CYCLIC POLYTOPES AND THE K-THEORY

OF TRUNCATED POLYNOMIAL ALGEBRAS

by Lars Hesselholt* and Ib Madsen

1. Introduction

This paper calculates the relative algebraic K-theory K_{∗}(k[x]/(x^{n}), (x)) of a truncated polynomial algebra over a perfect field k of positive characteristic p. Since the ideal generated by x is nilpotent, we can apply McCarthy’s theorem: the relative algebraic K-theory is isomorphic to the relative topological cyclic homology, [Mc], and it is the latter groups we actually evaluate.

The result is best expressed in terms of big Witt vectors. Let W_{m}(k) denote the big Witt vectors in k of length m, i. e. the multiplicative group

W_{m}(k) = (1 + xk[[x]])^{×}/(1 + x^{m+1}k[[x]])^{×},

and recall the Verschiebung map

V_{n}: W_{m}(k) → W_{mn}(k)

given by the substitution V_{n}(f(x)) = f(x^{n}). The relative K-theory K(k[x]/(x^{n}), (x)) is given by the fibration sequence

K(k[x]/(x^{n}), (x)) → K(k[x]/(x^{n})) → K(k),

with a corresponding exact sequence of homotopy groups

0 → K_{∗}(k[x]/(x^{n}), (x)) → K_{∗}(k[x]/(x^{n})) → K_{∗}(k) → 0.

The groups K_{∗}(k) were evaluted by Quillen in [Q] when k is a finite field. For a general perfect field of characteristic p > 0 one knows that the p-adic K-groups of k vanish in positive degrees by [K]. Theorem 4.2.10 below together with McCarthy’s theorem gives

Theorem A. Let k be a perfect field of positive characteristic. Then

K_{2m−1}(k[x]/(x^{n}), (x)) ≅ W_{mn}(k)/V_{n}W_{m}(k)

and the groups in even degrees are zero.

The result extends calculations by Aisbett and Stienstra of K_{3}(k[x]/(x^{n}), (x)). It was announced in [M], but was there stated as K_{2m−1}(k[x]/(x^{n}), (x)) ≅ W_{mn−1}(k)/V_{n}W_{m−1}(k).

In order to evaluate the topological cyclic homology TC(A) one begins with the topological Hochschild spectrum T (A). This is an S^{1}-equivariant spectrum. In the case where A = k[x]/(x^{n}), standard cyclic theory leads to an equivalence of spectra

T (A) ≃_{S^{1}} T (k) ∧ N^{cy}(Π_{n}),

Typeset by AMS-TEX

* Supported in part by NSF grant

where N^{cy}(Π_{n}) is the cyclic bar-construction of the pointed monoid Π_{n} = {0, 1, x, …, x^{n−1}}. Thus, for each closed subgroup C ⊂ S^{1}, the homotopy groups of the fixed sets T (A)^{C} may be viewed as the Cequivariant homology groups T (k)_{∗}^{C}(N^{cy}(Π_{n})). Knowledge of π_{∗}T (A)^{C} in turn determine the topological cyclic homology groups of A. There is an equivariant splitting

N^{cy}(Π_{n}) ≅ ⋁_{s⩾0} N^{cy}(Π_{n}; s),

where N^{cy}(Π_{n}; 0) = S^{0} and for s ⩾ 1 a homeomorphism

N^{cy}(Π_{n}; s) ≅ S_{+}^{1} ∧_{C_{s}} (Δ^{s−1}/C_{s} · Δ^{s−n}).

Here Δ^{s−1} is the standard s − 1 simplex with C_{s}-action given by cyclic permutation of the vertices, and Δ^{s−n} ⊂ Δ^{s−1} is the face spanned by the first s − n + 1 vertices. Given [HM], the main difficulty we have faced in this paper is to determine the S^{1}-equivariant homotopy type of the right hand side in the equality above. This is done in §3 below and is based on the concept of regular cyclic polytopes. One considers Δ^{s−1} as embedded in the regular representation ℝC_{s} by mapping the vertices to the group elements. Let d = [(s − 1)/n], and consider the projection

π_{d}: ℝC_{s} → V_{d}; V_{d} = ℂ(ξ_{s}) ⊕ ⋯ ⊕ ℂ(ξ_{s}^{d}).

The image of Δ^{s−1} is the regular cyclic polytope P_{s,d} in ℝ^{2d}. It contains Q_{s,d} = π_{d}(C_{s} · Δ^{s−n}), so we get a map

π_{d}: Δ^{s−1}/C_{s} · Δ^{s−n} → P_{s,d}/Q_{s,d}.

For dn < s < (d + 1)n it turns out that ∂P_{s,d} ⊂ Q_{s,d} and that 0 ∉ Q_{s,d}, and we can therefore compose with the projection

r: P_{s,d}/Q_{s,d} → P_{s,d}/∂P_{s,d} = S^{V_{d}};

let θ_{s,n} = r ∘π_{d} denote the composite. If s = (d+1)n, we project instead to V_{d+1}, and use another projection into the join of C_{n} and S^{V_{d}}

r: P_{s,d+1}/Q_{s,d+1} → C_{n} ∗ S^{V_{d}}.

Again, let θ_{s,n} = r ∘ π_{d+1}. The following result is proved in paragraph 3 by a mixture of geometry of cyclic polytopes and homological techniques based on results from [BAG].

Theorem B. There are S^{1}-equivariant equivalences

S_{+}^{1} ∧_{C_{s}} (Δ^{s−1}/C_{s} · Δ^{s−n}) ≃_{S^{1}} \{ \begin{matrix}S_{+}^{1} ∧_{C_{s}} S^{V_{d}} & if dn < s < (d + 1)n \\ S_{+}^{1} ∧_{C_{s}} (C_{n} ∗ S^{V_{d}}) & if s = (d + 1)n \end{matrix}

given by the maps S_{+}^{1} ∧_{C_{s}} θ_{s,n}.

We conjecture that the maps θ_{s,n} are themselves C_{s}-equivariant equivalences, such that

Δ^{s−1}/C_{s} · Δ^{s−n} ≃_{C_{s}} \{ \begin{matrix}S^{V_{d}} & , if dn < s < (d + 1)n \\ C_{n} ∗ S^{V_{d}} & , if s = (d + 1)n \end{matrix}

which of course implies theorem B. If θ_{s,n} is not an equivalence its cofiber X_{s,n} would be rather complicated in that the C_{s}-module H̃_{∗}(X_{s,n}) has vanishing group homology by theorem B.

It is the simplicity of the answer in theorem B which makes possible the calculation of T (k)_{∗}^{C}(N^{cy}(Π_{n})) and hence the proof of theorem A: V_{d} is an S^{1}-module so S_{+}^{1} ∧_{C_{s}} S^{V_{d}} ≅ S^{1}/C_{s+} ∧ S^{V_{d}} and T (k) ∧ S^{V_{d}} is the V_{d}th deloop of the S^{1}-spectrum T (k), so one really only needs to calculate the equivariant spectrum homology of the circles S^{1}/C_{s}.

It is a pleasure to acknowledge the help we have received from other mathematicians. First, we thank T. Geisser for urging us to use the topological cyclic homology techniques to prove theorem A, which he conjectured. Second, we are grateful to L. Bærentzen for some very helpful computer calculations at an early stage. Finally, we thank V. I. Arnold, M. Boij and B. Totaro for help with the key theorem 3.1.2. We had long ago conjectured it to be true, but had grave difficulties in proving it. Arnold and Totaro sent us two independent proofs, at the very same time as M. Boij supplied us with the final details in the proof we present.

2. Topological Hochschild homology of truncated polynomial algebras

2.1. This section describes the Hochschild homology and its topological extension for truncated polynomial algebras, and is mostly a recollection of results from [BAG] and [HM], but the reader is also referred to [H] and [M].

We begin with the linear case. For any ring A and A-bimodule M we write HH(A; M). for the simplicial abelian group with k-simplices

HH(A; M)_{k} = M ⊗ A^{⊗k}

and simplicial structure maps

d_{i}(m ⊗ a_{1} … ⊗ a_{k}) = m ⊗ … ⊗ a_{i}a_{i+1} ⊗ … ⊗ a_{k}  , 0 ⩽ i < k
= a_{k}m ⊗ a_{1} ⊗ … ⊗ a_{k−1}  , i = k
s_{i}(m ⊗ a_{1} … ⊗ a_{k}) = m ⊗ … ⊗ a_{i} ⊗ 1 ⊗ a_{i+1} ⊗ … ⊗ a_{k} , 0 ⩽ i ⩽ k.

The homotopy groups of the realization HH(A; M), or equivalently, the homology of the associated chain complex, are the Hochschild homology groups HH_{∗}(A; M). When M = A with its standard A-bimodule structure we shorten notation to HH(A).. The chain complex associated with HH(A; M). is derived from the two-sided bar construction by tensoring with M over A ⊗ A^{op}. Therefore

HH_{∗}(A; M) ≅ Tor_{∗}^{A⊗A^{op}}(A, M),

provided that A is flat over ℤ, i. e. torsion free. When the ring A is of the form ℤ[x]/(f(x)), where f(x) is a monic polynomial, one has the much smaller (periodic) resolution

(2.1.1)  ⋯ ⟶^{Δf/Δx} A ⊗ A ⟶^{Δx} A ⊗ A ⟶^{Δf/Δx} A ⊗ A ⟶^{Δx} A ⊗ A ⟶^{μ} A → 0,

where Δf = 1 ⊗ f(x) − f(x) ⊗ 1 and Δx = 1 ⊗ x − x ⊗ 1. Specializing to f(x) = x^{n}, where A is a truncated polynomial algebra, one gets for M = A,

(2.1.2)  HH_{i}(ℤ[x]/(x^{n})) ≅ \{ \begin{matrix}ℤ[x]/(x^{n}), & if i = 0 \\ xℤ[x]/(x^{n}), & if i > 0 and even \\ ℤ[x]/(x^{n−1}) ⊕ ℤ/nℤ⟨x^{n−1}⟩, & if i is odd \end{matrix}

The k-simplices HH(ℤ[x]/(x^{n}))_{k} has a ℤ-basis consisting of x^{i_{0}} ⊗ ⋯ ⊗ x^{i_{k}}. We give such a monomial the weight w = i_{0} + ⋯ + i_{k}, and note that the weight is preserved by the face and degeneracy operators. Hence we get a direct sum decomposition of simplicial abelian groups

(2.1.3)  HH(ℤ[x]/(x^{n})). ≅ ⨁_{s⩾0} HH(ℤ[x]/(x^{n}); s).

