-module structure
of V (1)*T HH(ku) for p = 3, as claimed in Theorem 8.15. Notice that P (~2) is a
subalgebra of E1*,*, and hence of V (1)*T HH(ku).
8.17. Remark. The proof for p = 3 given above almost determines the whole prod-
uct structure on V (1)*T HH(ku). In fact the permanent cycles in the spectral
sequence (8.16) are concentrated in filtration degrees 0 and -2, so there is not
much room for multiplicative extensions. By inspection all multiplicative relat*
*ions
T HH OF CONNECTIVE COMPLEX K-THEORY 33
of * can be read from this spectral sequence except for a0b2 = a1b1 = ua2 and
b1b2 = 0.
8.18. Remark. The proof given for p = 3 is also valid for primes p > 5, and pro*
*vides
an easy way to determine the additive structure of V (1)*T HH(ku). In that case
the differentials of (8.16) are given by
æ d2(p-2)(~ p-2
1) = zu
d2(p-1)(") = up-1 .
However, the permanent cycles are now scattered through p - 1 filtration degrees
and one is left with solving the multiplicative extensions.
At this point it is of course also possible to study the mod v1 Bockstein sp*
*ectral
sequence in order to recover V (0)*T HH(ku) from V (1)*T HH(ku). However, be-
cause of the relation up-1 = v1 in V (0)*ku, the mod u Bockstein spectral seque*
*nce
of x6 is more appropriate. Let us just describe the primary mod v1 Bockstein, w*
*hich
involves some of the generators of *. The following proposition is a consequen*
*ce
of Corollary 6.8.
8.19. Proposition. Let p > 3. In V (1)*T HH(ku) there are primary mod v1
Bocksteins
fi1,1(bi) = ai-1
for 1 6 i 6 p - 1.
9. On the extension ` ! ku
In this final section we analyze the homomorphism
V (1)*T HH(`) ! V (1)*T HH(ku)
induced by the S-algebra map ` ! ku defined in 1.1. We then interpret our com-
putations above in terms of number-theoretic properties of the extension ` ! ku.
The Fp-algebras V (1)*T HH(`) and V (1)*T HH(ku) were described in Theo-
rems 4.9 and 8.15, respectively. Let be the group defined in 1.1, and recall *
*the
notion of ffi-weight from Definition 7.2.
9.1. Proposition. The classes ~1 and ~2 have ffi-weight 0 in V (1)*T HH(ku), a*
*nd
the classes u, ai and bj have ffi-weight 1. The homomorphism V (1)*T HH(`) !
V (1)*T HH(ku) is given by ~1 7! ~1, ~2 7! up-2 ap-1 and ~2 7! ~2. In particular
it is injective with image the classes of ffi-weight 0, and induces an isomorph*
*ism
V (1)*T HH(`) = V (1)*T HH(ku) .
Proof. These statements are proven in homology, where they follow directly from
the definition of the various algebra generators.
Let us denote by T C(`; p) the topological cyclic homology spectrum of `, an*
*d by
K(`) its algebraic K-theory.
34 CHRISTIAN AUSONI
9.2. Theorem. Let p be an odd prime. There are homotopy equivalences
T HH(ku)h 'p T HH(`),
T C(ku; p)h 'p T C(`; p), and
K(ku)h 'p K(`).
Proof. Consider the homotopy fixed point spectral sequence
h
E2s,t= H-s , V (1)tT HH(ku) =) V (1)t+sT HH(ku) .
By Proposition 9.1, and since the order of is prime to p, its E2-term is give*
*n by
æ V (1) T HH(`) if s = 0,
E2s,t= t
0 if s 6= 0.
Thus the spectral sequence collapses and its edge homomorphism yields an isomor-
phism
V (1)*T HH(ku)h ~= V (1)*T HH(`).
The spectra T HH(ku)h and T HH(`) are both connective. In particular their
V (1)*-localization and their V (0)*-localization (or p-completion) agree. Thu*
*s we
have an equivalence of p-completed spectra T HH(ku)h 'p T HH(`).
The spectrum T C(ku; p) is defined as the homotopy limit
T C(ku; p) = holimF,RT HH(ku)Cpn
taken over the Frobenius and the restriction maps
F, R : T HH(ku)Cpn ! T HH(ku)Cpn-1
that are part of the cyclotomic structure of T HH. In particular we have an equ*
*iv-
alence
T C(ku; p)h ' holim (T HH(ku)Cpn )h .
