STABLE K-THEORY AND TOPOLOGICAL HOCHSCHILD HOMOLOGY. 185

Chapter III Reductions

3. STABLE K-THEORY AND TOPOLOGICAL HOCHSCHILD HOMOLOGY. 185

3.1 Stable K-theory

LetAbe a simplicial ring andP anA-bimodule. Recall the discussion of stable K-theory in section I.3.5, and in particular the equivalence between K^S(A, P) and the first differential of the functor C_A defined in I.3.4.4, and the homologyF(A, P)of I.3.3.

As beforeT(A, P)is theΩ-spectrum{k7→T HH(S^(k)P_A, S^(k)M_A(−,−⊗_AP))}.Notice that there is a map

D1C_A(P) ={n7→lim₋_→

Ω^k _

c∈obS⁽ⁿ⁾PA

S⁽ⁿ⁾HomA(c, c⊗AB^kP)}

→{n7→holim_−−→

x∈I

Ω^x _

c∈obS⁽ⁿ⁾P_A

S⁽ⁿ⁾HomA(c, c⊗AP ⊗Z[S˜ ^x])}=T(A, P)0

which is an equivalence by Bökstedt’s approximation lemma II.2.2.3.

Theorem 3.1.1 Let A be a simplicial ring and P a simplicial A bimodule. Then K^S(A, P)≃T(A, P)

and the equivalence is induced by

K^S(A, P)≃D1C_A(P) −−−→^≃ T(A, P)0

−−−→≃ T(A, P)

and this is compatible with the equivalence to the Fconstruction of of theorem I.3.5.2.

Proof: As both K-theory (of radical extensions) and T HH may be computed degreewise we may assume that A and P are discrete. Then the only thing which need verification is the compatibility. Recall that the equivalence K^S(A, P)≃F(A, P) of theorem I.3.5.2 was given by a chain

D₁C_A(−) −−−→^≃ D₁F₀(A,−) ←−−−^≃ F₀(A,−) −−−→^≃ F(A,−) of equivalences. Consider the diagram

D1C_A(P) ^≃ ^//

≃

T(A, P)0 ≃

≃

T(A, P)

≃

D1F₀(A,−)(P) ^//HH^Z(A, P)0

≃ //HH^Z(A, P)

F₀(A, P)

≃

hhRRRR

RRRRRRRRR

≃

≃ //F(A, P)

≃

where HH^Z(A, P) represents the spectrum HH^Z(ZSPA,SMA(−,− ⊗AP)) and HH^Z is the abelian group version of T HH as in IV.2.3.1. The right side of the diagram is simply the diagram of remark 1.2.1 (rotated), and the map from F0 to HH₀^Z is stabilization and so factors thorough the map to the differential.

3.2 T HH of split square zero extensions

LetA be an S-algebra andP anA-bimodule. Let A∨P be given the S-algebra structure we get by declaring that the multiplication P∧P → P is trivial. We want to study T HH(A∨ P) closer. If R is a simplicial ring and Q an R-bimodule, we get that the inclusion of wedge into product, HR∨HQ → H(R ⋉Q), is a stable equivalence of S-algebras, and so A∨P will cover all the considerations for split square zero extensions of rings.

The first thing one notices, is that the natural distributivity of smash and wedge give us a decomposition of T HH(A∨P;X), or more precisely a decomposition ofV(A∨P)(x) for every x∈ I^q+1, as follows. Let

V^(j)(A, P)(x) = _

φ∈∆m([j−1],[q])

0≤i≤q

Fi,φ(xi)

where ∆m([j−1],[q])is the set of order preserving injections [j−1]→[q]and Fi,φ(x) =

(A(S^x) if i /∈imφ P(S^x) if i∈imφ Then

V(A∨P)(x)∼= _

j≥0

V^(j)(A, P)(x) (note that V^(j)(A, P)(x) =∗ for j > q+ 1). Set

T HH^(j)(A, P;X)_q = holim_{−−−−−→}

x∈I^q+1

Ω^∨x(X∧V^(j)(A, P)(x))

and

T^(j)(A, P;X) ={k 7→T HH^(j)(A, P;S^k∧X)}.