The homology of the summand HH(ℤ[x]/(x^{n}); 0). is equal to ℤ concentrated in degree zero. To evaluate the homology of the remaining summands, we note that a chain homotopy equivalence from (2.1.1) to the two-sided bar construction must have weight (k/2)n, for k even, and ((k − 1)/2)n + 1, for k odd, cf. [HM], 6.3. Hence (2.1.2) shows that the non-zero groups are

(2.1.4)

HH_{2d}(ℤ[x]/(x^{n}); s) = HH_{2d+1}(ℤ[x]/(x^{n}); s) = ℤ, if n does not divide s
HH_{2d+1}(ℤ[x]/(x^{n}); s) = ℤ/nℤ,  if n divides s,

where d is the integer part

d = [\frac{s − 1}{n}].

The Buenos Aires Group in [BAG] worked out explicitely the chain homotopy equivalence from (2.1.1) to the (normalized) two-sided bar construction, following the standard procedure from e. g. [ML], p. 236. This gives us the following formulas for the cycles in the Hochschild complex which carry the homology in (2.1.4):

(2.1.5)  \begin{aligned}ι_{2d} & = & (−1)^{d}  ∑_{{k_{0}+···+k_{d}=s−d \atop 1⩽k_{i}⩽n−1}}x^{k_{0}} ⊗ x^{k_{1}} ⊗ x ⊗ x^{k_{2}} ⊗ x ⊗ ⋯ ⊗ x^{k_{d}} ⊗ x, \\ ι_{2d+1} & = & (−1)^{d}  ∑_{{k_{0}+···+k_{d}=s−d−1 \atop 0⩽k_{i}⩽n−1}}x^{k_{0}} ⊗ x ⊗ x^{k_{1}} ⊗ x ⊗ x^{k_{d}} ⊗ x, \end{aligned}

We note that since s > dn, the lower bound on k_{i} is redundant. The final result we shall need from [BAG] is the value of Connes’ B-operator. On the chain level,

(2.1.6)  B(ι_{2d}) = −sι_{2d+1},

and B is zero in odd degrees.

2.2. Let Π be a pointed monoid, i. e. a monoid with a designated base point such that the product map factors to μ: Π ∧ Π → Π, and let A be a ring. The pointed monoid algebra is the set A(Π) = A[Π]/A[∗] with the ring structure induced from that on A[Π]. So in particular,

(2.2.1)  A[x]/(x^{n}) = A(Π_{n}),

where Π_{n} = {0, 1, x, …, x^{n−1}} considered as a pointed monoid with 0 as base point and x^{n} = 0. The cyclic bar construction, introduced by Waldhausen, is the cyclic set N.^{cy}(Π) with k-simplices

(2.2.2)  N_{k}^{cy}(Π) = Π^{∧(k+1)}

and the Hochschild-like structure maps

d_{i}(π_{0}∧ … ∧π_{k}) = π_{0}∧ … ∧π_{i}π_{i+1}∧ … ∧π_{k}  , 0 ⩽ i < k
= π_{k}π_{0}∧π_{1}∧ … ∧π_{k−1}  , i = k
s_{i}(π_{0}∧ … ∧π_{k}) = π_{0}∧ … ∧π_{i}∧1∧π_{i+1}∧ … ∧π_{k} , 0 ⩽ i ⩽ k
τ_{k}(π_{0}∧ … ∧π_{k}) = π_{k}∧π_{0}∧ … ∧π_{k−1}.

Its geometric realization, which we denote N^{cy}(Π), has a continuous S^{1}-action. Moreover, the reduced chains ℤ(N.^{cy}(Π)) and HH(ℤ(Π)). are isomorphic simplicial abelian groups, so

(2.2.3)  H̃_{∗}(N^{cy}(Π)) ≅ HH_{∗}(ℤ(Π)).

Under this isomorphism, Connes’ B-operator is given by the composition

B: H̃_{∗}(N^{cy}(Π)) ⟶^{[S^{1}]} H̃_{∗+1}(S_{+}^{1} ∧ N^{cy}(Π)) ⟶^{μ_{∗}} H̃_{∗+1}(N^{cy}(Π)),

where the first map is exterior multiplication by [S^{1}] ∈ H_{1}(S^{1}), and the second map is induced from the action map, cf. [H], proposition 1.4.5.

The topological Hochschild spectrum T (A) of a ring A, is an S^{1}-equivariant spectrum. We refer the reader to [HM], §1, or [M], 2.3-2.4 for the definition and to [LMS] for the notion of an equivariant spectrum. In the case of a pointed monoid algebra, we have from [HM], theorem 6.1, an equivalence of S^{1}-equivariant spectra

(2.2.4)  T (A(Π)) ≃_{S^{1}} T (A) ∧ N^{cy}(Π),

where the smash product on the right is formed in the category of S^{1}-equivariant spectra.

We need to determine the S^{1}-space N^{cy}(Π_{n}). First, we have a decomposition analogous to (2.1.4) of the cyclic set

(2.2.5)  N.^{cy}(Π_{n}) = ⋁_{s⩾0} N.^{cy}(Π_{n}; s),

and the realization decomposes accordingly. The isomorphism (2.2.3) is compatible with these decompositions, so we know the reduced homology H̃(N^{cy}(Π_{n}; s)) by (2.1.4). However, to make use of (2.2.4), we need to determine the S^{1}-equivariant homotopy type of N^{cy}(Π_{n}; s). We note that N.^{cy}(Π_{n}; s) as a cyclic set is generated by the single simplex

x^{(s)} = x∧ … ∧x ∈ N_{s−1}^{cy}(Π_{n}; s).

Let Λ.^{s−1} be the cyclic (s − 1)-simplex, i. e. the cyclic set with k-simplices Λ([k], [s − 1]), cf. [L]. Its realization is homeomorphic to S^{1} × Δ^{s−1}. We choose the homeomorphism from [HM], 6.2, where the C_{s}- action induced from the C_{s}-action in the second factor in Λ([k], [s − 1]) becomes the diagonal action of S^{1} × Δ^{s−1} which rotates the first factor by 2π/s and cyclically permutes the vertices of Δ^{s−1}.

Lemma 2.2.6. The characteristic map ι_{s}: Λ.^{s−1} → N.^{cy}(Π_{n}; s) which represents x^{(s)} factors over C_{s} · Λ.^{s−n} and defines an S^{1}-equivariant homeomorphism

(Λ^{s−1}/C_{s} · Λ^{s−n})/C_{s} ≅ N^{cy}(Π_{n}; s).

Proof. The composition Λ.^{s−n} ⟶^{(d^{0})^{n−1}} Λ.^{s−1} ⟶^{ι_{s}} N.^{cy}(Π_{n}; s) is constant because d_{0}^{n−1}x^{(s)} = x^{n}∧x∧ … ∧x = 0 and therefore τ_{s−1}^{i}Λ.^{s−n} also maps to zero, for i = 0, 1, …, s − 1. Hence ι_{s} factors over Λ.^{s−1}/C_{s} · Λ.^{s−n}. On the other hand, x^{(s)} is invariant under cyclic permutation, so we get

(Λ^{s−1}/C_{s} · Λ^{s−n})/C_{s} → N^{cy}(Π_{n}; s).

This is surjective because N.^{cy}(Π_{n}; s) is generated by x^{(s)}, and it is injective because the only relation in N.^{cy}(Π_{n}; s) comes from x^{n} = 0. □

3. Cyclic polytopes and the structure of N^{cy}(Π_{n})

3.1. We embed Δ^{s−1} in the regular representation ℝC_{s} by letting the ith vertex map to t^{i}, where t ∈ C_{s} is the generator, and let Δ^{s−n} ⊂ Δ^{s−1} be the inclusion onto the first s + 1 − n vertices. The C_{s}-action on ℝC_{s} restricts to one on Δ^{s−1} and we set

C_{s} · Δ^{s−n} = ⋃_{i=0}^{s−1} t^{i} · Δ^{s−n} ⊂ Δ^{s−1}.

We may then restate lemma 2.2.6 as

(3.1.1)  N^{cy}(Π_{n}; s) ≅ S_{+}^{1} ∧_{C_{s}} (Δ^{s−1}/C_{s} · Δ^{s−n}),

where the S^{1}-action on N^{cy}(Π_{n}; s) corresponds to the induced S^{1}-action on the right hand. The purpose of this paragraph is to calculate explicitely the S^{1}-equivariant homotopy type of the right hand side of (3.1.1).

The trigonometric moment curve in 2d dimensions is

x(t) = (e^{it}, e^{2it}, …, e^{dit}) ∈ ℂ^{d}

or upon identifying ℂ^{d} with ℝ^{2d}

x(t) = (cos t, sin t, cos 2t, sin 2t, …, cos dt, sin dt) ∈ ℝ^{2d}.