Fh ,Rh
Thus the equivalence T C(ku; p)h 'p T C(`; p) will follow from the claim that*
* for
each n > 0, there is an equivalence (T HH(ku)Cpn )h 'p T HH(`)Cpn . We proceed
by induction on n, the case n = 0 having been proven above. Let n > 1 and assume
that the claim has been proven for m < n. Consider the homotopy commutative
diagram
T HH(`)hCpn _____N_____//_T HH(`)Cpn_____R______//T HH(`)Cpn-1
| | |
| | |
fflffl| h fflffl| h fflffl|
(T HH(ku)hCpn )h _N___//_(T HH(ku)Cpn )h _R____//(T HH(ku)Cpn-1 )h
where the top line is the norm-restriction fiber sequences for T HH(`) and the *
*bot-
tom line is obtained by taking the homotopy-fixed points of the one for T HH(*
*ku).
By induction hypothesis the right hand side vertical arrow is an equivalence. S*
*ince
T HH OF CONNECTIVE COMPLEX K-THEORY 35
T HH(ku)hCpn is p-complete and is of order prime to p, the homotopy norm map
(T HH(ku)hCpn )h ! (T HH(ku)hCpn )h is an equivalence. This implies that
(T HH(ku)hCpn )h ' (T HH(ku)h )hCpn .
The left hand side vertical arrow is therefore also an equivalence. Thus the mi*
*ddle
vertical arrow is an equivalence, which completes the proof of the claim.
Finally, by [HM1] and [Du] we have natural cofibre sequences
K(`)p ! T C(`; p) ! -1 HZp and K(ku)p ! T C(ku; p) ! -1 HZp
We take the -homotopy fixed points of the latter one and assemble these cofibre
sequences in a commutative diagram
K(`)p ___trc___//_T C(`;_p)_____// -1 HZp
| | |
| | |=
fflffl| h fflffl| |fflffl
K(ku)hp _trc__//T C(ku; p)h______// -1 HZp
The middle and right hand side vertical arrows are equivalences. Thus the map
K(`)p ! K(ku)hp is also an equivalence.
The computations given in this paper provide evidence for interesting specul*
*a-
tions on the properties of the extension ` ! ku, and on how these properties are
reflected in topological Hochschild homology.
Let us assume that we can make sense at a spectrum level of the formula
(9.3) ku = `[u]=(up-1 = v1)
which holds for the coefficients rings. The prime (v1) of ` ramifies as (u)p-1 *
*in ku,
so the extension ` ! ku should not qualify as an 'etale extension. And indeed, *
*the
computations of V (1)*T HH(`) and V (1)*T HH(ku) given in Theorems 4.9 and 8.15
imply that
ku ^` T HH(`) 6'p T HH(ku).
Compare with the algebraic situation, where the Geller-Weibel Theorem [GW]
states that if an extension A ! B of k-algebras is 'etale, this is reflected by*
* an
isomorphism
B A HHk*(A) ~=HHk*(B)
in Hochschild homology.
Inverting v1 in ` and ku, we obtain the periodic Adams summand L and the
periodic K-theory spectrum KU (both p-completed). The map L ! KU induces
on coefficients the inclusion
L* = Zp[v1, v-11] ,! Zp[u, u-1 ] = KU*.
Now (p-1)up-2 is invertible in KU and we expect the extension L ! KU to be 'eta*
*le
(compare with Remark 2.7). Evidence for this is provided by the computations in
topological Hochschild homology given in (6.11) and Proposition 6.12, which imp*
*ly
that we have an equivalence
KU ^L T HH(L) 'p T HH(KU).
36 CHRISTIAN AUSONI
9.4. Tame ramification. The extension ` ! ku is not unramified, but from
formula (9.3) we nevertheless expect it to be tamely ramified. In particular we
view Theorem 9.2 as an example of tamely ramified descent.
The behavior of topological Hochschild homology with respect to tamely rami-
fied extensions of discrete valuations rings was studied by Hesselholt and Mads*
*en
in [HM2]. Their results can be used to provide an interesting, at this point v*
*ery
speculative explanation of the structure of T HH(ku). It is due to Lars Hesselh*
*olt,
and I would like to thank him for sharing the ideas exposed in the remaining pa*
*rt
of this paper.
Let us briefly recall the results of [HM2] that are relevant here. Let A be*
* a
discrete valuation ring, K its quotient field (of characteristic 0) and k its p*
*erfect
residue field (of characteristic p). The localization cofibre sequence in algeb*
*raic K-
theory maps via the trace map to a localization sequence in topological Hochsch*
*ild
homology. We have a commutative diagram
! j
K(k) ____i____//K(A) __________//_K(K)
| | |
| | |
fflffl|i! fflffl|j fflffl|
T HH(k) ______//T HH(A) _____//_T HH(A|K)
whose rows are cofibre sequences. Here the map i! is the transfer, and j is a m*
*ap
of ring spectra. The cofibre T HH(A|K) is defined in [HM2, Definition 1.5.5] as
topological Hochschild homology of a suitable linear category.