We see that this defines cyclic objects (the transformations used to defineT HH respect the number of occurrences of the bimodule), when varying q. The inclusions and projections

V^(j)(A, P)(x)⊆V(A∨P)(x)→V⁽ⁱ⁾(A, P)(x) define cyclic maps

j≥0

T HH^(j)(A, P;X)→T HH(A∨P;X)→ Y

j≥0

T HH^(j)(A, P;X) The approximation lemma II.2.2.3 assures us that

holim

−−−−−→

x∈I^q+1

j≥0

Ω^∨x(X∧V^(j)(A, P)(x))→ Y

j≥0

T HH^(j)(A, P;X)q

is an equivalence. In effect, we have shown the first statement in the proposition below, and the second statement follows since T HH^(j)(A, P;X)is j−1 reduced.

3. STABLE K-THEORY AND TOPOLOGICAL HOCHSCHILD HOMOLOGY. 187 Proposition 3.2.1 Let Abe a connected S-algebra andP anA-bimodule. Then the cyclic map

T HH(A∨P;X) −−−→^∼ Q

0≤jT HH^(j)(A, P;X) is a weak equivalence.

If P isk−1 connected andX ism−1connected, then T HH^(j)(A, P;X)is jk+m−1 connected, and so

T HH(A∨P;X)→T HH(A;X)×T HH⁽¹⁾(A, P;X) is 2k+m connected.

This means that the space T HH⁽¹⁾(A, P;X)merits special attention as a first approxima-tion to the difference between T HH(A∨P;X) and T HH(A;X).

Also, since T HH^(j)(A, P;X) is j −1 reduced, the product is equivalent to the weak product, and we obtain

Corollary 3.2.2 Both maps in _

j≥0

T^(j)(A, P;X)→T(A∨P;X)→ Y

j≥0

T^(j)(A, P;X) are equivalences.

See also VII.1.2 for the effect on fixed points.

3.3 Free cyclic objects

In this section we review the little we need at this stage about free cyclic objects. See section VI.1.1 for a more thorough treatment. Recall that Λ is Connes’ cyclic category.

LetC be a category with finite coproducts. The forgetful functor from cyclic C objects to simplicial C objects has a left adjoint, the free cyclic functor j∗ defined as follows.

Ifφ∈Λwe can writeτ^−sφτ^s=ψτ^rin a unique fashion withψ ∈∆. IfX is a simplicial object, j∗X is given in dimension q by `

Cq+1Xq, and with φ^∗ sending x in the s ∈ Cq+1

summand to ψ^∗xin the r+sth summand.

Example 3.3.1 If X is a pointed set, then j∗(X) ∼= S₊¹∧X. If A is a commutative ring, then j∗(A)∼=HH(A).

Lemma 3.3.2 The map

j∗T(A, P;X)→T⁽¹⁾(A, P;X)

adjoint to the inclusion T(A, P;X) ⊆ T⁽¹⁾(A, P;X) is an equivalence. More precisely, if P is k−1 connected and X is m−1 connected, then

j∗T HH(A, P;X)→T HH⁽¹⁾(A, P;X) is a 2k+ 2m connected cyclic map.

Proof: Note that V(A, P)(x) ⊆ V⁽¹⁾(A, P)(x) defines the summand in which the P appears in the zeroth place. There are q other possibilities for placing P, and we may encode this by defining the map

Cq+1+∧T HH(A, P;X)q→T HH⁽¹⁾(A, P;X)q

taking tⁱ ∈C_q+1, x∈ I^q+1 and f: S^∨x →X∧V(A, P)(x)and sending it to

tⁱx,

S^∨tⁱ^x X∧V⁽¹⁾(A, P)(tⁱx)

∼=

x



x



S^∨x −−−→^f X∧V(A, P)(x) −−−→^⊆ X∧V⁽¹⁾(A, P)(x) Varying q, this is the cyclic map

j∗T HH(A, P;X)→T HH⁽¹⁾(A, P;X)

Let V^(1,i)(A, P)(x)⊂V⁽¹⁾(A, P)(x)be the summand with the P at theith place. The map may be factored as

tⁱ∈Cq+1_{−−−−−→}holim

x∈I^q+1

Ω^∨x(X∧V(A, P)(x)) −−−→^∼⁼ holim_{−−−−−→}

x∈I^q+1

tⁱ∈Cq+1Ω^∨x(X∧V^(1,i)(A, P)(x))

 y holim

−−−−−→

x∈I^q+1

Ω^∨x(X∧V⁽¹⁾(A, P)(x))

where the first map is given by the same formula with V^(1,i) instead ofV⁽¹⁾, and where the latter is induced by the inclusions