Consider the s points x_{j} = x(2πj/s), j = 0, 1, …, s − 1, on the trigometric moment curve. Their convex hull in ℝ^{2d} will be denoted P_{s,d}. It is the regular cyclic s-polytope in 2d-space, [G]. Note that P_{s,1} is the regular s-gon in the plane. If s ⩽ 2d + 1 then P_{s,d} is an (s − 1)-simplex.

The face structure of P_{s,d} is completely known. We shall only need the structure of facets, i. e. codimension 1 faces, where one has Gale’s evenness criterion: Picture the s points x_{j} on the circle (e. g. by projecting them onto P_{s,1}). Then a facet of P_{s,d} is the convex hull of d pairs of points in the cyclic ordering, that is,

F = ⟨x_{i_{1}}, x_{i_{1}+1}, …, x_{i_{d}}, x_{i_{d}+1}⟩,

where ⟨ ⟩ denotes the convex hull in ℝ^{2d}. Moreover, the facets are all (2d − 1)-simplices. The original and very readable account of this fact is [C]. See also [B] and [Z].

The cyclic group C_{s} acts on P_{s,d} by cyclically permuting the vertices. This action is linear. Indeed, let V_{d} denote the representation

V_{d} = ℂ(ξ_{s}) ⊕ ⋯ ⊕ ℂ(ξ_{s}^{d}).

Then the points x_{0}, x_{1}, …, x_{s−1} exactly form the orbit of x_{0} = (1, …, 1) under the action of C_{s}, and the action of C_{s} on V_{d} restricts to the C_{s}-action of P_{s,d} which permutes the vertices.

Let ⟨x_{0}, …, x_{k}⟩ be the convex hull of the first (k + 1) points x_{ν} = x(2πν/s), and let E be the convex hull

E = ⟨x(t) | 0 ⩽ t < 2πd/(d + 1)⟩.

We note that ⟨x_{0}, …, x_{k}⟩ ⊂ E when k/s < d/(d + 1).

Theorem 3.1.2. E does not contain 0. In particular, 0 ∉ ⟨x_{0}, …, x_{k}⟩ for k/s < d/(d + 1).

Proof. Let t_{0} = 2π/d + 1, and let H be the linear subspace

H = Span{x(0), x(t_{0}), …, x((d − 1)t_{0}); x′(t_{0}), …, x′((d − 1)t_{0})}.

Since x(0) + x(t_{0}) + ⋯ + x(dt_{0}) = 0, we have x(dt_{0}) ∈ H. Consider the distance (with sign) from H to x(t),

δ(t) = det(x(0), x(t_{0}), …, x((d − 1)t_{0}), x′(t_{0}), …, x′((d − 1)t_{0}), x(t))

It is non-zero, since otherwise we would have E ⊂ H, and this is impossible since E is 2d-dimensional, and H has codimension at least 1. Now δ(t) is a trigonometric polynomial of degree d,

δ(t) = ∑_{k=0}^{d}(a_{k} cos kt + b_{k} sin kt),

and therefore has at most 2d roots counted with multiplicity in the interval [0, dt_{0}] ⊂ [0, 2π). (Write it as a polynomial in e^{it}.) But 0 and dt_{0} are roots and t_{0}, …, (d − 1)t_{0} are double roots, so these are all roots. It follows that δ(t) has constant sign on [0, dt_{0}], δ(t) ⩾ 0 or δ(t) ⩽ 0. Finally,

E ∩ H ⊂ ⟨x(0), x(t_{0}), …, x((d − 1)t_{0})⟩.

This is a facet of the d-simplex ⟨x(0), …, x(dt_{0})⟩ and therefore does not contain 0, the barycenter. □

Remark 3.1.3. The proof above is basically the same as Totaro’s proof. There is a more combinatorial proof, pointed out by G. Ziegler and by B. Totaro, which runs as follows.

The combinatorial structure of P_{s,d} does not depend on the choice of points x_{j} = x(2πj/s). The convex hull of any s distinct points on the trigonometric moment curve is combinatorially equivalent to P_{s,d}. Let T consist of s points in the interval [0, dt_{0}] with {0, t_{0}, …, dt_{0}} ⊂ T and let P (T) = ⟨x(t) | t ∈ T ⟩. Then P (T) is a cyclic polytope and Gale’s evenness criterion shows that the set

F = ⟨x(0), x(t_{0}), …, x(dt_{0})⟩

is a face of P (T). Now 0 is the barycenter of F, so in particular, 0 is an interior point of F. It follows that 0 ∉ P (T − {dt_{0}}). □

We recall the structure of the regular representation ℝC_{s},

ℝC_{s} ≅ ℝ ⊕ V_{[(s−1)/2]} (⊕ℝ_{−}),

where the sign representation ℝ_{−} occurs only for even s. In particular, we have a projection

(3.1.4)  π_{d}: ℝC_{s} → V_{d}

whenever d ⩽ [(s − 1)/2]. By definition P_{s,d} = π_{d}Δ^{s−1}, and we now let Q_{s,d} = π_{d}(C_{s} · Δ^{s−n}), or

Q_{s,d} = C_{s} · ⟨x_{0}, …, x_{s−n}⟩ = ⋃_{i=0}^{s−1}⟨x_{i}, x_{i+1}, …, x_{i+s−n}⟩.

If s < (d + 1)n then the theorem shows that 0 ∉ Q_{s,d}. Hence radial projection away from 0 gives a C_{s}-equivariant map

(3.1.5)  r: P_{s,d}/Q_{s,d} → P_{s,d}/∂P_{s,d} = S^{V_{d}}

to the one-point compacification of the representation V_{d}.

Remark 3.1.6. If s > nd then Q_{s,d} contains the boundary ∂P_{s,d}. Indeed, the boundary is a union of the facets, given by Gale’s evenness criterion. A facet

F = ⟨x_{i_{1}}, x_{i_{1}+1}, …, x_{i_{d}}, x_{i_{d}+1}⟩

is contained in Q_{s,d} if there is a gap of at least n − 1 points between some of the vertices considered as points on the circle. The worst case is then the facet

F = ⟨x_{0}, x_{1}, x_{n}, x_{n+1}, …, x_{(d−1)n}, x_{(d−1)n+1}⟩,

or any of its cyclic consequences. But the gap between x_{(d−1)n+1} and x_{0} = x_{s} is s − (d − 1)n − 2 which is greater than or equal to n − 1 precisely when s > dn.

Suppose that dn < s < (d + 1)n. We expect that in this case radial projection away from 0 defines an equivariant strong deformation retract of Q_{s,d} ⊂ P_{s,d} onto the boundary ∂P_{s,d}, such that the map r above is a C_{s}-equivariant homotopy equivalence. We shall not need this however.

3.2. In this section, we consider the case where n does not divide s; dn < s < (d + 1)n. The facets of the convex polytope P_{s,d} are (2d − 1)-dimensional simplices, so we may triangulate P_{s,d} by taking cones from x_{0} of the facets which do not contain x_{0},

P_{s,d} = ⋃_{x_{0}∉F} cone_{x_{0}}(F).

This triangulation is closely related to the Buenos Aires formula (2.1.5). Indeed, let

(3.2.1)  \begin{aligned}X_{s,d} & = \{(k_{0}, …, k_{d}) | ∑k_{i} = s − d, 1 ⩽ k_{i}\} \\ Y_{s,d} & = \{(k_{0}, …, k_{d}) | ∑k_{i} = s − d, 1 ⩽ k_{i} ⩽ n − 1\}. \end{aligned}

A simple calculation in HH(ℤ[x]/(x^{n})). shows that

\begin{aligned}d_{k}x^{⊗s} & = d_{2d+1}^{k_{0}−1}d_{2d−1}^{k_{d}−1} … d_{1}^{k_{1}−1}x^{⊗s} \\ & = x^{k_{0}} ⊗ x^{k_{1}} ⊗ x ⊗ x^{k_{2}} ⊗ x ⊗ ⋯ ⊗ x^{k_{d}} ⊗ x, \end{aligned}

such that we may rewrite (2.1.5) as

(3.2.2)  ι_{2d} = ∑_{k∈Y_{s,d}} d_{k}x^{⊗s}.

Dually,

(3.2.3)  d^{k}: Δ^{2d} → Δ^{s−1}; d^{k} = (d^{1})^{k_{1}−1}(d^{3})^{k_{2}−1} … (d^{2d+1})^{k_{0}−1}

gives a 2d-face of Δ^{s−1}. If we embed Δ^{s−1} in ℝC_{s} as above, then d^{k} is the affine map given by the ordered set of vertices

(1, t^{k_{1}}, t^{k_{1}+1}, t^{k_{1}+k_{2}+1}, t^{k_{1}+k_{2}+2}, …, t^{k_{1}+···+k_{d}+d−1}, t^{k_{1}+···+k_{d}+d}).

The projection π_{d}: Δ^{s−1} → P_{s,d} of (3.1.4) maps t^{i} to x_{i}, so we get

π_{d}(d^{k}(Δ^{2d})) = cone_{x_{0}}(F_{k}),

where F_{k} is the facet of P_{s,d} = π_{d}(Δ^{s−1}) given by

F_{k} = ⟨x_{k_{1}}, x_{k_{1}+1}, x_{k_{1}+k_{2}+1}, x_{k_{1}+k_{2}+2}, …, x_{k_{1}+···+k_{d}+d−1}, x_{k_{1}+···+k_{d}+d}⟩.

Conversely, by Gale’s evenness criterion any facet, which does not contain x_{0}, has this form, and hence

(3.2.4)  P_{s,d} = ⋃_{k∈X_{s,d}} cone_{x_{0}}(F_{k}).

The numbers k_{0} − 1, …, k_{d} − 1 give the width of the gaps between the vertices in the simplex cone_{x_{0}}(F_{k}). Hence the simplex π_{d}d^{k}(Δ^{2d}) is in Q_{s,d} if and only if one of these gaps is at least n − 1, or equivalently, if and only if k ∈ X_{s,d} − Y_{s,d}.