Let M = A \ Kx , and consider the log ring (A, M) with pre-log structure giv*
*en
by the inclusion ff : M ,! A. Then the homotopy groups (ß*T HH(A|K), M) form
a log differential graded ring over (A, M). The universal example of such a log
differential graded ring is the de Rham-Witt complex with log poles !*(A,M), and
there is a canonical map !*(A,M)! ß*T HH(A|K). Hesselholt and Madsen define
an element ~ 2 V (0)*T HH(A|K) and prove in [HM2, Theorem 2.4.1] that there is
a natural isomorphism
~=
(9.5) !*(A,M) Z P (~) -! V (0)*T HH(A|K).
Let L be a finite, tamely ramified extension of K, and B be the integral closur*
*e of
A in L. If follows from [HM2, Lemma 2.2.4 and 2.2.6] that there is an isomorphi*
*sm
~= *
(9.6) B A !*(A,MA ) Z Fp -! !(B,MB ) Z Fp.
In fact a tamely ramified extension A ! B has the formal property of an 'etale
extension in the context of log rings (i.e. it is log-'etale), and this isomor*
*phism
is analogous to the Geller-Weibel Theorem mentioned above. Assembling (9.5)
and (9.6) we obtain an isomorphism
~=
B A V (0)*T HH(A|K) -! V (0)*T HH(B|L).
In particular we have an equivalence
(9.7) HB ^HA T HH(A|K) 'p T HH(B|L)
T HH OF CONNECTIVE COMPLEX K-THEORY 37
of p-completed spectra.
Let us now optimistically assume that these results hold also in the general*
*ity of
commutative S-algebra. The ring ` has a maximal ideal (v1), with residue ring H*
*Zp
and quotient ring L. Similarly, ku has a maximal ideal (u), with residue ring H*
*Zp
and quotient ring KU. The localization cofibre sequences in topological Hochsch*
*ild
homology fit into a commutative diagram
! j
T HH(HZp) __i___//_T HH(`)________//_T HH(`|L)
(9.8) ' || || ||
fflffl| i! fflffl| j fflffl|
T HH(HZp) _____//_T HH(ku)______//T HH(ku|KU).
This requires that we can identify by d'evissage T HH(HZp) with the topological
Hochschild homology spectrum of a suitable category of finite v1-torsion `-modu*
*les
or ku-modules. Since ` ! ku is tamely ramified, we expect that there is an equi*
*v-
alence
(9.9) ku ^` T HH(`|L) ' T HH(ku|KU)
analogous to (9.7). The top cofibration of (9.8) induces a long exact sequence
! j* @
. . .@*-!V (1)nT HH(HZp) -i*!V (1)nT HH(`) -! V (1)nT HH(`|L) -!* . . .
in V (1) homotopy. There are isomorphisms
V (1)*T HH(HZp) ~= E(~1, ") P (~1),
V (1)*T HH(`) ~= E(~1, ~2) P (~2).
Here " has degree 2p - 1 and supports a primary v1-Bockstein fi1,1(") = 1. From
the structure of the higher v1-Bocksteins we know that i!*E(~1) P (~1) = 0 a*
*nd
that P (~2) injects into V (1)*T HH(`|L) via j*. Thus we expect that
V (1)*T HH(`|L) = E(d, ~1) P (~1),
where d 2 V (1)1T HH(`|L) satisfies @*(d) = 1. We should have @*(d~1) = ~1,
@*(d~j1) = ~j1, @*(~1) = ", j*(~1) = ~1, j*(~k2) = ~pk1and i!("~p-11) = ~2. How
the remaining classes map under @*, i!*or j* is then forced by the grading and *
*the
exactness. We deduce from (9.9) that
V (1)*T HH(ku|KU) ~=Pp-1 (u) E(~1, d) P (~1).
Assembling these computations in diagram (9.8) and chasing, we obtain an (addi-
tive) isomorphism
p-2
V (1)*T HH(ku) ~= E(~1, ~2) P (~2) E(~1, d) P (~1) Fp{u, . .,.u }*
* .
Under the identifications ai = du~i1and bj = u~j1, this is compatible with Theo-
rem 8.15.
38 CHRISTIAN AUSONI
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SFB 478, University of Münster, Hittorfstr. 27, 48149 Münster, Germany
E-mail address: ausoni@math.uni-muenster.de