V^(1,j)(A, P)(x)⊆ _

tⁱ∈Cq+1

V^(1,i)(A, P)(x)∼=V⁽¹⁾(A, P)(x)

We may exchange the wedges by products holim

−−−−−→

x∈I^q+1

tⁱ∈Cq+1Ω^∨x(X∧V^(1,i)(A, P)(x)) −→ _{−−−−−→}holim

x∈I^q+1

Ω^∨x(X∧V⁽¹⁾(A, P)(x))



y ^≃y

holim

−−−−−→

x∈I^q+1

tⁱ∈Cq+1Ω^∨x(X∧V^(1,i)(A, P)(x)) −→^∼⁼ holim_{−−−−−→}

x∈I^q+1

Ω^∨x(X∧Q

tⁱ∈Cq+1V^(1,i)(A, P)(x)) and the left vertical arrow is 2(k+m)connected and the right vertical arrow is an equiv-alence by Blakers–Massey.

When Ais a discrete ring andP anA-bimodule (not necessarily discrete), these consid-erations carry over to theT(A⋉P)spectra. Recall the notation from I.2.5 where we defined a categoryDAP with objectsobSPA, and whereDAP(c, d)∼=SPA(c, d)⊕SMA(c, d⊗AP)

3. STABLE K-THEORY AND TOPOLOGICAL HOCHSCHILD HOMOLOGY. 189 (where we have suppressed the index n running in the spectrum direction, and identified the morphism objects via the lemma I.2.5.1 and I.2.5.2).

We saw in I.2.5.4 that D^(m)_A P ⊆S^(m)PA⋉P is a degreewise equivalence of categories, so T HH(DAP) ^∼ ^//T(A⋉P). Furthermore, recall that the objects of DAP were obSPA, andDAP(c, d) =SPA(c, d)⊕SMA(c, d⊗AP). SubstitutingX 7→ DAP(c, d)⊗ZZ[X]˜ with the stably equivalent X 7→SP_A(c, d)⊗_ZZ[X]˜ ∨SM_A(c, d⊗_AP)⊗_ZZ[X]˜ we may define T^(j)(A, P) as we did in 3.2, and we get that the cyclic map

j≥0

T^(j)(A, P;X)→T HH(D_AP)→T(A⋉P) is an equivalence. If P isk−1 connected then

T(A;X)∨T⁽¹⁾(A, P;X)→T(A⋉P;X)

is 2k −1 connected. Furthermore, as j∗ preserves equivalences (see lemma VI.1.1.3), we have that the composite

S₊¹∧T0(A, P;X) =j∗(T0(A, P;X))→j∗T(A, P;X)→T⁽¹⁾(A⋉P)

is an equivalence, and so the weak natural transformationj∗T(A, P;X)→S₊¹∧T(A, P;X) is an equivalence.

3.4 Relations to the trace K ˜ (A ⋉ P ) → T ˜ (A ⋉ P )

Our definition of the (“nerveless”) trace K(A˜ ⋉P)→T HH^(A⋉P) in 1.1.7 is the map K(A˜ ⋉P) = obSPe A⋉P tr

−−−→ T HH(SP^ A⋉P) = ˜T(A⋉P).

Recall that, by I.3.4.3K(A˜ ⋉P)≃C_A(BP), so another definition of this map could be via C_A(BP) −−−→ C_A(B^cyP)∼=Bg^cytDAP −−−→ T HH(D^ AP) −−−→^≃ T HH^(SPA⋉P) The two are related by the diagram

C_A(BP) −−−→^∼ N tSP˜ _A⋉P

←−−−∼ obSP˜ _A⋉P

 y

 y C_A(B^cyP) −−−→^∼ gB^cytSP_A⋉P −−−→ T HH^(SP_A⋉P)

(3.4.0)

Lemma 3.4.1 If P is k−1 connected, and X a ﬁnite pointed simplicial set, then X∧C_A(P)→C_A(P ⊗ZZ[X])˜

is 2k-connected.

Proof: It is enough to prove it for a finite set X. The smash moves past the wedges in the definition of C_A, and the map is simply W

c∈obSq^(m)PA of the inclusion X∧Sq^(m)M(c, c⊗_AP) −−−→^∼⁼ W

X−∗Sq^(m)M(c, c⊗_AP)

⊆

 y Z[X]˜ ⊗ZSq^(m)M(c, c⊗_AP) ←−−−^∼⁼ Q

X−∗Sq^(m)M(c, c⊗_AP)

which is 2k-connected by Blakers Massey. The usual considerations aboutm-reducedness in the q direction(s), give the lemma.

Lemma 3.4.2 If P is k−1 connected, then the composite

C_A(BP) −−−→ C_A(B^cyP) ←−−− S₊¹∧C_A(P) −−−→ S¹∧C_A(P)

is 2k-connected (i.e., induces isomorphisms on homotopy groups in the expected range).

Proof: Follows from lemma 3.4.1, and the commuting diagram C_A(BP) ^//

MM MM MM MM MM M

C_A(B^cyP)

S₊¹∧C_A(P)

C_A(BP)^oo S¹∧C_A(P) .

Consider the diagram (of bispectra)

T(A˜ ⋉P) −−−→^∼ T(A˜ ⋉P) ←−−−^∼ T˜(A⋉P) x

 x x

j∗T(A, P) −−−→^∼ j∗T(A, P) ←−−−^∼ j∗T(A, P)

≃

x



x



x

 S₊¹∧T(A, P)0

−−−→∼ S₊¹∧T(A, P)0 ←−−− S₊¹∧T(A, P)0

≃



y y y

S₊¹∧T(A, P) −−−→^∼ S₊¹∧T(A, P) ←−−−^∼ S₊¹∧T(A, P)



y y y

S¹∧T(A, P) −−−→^∼ S¹∧T(A, P) ←−−−^∼ S¹∧T(A, P)

The upwards pointing arrows are induced by the inclusion V(A, P)(x) ⊆ V(A ⋉P)(x) (likewise with V(SPA, P) instead of V(A, P)). The rightmost upper vertical map is 2k-connected by the considerations in 3.2, and so all up-going arrows are 2k-connected. Note that the middle layer of 0-simplices could have been skipped if we preformed geometric realization all over the place, using the well-known equivalence |j∗X| ≃ |S₊¹∧X|.

3. STABLE K-THEORY AND TOPOLOGICAL HOCHSCHILD HOMOLOGY. 191 Proposition 3.4.3 If P is k−1 connected, then the composites

K(A˜ ⋉P) −−−→ T(A˜ ⋉P) ←−−− S₊¹∧T(A, P)0 −−−→ S¹∧T(A, P) and

K˜(A⋉P) −−−→ T˜(A⋉P) ←−−−^∼ S₊¹∧T(A, P)0 −−−→ S¹∧T(A, P) ←−−−^∼ S¹∧T(A, P) are 2k-connected (i.e., induce isomorphism on homotopy groups in the expected range).

Proof: The second statement follows from the first. As C_A(P)→(D1C_A)(P)≃T(A, P)0

is2k-connected (I.3.5.2), the lemma gives that all composites from top left to bottom right in

C_A(BP) −−−→ C_A(B^cyP) ←−−− S₊¹∧C_A(P) −−−→ S¹∧C_A(P)



y y y

T˜(A⋉P) ←−−− S₊¹∧T(A, P)0 −−−→ S¹∧T(A, P)0

≃



y ^≃

 y S₊¹∧T(A, P) −−−→ S¹∧T(A, P) are 2k-connected.

3.5 Stable K-theory and T HH for S -algebras

The functor S 7→Aⁿ_S from section III.3.1.9, can clearly be applied toA-bimodules as well, and S 7→ P_Sⁿ will be a cube of S 7→ Aⁿ_S bimodules, which ultimately gives us a cube S 7→Aⁿ_S∨P_Sⁿ ofS-algebras. If P is anA bimodule, so is X 7→Σ^mP(X) =P(S^m∧X). We defined

K^S(A, P) = holim_→₋

Ω^kfiber{K(A∨Σ^k−1P)→K(A)}

The trace map induces a map to holim₋_→

Ω^kfiber{T HH(A∨Σ^k−1P)→T HH(A)}

and we may compose with the weak map to holim₋_→

Ω^k(S¹∧T HH(A,Σ^k−1P))

given by the discussion of the previous section. We know that this is an equivalence forA a ring and P a simplicial A-bimodule.

Theorem 3.5.1 Let A be an S-algebra and P an A-bimodule. Then the trace induces an equivalence K^S(A, P)≃T HH(A, P).