Lemma 3.2.5. The chain c = ∑_{k∈Y_{s,d}} d^{k} is a cycle in C_{∗}(Δ^{s−1}, C_{s} · Δ^{s−n}), and r_{∗}π_{d∗}(c) generates H_{2d}(S^{V_{d}}).

Proof. We note that the sum k_{i} + k_{j} of any pair of indices is strictly greater than n − 1. For on the one hand k_{0} + ⋯ + k_{d} = s − d > (n − 1)d, and on the other hand each k_{r} ⩽ n − 1, and hence

k_{i} + k_{j} > (n − 1)d − (n − 1)(d − 1) = n − 1.

In particular, d_{0}(d^{k}) ∈ C_{∗}(C_{s} · Δ^{s−n}), since the gap between t^{k_{1}+···+k_{d}+d} and t^{k_{1}} is k_{0} + k_{1} − 1 and hence greater than or equal to n − 1.

Now consider d_{1}d^{k}. If k_{1} = n−1 then the gap between 1 and t^{k_{1}+1} is n−1 and hence d_{1}(d^{k}) ∈ C_{∗}(C_{s}Δ^{s−n}). If k_{1} ⩽ n − 2 then the remaining k_{i} are at least 2, so we also have l ∈ Y_{s,d}, where

l = \{ \begin{matrix}(k_{0}, k_{1} + 1, k_{2} − 1, k_{3}, …, k_{d}), & if d ⩾ 2 \\ (k_{0} − 1, k_{1} + 1), & if d = 1. \end{matrix}

But d_{1}(d^{k}) = d_{2}(d^{l}) and the two faces appear with opposite sign in the boundary of the chain c. The other faces d_{i}(d^{k}) are treated analogously. □

By Poincaré duality the 2dth homology of P_{s,d}/Q_{s,d} is free abelian of rank equal to the number of connected components of P_{s,d} − Q_{s,d}. The simplicial chain π_{d∗}(c) consists of simplices of P_{s,d} which covers P_{s,d} − Q_{s,d}. Thus it represents the sum of the generators of the summands in H_{2d}(P_{s,d}, Q_{s,d}). The map r maps the connected component of P_{s,d} − Q_{s,d} which contains 0 to S^{V_{d}} by a degree one map, and it follows that r_{∗}π_{d∗}(c) generates H_{2d}(S^{V_{d}}).

Proposition 3.2.6. The map S_{+}^{1} ∧_{C_{s}} (Δ^{s−1}/C_{s}Δ^{s−n}) → S_{+}^{1} ∧_{C_{s}} S^{V_{d}}, induced from the composition of r and π_{d} is a homotopy equivalence.

Proof. Both spaces are simply connected when d ⩾ 1, so it suffices to argue that the map is a homology equivalence. To this end, we consider the commutative diagram

Δ^{s−1}/C_{s} · Δ^{s−n} ⟶^{f} S_{+}^{1} ∧_{C_{s}} Δ^{s−1}/C_{s} · Δ^{s−n} ⟶^{ι_{s}} N^{cy}(Π_{n}; s)
↓^{π_{d}}  ↓π̂_{d}
P_{s,d}/Q_{s,d}  ⟶ S_{+}^{1} ∧_{C_{s}} P_{s,d}/Q_{s,d}
↓^{r}  ↓r̂
S^{V_{d}}  ⟶^{g}  S_{+}^{1} ∧_{C_{s}} S^{V_{d}},

where f and g maps u to [1, u]. We remarked above that r_{∗}π_{d∗}(c) represents the generator. Since V_{d} extends to an S^{1}-module,

S_{+}^{1} ∧_{C_{s}} S^{V_{d}} ≅ S^{1}/C_{s+} ∧ S^{V_{d}}; [z, v] ↦ [z, zv].

This implies that g induces an isomorphism on homology in dimension 2d, and that the composite

H̃_{2d}(S_{+}^{1} ∧_{C_{s}} S^{V_{d}}) ⟶^{[S^{1}]} H̃_{2d+1}(S_{+}^{1} ∧ (S_{+}^{1} ∧_{C_{s}} S^{V_{d}})) ⟶^{μ_{∗}} H̃_{2d+1}(S_{+}^{1} ∧_{C_{s}} S^{V_{d}}),

is multiplication by s, cf. (2.2.3). On the other hand, since ι_{s} is a homeomorphism by lemma 2.2.6, the Buenos Aires formula (2.1.5) and the calculation (3.2.2) shows that f_{∗}(c) is the generator of H_{2d}(S_{+}^{1} ∧_{C_{s}} (Δ^{s−1}/C_{s} · Δ^{s−n})). Commutativity of the diagram shows that r̂ ∘ π̂_{d} induces an isomorphism on homology in dimension 2d. To prove the same in dimension 2d + 1 we use the commutative diagram

H̃_{2d}(N^{cy}(Π_{n}; s)) ⟶^{B} H̃_{2d+1}(N^{cy}(Π_{n}; s))
↓≅  ↓
H̃_{2d}(S_{+}^{1} ∧_{C_{s}} S^{V_{d}}) ⟶ H̃_{2d+1}(S_{+}^{1} ∧_{C_{s}} S^{V_{d}})

and the fact that the upper horizontal morphism is Connes’ B-operator, hence is multiplication by −s according to (2.1.6). Since the lower map multiplies by s, the right hand vertical map is an isomorphism. □

3.3. We next treat the case where n divides s; s = (d + 1)n and V_{d} = ℂ(ξ_{s}) ⊕ ⋯ ⊕ ℂ(ξ_{s}^{d}). There is a cofibration sequence of C_{s}-spaces

(3.3.1)  C_{n+} ∧ S^{V_{d}} ⟶^{pr_{2}} S^{V_{d}} ⟶ C_{n} ∗ S^{V_{d}},

where C_{s} acts via the projection onto C_{n} and diagonally in the smash and join. Indeed, the mapping cone of pr_{2} is homeomorphic to C_{n} ∗ S^{V_{d}}/C_{n} ∗ ∞ and C_{n} ∗ ∞ is contractible. Hence

(3.3.2)  S_{+}^{1} ∧_{C_{s}} (C_{n} ∗ S^{V_{d}}) ≅ (S^{1} ∪_{n} e^{2}) ∧ S^{V_{d}}.

Indeed, the left hand side is the cofiber

cof(S_{+}^{1} ∧_{C_{s}} (C_{n+} ∧ S^{V_{d}}) → S_{+}^{1} ∧_{C_{s}} S^{V_{d}}) ≅ cof(S_{+}^{1} ∧_{C_{d+1}} S^{V_{d}} → S_{+}^{1} ∧_{C_{s}} S^{V_{d}})

≅ cof(S^{1}/C_{(d+1)+} ∧ S^{V_{d}} → S^{1}/C_{s+} ∧ S^{V_{d}}) ≅ cof(S^{1}/C_{(d+1)+} → S^{1}/C_{s+}) ∧ S^{V_{d}} ≅ (S^{1} ∪_{n} e^{2}) ∧ S^{V_{d}}.

Here the second homeomorphism uses that the C_{s}-module V_{d} is the restriction of an S^{1}-module so that

S_{+}^{1} ∧_{C_{s}} S^{V_{d}} ≅ S^{1}/C_{s+} ∧ S^{V_{d}}, [z, v] ↦ [z, zv],

where now on the right hand side the S^{1}-action is diagonal. In particular, H_{2d+1}(S_{+}^{1} ∧_{C_{s}} (C_{n} ∗S^{V_{d}})) = ℤ/nℤ. It follows from (3.3.1) that

(3.3.3)  H_{2d+1}(C_{n} ∗ S^{V_{d}}) = IC_{n},

the augmentation ideal in the integral group ring ℤC_{n}. We have

IC_{n}/IC_{n}^{2} ≅ C_{n},

since the left hand side is H_{0}(C_{n}; IC_{n}) which is isomorphic to H_{1}(C_{n}; ℤ) by the exact homology sequence associated with 0 → IC_{n} → ℤC_{n} ⟶^{ϵ} ℤ → 0. The isomorphism associates to g ∈ C_{n} the class g − 1 in IC_{n}/IC_{n}^{2}.

Lemma 3.3.4. The map g: C_{n} ∗ S^{V_{d}} → S_{+}^{1} ∧_{C_{s}} (C_{n} ∗ S^{V_{d}}), g(u) = [1, u], induces the natural map

IC_{n} → IC_{n}/IC_{n}^{2} ≅ ℤ/nℤ

on homology in dimension 2d + 1.

Proof. Since the action by C_{s} on S_{+}^{1} ∧ (C_{n} ∗ S^{V_{d}}) is free in the pointed sense, orbits and homotopy orbits agree, so we have a spectral sequence

E^{2} = H_{∗}(C_{s}, H̃_{∗}(S_{+}^{1} ∧ (C_{n} ∗ S^{V_{d}}))) ⇒ H̃_{∗}(S_{+}^{1} ∧_{C_{s}} (C_{n} ∗ S^{V_{d}})).

It is concentrated on two horizontal lines, since

H̃_{2d+1}(S_{+}^{1} ∧ (C_{n} ∗ S^{V_{d}})) = H̃_{2d+2}(S_{+}^{1} ∧ (C_{n} ∗ S^{V_{d}}))

are the only non-zero groups. Thus both lines are H_{∗}(C_{s}; IC_{n}). But

H_{∗}(C_{s}; ℤC_{n}) ≅ H_{∗}(C_{d+1}; ℤ)

by Frobenius reciprocity, cf. [Br], and H_{∗}(C_{s}; IC_{n}) can be obtained from the exact sequence

⋯ → H_{∗}(C_{s}; IC_{n}) → H_{∗}(C_{s}; ℤC_{n}) → H_{∗}(C_{s}; ℤ) → ⋯

It follows that H_{∗}(C_{s}; IC_{n}) vanishes in odd degrees and is equal to ℤ/nℤ in even degrees.

By (3.3.2) the spectral sequence converges to a single copy of ℤ/nℤ in degree 2d + 1. Hence all d^{2}- differentials must be isomorphisms, and the edge homomorphism

H_{0}(C_{s}; H̃_{2d+1}(S_{+}^{1} ∧ (C_{n} ∗ S^{V_{d}}))) → H̃_{2d+1}(S_{+}^{1} ∧_{C_{s}} (C_{n} ∗ S^{V_{d}}))

is an isomorphism. It follows that the induced map g_{∗} of the lemma can be identified with

H̃_{2d+1}(C_{n} ∗ S^{V_{d}}) → H_{0}(C_{s}; H̃_{2d+1}(C_{n} ∗ S^{V_{d}}))

which is precisely the map IC_{n} → IC_{n}/IC_{n}^{2}. □

We are now ready to determine S_{+}^{1} ∧_{C_{s}} (Δ^{s−1}/C_{s} · Δ^{s−n}) when s = n(d + 1) by an argument similar to that of section 3.2. This makes use to the projection

π_{d+1}: Δ^{s−1}/C_{s} · Δ^{s−n} → P_{s,d+1}/Q_{s,d+1},

and of the associated π̂_{d+1} = S_{+}^{1} ∧_{C_{s}} π_{d+1}. We still have 0 ∉ Q_{s,d+1}, but this time Q_{s,d+1} does not contain all of ∂P_{s,d+1}. Indeed,

\begin{aligned}∂_{1}P_{s,d+1} & = ∂P_{s,d+1} − Q_{s,d+1} ∩ ∂P_{s,d+1} = C_{n} · ⟨x_{0}, x_{1}, x_{n}, x_{n+1}, …, x_{dn}, x_{dn+1}⟩ \\ & = ⋃_{i=0}^{n−1}⟨x_{i}, x_{i+1}, …, x_{i+dn}, x_{i+dn+1}⟩, \end{aligned}

which in turn is homotopy equivalent to a circle. This follows because each of the involved facets are joins of two (n − 1)-simplices which each may be retracted on their barycenters, and the resulting line segments form a circle.

We write V_{d+1} = V_{d} ⊕ ℂ(ξ_{s}^{d+1}); ξ_{s}^{d+1} = ξ_{n}. Then

(3.3.5)  Q_{s,d+1} ∩ ℂ(ξ_{s}^{d+1}) = {ξ_{n}^{i} | i = 0, 1, …, n − 1} = C_{n} · {1}

where ℂ(ξ_{s}^{d+1}) = 0 ⊕ ℂ(ξ_{s}^{d+1}) ⊂ V_{d+1}. Indeed, projecting to V_{d} we have 0 ∈ ⟨x_{0}, …, x_{nd}⟩ ⊂ V_{d}, namely,

0 = \frac{1}{d}(1 + ξ_{s}^{ni} + ⋯ + ξ_{s}^{ndi}).

The proof of (3.1.2) shows that this is the only representation of 0 in the cyclic polytope ⟨x_{0}, …, x_{nd}⟩ ⊂ V_{d}, and (3.3.5) follows.

Let n = \{\frac{1}{2}(ξ_{n}^{i} + ξ_{n}^{i+1}) | i = 0, 1, …, n − 1\}, the set of midpoints in the regular n-gon in ℂ(ξ_{n}). Radial projection from 0 ∈ V_{d+1} gives a map

r: (P_{s,d+1}, Q_{s,d+1}) → (P_{s,d+1}, ∂P_{s,d+1} − n).

On the other hand,

∂P_{s,d+1} ≅ S(V_{d}) ∗ S(ℂ(ξ_{n}))

and the general formula X ∗Y −Y_{0} ≃ X ∗(Y −Y_{0}) together with the C_{s}-homotopy equivalence S(ℂ(ξ_{n}))−n ≃ C_{n} · {1} show that

(3.3.6)  ∂P_{s,d+1} − n ≃_{C_{s}} C_{n} ∗ S(V_{d}).

Thus we get a C_{s}-equivariant map

(3.3.7)  r: P_{s,d+1}/Q_{s,d+1} → ΣC_{n} ∗ S(V_{d}) ≅ C_{n} ∗ S^{V_{d}}.

This may very well be an equivariant homotopy equivalence, but we shall do with less.

The Buenos Aires formula, giving the generator ι_{2d+1} of H̃_{2d+1}(N^{cy}(Π_{n}; s)) = ℤ/nℤ, has only one term, namely

x^{n−1} ⊗ x ⊗ ⋯ ⊗ x^{n−1} ⊗ x ∈ ℤ(N_{2d+1}^{cy}(Π_{n}; s)).

This is the iterated face d_{2d+2}^{n−1}d_{2d}^{n−1} … d_{2}^{n−1} applied to the generator x^{⊗s}, so dually consider the face

c: Δ^{2d+1} → Δ^{s−1}; c = (d^{2})^{n−1} … (d^{2d+2})^{n−1}.

This is the affine map given by the ordered set of vertices

(1, t, t^{n}, t^{n+1}, …, t^{dn}, t^{dn+1})

in Δ^{s−1} ⊂ ℝC_{s}, and maps to a facet of P_{s,d+1} under the projection π_{d+1}. The argument of lemma 3.2.5 implies that c becomes a cycle in C_{∗}(Δ^{s−1}, C_{s} · Δ^{s−n}). Thus

π_{d+1}(c) ∈ C_{2d+1}(P_{s,d+1}, Q_{s,d+1}), rπ_{d+1}(c) ∈ C_{2d+1}(P_{s,d+1}, ∂P_{s,d+1} − n)

are also cycles. In order to determine their homology class, we may instead determine the homology class of

∂rπ_{d+1}(c): ∂Δ^{2d+1} → ∂P_{s,d+1} − n ≅ S(V_{d+1}) − n.

We know from (3.3.4) and (3.3.6) that

H_{2d+1}(S(V_{d+1}) − n) = IC_{n}.

The augmentation ideal IC_{n} is isomorphic to ℤC_{n}/(N), where N is the norm element. We saw above that H_{0}(C_{s}; IC_{n}) ≅ ℤ/nℤ, and hence that H_{0}(C_{s}; ℤC_{n}/(N)) ≅ ℤ/nℤ. Indeed a specific isomorphism is induced from the augmentation ϵ: ℤC_{n}/(N) → ℤ/nℤ.

Lemma 3.3.8. The cycle ∂rπ_{d+1}(c) represents an element of H_{2d}(S(V_{d+1})−n) = ℤC_{n}/(N) which augments to 1 + nℤ.

Proof. The boundary of rπ_{d+1}(c) is a 2d-dimensional sphere S^{2d} in S^{2d+1} = S(V_{d+1}) which does not intersect n. Let Ḋ_{+} and Ḋ_{−} be the two components of S(V_{d+1}) − S^{2d}, say, rπ_{d+1}(c) = D_{+}. The inclusion S^{2d} ⊂ S(V_{d+1}) − n induces an injection on homology if and only if n_{+} = D_{+} ∩ n and n_{−} = D_{−} ∩ n are both non-empty, and [S^{2d}] ∈ H_{2d}(S(V_{d+1}) − n) augments to n_{+} ∈ ℤ/nℤ, n_{+} = #n_{+}. This follows from Poincaré duality. The dual of the homology sequence

⋯ → H_{2d+1}(S(V_{d+1}) − n, S^{2d}) → H_{2d}(S^{2d}) → H_{2d}(S(V_{d+1})) → …

is the sequence

⋯ → H^{0}(S(V_{d+1}) − S^{2d}, n) → H^{1}(S(V_{d+1}), S(V_{d+1}) − S^{2d}) → H^{1}(S(V_{d+1}), n) → …

Now S(V_{d+1}) − S^{2d} = Ḋ_{+} ⊔ Ḋ_{−} and the lower sequence is isomorphic to

⋯ → H^{0}(Ḋ_{+}, n_{+}) ⊕ H^{0}(Ḋ_{−}, n_{−}) → (H^{0}(Ḋ_{+}) ⊕ H^{0}(Ḋ_{−}))/Δ → (H^{0}(n_{+}) ⊕ H^{0}(n_{−}))/Δ → …

where Δ is the image of H^{0}(S(V_{d+1})). The element (1, 0) ∈ H^{0}(Ḋ_{+}) ⊕ H^{0}(Ḋ_{−}) represents the generator of H_{2d}(S^{2d}) under duality and its image in

ℤC_{n}/(N) = (H^{0}(n_{+}) ⊕ H^{0}(n_{−}))/Δ

has augmentation #n_{+} ∈ ℤ/nℤ as claimed.

We have left to check that n_{+} consists of one point. Note that

π_{d+1}(c) = ⟨x_{0}, x_{1}, x_{n}, x_{n+1}, …, x_{dn}, x_{dn+1}⟩ = ⟨x_{0}, x_{n}, …, x_{dn}⟩ ∗ ⟨x_{1}, x_{n+1}, …, x_{dn+1}⟩

in V_{d+1}, and moreover,

⟨x_{0}, x_{n}, …, x_{dn}⟩ ∩ ℂ(ξ_{s}^{d+1}) = {1}, ⟨x_{1}, x_{n+1}, …, x_{dn+1}⟩ ∩ ℂ(ξ_{s}^{d+1}) = {ξ_{s}^{d+1}}.

Thus π_{d+1}(c) is the line segment which joins 1 and ξ_{s}^{d+1}, and the midpoint is the only element of this line segment which also belongs to n. This shows that π_{d+1}(c) ∩ ℂ(ξ_{s}^{d+1}) ∩ n consists of a single point, and n ⊂ ℂ(ξ_{s}^{d+1}) so π_{d+1}(c) ∩ n is a one point space. The linear projection r does not change this fact, i. e. D_{+} ∩ n is also a one point space. □

Proposition 3.3.9. For s = n(d + 1), the map r̂ ∘ π̂_{d+1} gives a homotopy equivalence

S_{+}^{1} ∧_{C_{s}} (Δ^{s−1}/C_{s} · Δ^{s−n}) ≃ S_{+}^{1} ∧_{C_{s}} (C_{n} ∗ S^{V_{d}}),

where V_{d} = ℂ(ξ_{s}) ⊕ ⋯ ⊕ ℂ(ξ_{s}^{d}).

Proof. Consider the diagram

Δ^{s−1}/C_{s} · Δ^{s−n} ⟶^{f} S_{+}^{1} ∧_{C_{s}} (Δ^{s−1}/C_{s} · Δ^{s−n}) ⟶_{≅}^{ι_{s}} N^{cy}(Π_{n}; s)
↓^{π_{d+1}}  ↓π̂_{d+1}
P_{s,d+1}/Q_{s,d+1} ⟶ S_{+}^{1} ∧_{C_{s}} (P_{s,d+1}/Q_{s,d+1})
↓^{r}  ↓r̂
C_{n} ∗ S^{V_{d}}  ⟶^{g} S_{+}^{1} ∧_{C_{s}} (C_{n} ∗ S^{V_{d}}).

The cohomology class of the cycle c ∈ C_{2d+1}(Δ^{s−1}, C_{s} · Δ^{s−n}) maps to the generator of H_{2d+1}(S_{+}^{1} ∧_{C_{s}} (C_{n} ∗ S^{V_{d}})) by lemmas 3.3.5 and 3.3.8. Thus f_{∗}[c] maps onto this generator under the right hand vertical map. Since H_{∗}(N^{cy}(Π_{n}; s)) = ℤ/nℤ in degree 2d+1, r̂∘π̂_{d+1} is a homology isomorphism. Both domain and target are simply connected for d > 1. For d = 1, a direct geometric argument gives the equivalence. □

Proof of theorem B. The equivariant Whitehead theorem states that a G-map f: X → Y between G-CW-complexes (G a compact Lie group) is a G-homotopy equivalence if and only if the induced map f^{H}: X^{H} → Y^{H} is a weak equivalence for every closed subgroup H ⊂ G, cf. [A], [LMS]. In the case at hand, we have constructed C_{s}-equivariant maps

θ_{s,n}: Δ^{s−1}/C_{s} · Δ^{s−n} → \{ \begin{matrix}S^{V_{d}} & , if dn < s < (d + 1)n \\ C_{n} ∗ S^{V_{d}} & , if s = (d + 1)n, \end{matrix}

and proved that the induced S^{1}-equivariant maps S_{+}^{1} ∧_{C_{s}} θ_{s,n} are weak homotopy equivalences. Therefore, to prove theorem B we must show that the same holds on C-fixed sets for every closed C ⊂ S^{1}. We note that if G is an abelian group, H ⊂ G a subgroup and X an H-space, then

(G_{+} ∧_{H} X)^{K} = \{ \begin{matrix}G/K_{+} ∧_{H/K} X^{K} & , if K ⊂ H \\ * & , if K ⊄ H, \end{matrix}

so it suffices to consider the subgroups C_{r} ⊂ C_{s} ⊂ S^{1}. We shall prove that in this case θ_{s,n}^{C_{r}} may be identified with θ_{s/r,n}. Then theorem B follows from the equivariant Whitehead theorem.

Let N_{r} ∈ ℝC_{s} be the norm element of the subgroup C_{r} ⊂ C_{s}, i.e.

N_{r} = 1 + t^{s/r} + t^{2s/r} + ⋯ + t^{(r−1)s/r},

where t ∈ C_{s} is the generator. Then

(Δ^{s−1})^{C_{r}} = ⟨\frac{1}{r}N_{r}, \frac{1}{r}tN_{r}, …, \frac{1}{r}t^{s/r−1}N_{r}⟩ ≅ Δ^{s/r−1}.

Moreover (C_{s} · Δ^{s−n})^{C_{r}} = C_{s} · (Δ^{s−n})^{C_{r}} and

(Δ^{s−n})^{C_{r}} = (Δ^{s−1})^{C_{r}} ∩ Δ^{s−n} ≅ Δ^{s/r−n} ⊂ Δ^{s/r−1}

since the representation of an element in a simplex as a convex combination of the vertices is unique. Hence

(Δ^{s−1}/C_{s} · Δ^{s−n})^{C_{r}} ≅ Δ^{s/r−1}/C_{s/r} · Δ^{s/r−n}.

One readily verifies that as C_{r/s}-spaces

(S^{V_{[(s−1)/n]}})^{C_{r}} ≅ S^{V_{[(s/r−1)/n]}}

and since (X ∗ Y)^{G} = X^{G} ∗ Y^{G} that

(C_{n} ∗ S^{V_{[(s−1)/n]}})^{C_{r}} ≅ \{ \begin{matrix}C_{n} ∗ S^{V_{[(s/r−1)/n]}} & , if n divides s/r \\ S^{V_{[(s/r−1)/n]}} & , if n does not divide s/r. \end{matrix}

Hence the maps θ_{s,n}^{C_{r}} and θ_{s/r,n} have the same domain and range, and one sees similarly that the maps are equal. □

4. Topological cyclic homology of k[x]/(x^{n})

4.1. In this paragraph, we give a general formula for the topological cyclic homology of a truncated polynomial algebra A[x]/(x^{n}). We then calculate the groups TC_{∗}(k[x]/(x^{n})) explicitely, when the coefficient ring is a perfect field of characteristic p > 0. First, we recall some facts about big Witt vectors and refer the reader to Bergman’s lecture in [Mu] for details.

Let A be a commutative ring. The big Witt ring of A is the set W(A) = A^{ℕ} equipped with a new ring structure characterized by the requirement that the ghost map

(4.1.1)  w: W(A) → A^{ℕ}

given by the Witt polynomials

w_{n} = ∑_{d|n} da_{d}^{n/d}, n ⩾ 1,

be a natural transformation of functors from rings to rings, when the range is given the componentwise ring structure. One has,

(4.1.2)  \begin{aligned}(a_{1}, a_{2}, …) + (b_{0}, b_{1}, …) & = (s_{1}(a_{1}, b_{1}), s_{2}(a_{1}, a_{2}, b_{1}, b_{2}), …) \\ (a_{1}, a_{2}, …) · (b_{0}, b_{1}, …) & = (p_{1}(a_{1}, b_{1}), p_{2}(a_{1}, a_{2}, b_{1}, b_{2}), …) \end{aligned}

where s_{n} and p_{n} are certain polynomials, which only depend on the a_{d} and b_{d} where d divides n. To see this, one proves that in the universal case A = ℤ[a_{1}, a_{2}, …; b_{1}, b_{2}, …] the ghost map identifies W(A) with a subring of A^{ℕ}. The polynomials s_{n} and p_{n} are then defined by the formula (4.1.2), with the sum and product on the left formed in this subring structure. Since the ring axioms hold in the universal case, they hold in general.

There are operators

(4.1.3)

F_{n}: W(A) → W(A), (Frobenius)
V_{n}: W(A) → W(A), (Verschiebung)

one for each n ⩾ 1, characterized by

F_{n}(w_{m}) = w_{mn}
V_{n}(a_{m}) = \{ \begin{matrix}a_{m/n} & if n divides m \\ 0 & else. \end{matrix}

By the same line of reasoning as in the case of the ring structure, one shows that F_{n} is well-defined and a ring homomorphism, that V_{n} is addivitive and that the following formulas hold

(4.1.4)  \begin{matrix}V_{n}(F_{n}(x)y) = xV_{n}(y) \\ F_{n}V_{n} = n, V_{n}F_{n} = mult_{V_{n}(1)} \\ F_{n}V_{m} = V_{m}F_{n}, if (m, n) = 1. \end{matrix}

As noted above, the polynomials s_{n} and p_{n} depend only on the a_{d} and b_{d}, where d divides n. We can therefore, in the above discussion, replace ℕ by any subset S ⊂ ℕ which is stable under division and get a ring W_{S}(A). We call W_{S}(A) the truncated ring of big Witt vectors and S the truncation set. For example, we have the ring of big Witt vectors of length m defined by

(4.1.5)  W_{m}(A) = W_{{1,2,... ,m}}(A).

Evidently, if S ⊂ S′ are two truncation sets, then the projection W_{S′}(A) → W_{S}(A) is a ring homomorphism, and moreover, the Frobenius and Verschiebung maps restrict to operators

(4.1.6)  \begin{aligned}F_{n} & : W_{S}(A) → W_{S/n}(A), \\ V_{n} & : W_{S/n}(A) → W_{S}(A), \end{aligned}

where S/n is the truncation set S/n = {m ∈ ℕ | nm ∈ S}. When S is the empty set, it is understood that W_{S}(A) is the trivial ring.

Now let p be a prime. The ring of p-typical Witt vectors is defined as the truncated Witt ring

(4.1.7)  W (A) = W_{{1,p,p^{2},... }}(A).

If A is a ℤ_{(p)}-algebra, one has a ring isomorphism

(4.1.8)  W(A) ≅ ∏_{(d,p)=1} W (A),

which on the dth factor is given by the composite ring map

I_{d}: W(A) ⟶^{F_{d}} W(A) ⟶^{pr} W (A);

see [Mu]. More generally, we have the ring of p-typical Witt vectors of length s,

(4.1.9)  W_{s}(A) = W_{{1,p,... ,p^{s−1}}}(A),

and, when A is a ℤ_{(p)}-algebra, a ring isomorphism

(4.1.10)  W_{m}(A) ≅ ∏_{(d,p)=1} W_{s}(A), p^{s−1}d ⩽ m < p^{s}d,

whose dth factor is the composite

I_{d}: W_{m}(A) ⟶^{F_{d}} W_{[m/d]}(A) ⟶^{pr} W_{s}(A).

Suppose that n = p^{i}k with (p, k) = 1 and let d = ek. Then one readily shows that the formulas (4.1.4) give a commutative square

(4.1.11)  \begin{matrix}W_{m}(A) & ⟶^{I_{e}} & W_{s}(A) \\ ↓^{V_{n}} & & ↓^{kV_{p^{i}}} \\ W_{mn}(A) & ⟶^{I_{d}} & W_{s+i}(A), \end{matrix}

which describes the Verschiebung map under the splitting in (4.1.10).

Finally, we recall that the underlying additive group of the ring W(A) is isomorphic to the multiplicative group of power series in A with constant term 1,

(4.1.12)  W(A) ≅ (1 + XA[[X]])^{×}, (a_{1}, a_{2}, …) ↦ ∏_{i⩾1}(1 − a_{i}X^{i}),

and under this isomorphism

F_{n}(f(X)) = ∏_{i=1}^{n} f(ξ_{i}), V_{n}(f(X)) = f(X^{n}),

where ξ_{1}, …, ξ_{n} are the formal nth roots of X. Similarly,

W_{m}(A) ≅ (1 + XA[[X]])^{×}/(1 + X^{m+1}A[[X]])^{×}.

4.2. Given a ring A and a two-sided ideal I, we let T (A, I) denote the relative topological Hochschild spectrum, that is, the homotopy fiber of the map T (A) → T (A/I). We recall the equivalence of S^{1}-spectra

T (A[x]/(x^{n})) ≃_{S^{1}} T (A) ∧ N^{cy}(Π_{n}),

where the smash product on the right is formed in the category of S^{1}-spectra. Thus for C ⊂ S^{1},

π_{∗}(T (A[x]/(x^{n}))^{C}) ≅ T (A)_{∗}^{C}(N^{cy}(Π_{n})),

the C-equivariant homology of the S^{1}-space N^{cy}(Π_{n}) with respect to the S^{1}-spectrum T (A). The S^{1}-space N^{cy}(Π_{n}) splits equivariantly as a wedge

N^{cy}(Π_{n}) = ⋁_{s⩾0} N^{cy}(Π_{n}; s)

and the summand N^{cy}(Π_{n}; 0) = S^{0} corresponds to the copy of T (A) in T (A[x]/(x^{n})). The equivariant homotopy type of the remaining summands is given by theorem B which we proved in §3. Let again V_{d} denote the representation V_{d} = ℂ(ξ) ⊕ ⋯ ⊕ ℂ(ξ^{d}) and let [x] denote the largest integer less than or equal to x. Then we have a cofibration sequence of S^{1}-spectra

(4.2.1)  ⋁_{s⩾1,n|s} T (A)_{V_{[ \frac{s−1}{n}]}} ∧ S^{1}/C_{s/n+} ⟶^{pr} ⋁_{s⩾1} T (A)_{V_{[ \frac{s−1}{n}]}} ∧ S^{1}/C_{s+} ⟶ T (A[x]/(x^{n}), (x)).

Indeed, for any finite dimensional S^{1}-representation V, T (A) ∧ S^{V} ≃_{S^{1}} T (A)_{V}, the V th deloop of the S^{1}-spectrum T (A), cf. [LMS], theorem I.7.9, and (4.2.1) follows from theorem B.

Recall from [HM], §1, that T (A) is a cyclotomic spectrum so that we have restriction maps

R_{r}: T (A)_{V}^{C_{rs}} → T (A)_{ρ_{Cr}^{∗} V^{Cr}}^{C_{s}} ,

where ρ_{C_{r}}^{∗}V^{C_{r}} indicates V^{C_{r}} considered as an S^{1}-representation through the rth root isomorphism ρ_{C_{r}}: S^{1} → S^{1}/C_{r}. In our case

ρ_{C_{r}}^{∗}V_{[ \frac{rs−1}{n}]}^{C_{r}} ≅ V_{[ \frac{s−1}{n}]},

such that the restriction maps are

(4.2.2)  R_{r}: T (A)_{V_{[ \frac{rs−1}{n}]}}^{C_{rs}} → T (A)_{V_{[ \frac{s−1}{n}]}}^{C_{s}}.

We can then repeat the argument of [HM], §8, to get

Proposition 4.2.3. After profinite completion there is a cofibration sequence of spectra

Σ holim_{\overleftarrow{R}} T (A)_{V_{[ \frac{s−1}{n}]}}^{C_{s/n}} ⟶^{V_{n}} Σ holim_{\overleftarrow{R}} T (A)_{V_{[ \frac{s−1}{n}]}}^{C_{s}} ⟶ TC(A[x]/(x^{n}), (x)).

The homotopy limits runs over the natural numbers ordered by division and T (A)_{V_{[ \frac{s−1}{n}]}}^{C_{s/n}} is understood to be the trivial spectrum when n does not divide s. □

By cofinality, the restriction of the homotopy limit on the left in (4.2.3) to the natural numbers divisible by n gives an equivalence

(4.2.4)  holim_{\overleftarrow{R}} T (A)_{V_{[ \frac{s−1}{n}]}}^{C_{s/n}} ⟶^{∼} holim_{\overleftarrow{R}} T (A)_{V_{r−1}}^{C_{r}}.

We now fix a prime p and consider, for every d with (d, p) = 1, the composition

holim_{\overleftarrow{R}} T (A)_{V_{[ \frac{s−1}{n}]}}^{C_{s}} ⟶^{pr_{d}} holim_{\overleftarrow{R}} T (A)_{V_{[ \frac{p^{r} d−1}{n}]}}^{C_{p^{r} d}} ⟶^{F_{d}} holim_{\overleftarrow{R}} T (A)_{V_{[ \frac{p^{r} −1}{n}]}}^{C_{p^{r}}}.

The first map is the restriction of the homotopy limit to the natural numbers of the form p^{r}d, r ⩾ 0, and the second map is the map of homotopy limits induced from the inclusion map

F_{d}: T (A)_{V_{[ \frac{p^{r} d−1}{n}]}}^{C_{p^{r} d}} → T (A)_{V_{[ \frac{p^{r} d−1}{n}]}}^{C_{p^{r}}}.

We then have the following analog of (4.1.8).

Proposition 4.2.5. Suppose that A is a commutative ℤ_{(p)}-algebra. Then the map

holim_{\overleftarrow{R}} T (A)_{V_{[ \frac{s−1}{n}]}}^{C_{s}} ⟶^{∼} ∏_{(d,p)=1} holim_{\overleftarrow{R}} T (A)_{V_{[ \frac{p^{r} d−1}{n}]}}^{C_{p^{r}}},

whose dth factor is F_{d} ∘ pr_{d}, is an equivalence.

Proof. Let F be the category associated with the poset of natural numbers ordered after division, and let F_{p} and F′ be the full subcategories of natural numbers which are powers of p and prime to p, respectively. Then the homotopy limit on the left in (4.2.5) is indexed by F and the homotopy limits on the right are indexed by F_{p}. As categories F = F_{p} × F′ and hence we have a homeomorphism

holim_{\overleftarrow{F}} T (A)_{V_{[ \frac{s−1}{n}]}}^{C_{s}} ≅ holim_{\overleftarrow{F′}} ( holim_{\overleftarrow{F_{p}}} T (A)_{V_{[ \frac{p^{r} d−1}{n}]}}^{C_{p^{r} d}}).

We claim that the map

T (A)_{V_{[ \frac{p^{r} d−1}{n}]}}^{C_{p^{r} d}} → ∏_{e|d} T (A)_{V_{[ \frac{p^{r} e−1}{n}]}}^{C_{p^{r}}},

which on the eth factor is given by R_{d/e}F_{e}, is an equivalence. Given this, the proposition follows. To prove the claim we need the following generalization of [HM], theorem 2.2: let T be a cyclotomic spectrum, let q be a prime and let l be a natural number prime to q. Then there is a cofibration sequence of (non-equivariant) spectra

(4.2.6)  (ρ_{C_{l}}^{#}T_{V}^{C_{l}})_{hC_{q}^{i}} ⟶^{N} T_{V}^{C_{q^{i}l}} ⟶^{R} T_{ρ_{Cq}^{∗} V^{Cq}}^{C_{q^{i−1}l}} .

If V = 0 this is because T^{C_{l}} is a q-cyclotomic spectrum. In general, it follows from loc. cit., proposition 2.1 and the proof of proposition 2.3. We write d = q_{1}^{i_{1}} … q_{m}^{i_{m}}, where q_{1}, …, q_{m} are primes. Then (4.2.6) gives a cofibration sequence

(ρ_{C_{l}}^{#}T (A)_{V_{[ \frac{p^{r} d−1}{n}]}}^{C_{l}})_{hC_{q^{i}}} ⟶^{N} T (A)_{V_{[ \frac{p^{r} d−1}{n}]}}^{C_{p^{r} d}} ⟶^{R_{q}} T (A)_{V_{[ \frac{p^{r} d/q−1}{n}]}}^{C_{p^{r} d/q}},

where, for notational reasons, we have written q for q_{m}, i for i_{m} and l for p^{r}d/q^{i}. Moreover, we have a map of cofibration sequences

(ρ_{C_{l}}^{#}T (A)_{V_{[ \frac{p^{r} d−1}{n}]}}^{C_{l}})_{hC_{q^{i}}}  ⟶^{N} T (A)_{V_{[ \frac{p^{r} d−1}{n}]}}^{C_{p^{r} d}} ⟶^{R_{q}} T (A)_{V_{[ \frac{p^{r} d/q−1}{n}]}}^{C_{p^{r} d/q}}
↓^{tr}  ↓^{F_{q}}  ↓^{F_{q}}
(ρ_{C_{l}}^{#}T (A)_{V_{[ \frac{p^{r} d−1}{n}]}}^{C_{l}})_{hC_{q^{i−1}}} ⟶^{N} T (A)_{V_{[ \frac{p^{r} d−1}{n}]}}^{C_{p^{r} d/q}} ⟶^{R_{q}} T (A)_{V_{[ \frac{p^{r} d/q−1}{n}]}}^{C_{p^{r} d/q^{2}}},

where the left hand vertical map is the transfer of the projection going in the opposite direction. We shall prove that tr is an equivalence such that the square on the right is homotopy cartesian. The claim then readily follows by induction on m and i_{k}, k = 1, …, m.

Let us write V for V_{[(p^{r}d/q−1)/n]}. We note that multiplication by q on π_{∗}T (A)_{V}^{C_{l}} is an isomorphism. For T (A)_{V}^{C_{l}} is a module spectrum over the ring spectrum T (A)^{C_{l}}. In particular, the homotopy groups π_{∗}T (A)_{V}^{C_{l}} are modules over the ring π_{0}T (A)^{C_{l}}. By [HM], addendum 3.3, this ring is isomorphic to the truncated big Witt ring W_{⟨l⟩}(A), ⟨l⟩ = {d | d divides l}, and q is a unit in W_{⟨l⟩}(A). Now, for any S^{1}-spectrum T, one has a first quadrant homology type spectral sequence

E_{s,t}^{2} = H_{s}(C_{q^{i}}; π_{t}T) ⇒ π_{s+t}(T_{hC_{q^{i}}}),

where the homotopy groups π_{∗}T are trivial C_{q^{i}}-modules because the C_{q^{i}}-action extends to an S^{1}-action. The edge homomorphism of the spectral sequence is the map on homotopy groups induced from the projection T → T_{hC_{q^{i}}}. When multiplication by q is an isomorphism on π_{∗}T, the spectral sequence collapses onto the axis s = 0, and the edge homomorphism is an isomorphism. This shows that the projection T_{hC_{q^{i−1}}} → T_{hC_{q^{i}}} is an equivalence. But the composition

T_{hC_{q^{i}}} ⟶^{tr} T_{hC_{q^{i−1}}} ⟶^{pr} T_{hC_{q^{i}}}

induces multiplication by q on homotopy groups and is therefore also an equivalence. Hence tr is an equivalence. This finishes the proof. □

We note that the individual terms in the homotopy limits on the right hand side of (4.2.5) approximate the limit. More precisely, the fundamental cofibration sequence, [HM], theorem 2.2, and the fact that taking homotopy orbits preserves connectivity, shows that the projection

(4.2.7)  holim_{\overleftarrow{R}} T (A)_{V_{[ \frac{p^{r} d−1}{n}]}}^{C_{p^{r}}} → T (A)_{V_{[ \frac{p^{s}d−1}{n}]}}^{C_{p^{s}}}

is 2[(p^{s+1}d − 1)/n]-connected. We also note that the proposition gives a splitting of the left hand spectrum in (4.2.3). Indeed, if we write n = p^{i}k with (k, p) = 1, then (4.2.4) and (4.2.5) show that the map

(4.2.8)  holim_{\overleftarrow{R}} T (A)_{V_{[ \frac{s−1}{n}]}}^{C_{s/n}} ⟶^{∼} ∏_{(e,p)=1} holim_{\overleftarrow{R}} T (A)_{V_{[ \frac{p^{r} ek−1}{n}]}}^{C_{p^{r−i}}},

which on the eth factor is given by F_{e} ∘ pr_{ek}, is an equivalence.

Addendum 4.2.9. Suppose that n = p^{i}k with (p, k) = 1 and let d = ek. Then there is a commutative diagram

π_{∗}holim_{\overleftarrow{R}} T (A)_{V_{[ \frac{s−1}{n}]}}^{C_{s/n}} ⟶^{F_{e}∘pr_{d}} π_{∗}holim_{\overleftarrow{R}} T (A)_{V_{[ \frac{p^{r} d−1}{n}]}}^{C_{p^{r−i}}}
↓^{V_{n}}  ↓^{kV_{p^{i}}}
π_{∗}holim_{\overleftarrow{R}} T (A)_{V_{[ \frac{s−1}{n}]}}^{C_{s}} ⟶^{F_{d}∘pr_{d}} π_{∗}holim_{\overleftarrow{R}} T (A)_{V_{[ \frac{p^{r} d−1}{n}]}}^{C_{p^{r}}}.

Proof. By naturality, we have a commutative square

holim_{\overleftarrow{R}} T (A)_{V_{[ \frac{s−1}{n}]}}^{C_{s/n}} ⟶^{pr_{d}} holim_{\overleftarrow{R}} T (A)_{V_{[ \frac{p^{r} d−1}{n}]}}^{C_{p^{r} d/n}}
↓^{V_{n}}  ↓^{V_{n}}
holim_{\overleftarrow{R}} T (A)_{V_{[ \frac{s−1}{n}]}}^{C_{s}} ⟶^{pr_{d}} holim_{\overleftarrow{R}} T (A)_{V_{[ \frac{p^{r} d−1}{n}]}}^{C_{p^{r} d}}.

We recall from [HM], lemma 2.3.1, that on the level of homotopy groups F_{s}V_{r} = V_{r}F_{s}, provided that (r, s) = 1. Since we have

π_{∗} holim_{\overleftarrow{R}} T (A)_{V_{[ \frac{p^{r} d−1}{n}]}}^{C_{p^{r}}} ≅ \underleftarrow{lim}_{R} π_{∗}T (A)_{V_{[ \frac{p^{r} d−1}{n}]}}^{C_{p^{r}}}

from (4.2.7), we get a commutative square

π_{∗} holim_{\overleftarrow{R}} T (A)_{V_{[ \frac{p^{r} d−1}{n}]}}^{C_{p^{r} d/n}} ⟶^{F_{d/k}} π_{∗} holim_{\overleftarrow{R}} T (A)_{V_{[ \frac{p^{r} d−1}{n}]}}^{C_{p^{r−i}}}
↓^{V_{p^{i}}}  ↓^{V_{p^{i}}}
π_{∗} holim_{\overleftarrow{R}} T (A)_{V_{[ \frac{p^{r} d−1}{n}]}}^{C_{p^{r} d/k}} ⟶^{F_{d/k}} π_{∗} holim_{\overleftarrow{R}} T (A)_{V_{[ \frac{p^{r} d−1}{n}]}}^{C_{p^{r}}}.

Finally, we can write F_{d} ∘ V_{n} as the composite

holim_{\overleftarrow{R}} T (A)_{V_{[ \frac{p^{r} d−1}{n}]}}^{C_{p^{r} d/n}} ⟶^{V_{p^{i}}} holim_{\overleftarrow{R}} T (A)_{V_{[ \frac{p^{r} d−1}{n}]}}^{C_{p^{r} d/k}} ⟶^{V_{k}} holim_{\overleftarrow{R}} T (A)_{V_{[ \frac{p^{r} d−1}{n}]}}^{C_{p^{r} d}}

⟶^{F_{k}} holim_{\overleftarrow{R}} T (A)_{V_{[ \frac{p^{r} d−1}{n}]}}^{C_{p^{r} d/k}} ⟶^{F_{d/k}} holim_{\overleftarrow{R}} T (A)_{V_{[ \frac{p^{r} d−1}{n}]}}^{C_{p^{r}}}

and F_{k} ∘ V_{k} induces multiplication by k on homotopy groups. □

We now assume that the coefficient ring is a perfect field k of positive characteristic p and proceed to prove theorem A of the introduction.

Theorem 4.2.10. Let k be a perfect field of characteristic p > 0. Then the long exact homotopy sequence associated with the cofibration sequence of (4.2.3) takes the form

0 → W_{m}(k) ⟶^{V_{n}} W_{mn}(k) ⟶ TC_{2m−1}(k[x]/(x^{n}), (x)) → 0,

the groups in even degrees being zero.

Proof. We have from [HM], proposition 9.1, and (4.2.7) that

(4.2.11)  π_{2j} holim_{\overleftarrow{R}} T (k)_{V_{[ \frac{p^{r} d−1}{n}]}}^{C_{p^{r}}} ≅ W_{s}(k),

with s determined by the inequalities

[(p^{s−1}d − 1)/n] ⩽ j < [(p^{s}d − 1)/n].

One readily checks that these inequalities are equivalent to the inequalities

p^{s−1}d ⩽ n(j + 1) < p^{s}d,

which appear in the decomposition in (4.1.10). Therefore, by (4.2.5)

π_{2j} holim_{\overleftarrow{R}} T (k)_{V_{[ \frac{s−1}{n}]}}^{C_{s}} ≅ W_{n(j+1)}(k)

and the groups in odd degrees are zero. Similarly, we have

(4.2.12)  π_{2j} holim_{\overleftarrow{R}} T (k)_{V_{[ \frac{p^{r} d−1}{n}]}}^{C_{p^{r−i}}} ≅ W_{s−i}(k),

with s determined by the same inequalities as above. Suppose that d = ek and let r = s − i. Then one immediately shows that r is given by the inequalities

p^{r−1}e ⩽ j + 1 < p^{r}e

and hence

π_{2j} holim_{\overleftarrow{R}} T (k)_{V_{[ \frac{s−1}{n}]}}^{C_{s/n}} ