ALGEBRAIC K-THEORY OF TOPOLOGICAL K-THEORY
Christian Ausoni and John Rognes
Abstract. Let `p be the p-complete connective Adams summand of topologica*
*l K-
theory, with coefficient ring (`p)* = Zp[v1], and let V (1) be the Smith-*
*Toda complex,
with BP*(V (1)) = BP*=(p; v1). For p 5 we explicitly compute the V (1)-h*
*omotopy
of the algebraic K-theory spectrum of `p, denoted V (1)*K(`p). In partic*
*ular we
find that it is a free finitely generated module over the polynomial alge*
*bra P (v2),
except for a sporadic class in degree 2p - 3. Thus also in this case alg*
*ebraic K-
theory increases chromatic complexity by one. The proof uses the cyclotom*
*ic trace
map from algebraic K-theory to topological cyclic homology, and the calcu*
*lation is
actually made in the V (1)-homotopy of the topological cyclic homology of*
* `p.
Contents
Introduction
1.Classes in algebraic K-theory
2.Topological Hochschild homology
3.Topological cyclotomy
4.Circle homotopy fixed points
5.The homotopy limit property
6.Higher fixed points
7.The restriction map
8.Topological cyclic homology
9.Algebraic K-theory
Introduction
We are interested in the arithmetic of ring spectra.
To make sense of this we must work with structured ring spectra, such as S-
algebras [EKMM], symmetric ring spectra [HSS] or -rings [Ly]. We will refer to
these as S-algebras. The commutative objects are then commutative S-algebras.
The category of rings is embedded in the category of S-algebras by the Eilen*
*berg-
Mac Lane functor R 7! HR. We may therefore view an S-algebra as a generalization
of a ring in the algebraic sense. The added flexibility of S-algebras provides *
*room
for new examples and constructions, which may eventually also shed light upon t*
*he
category of rings itself.
In algebraic number theory the arithmetic of the ring of integers in a numbe*
*r field
is largely captured by its Picard group, its unit group and its Brauer group. T*
*hese
are in turn reflected in the algebraic K-theory of the ring of integers. Algeb*
*raic
Typeset by AM S-T*
*EX
1
2 CHRISTIAN AUSONI AND JOHN ROGNES
K-theory is defined also in the generality of S-algebras. We can thus view the
algebraic K-theory of an S-algebra as a carrier of some of its arithmetic prope*
*rties.
The algebraic K-theory of (connective) S-algebras can be closely approximated
by diagrams built from the algebraic K-theory of rings [Du]. Hence we expect th*
*at
global structural properties enjoyed by algebraic K-theory as a functor of rings
should also have an analogue for algebraic K-theory as a functor of S-algebras.
We have in mind, in particular, theetale descent property of algebraic K-the*
*ory
conjectured by Lichtenbaum [Li] and Quillen [Qu], which has been established for
several classes of commutative rings [V], [RW], [HM2]. We are thus led to ask w*
*hen
a map of commutative S-algebras A ! B should be considered as anetale covering
with Galois group G. In such a situation we may further ask whether the natural
map K(A) ! K(B)hG to the homotopy fixed point spectrum for G acting on K(B)
induces an isomorphism on homotopy in sufficiently high degrees. These questions
will be considered in more detail in [Ro2].
One aim of this line of inquiry is to find a conceptual description of the a*
*lge-
braic K-theory of the sphere spectrum, K(S0) = A(*), which coincides with Wald-
hausen's algebraic K-theory of the one-point space *. In [Ro1] the second auth*
*or
computed the mod 2 spectrum cohomology of A(*) as a module over the Steenrod
algebra, providing a very explicit description of this homotopy type. However, *
*this
result is achieved by indirect computation and comparison with topological cycl*
*ic
homology, rather than by a structural property of the algebraic K-theory functo*
*r.
What we are searching for here is a more memorable intrinsic explanation for the
homotopy type appearing as the algebraic K-theory of an S-algebra.
More generally, for a simplicial group G with classifying space X = BG there*
* is
an S-algebra S0[G], which can be thought of as a group ring over the sphere spe*
*c-
trum, and K(S0[G]) = A(X) is Waldhausen's algebraic K-theory of the space X.
When X has the homotopy type of a manifold, A(X) carries information about
the geometric topology of that manifold. Hence anetale descent description of
K(S0[G]) will be of significant interest in geometric topology, reaching beyond*
* al-
gebraic K-theory itself.
In the present paper we initiate a computational exploration of this `brave *
*new
world' of ring spectra and their arithmetic.
We begin by considering some interesting examples of (pro-)etale coverings in
the category of commutative S-algebras. For convenience we will choose to work
locally, with S-algebras that are complete at a prime p. For the purpose of alg*
*ebraic
K-theory this is less of a restriction than it may seem at first. What we have*
* in
mind here is that the square diagram
K(A) ________//K(Ap)
| |
| |
fflffl| fflffl|
K(ss0A) _____//_K(ss0Ap)
is homotopy cartesian after p-adic completion [Du], when A is a connective S-
algebra, Ap its p-completion, ss0A its ring of path components and ss0(Ap) ~=(s*
*s0A)p.
ALGEBRAIC K-THEORY OF TOPOLOGICAL K-THEORY 3
This reduces the p-adic comparison of K(A) and K(Ap) to the p-adic comparison
of K(ss0A) and K(ss0Ap), i.e., to a question about ordinary rings, which we view
as a simpler question, or at least as one lying in better explored territory.
This leads us to study p-complete S-algebras, or algebras over the p-complete
sphere spectrum S0p. This spectrum is approximated in the category of commutati*
*ve
S-algebras (or E1 ring spectra) by a tower of chromatic localizations [Ra1]
S0p! . .!.LnS0p! . .!.L1S0p! L0S0p= HQp :
Here Ln = LE(n) is Bousfield's localization functor [Bo], [EKMM] with respect to
the nth Johnson-Wilson theory with coefficient ring E(n)* = Z(p)[v1; : :;:vn; v*
*-1n].
By the Hopkins-Ravenel chromatic convergence theorem [Ra3, x8], the natural map
S0p! holim nLnS0pis a homotopy equivalence. For each n 1 there is a further
map of commutative S-algebras LnS0p! LK(n)S0pto the Bousfield localization with
respect to the nth Morava K-theory with coefficient ring K(n)* = Fp[vn; v-1n]. *
*This
is an equivalence for n = 0, and LK(1)S0p' Jp is the non-connective p-complete
image-of-J spectrum.
There is a highly interesting sequence of commutative S-algebras En construc*
*ted
by Morava as spectra [Mo], by Hopkins and Miller [Re] as S-algebras (or A1 ring
spectra) and by Goerss and Hopkins [GH] as commutative S-algebras (or E1 ring
spectra). The coefficient ring of En is (En)* ~=W Fpn[[u1; : :;:un-1 ]][u; u-1 *
*]. As a
special case E1 ' KUp is the p-complete complex topological K-theory spectrum.
The cited authors also construct a group action on En through commutative S-
algebra maps, by a semidirect product Gn = Sn oCn where Sn is the nth (profinit*
*e)
Morava stabilizer group [Mo] and Cn = Gal(Fpn=Fp) is the cyclic group of order *
*n.
There is a homotopy equivalence LK(n)S0p' EhGnn, where the homotopy fixed point
spectrum is formed in a continuous sense [DH], which reflects the Morava change
of rings theorem [Mo].
Furthermore, the space of self equivalences of En in the category of commu-
tative S-algebras is weakly equivalent to its group of path components, which is
precisely Gn. In fact the extension LK(n)S0p! En qualifies as a pro-etale cover*
*ing
in the category of commutative S-algebras, with Galois group weakly equivalent *
*to
Gn. The weak contractibility of each path component of the space of self equiv-
alences of En (over either S0por LK(n)S0p) serves as the commutative S-algebra
version of the unique lifting property foretale coverings. Also the natural map
i :En ! T HH(En) is a K(n)-equivalence, cf. [MS1, 5.1], implying that the space
of relative K"ahler differentials of En over LK(n)S0pis contractible. See [Ro2*
*] for
further discussion.
There are furtheretale coverings of En. For example there is one with coeff*
*i-
cient ring W Fpm [[u1; : :;:un-1 ]][u; u-1 ] for each multiple m of n. Let Enr*
*nbe the
colimit of these, with Enrn*= W Fp[[u1; : :;:un-1 ]][u; u-1 ]. Then Gal(Enrn=LK*
*(n)S0p)
is weakly equivalent to an extension of Sn by the profinite integers ^Z= Gal(Fp*
* =Fp).
Let En be a maximal pro-etale covering of En, and thus of LK(n)S0p. What is the
absolute Galois group Gal(En =LK(n)S0p) of LK(n)S0p?
The tower of commutative S-algebras induces a tower of algebraic K-theory sp*
*ec-
tra
K(S0p) ! . .!.K(LnS0p) ! . .!.K(Jp) ! K(Qp)
4 CHRISTIAN AUSONI AND JOHN ROGNES
studied in the p-local case by Waldhausen [Wa2]. The natural map K(S0p) !
holim nK(LnS0p) may well be an equivalence, see [MS2]. We are thus led to study
the spectra K(LnS0p), and their relatives K(LK(n)S0p). (More precisely, Waldhau*
*sen
studied finite localization functors Lfncharacterized by their behavior on fini*
*te CW-
spectra. However, for n = 1 the localization functors L1 and Lf1agree, and this*
* is
the case that we will explore in the body of this paper. Hence we will suppress*
* this
distinction in the present discussion.)
Granting that LK(n)S0p! En qualifies as anetale covering in the category of
commutative S-algebras, the descent question concerns whether the natural map
(0.1) K(LK(n)S0p) ! K(En)hGn
is a p-adic homotopy equivalence in sufficiently high dimensions. We conjecture
that it does.
To analyze K(En) we expect to use a localization sequence in algebraic K-the*
*ory
to reduce to the algebraic K-theory of connective commutative S-algebras, and
to use the B"okstedt-Hsiang-Madsen cyclotomic trace map to topological cyclic
homology to compute these [BHM]. The ring spectra En and E(n)p are closely
related, and for n 1 we expect that there is a cofiber sequence of spectra
(0.2) K(BP p) ! K(BP p) ! K(E(n)p)
analogous to the localization sequence K(Fp) ! K(Zp) ! K(Qp) in the case n = 0.
Something similar should work for En.
The cyclotomic trace map
trc: K(BP p) ! T C(BP p; p) ' T C(BP ; p)
induces a p-adic homotopy equivalence upon replacing the target with its connec*
*tive
cover [HM1]. Hence a calculation of T C(BP ; p) is as good as a calculation*
* of
K(BP p), after p-adic completion. In this paper we present computational tec*
*h-
niques which are well suited for calculating T C(BP ; p), at least when BP <*
*n>p
is a commutative S-algebra and the Smith-Toda complex V (n) exists as a ring
spectrum. In the algebraic case n = 0, with BP <0> = HZ(p), these techniques si-
multaneously provide a simplification of the argument in [BM1], [BM2] computing
T C(Z; p) and K(Zp) for p 3. Presumably the simplification is related to that
appearing in different generality in [HM2].
It is also plausible that variations on these techniques apply when replacin*
*g V (n)
by another finite type n + 1 ring spectrum, and the desired commutative S-algeb*
*ra
structure on BP p is weakened to the existence of an S-algebra map from a
related commutative S-algebra, such as MU or BP .
The first non-algebraic case occurs for n = 1. Then E1 ' KUp has an action by
G1 = Zxp~= x . Here Zp ~= = 1 + pZp Zxp, Z=(p - 1) ~= Zxpand k 2 Zxp
acts on E1 like the p-adic Adams operation k on KUp.
Let Lp = Eh1 be the p-complete Adams summand with coefficient ring (Lp)* =
Zp[v1; v-11], so Lp ' E(1)p. Then acts continuously on Lp with Jp ' Lhp . Let *
*`p
ALGEBRAIC K-THEORY OF TOPOLOGICAL K-THEORY 5
be the p-complete connective Adams summand with coefficient ring (`p)* = Zp[v1],
so `p ' BP <1>p. We expect that there is a cofiber sequence of spectra
K(Zp) ! K(`p) ! K(Lp) :
The previous calculation of T C(Z; p) [BM1], [BM2], and the calculation of T C(*
*`; p)
presented in this paper, identify the p-adic completions of K(Zp) and K(`p), re-
spectively. Given an evaluation of the transfer map between them, this identif*
*ies
K(Lp). The homotopy fixed points for the -action on K(Lp) induced by the
Adams operations k for k 2 1 + pZp should then model K(Jp) = K(LK(1)S0p).
This brings us to the contents of the present paper. In x1 we produce two us*
*eful
classes K1 and K2 in the algebraic K-theory of `p. In x2 we compute the V (1)-
homotopy of the topological Hochschild homology of `, simplifying the argument
of [MS1]. In x3 we present notation concerning topological cyclic homology and
the cyclotomic trace map of [BHM]. In x4 we make preparatory calculations in
the spectrum homology of the S1-homotopy fixed points of T HH(`). These are
applied in x5 to prove that the canonical map from the Cpn fixed points to the *
*Cpn
homotopy fixed points of T HH(`) induces an equivalence on V (1)-homotopy above
dimension (2p - 2), using [Ts]. In x6 we inductively compute the V (1)-homotopy
of all these (homotopy) fixed point spectra, and their homotopy limit T F (`; p*
*).
The action of the restriction map on this limit is then identified in x7. The p*
*ieces
of the calculation are brought together in Theorem 8.4 of x8, yielding an expli*
*cit
computation of the V (1)-homotopy of T C(`; p):
Theorem 0.3. Let p 5. There is an isomorphism of E(1; 2) P (v2)-modules
V (1)*T C(`; p)~=E(1; 2; @) P (v2)
E(2) P (v2) Fp{1te | 0 < e < p}
E(1) P (v2) Fp{2tep | 0 < e < p}
with |1| = 2p - 1, |2| = 2p2 - 1, |v2| = 2p2 - 2, |@| = -1 and |t| = -2.
The p-completed cyclotomic trace map K(`p)p ! T C(`p; p) ' T C(`; p) identif*
*ies
K(`p)p with the connective cover of T C(`; p). This yields the following expres*
*sion
for the V (1)-homotopy of K(`p), given in Theorem 9.1 of x9:
Theorem 0.4. Let p 5. There is an exact sequence of E(1; 2)P (v2)-modules
0 ! 2p-3Fp -! V (1)*K(`p) -trc-!V (1)*T C(`; p) ! -1 Fp ! 0
taking the degree 2p - 3 generator in 2p-3HFp to a class a 2 V (1)2p-3K(`p), and
taking the class @ in V (1)-1 T C(`; p) to the degree -1 generator in -1 HFp.
The V (1)-homotopy of any spectrum is a P (v2)-module, but we emphasize that
V (1)*T C(`; p) is a free finitely generated P (v2)-module, and V (1)*K(`p) is *
*free
and finitely generated except for the summand Fp{a} in degree 2p - 3. Hence both
K(`p)p and T C(`; p) are fp-spectra in the sense of [MR], with finitely present*
*ed
6 CHRISTIAN AUSONI AND JOHN ROGNES
mod p cohomology as a module over the Steenrod algebra. They both have fp-
type 2, because V (1)*K(`p) is infinite while V (2)*K(`p) is finite, and simila*
*rly for
T C(`; p). In particular, K(`p) is closely related to elliptic cohomology.
More generally, at least if BP p is a commutative S-algebra and V (n) exi*
*sts
as a ring spectrum, similar calculations to those presented in this paper show *
*that
V (n)*T C(BP ; p) is a free P (vn+1 )-module on 2n+2 + 2n(n + 1)(p - 1) gene*
*ra-
tors. So algebraic K-theory takes such fp-type n commutative S-algebras to fp-t*
*ype
(n + 1) commutative S-algebras. If our ideas about localization sequences are c*
*or-
rect then also K(En)p will be of fp-type (n + 1), and ifetale descent holds in
algebraic K-theory for LK(n)S0p! En with cdp(Gn) < 1 then also K(LK(n)S0p)p
will be of fp-type (n + 1). The moral is that algebraic K-theory in many cases
increments chromatic complexity by one, i.e., it produces a constant red-shift *
*in
stable homotopy theory.
Notations and conventions. 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 , re*
*spec-
tively. When V has a basis {x1; : :;:xn}, we write E(x1; : :;:xn), P (x1; : :;:*
*xn) and
(x1; : :;:xn) for these algebras. So (x) = Fp{flj(x) | j 0} with fli(x) . flj(*
*x) =
(i; j)fli+j(x). Let Ph(x) = P (x)=(xh = 0) be the truncated polynomial algebra *
*of
height h. For a b 1 let Pab(x) = Fp{xk | a k b} as a P (x)-module.
By an infinite cycle in a spectral sequence we mean a class x such that dr(x*
*) = 0
for all r. By a permanent cycle we mean an infinite cycle which is not a bounda*
*ry,
i.e., a class that survives to represent a nonzero class at E1 . Differentials *
*are often
only given up to multiplication by a unit.
1. Classes in algebraic K-theory
1.1. E1 ring spectrum models. Let p be an odd prime. Let ` = BP <1> be the
Adams summand of p-local connective topological K-theory. Its homotopy groups
are `* ~=Z(p)[v1], with |v1| = q = 2p - 2.
Its p-completion `p with `p* ~=Zp[v1] admits a model as an E1 ring spectrum,
which can be constructed as the algebraic K-theory spectrum of a perfect field
k0. Let g be a prime power topologically generating the p-adic units and let k0*
* =
colimn0 Fgpn k be a Zp-extension of k = Fg. Then `p = K(k0)p is an E1 ring
spectrum model for the p-completed Adams summand.
Likewise jp = K(k)p and kup = K(k)p are E1 ring spectrum models for the
p-completed image-of-J spectrum and the p-completed connective topological K-
theory spectrum, respectively. The Frobenius automorphism oeg(x) = xg induces
the Adams operation g on both `p and kup. Then k is the fixed field of oeg, and
jp is the connective cover of the homotopy fixed point spectrum for g acting on
either one of `p or kup.
The E1 ring spectrum maps S0p! jp ! `p ! kup ! HZp induce E1 ring
spectrum maps on algebraic K-theory:
K(S0p) -! K(jp) -! K(`p) -! K(kup) -! K(Zp) :
In particular these are H1 ring spectrum maps.
ALGEBRAIC K-THEORY OF TOPOLOGICAL K-THEORY 7
1.2. A first class in algebraic K-theory. The B"okstedt trace map tr :K(Zp) !
T HH(Zp) maps onto the first p-torsion in the image, which is T HH2p-1(Zp) ~=
Z=p{e} [BM1]. Let eK 2 K2p-1(Zp) be a class with tr(eK ) = e.
There is a (2p - 2)-connected linearization map `p ! HZp of E1 ring spectra,
which induces a (2p - 1)-connected map K(`p) ! K(Zp).
Definition 1.3. Let K1 2 K2p-1(`p) be a chosen class mapping to eK 2 K2p-1(Zp)
under the map induced by linearization `p ! HZp.
The image tr(K1) 2 T HH2p-1(`p) of this class under the trace map tr :K(`p) !
T HH(`p) will map under linearization to e 2 T HH2p-1(Zp).
Remark 1.4. The class K1 2 K2p-1(`p) does not lift further back to K2p-1(S0p),
since eK has a nonzero image in ss2p-2 of the homotopy fiber of K(S0p) ! K(Zp)
[Wa1]. Thus K1 does also not lift to K2p-1(jp), because the map S0p! jp is
(pq - 2)-connected. It is not clear if the induced action of g on K(`p) leaves*
* K1
invariant.
1.5. Homotopy and homology operations. For a spectrum X, let DpX =
Ep np X^p be its pth extended power. Part of the structure defining an H1
ring spectrum E is a map :DpE ! E. Then a mod p homotopy class 2
ssm (DpSn ; Fp) determines a mod p homotopy operation
* :ssn(E) -! ssm (E; Fp)
natural for maps of H1 ring spectra E. Its value on the homotopy class represe*
*nted
by a map a: Sn ! E is the image of under the composite map
ssm (DpSn ; Fp) -Dp(a)---!ssm (DpE; Fp) -! ssm (E; Fp) :
Likewise the Hurewicz image h() 2 Hm (DpSn ; Fp) induces a homology operation
h()* :Hn(E; Fp) -! Hm (E; Fp) ;
and the two operations are compatible under the Hurewicz homomorphisms.
For Sn with n = 2k - 1 an odd dimensional sphere the bottom two cells of DpSn
are in dimensions pn + (p - 2) and pn + (p - 1), and are connected by a mod p
Bockstein. Hence the bottom two mod p homotopy classes of DpSn are in these
two dimensions, and are called fiP kand P k, respectively. Their Hurewicz images
induce the Dyer-Lashof operations denoted fiQk and Qk in homology, cf. [Br].
1.6. A second class in algebraic K-theory. We use the H1 ring spectrum
structure on K(`p) to produce a further element in its mod p homotopy.
Definition 1.7. Let K2 = (P p)*(K1) 2 K2p2-1(`p; Fp) be the image under the
mod p homotopy operation
(P p)* :K2p-1(`p) -! K2p2-1(`p; Fp)
of K1 2 K2p-1(`p) .
8 CHRISTIAN AUSONI AND JOHN ROGNES
Since the trace map tr :K(`p) ! T HH(`p) is an E1 ring spectrum map, it fol-
lows that tr(K2) 2 T HH2p2-1(`p; Fp) equals the image of tr(K1) 2 T HH2p-1(`p)
under the mod p homotopy operation (P p)*. We shall identify this image in the
next section.
Remark 1.8. It is not clear whether K2 lifts to an integral homotopy class in
K2p2-1(`p). The image of eK 2 K2p-1(Zp) in K2p-1(Qp; Fp) is v1d logp for a cla*
*ss
d logp 2 K1(Qp; Fp) mapping to the generator of K0(Fp; Fp) in the K-theory loca*
*l-
ization sequence for Zp, [HM2]. It appears that the image of K2 in V (1)2p2-1(L*
*p) is
v2d logv1 for a class d logv1 2 V (1)1K(Lp) mapping to the generator of V (0)0K*
*(Zp)
in the expected K-theory localization sequence for `p. The classes K1 and K2 are
therefore related to logarithmic differentials for poles at p and v1, respectiv*
*ely,
which partially motivates the choice of the letter `'.
2. Topological Hochschild homology
The topological Hochschild homology functor T HH(-), as well as its refined
versions T HH(-)hS1, T HH(-)Cpn , T F (-; p) and T C(-; p), preserve p-adic equ*
*iv-
alences. Hence we will tend to write T HH(Z) and T HH(`) in place of T HH(Zp)
and T HH(`p), and similarly for the other related functors.
2.1. Homology of T HH(`). The ring spectrum map ` ! HFp induces an injec-
tion on mod p homology, identifying H*(`; Fp) with the subalgebra
H*(`; Fp) = E(ok | k 2) P (k | k 1)
of the dual Steenrod algebra A*.
There is a B"okstedt spectral sequence
(2.2) E2**= HH*(H*(`; Fp)) =) H*(T HH(`); Fp)
with
E2**= H*(`; Fp) (oeok | k 2) E(oek | k 1) :
Here oex 2 HH1(-) is represented by 1 x in degree 1 of the Hochschild complex.
It corresponds to the map oe :` ! T HH(`) induced by the S1 action on T HH(`)
and the inclusion of 0-simplices ` ! T HH(`).
By naturality with respect to the map ` ! HFp, the differentials
dp-1 (flj(oeok)) = oek+1 . flj-p(oeok)
for j p, found in the B"okstedt spectral sequence for T HH(Fp), lift to the sp*
*ectral
sequence (2.2) above. Hence
Ep**= H*(`; Fp) Pp(oeok | k 2) E(oe1 ; oe2 )
and this equals the E1 -term for bidegree reasons.
In H*(T HH(`); Fp) there are Dyer-Lashof operations acting, and (oeok)p =
Qpk(oeok) = oe(Qpk(ok)) = oeok+1 for all k 2 [St]. Thus as an algebra
(2.3) H*(T HH(`); Fp) ~=H*(`; Fp) E(1; 2) P ()
where 1, 2 and are represented by oe1 , oe2 and oeo2, respectively, in the
B"okstedt spectral sequence. Here |1| = 2p - 1, |2| = 2p2 - 1 and || = 2p2.
Furthermore Qp(1) = Qp(oe1 ) = oe(Qp(1 )) = oe2 , so we may assume that we
have chosen 2 = Qp(1).
ALGEBRAIC K-THEORY OF TOPOLOGICAL K-THEORY 9
2.4. V (1)-homotopy of T HH(`). Let V (n) be the nth Smith-Toda complex,
with homology H*(V (n); Fp) ~= E(o0; : :;:on). Thus V (-1) = S0, V (0) is the
mod p Moore spectrum, and V (1) is the cofiber of the multiplication by v1-map
qV (0) ! V (0). There are cofiber sequences
S0 -p!S0 -i0!V (0) -j0!S1
and
qV (0) -v1!V (0) -i1!V (1) -j1!q+1V (0)
defining the maps labeled i0, j0, i1 and j1. When p 5, V (1) is a commutative
ring spectrum [Ok].
For a spectrum X the rth (partially defined) v1-Bockstein homomorphism fi1;r
is defined on the classes x 2 V (1)*(X) with j1(x) 2 V (0)*(X) divisible by vr-*
*11.
Then for y 2 V (0)*(X) with vr-11. y = j1(x) let fi1;r(x) = i1(y) 2 V (1)*(X). *
* So
fi1;rdecreases degrees by rq + 1.
Definition 2.5. Let r(n) = 0 for n 0, and let r(n) = pn + r(n - 2) for all n *
* 1.
Thus r(2n - 1) = p2n-1 + . .+.p (n odd powers of p) and r(2n) = p2n + . .+.p2 (n
even powers of p). Note that (p2 - 1)r(2n - 1) = p2n+1 - p, while (p2 - 1)r(2n)*
* =
p2n+2 - p2.
Proposition 2.6 (McClure-Staffeldt). There is an algebra isomorphism
V (1)*T HH(`) ~=E(1; 2) P ()
with |1| = 2p - 1, |2| = 2p2 - 1 and || = 2p2. There are v1-Bocksteins fi1;p() =
1, fi1;p2(p) = 2 and generally fi1;r(n)(pn-1) 6= 0 for n 1.
Proof. One proof proceeds as follows, leaving the v1-Bockstein structure to the
more detailed work of [MS1].
H*(T HH(`); Fp) is an A*-comodule algebra. The coaction
:H*(T HH(`); Fp) ! A* H*(T HH(`); Fp)
agrees with the coproduct :A* ! A* A* when both are restricted to the
subalgebra H*(`; Fp) A*. Furthermore (oex) = (1 oe) (x), so (1) = 1 1,
(2) = 1 2 and () = 1 + o0 2.
There are change-of-rings isomorphisms of A*-comodule algebras
H*(V (1) ^ T HH(`); Fp) ~=(A*H*(`;Fp)Fp) (H*(`; Fp) E(1; 2) P ())
~=A*H*(`;Fp)(H*(`; Fp) E(1; 2) P ())
~=A* E(1; 2) P () :
The homology classes 1 ^ 1, 1 ^ 2 and 1 ^ - o0^ 2 in H*(V (1) ^ T HH(`); Fp)
are primitive, and correspond to the classes 1, 2 and under the isomorphisms
above. Hence these three homology classes are in the Hurewicz image from spheri*
*cal
classes 1 2 V (1)2p-1T HH(`), 2 2 V (1)2p2-1T HH(`) and 2 V (1)2p2T HH(`),
respectively.
10 CHRISTIAN AUSONI AND JOHN ROGNES
Proposition 2.7. The classes K1 2 K2p-1(`p) and K2 2 K2p2-1(`p; Fp) map
under the trace map to integral and mod p lifts of 1 2 V (1)2p-1T HH(`) and
2 2 V (1)2p2-1T HH(`), respectively.
Proof. The linearized image in V (1)2p-1T HH(Z) of 1 2 V (1)2p-1T HH(`) equals
the mod p and v1 reduction i1i0(e) of the class e 2 T HH2p-1(Z), since both cla*
*sses
have the same Hurewicz image 1 ^ oe1 in H2p-1(V (1) ^ T HH(Z); Fp). Thus the
mod p and v1 reduction of the trace image tr(K1) equals 1.
The Hurewicz image in H2p2-1(T HH(`); Fp) of tr(K2) = (P p)*(tr(K1)) equals
the image of the homology operation Qp on the Hurewicz image 1 = oe1 in
H2p-1(T HH(`); Fp) of tr(K1). This is Qp(oe1 ) = oeQp(1 ) = oe2 = 2. So the
mod v1 image in V (1)2p2-1T HH(`) of tr(K2) equals 2, since both classes have
the same Hurewicz image 1 ^ 2 in H2p2-1(V (1) ^ T HH(`); Fp).
3. Topological cyclotomy
We now review some terminology and notation concerning topological cyclic
homology and the cyclotomic trace map.
3.1. Frobenius, restriction, Verschiebung. As already indicated, T HH(`) is
an S1-equivariant spectrum. Let Cpn S1 be the cyclic group of order pn. The
Frobenius maps F :T HH(`)Cpn ! T HH(`)Cpn-1 are the usual inclusions of fixed
point spectra that forget part of the invariance. Their homotopy limit defines
T F (`; p) = holimn;FT HH(`)Cpn :
There are also restriction maps R :T HH(`)Cpn ! T HH(`)Cpn-1 , defined using
the cyclotomic structure of T HH(`), cf. [HM1]. They commute with the Frobenius
maps, and thus induce a self map R :T F (`; p) ! T F (`; p). Its homotopy equa*
*l-
izer with the identity map defines the topological cyclic homology of `, which *
*was
introduced in [BHM]:
___R__//
T C(`; p)__ss_//_T F (`;_p)__//T F (`; p) :
1
Hence there is a cofiber sequence
-1 T F (`; p) -@!T C(`; p) -ss!T F (`; p) -1-R--!T F (`; p) ;
which we shall use to compute V (1)*T C(`; p). There are also Verschiebung maps
V :T HH(`)Cpn-1 ! T HH(`)Cpn , defined up to homotopy in terms of the S1-
equivariant transfer.
3.2. The cyclotomic trace map. The B"okstedt trace map admits lifts
trn :K(`p) ! T HH(`)Cpn
for all n 0, with tr = tr0, which commute with the Frobenius maps and homotopy
commute with the restriction maps up to preferred homotopy. Hence the limiting
map trF :K(`p) ! T F (`; p) homotopy equalizes R and the identity map, and the
resulting lift
trc: K(`p) ! T C(`; p)
is the B"okstedt-Hsiang-Madsen cyclotomic trace map [BHM].
ALGEBRAIC K-THEORY OF TOPOLOGICAL K-THEORY 11
3.3. The norm-restriction sequences. For each n 1 there is a homotopy
commutative diagram
K(`p) R
RRRR
trn|| RRtrn-1RRRRR
N fflffl| R RR((
(3.4) T HH(`)hCpn _____//_T HH(`)Cpn________//T HH(`)Cpn-1
|| | ^|
|| |n n|
|| h fflffl| h fflffl|
T HH(`)hCpn _N___//_T HH(`)hCpn_R___//_^H(Cpn; T HH(`)) :
The lower part is the map of cofiber sequences that arises by smashing the S1-
equivariant cofiber sequence ES1+ ! S0 ! EeS1 with the S1-equivariant map
T HH(`) ! F (ES1+; T HH(`)) and taking Cpn fixed point spectra. For closed
subgroups G S1 recall that T HH(`)hG = F (ES1+; T HH(`))G is the G homotopy
fixed point spectrum of T HH(`), and ^H(G; T HH(`)) = [EeS1 ^ F (ES1+; T HH(`)]G
is the G Tate construction on T HH(`). The remaining terms of the diagram are
then identified by the homotopy equivalences
T HH(`)hCpn ' [ES1+^ T HH(`)]Cpn ' [ES1+^ F (ES1+; T HH(`))]Cpn
and
T HH(`)Cpn-1 ' [EeS1 ^ T HH(`)]Cpn :
We call N, R, Nh and Rh the norm, restriction, homotopy norm and homotopy re-
striction maps, respectively. We call n and ^n the canonical maps. The middle a*
*nd
lower cofiber sequences are the norm-restriction and homotopy norm-restriction *
*se-
quences, respectively.
By passage to homotopy limits over Frobenius maps, there is also a limiting
diagram
K(`p) Q
QQQQ
trF || QtrFQQQQQ
N fflffl| R QQ((
(3.5) T HH(`)hS1 _______//_T F (`;_p)________//T F (`; p)
|| | |
|| | |^
|| Nh fflffl|1Rh fflffl|
T HH(`)hS1 _____//_T HH(`)hS _____//_^H(S1; T HH(`)) :
Implicit here are the p-adic homotopy equivalences
T HH(`)hS1 ' holimn;FT HH(`)hCpn
1 hC n
T HH(`)hS ' holimn;FT HH(`) p
^H(S1; T HH(`))' holim ^H(Cpn; T HH(`)) :
n;F
12 CHRISTIAN AUSONI AND JOHN ROGNES
4. Circle homotopy fixed points
4.1. The circle trace map. The circle trace map
1 1 S1
trS1 = O trF :K(`p) ! T HH(`)hS = F (ES+ ; T HH(`))
is a preferred lift of the trace map tr :K(`p) ! T HH(`). We take S1 as our mo*
*del
for ES1. Let 1
T n= F (S1 =S2n-1 ; T HH(`))S
for n 0, so that there is a descending filtration {T n}n on T 0= T HH(`)hS1, w*
*ith
layers T n=T n+1 ~=F (S2n+1 =S2n-1 ; T HH(`))S1 ~=-2n T HH(`).
4.2. The homology spectral sequence. Placing T n in filtration s = -2n and
applying homology, we obtain a (not necessarily convergent) homology spectral
sequence
1
(4.3) E2s;t= H-s (S1; Ht(T HH(`); Fp)) =) Hs+t(T HH(`)hS ; Fp)
with
E2**= P (t) H*(`; Fp) E(1; 2) P () :
Here t has bidegree (-2; 0) while the other generators are located on the verti-
cal axis. (No confusion should arise from the double usage of t as a polynomial
cohomology class and the vertical degree in this or other spectral sequences.)
Proposition 4.4. There are differentials d2(1 ) = t1, d2(2 ) = t2, d2(o2) = t
and d2p(p1) = tp2 in the spectral sequence (4.3).
Proof. The d2-differential
d20;t:E20;t~=Ht(T HH(`); Fp){1} -! E2-2;t+1~=Ht+1(T HH(`); Fp){t}
is adjoint to the S1-action on T HH(`), hence restricts to oe on Ht(`; Fp). Th*
*us
d2(i ) = toei = ti for i = 1 and 2, and d2(o2) = toeo2 = t.
Write q = 2p-2 and let x 2 Cq(T 0; T 1; Fp) be a chain representing the diff*
*erential
d2(1 ) = t1, i.e., x maps to a cycle representing 1 in Hq(T HH(`); Fp) = E20;2*
*p-2,
and has boundary dx 2 Cq-1 (T 1; Fp) mapping to a cycle representing t1 in
Hq-1 (-2 T HH(`); Fp) = E2-2;2p-1. Let :DpT 0 ! T 0 be the pth H1 struc-
ture map for T 0= T HH(`)hS1, and form the chain *(e0 xp ) 2 Cpq(T 0; T p; Fp*
*).
It maps to a cycle representing p1in Hpq(T HH(`); Fp) = E20;2p2-2p, and has bou*
*nd-
ary *(e0 d(xp )) 2 Cpq-1(T p; Fp). By a chain level calculation [Br, 3.4] in *
*the
extended pth power of the pair (D2p-2; S2p-3 ), this boundary is homologous to
a unit multiple of *(ep-1 (dx)p ), representing Qp-1 (t1) = tpQp(1) = tp2
in Hpq-1(-2p T HH(`); Fp) = E2-2p;2p2-1. Hence there is a nontrivial different*
*ial
d2p(p1) = tp2 in the spectral sequence above.
ALGEBRAIC K-THEORY OF TOPOLOGICAL K-THEORY 13
4.5. The V (1)-homotopy spectral sequence. Applying V (1)-homotopy to the
filtration {T n}n, in place of homology, we obtain a conditionally convergent V*
* (1)-
homotopy spectral sequence
1
(4.6) E2s;t(S1) = H-s (S1; V (1)tT HH(`)) =) V (1)s+tT HH(`)hS
with
E2**(S1) = P (t) E(1; 2) P () :
Again t has bidegree (-2; 0) while the other generators are located on the vert*
*ical
axis.
Definition 4.7. Let
ff1 2 ss2p-3(S0), fi012 ss2p2-2p-1V (0) and v2 2 ss2p2-2V (1)
be the classes represented in their respective Adams spectral sequences by the
cobar cycles h10 = [1 ], h11 = [p1] and [o2]. So j1(v2) = fi01and j0(fi01) = f*
*i1 2
ss2p2-2p-2(S0).
Consider the unit map S0 ! K(`p) ! T HH(`)hS1, which is well defined after
p-adic completion.
Proposition 4.8. The classes i1i0(ff1) 2 ss2p-3V (1), i1(fi01) 2 ss2p2-2p-1V (*
*1) and
v2 2 ss2p2-2V (1) map under the unit map V (1)*S0 ! V (1)*T HH(`)hS1 to classes
represented in E1 (S1) by t1, tp2 and t, respectively.
Proof. Consider first the filtration subquotient T 0=T 2= F (S3+; T HH(`))S1. *
*The
unit map V (1) ! V (1) ^ (T 0=T 2) induces a map of Adams spectral sequences,
taking the permanent cycles [1 ] and [o2] in the source Adams spectral sequence
to infinite cycles with the same cobar names in the target Adams spectral se-
quence. These are not boundaries in the cobar complex for the A*-comodule
H*(T 0=T 2; Fp), because of the differentials d2(1 ) = t1 and d2(o2) = t that
are present in the two-column spectral sequence converging to H*(T 0=T 2; Fp). *
*In
detail, H2p-2(T 0=T 2; Fp) = 0 and H2p2-1(T 0=T 2; Fp) is spanned by the primit*
*ives
2 and 1p1.
Thus [1 ] and [o2] are nonzero infinite cycles in the target Adams E2-term. *
*They
have Adams filtration one, hence cannot be boundaries. Thus they are permanent
cycles, and the classes i1i0(ff1) and v2 have nonzero images under the composite
V (1)* ! V (1)*(T 0) ! V (1)*(T 0=T 2). Thus they are also detected in V (1)*(T*
* 0),
in filtration s -2. For bidegree reasons the only possibility is that i1i0(ff*
*1) is
detected in the V (1)-homotopy spectral sequence E1 (S1) as t1, and v2 is detec*
*ted
as t. 1
Next consider the filtration subquotient T 0=T p+1 = F (S2p+1+; T HH(`))S . *
*The
unit map V (1) ! V (1) ^ (T 0=T p+1) again induces a map of Adams spectral se-
quences, taking the permanent cycle [p1] in the source Adams spectral sequence
to an infinite cycle with the same name. Again this is not a boundary in the
cobar complex for the A*-comodule H*(T 0=T p+1; Fp), because of the differential
14 CHRISTIAN AUSONI AND JOHN ROGNES
d2p(p1) = tp2 that is present in the (p + 1) column spectral sequence converging
to H*(T 0=T p+1; Fp).
Thus [p1] is a nonzero infinite cycle in the target Adams E2-term, of Adams
filtration one. Hence the class i1(fi01) has a nonzero image under the composi*
*te
V (1)* ! V (1)*(T 0) ! V (1)*(T 0=T p+1). Thus it is also detected in V (1)*(T *
*0) in
filtration s -2p. Again for bidegree reasons the only possibility is that i1(f*
*i01) is
detected in the V (1)-homotopy spectral sequence E1 (S1) as tp2.
5. The homotopy limit property
5.1. Homotopy fixed point and Tate spectral sequences. For closed sub-
groups G S1 we will consider the (second quadrant) G homotopy fixed point
spectral sequence
E2s;t(G)= H-s (G; V (1)tT HH(`))
=) V (1)s+tT HH(`)hG :
We also consider the (upper half plane) G Tate spectral sequence
^E2s;t(G)= ^H-s(G; V (1)tT HH(`))
=) V (1)s+tH^(G; T HH(`)) :
When G = S1 we have
E2**(S1) = E(1; 2) P (t; )
since H*(S1; Fp) = P (t), and
^E2**(S1) = E(1; 2) P (t; t-1 ; )
since H^*(S1; Fp) = P (t; t-1 ). When G = Cpn we have
E2**(Cpn) = E(un; 1; 2) P (t; )
since H*(Cpn; Fp) = E(un) P (t), while
E^2**(Cpn) = E(un; 1; 2) P (t; t-1 ; )
since H^*(Cpn; Fp) = E(un) P (t; t-1 ).
The homotopy restriction map Rh induces a map of spectral sequences
E*(Rh): E*(G) ! ^E*(G) ;
which on E2-terms identifies E2(G) with the restriction of E^2(G) to the second
quadrant.
ALGEBRAIC K-THEORY OF TOPOLOGICAL K-THEORY 15
The Frobenius and Verschiebung maps F and V are compatible under n and
n-1 with homotopy Frobenius and Verschiebung maps F h and V h that induce
maps of homotopy fixed point spectral sequences
E*(F h): E*(Cpn) ! E*(Cpn-1)
and
E*(V h): E*(Cpn-1) ! E*(Cpn) :
Here E2(F h) maps the even columns isomorphically and the odd columns trivially.
On the other hand, E2(V h) maps the odd columns isomorphically and the even
columns trivially. This pattern persists to higher Er-terms, until a different*
*ial
of odd length appears in either spectral sequence. Thus the spectral sequences
E*(Cpn) and E*(Cpn-1) are abstractly isomorphic up to and including the Er-
term, where r is the length of the first odd differential in either spectral se*
*quence.
The same remarks apply for the Tate spectral sequences.
5.2. Input for Tsalidis' theorem.
Definition 5.3. A map A* ! B* of graded groups is k-coconnected if it is an
isomorphism in all dimensions > k and injective in dimension k.
Theorem 5.4. The canonical map
^1: T HH(`) ! ^H(Cp; T HH(`))
induces a (2p - 2)-coconnected map on V (1)-homotopy.
Proof. Consider diagram (3:4) in the case n = 1. The classes i1i0(ff1), i1(fi01*
*) and
v2 in V (1)* map through V (1)*K(`p) and 1 O tr1 to classes in V (1)*T HH(`)hCp
that are detected by t1, tp2 and t in E1 (Cp), respectively. Continuing by Rh
to V (1)*H^(Cp; T HH(`)) these classes factor through V (1)*T HH(`), where they
pass through zero groups. Hence the images of t1, tp2 and t in E^1 (Cp) must
be zero, i.e., these infinite cycles in E^2(Cp) are boundaries. For dimension r*
*easons
the only possibilities are
d2p(t1-p )= t1
2 p-p2 p
d2p (t )= t 2
2+1 -p2
d2p (u1t )= t :
The classes i1i0(K1) and i1(K2) in V (1)*K(`p) map by 1 O tr1 to classes in
V (1)*T HH(`)hCp that have Frobenius images 1 and 2 in V (1)*T HH(`), and
hence survive as permanent cycles in E10;*(Cp). Thus their images 1 and 2 in
E^*(Cp) are infinite cycles.
Hence the various Er-terms of the Cp Tate spectral sequence are:
^E2(Cp)= E(u1; 1; 2) P (t; t-1 ; t)
E^2p+1(Cp) = E(u1; 1; 2) P (tp; t-p ; t)
E^2p2+1(Cp) = E(u1; 1; 2) P (tp2; t-p2 ; t)
E^2p2+2(Cp) = E(1; 2) P (tp2; t-p2 ) :
16 CHRISTIAN AUSONI AND JOHN ROGNES
For bidegree reasons there are no further differentials, so E^2p2+2(Cp) = E^1 (*
*Cp)
and the classes 1, 2 and tp2 are permanent cycles.
The map ^1 :T HH(`) ! ^H(Cp; T HH(`)) induces on V (1)-homotopy the homo-
morphism 2 2
E(1; 2) P () -! E(1; 2) P (tp ; t-p )
that maps 1 7! 1, 2 7! 2 and 7! t-p2 . For the classes i1i0(K1) and i1(K2)
in V (1)*K(`p) map by tr to 1 and 2 in V (1)*T HH(`), and by Rh O 1 O tr1
to the classes in V (1)*H^(Cp; T HH(`)) represented by 1 and 2. The class
in V (1)*T HH(`) must have nonzero image in V (1)*H^(Cp; T HH(`)), since its pth
v1-Bockstein fi1;p() = 1 has nonzero image there. Thus maps to the class
represented by t-p2 , up to a unit multiple which we ignore. So V (1)*^1 is an
isomorphism in dimensions greater than |12tp2| = 2p - 2, and is injective in
dimension 2p - 2.
5.5. The homotopy limit property.
Theorem 5.6. The canonical maps
n : T HH(`)Cpn ! T HH(`)hCpn
^n : T HH(`)Cpn-1 ! ^H(Cpn; T HH(`))
and
1
: T F (`; p) ! T HH(`)hS
^: T F (`; p) ! ^H(S1; T HH(`))
all induce (2p - 2)-coconnected maps on V (1)-homotopy.
Proof. The claims for n and ^n follows from 5.4 and a theorem of Tsalidis [Ts].
The claims for and ^ follow by1passage to homotopy limits, using the p-adic
homotopy equivalence T HH(`)hS ' holim n;FT HH(`)hCpn and its analogue for
the Tate constructions.
6. Higher fixed points
Let [k] = 1 when k is odd, and [k] = 2 when k is even. Let 0[k]= [k+1], so t*
*hat
{[k]; 0[k]} = {1; 2} for all k. We write vp(k) for the p-valuation of k, i.e.,*
* the
exponent of the greatest power of p that divides k. By convention, vp(0) = +1.
Recall the integers r(n) from 2.5.
Theorem 6.1. In the Cpn Tate spectral sequence E^*(Cpn) there are differentials
k-1-pk pk-1 r(k-2)
d2r(k)(tp ) = [k]t (t)
for all 1 k 2n, and
2n r(2n-2)+1
d2r(2n)+1(unt-p ) = (t) :
ALGEBRAIC K-THEORY OF TOPOLOGICAL K-THEORY 17
The classes 1, 2 and t are infinite cycles.
We shall prove this by induction on n, the case n = 1 being settled in the p*
*revious
section. Hence we assume the theorem holds for one n 1 and we will establish i*
*ts
assertions for n + 1.
The terms of the Tate spectral sequence are
E^2r(m)+1(Cpn) = E(un; 1; 2) P (tpm ; t-pm ; t)
Mm
E(un; 0[k]) Pr(k-2)(t) Fp{[k]ti | vp(i) = k - 1}
k=3
for 1 m 2n. Next
^E2r(2n)+2(Cpn)= E(1; 2) Pr(2n-2)+1(t) P (tp2n; t-p2n )
2nM
E(un; 0[k]) Pr(k-2)(t) Fp{[k]ti | vp(i) = k - 1} :
k=3
For bidegree reasons the remaining differentials are zero, so E^2r(2n)+2(Cpn) =
E^1 (Cpn), and the classes tp2n are permanent cycles.
Proposition 6.2. The associated graded of V (1)*H^(Cpn; T HH(`)) is
E^1 (Cpn) = E(1; 2) Pr(2n-2)+1(t) P (tp2n; t-p2n )
2nM
E(un; 0[k]) Pr(k-2)(t) Fp{[k]ti | vp(i) = k - 1} :
k=3
Comparing E*(Cpn) with E^*(Cpn) via the homotopy restriction map Rh, we
obtain the following:
Proposition 6.3. In the Cpn homotopy fixed point spectral sequence E*(Cpn) the*
*re
are differentials
k-1 pk+pk-1 r(k-2)
d2r(k)(tp ) = [k]t (t)
for all 1 k 2n, and
2n r(2n-2)+1
d2r(2n)+1(un) = tp (t) :
The classes 1, 2 and t are infinite cycles.
We also consider the -inverted spectral sequences -1 E*(G) for G closed in S*
*1,
obtained by inverting t in E^*(G) and restricting to the left half plane. The E*
*2-
term -1 E2(G) is obtained from E2(G) by inverting . At each term, the natural
map E*(G) ! -1 E*(G) is an isomorphism in total degrees greater than 2p - 2,
and an injection in total degree 2p - 2.
18 CHRISTIAN AUSONI AND JOHN ROGNES
Proposition 6.4. In the -inverted spectral sequence -1 E*(Cpn) there are dif-
ferentials
k-pk-1 r(k)-pk-1
d2r(k)(p ) = [k](t)
for all 1 k 2n, and
2n r(2n)+1
d2r(2n)+1(unp ) = (t) :
The classes 1, 2 and t are infinite cycles.
The terms of the -inverted spectral sequence are
m -pm
-1 E2r(m)+1(Cpn) = E(un; 1; 2) P (p ; ; t)
Mm
E(un; 0[k]) Pr(k)(t) Fp{[k]j | vp(j) = k - 1}
k=1
for 1 m 2n. Next
2n -p2n
-1 E2r(2n)+2(Cpn) = E(1; 2) Pr(2n)+1(t) P (p ; )
2nM
E(un; 0[k]) Pr(k)(t) Fp{[k]j | vp(j) = k - 1} :
k=1
Again -1 E2r(2n)+2(Cpn) = -1 E1 (Cpn) for bidegree reasons, and the classes
p2n are permanent cycles.
Proposition 6.5. The associated graded E1 (Cpn) of V (1)*T HH(`)hCpn maps by
a (2p - 2)-coconnected map to
2n -p2n
-1 E1 (Cpn) = E(1; 2) Pr(2n)+1(t) P (p ; )
2nM
E(un; 0[k]) Pr(k)(t) Fp{[k]j | vp(j) = k - 1} :
k=1
Proof of Theorem 6.1. By our inductive hypothesis, the abutment -1 E1 (Cpn)
contains summands
2n-2 p2n-1 p2n
Pr(2n-1)(t){1p }, Pr(2n)(t){2 } and Pr(2n)+1(t){ }
representing elements in V (1)*T HH(`)Cpn . By inspection there are no perma-
nent cycles in the same total degree and of lower s-filtration in -1 E1 (Cpn) t*
*han
(t)r(2n-1). 1p2n-2, (t)r(2n). 2p2n-1 and (t)r(2n)+1. p2n, respectively. So the
three homotopy classes represented by 1p2n-2, 2p2n-1 and p2n are v2-torsion
classes of orders precisely r(2n - 1), r(2n) and r(2n) + 1, respectively.
ALGEBRAIC K-THEORY OF TOPOLOGICAL K-THEORY 19
Consider the commutative diagram
n ^n+1
T HH(`)hCpn oo____T HH(`)Cpn _____//_^H(Cpn+1; T HH(`))
|Fn| |Fn| |Fn|
|fflffl 0 fflffl| ^1 fflffl|
T HH(`) oo__=_____T HH(`) ________//_^H(Cp; T HH(`)) :
Here F n is the n-fold Frobenius map forgetting Cpn-invariance. The right hand
diagram commutes because ^n+1 is constructed as the Cpn-invariant part of an
S1-equivariant model for ^1.
The above three homotopy classes in V (1)*T HH(`)Cpn map by the middle F n
to homotopy classes in V (1)*T HH(`) with the same names, and by ^1 to homotopy
classes in V (1)*H^(Cp; T HH(`)) represented by 1t-p2n , 2t-p2n+1 and t-p2n+2 in
E^1 (Cp), respectively. Hence they map by ^n+1 to permanent cycles in ^E*(Cpn+*
*1)
with these images under the right hand F n.
By comparison over homotopy Frobenius and Verschiebung maps, there are iso-
morphisms E^r(Cpn) ~=E^r(Cpn+1) for all r 2r(2n) + 1, taking un to un+1 . This
determines the dr-differentials and Er-terms of ^E*(Cpn+1) up to and including *
*the
Er-term with r = 2r(2n) + 1:
E^2r(2n)+1(Cpn+1) = E(un+1 ; 1; 2) P (tp2n; t-p2n ; t)
2nM
E(un+1 ; 0[k]) Pr(k-2)(t) Fp{[k]ti | vp(i) = k - 1} :
k=3
By inspection there are no permanent2cyclesnin2thensame+total1degree2andnof+hig*
*her2
s-filtration in E^*(Cpn+1) than 1t-p , 2t-p and t-p , respectively. So
also the equivalence ^n+1 -1ntakes the homotopy classes represented by 1p2n-2,
2p2n-12andnp2n+to2homotopy classes represented by 1t-p2n , 2t-p2n+1 and
t-p , respectively.
Since ^n+1 -1ninduces an isomorphism on V (1)-homotopy in dimensions > (2p-
2), it preserves the v2-torsion order of these classes. Thus the infinite cycles
2n r(2n) -p2n+1 r(2n)+1 -p2n+2
(t)r(2n-1). 1t-p , (t) . 2t and (t) . t
are all boundaries in ^E*(Cpn+1). These are all t-periodic classes in ^Er(Cpn+1*
*) for
r = 2r(2n) + 1, hence cannot be hit by differentials on the t-torsion classes i*
*n this
Er-term. 2n 2n
This leaves the t-periodic part E(un+1 ; 1; 2) P (tp ; t-p ; t), where a*
*ll
the generators above the horizontal2axisnare2infinitencycles.+Hence1the2differe*
*ntialsn+2
hitting (t)r(2n-1) . 1t-p , (t)r(2n). 2t-p and (t)r(2n)+1 . t-p must
originate on the horizontal axis, and the only possibilities are
2n-p2n+1 r(2n-1) -p2n
d2r(2n+1)(t-p )= (t) . 1t
2n+1-p2n+2 r(2n) -p2n+1
d2r(2n+2)(t-p )= (t) . 2t
2n+2 r(2n)+1 -p2n+2
d2r(2n+2)+1(un+1 t-2p )= (t) . t :
20 CHRISTIAN AUSONI AND JOHN ROGNES
The algebra structure on E^*(Cpn+1) allows us to rewrite these differentials as*
* the
remaining differentials asserted by case n + 1 of 6.1.
Passing to the limit over the Frobenius maps, we obtain:
Theorem 6.6. The associated graded of V (1)*H^(S1; T HH(`)) is
^E1(S1) = E(1; 2) P (t)
M
E(0[k]) Pr(k-2)(t) Fp{[k]ti | vp(i) = k - 1} :
k3
Theorem 6.7. The associated graded E1 (S1) of V (1)*T HH(`)hS1 maps by a
(2p - 2)-coconnected map to
-1 E1 (S1) = E(1; 2) P (t)
M
E(0[k]) Pr(k)(t) Fp{[k]j | vp(j) = k - 1} :
k1
Each of these E1 terms compute V (1)*T F (`; p) in dimensions > (2p - 2), by
way of the (2p - 2)-coconnected maps and ^, respectively.
7. The restriction map
We now evaluate the homomorphism
R* :V (1)*T F (`; p) ! V (1)*T F (`; p)
induced on V (1)-homotopy by the restriction map R, in dimensions > (2p - 2).
The source and target are both identified with V (1)*T HH(`)hS1 via *. Then
R* is identified with the composite homomorphism (^-1 )* O Rh*, where Rh is the
homotopy restriction map. The latter induces a map of spectral sequences
E*(Rh): E*(S1) ! ^E*(S1) ;
where the E1 terms are given in 6.6 and 6.7.
Proposition 7.1. In dimensions > (2p - 2) the homomorphism E1 (Rh) maps:
(a) E(1; 2)P (t) in E1 (S1) isomorphically to E(1; 2)P (t) in ^E1 (S1).
(b) E(0[k])Pr(k)(t)Fp{[k]-epk-1} in E1 (S1) onto E(0[k])Pr(k-2)(t)
Fp{[k]tepk-1} in E^1 (S1), for k 3 and 0 < e < p.
(c) the remaining terms in E1 (S1) to zero.
Proof. Case (a) is clear. For (b) and (c) note that E1 (Rh) maps the term
k-1
E(0[k]) Pr(k)(t) Fp{[k]-ep }
in E1 (S1) to the term
k-1
E(0[k]) Pr(k-2)(t) Fp{[k]tep }
in E^1 (S1). Here e is prime to p. For e > p the source and target are in negat*
*ive
dimensions, while for e < 0 the source and target are concentrated in disjoint
dimensions. The cases 0 < e < p remain, when the map is a surjection since
r(k) - epk-1 > r(k - 2).
This identifies the image of Rh*, by the following lemma from [BM1, x2].
ALGEBRAIC K-THEORY OF TOPOLOGICAL K-THEORY 21
Lemma 7.2. The representatives in E1 (S1) of the kernel of Rh*equal the kernel
of E1 (Rh). Hence the image of Rh*is isomorphic to the image of E1 (Rh).
The composite equivalence ^-1 does not induce a map of spectral sequences.
Nonetheless it induces an isomorphism of E(1; 2) P (v2)-modules on V (1)-
homotopy in dimensions > (2p - 2). Here v2 acts by multiplication in V (1)*,
while multiplications by 1 and 2 are realized by the images of K1 and K2, since
both and ^ are ring spectrum maps.
Proposition 7.3. In dimensions > (2p - 2) the composite equivalence ^-1 in-
duces an isomorphism
1
V (1)*H^(S1; T HH(`)) ~=V (1)*T HH(`)hS
of P (v2)-modules, taking all classes represented by ffl11ffl22(t)m ti in E^1 (*
*S1) to
classes represented by ffl11ffl22(t)m j in E1 (S1) with i+p2j = 0. Here 0 ffl1*
*; ffl2
1 and m 0.
Proof. Let T^ = ^H(Cp; T HH(`)). The S1-equivariant map ^1 :T HH(`) ! T^ in-
duces a map of S1 homotopy fixed points, which realizes the localization homo-
morphism E*(S1) ! -1 E*(S1) on the level of spectral sequences. It follows that
there is a homotopy equivalence V (1)^H^(S1; T HH(`)) ' V (1)^T^hS1 which agrees
with ^-1 on (2p - 2)-connected covers.
The v2-indivisible elements in V (1)*H^(S1; T HH(`)) are represented in E^1 *
*(S1)
by the classes
E^1 (S1)=(t) = E(1; 2)
M
E(0[k]) Fp{[k]ti | vp(i) = k - 1} :
k3
The v2-indivisible elements in V (1)*T^hS1 are represented in -1 E1 (S1) by the
classes
-1 E1 (S1)=(t) = E(1; 2)
M
E(0[k]) Fp{[k]j | vp(j) = k - 1} :
k1
The asserted homotopy equivalence induces an isomorphism between these two
terms, which by a dimension count must be given by
ffl11ffl22ti 7- ! ffl11ffl22j
with i + p2j = 0. Hence the same formulas hold modulo multiples of v2 on V (1)-
homotopy. Taking the P (v2)-module structure into account, the corresponding
formulas including factors (t)m also hold, and express the isomorphism
1
V (1)*H^(S1; T HH(`)) ~=V (1)*T^hS
which agrees with ^-1 in dimensions > (2p - 2).
22 CHRISTIAN AUSONI AND JOHN ROGNES
Definition 7.4. Let A = E(1; 2) P (t),
k-1
Bk = E(0[k]) Pr(k)(t) Fp{[k]-ep | 0 < e < p}
L
and B = k1 Bk. Let C be the span of the remaining monomial terms in
-1 E1 (S1). Then E1 (S1) = A B C in dimensions > (2p - 2).
Theorem 7.5. In dimensions > (2p-2) there are subgroups eA= E(1; 2)P (v2),
Bek and Ce of V (1)*T F (`; p) represented by A, Bk and C in E1 (S1), respectiv*
*ely,
such that
(a) R* is the identity on Ae.
(b) R* maps Bek+2 onto Bek for all k 1.
(c) R* is zero on Be1, Be2 and eC. Q
In these dimensions V (1)*T F (`; p) = eA eB eC, with Be= k1 eBk.
Proof. At the level of E1 (S1), the composite map ^-1 O E1 (Rh) is the identity
on A, maps Bk+2 onto Bk for all k 1 and is zero on B1, B2 and C, by 7.1 and 7.*
*3.
The task is to find lifts of these groups to V (1)*T F (`; p) such that R* has *
*similar
properties.
Let eA= E(1; 2) P (v2) V (1)*T F (`; p) be the subalgebra generated by the
images of the classes K1, K2 and v2 in V (1)*K(`p). Then eAlifts A and consists*
* of
classes in the image from V (1)*K(`p). Hence R* is the identity on Ae.
By 7.1 we have C kerE1 (Rh). Thus by 7.2 there is a subgroup eCin ker(R*) ~=
ker(Rh*) represented by C. Then R* is zero on eC.
Note that im(R*) and ker(R*) span V (1)*T F (`; p). For by 7.1 the represent*
*atives
of im(R*) span A B, and the representatives of the subgroup eCin ker(R*) span
C. Thus the classes in im(R*) and ker(R*) have representatives spanning E1 (S1),
and therefore span all of V (1)*T F (`; p). Hence the image of R* on V (1)*T F *
*(`; p)
equals the image of its restriction to im(R*).
Consider the subgroup
B0k= Bk \ kerE1 (Rh)
M r(k)-1 k-1
= E(0[k]) Pr(k-2)+epk-1(t) Fp{[k]-ep } :
0 (2p - 2) there are isomorphisms
ker(R* - 1) ~= E(1; 2) P (v2)
E(2) P (v2) Fp{1te | 0 < e < p}
E(1) P (v2) Fp{2tep | 0 < e < p}
and
cok(R* - 1) ~= E(1; 2) P (v2) :
Proof. By 7.5 the homomorphism R* - 1 is zero on Ae= E(1; 2) P (v2) and an
isomorphism on eC. The remainder of V (1)*T F (`; p) decomposes as
Y Y
eB= eBk Bek
k odd k even
and R* takes Bek+2 to Bek for k 1, forming two sequential limit systems. Hence
there is an exact sequence
Y R*-1 Y
0 ! klimoddeBk! eBk- --! eBk! lim1 eBk! 0
k odd k odd k odd
and a corresponding one for k even. The right derived limit vanishes since each*
* eBk
has finite type. Hence it remains to prove that in dimensions > (2p - 2),
klimoddeBk~=E(2) P (v2) Fp{1te | 0 < e < p}
and
klimeveneBk~=E(1) P (v2) Fp{2tep | 0 < e < p} :
Each Bek ~= Bk is a sum of (2p - 2) finite cyclic P (v2)-modules. The restrict*
*ion
homomorphisms R* respect this sum decomposition, and map each cyclic module
24 CHRISTIAN AUSONI AND JOHN ROGNES
surjectively onto the next. Hence their limit is a sum of (2p - 2) cyclic modu*
*les,
and it remains to check that these are infinitekcyclic,-i.e.,1not bounded above.
For k odd the `top' class 12(t)r(k)-1-ep in Bk is in dimension 2pk+1 (p-e*
*).
For k even the corresponding class in Bk is in dimension 2pk+1 (p - e) + 2p - 2*
*p2.
In both cases the dimension grows to +1 for 0 < e < p as k grows.
For k odd each infinite cyclic P (v2)-module contains a class in non-negative
degree with nonzero image in Be1 ~= B1, namely the classes 1te and 12te for
0 < e < p. Hence we take these as generators for limk oddBek. Likewise there *
*are
generators in non-negative degrees for limk eveneBkwith nonzero image in eB2~= *
*B2,
namely the classes 2tep and 12tep for 0 < e < p.
Let e 2 ss2p-1T C(Z; p) be the image of eK 2 K2p-1(Zp), and let @ 2 ss-1 T *
*C(Z; p)
be the image of 1 2 ss0T F (Z; p) under @ :-1 T F (Z; p) ! T C(Z; p). We recall*
* from
[BM1], [BM2] the calculation of the mod p homotopy of T C(Z; p).
Theorem 8.3 (B"okstedt-Madsen).
V (0)*T C(Z; p) ~=E(e; @) P (v1) P (v1) Fp{eti | 0 < i < p} :
Hence
V (1)*T C(Z; p) ~=E(e; @) Fp{eti | 0 < i < p} :
The (2p - 2)-connected map `p ! HZp induces a (2p - 1)-connected map
K(`p) ! K(Zp), and thus a (2p - 1)-connected map T C(`; p) ! T C(Z; p) after
p-adic completion, by [Du]. This brings us to our main theorem.
Theorem 8.4. There is an isomorphism of E(1; 2) P (v2)-modules
V (1)*T C(`; p)~=E(1; 2; @) P (v2)
E(2) P (v2) Fp{1te | 0 < e < p}
E(1) P (v2) Fp{2tep | 0 < e < p}
with |1| = 2p - 1, |2| = 2p2 - 1, |v2| = 2p2 - 2, |@| = -1 and |t| = -2.
Proof. This follows in dimensions > (2p - 2) from 8.2 and the exact sequence (8*
*.1).
It follows in dimensions (2p - 2) from 8.3 and the (2p - 1)-connected map
V (1)*T C(`; p) ! V (1)*T C(Z; p). It remains to check that the module structu*
*res
are compatible for multiplications crossing dimension (2p - 2).
The classes E(1)Fp{1te | 0 < e < p} in V (1)*T C(`; p) map to E(e)Fp{eti |
0 < i < p} in V (1)*T C(Z; p), and map by O ss to classes with the same names
in the S1 homotopy fixed point spectral sequence for T HH(Z). By naturality,
the given classes in V (1)*T C(`; p) map by O ss to classes with the same names
in E1 (S1). Here these classes generate free E(2) P (t)-modules. For degree
reasons multiplication by 1 is zero on each 1te. Hence the E(1; 2) P (v2)-
module structure on the given classes is as claimed.
Finally the class @ in V (1)-1 T C(`; p) is the image under the connecting h*
*omo-
morphism @ of the class 1 in V (1)*T F (`; p), which generates the free E(1; 2)
P (v2)-module cok(R* - 1) of 8.2. Hence also the module structure on @ and 1@ is
as claimed.
ALGEBRAIC K-THEORY OF TOPOLOGICAL K-THEORY 25
A very important feature of this calculational result is that V (1)*T C(`; p*
*) is
a finitely generated free P (v2)-module. Thus T C(`; p) is a fp-spectrum of fp-
type 2 in the sense of [MR]. Notice that V (1)*T F (`; p) is not a free P (v2)-*
*module.
On the other hand we have the following calculation for the companion functor
T R(`; p) = holim n;RT HH(`)Cpn , showing that V (1)*T R(`; p) is a free but not
finitely generated P (v2)-module.
Theorem 8.5. There is an isomorphism of E(1; 2) P (v2)-modules
V (1)*T R(`; p)~=E(1; 2) P (v2)
M
E(u; 2) P (v2) Fp{1te | 0 < e < p}
n1
M
E(u; 1) P (v2) Fp{2tep | 0 < e < p} :
n1
The nth summand classes uffi1te and uffi2tep for 0 ffi 1 and 0 < e < p are
detected in V (1)*T HH(`)Cpn by the classes representing uffin1te and uffin2te*
*p in
E1 (Cpn), respectively.
We omit the proof.
9. Algebraic K-theory
We are now in a position to describe the V (1)-homotopy of the algebraic K-
theory spectrum of the p-completed Adams summand of connective topological
K-theory, i.e., V (1)*K(`p). We use the cyclotomic trace map to largely identi*
*fy
it with the corresponding topological cyclic homology. Hence we will identify *
*the
algebraic K-theory classes K1 and K2 with their cyclotomic trace images 1 and
2, in this section.
Theorem 9.1. There is an exact sequence of E(1; 2) P (v2)-modules
0 ! 2p-3Fp -! V (1)*K(`p) -trc-!V (1)*T C(`; p) ! -1 Fp ! 0
taking the degree 2p - 3 generator in 2p-3HFp to a class a 2 V (1)2p-3K(`p), and
taking the class @ in V (1)-1 T C(`; p) to the degree -1 generator in -1 HFp. H*
*ence
V (1)*K(`p) ~= E(1; 2) P (v2)
P (v2) Fp{@1; @v2; @2; @12}
E(2) P (v2) Fp{1te | 0 < e < p}
E(1) P (v2) Fp{2tpe | 0 < e < p}
Fp{a} :
Proof. By [HM1] the map `p ! HZp induces a map of horizontal cofiber sequences
of p-complete spectra:
K(`p)p __trc//_T C(`; p)____//_-1 HZp
| | ||
| | ||
fflffl|trc fflffl| ||
K(Zp)p _____//_T C(Z; p)___//_-1 HZp :
26 CHRISTIAN AUSONI AND JOHN ROGNES
Here V (1)*-1 HZp is Fp in degrees -1 and 2p - 2, and 0 otherwise. Clearly @
in V (1)*T C(`; p) maps to the generator in degree -1, since K(`p)p is a connec*
*tive
spectrum. The connecting map in V (1)-homotopy for the lower cofiber sequence
takes the generator in degree (2p-2) to the nonzero class i1(@v1) in V (1)2p-3K*
*(Zp).
By naturality it factors through V (1)2p-3K(`p), where we let a be its image.
Hence also K(`p)p is an fp-spectrum of fp-type 2. By [MR, 3.2] its mod p
spectrum cohomology is finitely presented as a module over the Steenrod algebra,
hence is induced up from a finite module over a finite subalgebra of the Steenr*
*od
algebra. In particular, K(`p) is closely related to elliptic cohomology.
We conclude with some comments on the v1-Bockstein spectral sequence leading
from the V (1)-homotopy of K(`p) to its V (0)-homotopy, i.e., its mod p homotop*
*y.
For any X, classes in the image of i1: V (0)*X ! V (1)*X are called mod p class*
*es,
while classes in the image of i1i0: ss*Xp ! V (1)*X are called integral classes.
Lemma 9.2. The classes 1, @1, 1 and 1te for 0 < e < p are integral classes
both in V (1)*K(`p) and V (1)*T C(`; p). Also @ is integral in V (1)*T C(`; p),*
* while
a is integral in V (1)*K(`p).
The classes @2, 2, @12, 12, 12te, 2tep and 12tep for 0 < e < p are
mod p classes in both V (1)*K(`p) and V (1)*T C(`; p).
We are not excluding the possibility that some of the mod p classes are actu*
*ally
integral classes.
Proof. Each v1-Bockstein fi1;rlands in a trivial group when applied to the clas*
*ses
@, 1, a and 1te for 0 < e < p in V (1)*K(`p) or V (1)*T C(`; p). Hence these ar*
*e at
least mod p classes.
Since 1 maps to an element of infinite order in ss0T C(Z; p) ~=Zp and the ot*
*her
classes sit in odd degrees, all mod pr Bocksteins on these classes are zero. H*
*ence
they are integral classes. The class 1 is integral by construction, hence so i*
*s the
product @1.
The mod p homotopy operation (P p-e)* takes 1te in integral homotopy to 2tep
in mod p homotopy, for 0 < e < p. Hence these are all mod p classes, as is 2 by
construction. The remaining classes listed are then products of established int*
*egral
and mod p classes, and are therefore mod p classes.
The classes listed in this lemma generate V (1)*K(`p) and V (1)*T C(`; p) as
P (v2)-modules. But v2 itself is not a mod p class.
Lemma 9.3. Let x be a mod p (or integral) class of V (1)*K(`p) or V (1)*T C(`;*
* p)
and let t 0. Then
fi1;1(vt2. x) = tvt-12i1(fi01) . x :
In particular i1(fi01) . 1 = tp2 and i1(fi01) . 1 = tp12.
We expect that i1(fi01) . tp2-p2 = @2 and i1(fi01) . tp2-p12 = @12, by sym-
metry considerations.
Proof. The v1-Bockstein fi1;1 = i1j1 acts as a derivation by [Ok]. By definiti*
*on
j1(v2) = fi01= [h11], which is detected as tp2 by 4.8. Clearly j1(x) = 0 for mo*
*d p
classes x.
ALGEBRAIC K-THEORY OF TOPOLOGICAL K-THEORY 27
In V (1)* the powers vt2support nonzero differentials fi1;1(vt2) = tvt-12i1(*
*fi01) for
p - t. The first nonzero differential on vp2is fi1;p:
Lemma 9.4. fi1;p(vp2) = [h12] 6= 0 in V (1)*.
We refer to [Ra2, x4.4] for background for the following calculation.
Proof. In the BP -based Adams-Novikov spectral sequence for V (0) the relation2
j1(vp2) = vp-11fi0p=pholds, where fi0p=pis the class represented by h12 + vp1-p*
* h11 in
degree 1 of the cobar complex. Its integral image fip=p = j0(fi0p=p) is represe*
*nted by
b11, and supports the Toda differential d2p-1(fip=p) = ff1fip1. This differenti*
*al lifts to
d2p-1(fi0p=p) = v1fip1in the Adams-Novikov spectral sequence for V (0). Conside*
*r the
image of fi0p=punder i1 in the Adams-Novikov spectral sequence for V (1), which*
* is
represented by h12 in the cobar complex. Then d2p-1(i1(fi0p=p)) = i1(v1fip1) = *
*0. By
sparseness and the vanishing line there are no further differentials, and i1(fi*
*0p=p) =
[h12] represents a nonzero element of V (1)*. Hence fi1;p(vp2) = [h12], as clai*
*med.
Remark 9.5. We would like to obtain the mod p homotopy groups V (0)*T C(`; p) by
means of the v1-Bockstein spectral sequence. This requires, first, that we comp*
*ute
the product with fi1;1(v2) = i1(fi01) in the remaining V (1)-homotopy groups. N*
*ext
we must identify the image of fi1;p(vp2) = [h12] in V (1)*T C(`; p). Imaginably*
* there is
a homology differential in (4.3) on [p2 ] hitting (t)p1tp2, if (P p2)*(2) = (t)*
*p1.
Then most likely [h12] is detected by (t)p1tp2 in V (1)*T F (`; p), which is id*
*entified
under R with the class (t)p1-1 generating V (1)*T C(`; p) in this degree. The
general picture appears to be complicated.
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Department of Mathematics, ETH Z"urich, Switzerland
E-mail address: ausoni@math.ethz.ch
Department of Mathematics, University of Oslo, Norway
E-mail address: rognes@math.uio.no