Proof: If A is discrete and P a simplicial A-bimodule this has already been covered. If A is a simplicial ring P a simplicialA-bimodule this follows by considering each degree at a time, using that K-theory of simplicial radical extensions may be calculated degreewise, I.1.4.2. In the general case we reduce to the simplicial case as follows. There is a stable equivalence Aⁿ_S∨P_Sⁿ →(A∨P)ⁿ_S, consisting of repeated applications of the 2k-connected map Z[A(S˜ ^k)]∨Z[P˜ (S^k)] → Z[A(S˜ ^k)]⊕Z[P˜ (S^k)] ∼= ˜Z[A(S^k)∨P(S^k)]. The noninitial nodes in these cubes are all equivalent to a simplicial ring case, and is hence taken care of by theorem 3.1.1 (or rather proposition 3.4.3 since the identification of the equivalence in theorem 3.1.1 with the trace map is crucial in order to have functoriality for S-algebras), and all we need to know is that

K(A∨P)→ holim_←−−

S6=∅

K(Aⁿ_S∨P_Sⁿ) in n+ 1 connected, and that

T HH(A∨P)→ holim_←−−

S6=∅

T HH(Aⁿ_S∨P_Sⁿ) and

T HH(A, P)→ holim_←−−

S6=∅

T HH(Aⁿ_S, P_Sⁿ)

are n-connected. These follow from the theorems III.3.2.2 and IV.1.4.3.

Chapter VI

Topological Cyclic homology

A motivation for the definitions to come can be found by looking at the example of a ΓS∗-categoryC. Consider the trace map

obC →T HH(C)

Topological Hochschild homology is a cyclic space, obC is merely a set. However, the trace IV.2.2 is universal in the sense that obC = lim^←_Λ⁻oT HH(C). A more usual way of putting this, is to say thatobC → |T HH(C)|is the inclusion of the T-fixed points, which also makes sense since the realization of a cyclic space is a topological space with a circle action (see 1.1 below).

In particular, the trace from K-theory has this property. The same is true for the other definition of the trace (IV.1.5), but this follows more by construction than by fate. In fact, any reasonable definition of the trace map should factor through the T-fixed point space, and so, if one wants to approximate K-theory one should try to mimic the T-fixed point space by any reasonable means. The awkward thing is that forming theT-fixed point space as such is really not a reasonable thing to do, in the sense that it does not preserve weak equivalences. Homotopy fixed point spaces are nice approximations which are well behaved, and strangely enough it turns out that so are the actual fixed point spaces with respect to finite subgroups of the circle. The aim is now to assemble as much information from these nice constructions as possible.

0.1 Connes’ Cyclic homology

The first time the circle comes into action for trace maps, is when Alain Connes defines his cyclic cohomology [41]. We are mostly concerned with homology theories, and in one of its many guises, cyclic homologyis just theT-homotopy orbits of the Hochschild homology spectrum. This is relevant to K-theory for several reasons, and one of the more striking reasons is the fact discovered by Loday and Quillen [138] and Tsygan [226]: just as the K-groups are rationally the primitive part of the group homology ofGL(A), cyclic homology is rationally the primitive part of the Lie-algebra homology of gl(A).

193

However, in the result above there is a revealing dimension shift, and, for the purposes of comparison with K-theory via trace maps, it is not the homotopy orbits, but the homotopy fixed points which play the central rôle. The homotopy fixed points of Hochschild homology give rise to Goodwillie and J. D. S. Jones’ negative cyclic homology HC⁻(A). In [81]

Goodwillie proves that if A → B is a map of simplicial Q-algebras inducing a surjection π₀(A)→π₀(B) with nilpotent kernel, then the relative K-theory K(A→B)is equivalent to the relative negative cyclic homology HC⁻(A →B).

All told, the cyclic theories associated with Hochschild homology seem to be right rationally, but just as for the comparison with stable K-theory, we must replace Hochschild homology by topological Hochschild homology to obtain integral results.

0.2 Bökstedt, Hsiang, Madsen and T C b

Topological cyclic homology, also known as T C, appears for the first time in Bökstedt, Hsiang and Madsen’s proof on the algebraic K-theory analog of the Novikov conjecture [18], and is something of a surprise. The obvious generalization of negative cyclic homology would be the homotopy fixed point space of the circle action on topological Hochschild homology, but this turns out not to have all the desired properties. Instead, they consider actual fixed points under the actions of the finite subgroups of T.

After completing at a prime, looking only at the action of the finite subgroups is not an unreasonable thing to do, since you can calculate the homotopy fixed points of the entire circle action by looking at a tower of homotopy fixed points with respect to cyclic groups of prime power order (see example A.6.6.5). The equivariant nature of Bökstedt’s formulation ofT HH is such that the actual fixed point spaces under the finite groups are nicely behaved 1.4.7, and in one respect they are highly superior to the homotopy fixed point spaces: The fixed point spaces with respect to the finite subgroups ofTare connected by more maps than you would think of by considering the homotopy fixed points or the linear analogs, and the interplay between these maps can be summarized in topological cyclic homology to give an amazingly good approximation of K-theory.

Topological cyclic homology, as we define it, is a non-connective spectrum, but its completionsT C(−)b^pare all−2-connected. As opposed to topological Hochschild homology, the topological cyclic homology of a discrete or simplicial ring is generally not an Eilenberg-MacLane spectrum.

In [18] the problem at hand is reduced to studying topological cyclic homology and trace maps of S-algebras of the form S[G], where S is the sphere spectrum and G is some simplicial group (see example II.1.4.4), i.e., the S-algebras associated to Waldhausen’s A theory of spaces (see section III.2.3.4). In this case, T C is particularly easy to describe:

for each prime p, there is a cartesian square

T C(S[G])b^p −−−→ (ΣT(S[G])hS¹)b^p

 y

 y T(S[G])b^p −−−→ T(S[G])b^p

195 (in the homotopy category) where the right vertical map is the “circle transfer”, and the lower horizontal map is analogous to something like the difference between the identity and a pth power map.

0.3 T C of the integers

Topological cyclic homology is much harder to calculate than topological Hochschild ho-mology, but – and this is the main point of this book – it exhibits the same “local” behavior as algebraic K-theory, and so is well worth the extra effort. The first calculation to appear is in fact one of the hardest ones produced to date, but also the most prestigious: in [19]

Bökstedt and Madsen set forth to calculate T C(Z)b^p for p >2, and found that they could describe T C(Z)b^p in terms of objects known to homotopy theorists:

T C(Z)b^p ≃imJb^p×BimJb^p×SUb^p,

where imJ is the image of J [3] and SU is the infinite special unitary group, provided a certain spectral sequence behaved as they suspected it did. In his thesis “The equivariant structure of topological Hochschild homology and the topological cyclic homology of the integers”, [Ph.D. Thesis, Brown Univ., Providence, RI, 1994] Stavros Tsalidis proved that the spectral sequence was as Bökstedt and Madsen had supposed, by adapting an argument in G. Carlsson’s proof of the Segal conjecture [37] to suit the present situation. Using this Bökstedt and Madsen calculates in [20]T C(A)bpforAthe Witt vectors of finite fields of odd characteristic, and in particular get the above formula for T C(Z)b^p ≃ T C(Zb^p)b^p. See also Tsalidis’ papers [224] and [225]. Soon after J. Rognes showed in [189] that an analogous formula holds for p = 2 (you do not have the splitting, and the image of J should be substituted with the complex image of J).

A bit more on the story behind this calculation, and also the others briefly presented in this introduction, can be found in section VII.3.

0.4 Other calculations of T C

All but the last of the calculations below are due to the impressive effort of Hesselholt and Madsen. As the calculations below were made after the p-complete version of theo-rem VII.0.0.2 on the correspondence between K-theory and T C was known for rings, they were stated for K-theory whenever possible, even though they were actually calculations of T C.

For a ring A, let W(A) be the p-typical Witt vectors, see [202] for the commutative case and [98] for the general case. Let W(A)F be the coinvariants under the Frobenius action, i.e., the cokernel of 1−F: W(A)→W(A). Note that W(F_p) =W(F_p)_F =Zbp.

1. Hesselholt [98] π−1T C(A)b^p ∼=W(A)F.

2. Hesselholt and Madsen (cf. [100] and [147]) Let k be a perfect field of characteristic p >0. Then T C(A) is an Eilenberg-MacLane spectrum for any k-algebra A, and

πiT C(k)b^p =







W(k)F if i=−1 Zb^p if i= 0

0 otherwise

and

πiT C(k[t]/tⁿ)b^p =







πiT C(k)b^p if i=−1 ori= 0 W_nm−1/VnW_m−1 if i= 2m−1>0

0 otherwise

where W_j = (1 + tk[[t]])^×/(1 + t^j+1k[[t]])^× is the truncated Witt vectors, and V_n:W_m−1 →W_nm−1 is the Vershibung map sending f(t) = 1 +tP∞

i=0a_itⁱ to f(tⁿ).

LetC be the cyclic group of order p^N. Then

πiT C(k[C])b^p =







πiT C(k)b^p if i=−1 or i= 0 K₁^⊕ⁿ if i= 2n−1

0 otherwise

whereK1 is the p-part of the units k[C]^∗.

3. Hesselholt ([98]). Let A be a free associative F_p-algebra. Then

πiT C(A)b^p =







W(A)F if i=−1 Zb^p if i= 0

0 otherwise

On the other hand,

π_iT C(F_p[t₁, . . . t_n])bp = ((L

g∈GmZb^p)b^p for −1≤i≤n−2

0 otherwise

whereGm is some explicit (non-empty) set (see [98, page 140])

4. Hesselholt and Madsen [103]. Let K be a complete discrete valuation field of char-acteristic zero with perfect residue field k of characteristic p > 2. Let A be the valuation ring of K. Hesselholt and Madsen analyze T C(A)bp, and in particular they give very interesting algebraic interpretations of the relative term of the trans-fer map T C(k)b^p → T C(A)b^p (obtained by inclusion of the k-vector spaces into the torsion modules ofA). See [103].

5. Rognes and Ausoni [8]. As a first step towards calculating the algebraic K-theory of connective complex K-theory ku, Ausoni and Rognes calculate topological cyclic homology of the Adams summand ℓp.

197

0.5 Where to read

The literature on T C is naturally even more limited than on T HH. Böksted, Hsiang and Madsen’s original paper [18] is still very readable. The first chapters of Hesselholt and Madsen’s [101] can serve as a streamlined introduction for those familiar with equivariant G-spectra. For more naïve readers, the unpublished lecture notes of Goodwillie can be of great help. Again, the survey article of Madsen [147] is recommendable.

1 The fixed point spectra of T HH .

We will defineT C by means of a homotopy cartesian square of the type (i.e., it will be the homotopy limit of the rest of the diagram)

T C(−) −−−→ T HH(−)^hS¹

 y

 Q y

pprimeT C(−;p)b^p −−−→ (Q

pprimeT HH(−)b^p)^hS¹

(as it stands, this strictly does not make sense: there are some technical adjustments we shall return to). The S¹-homotopy fixed points are formed with respect to the cyclic structure.

In this section we will mainly be occupied with preparing the ground for the lower left hand corner of this diagram. Let Cn ⊆ S¹ be the subgroup consisting of the n-th roots of unity. We choose our generator of the cyclic group Cn to be tn−1 = t = e^2πi/n. For each prime numberp, the functor T C(−;p)is defined as the homotopy limit of a diagram of fixed point spaces |T HH(−)|^C^pn. The maps in the diagrams are partially inclusion of fixed points |T HH(−)|^C^pn⁺¹ ⊆ |T HH(−)|^C^pn, and partially some more exotic maps - the “restriction maps” - which we will describe below. The contents of this section is mostly fetched from the unpublished MSRI notes [83]. If desired, the reader can consult appendix A.8 for some facts on group actions.

1.1 Cyclic spaces and the edgewise subdivision

Recall Connes’ category Λ (see e.g., IV.1.1.2). Due to the inclusion j: ∆⊂ Λ, any cyclic object X gives rise to a simplicial object j^∗X.

As noted by Connes [40], cyclic objects are intimately related to objects with a circle action (see also [113], [56] and [18]). In analogy with the standard n-simplices ∆[n] = {[q]7→∆([q],[n])}, we define the cyclic sets

Λ[n] = Λ(−,[n]) : Λ^o → Ens.

Lemma 1.1.1 For alln, |j^∗Λ[n]|is a T-space, whereT=|S¹|is the circle group, naturally (in [n]∈obΛ^o) homeomorphic to T× |∆[n]|.

Sketch proof: See e.g., [56, 2.7]. '!..&"%#$

This gives us the building blocks for a realization/singular functor pair connecting T-spaces with (pointed) cyclic sets:

T−T op∗

|−|_Λ

⇆

sinΛ

Ens∗Λ^o

1. THE FIXED POINT SPECTRA OF T HH. 199

In document The local structure of algebraic K-theory (sider 185-199)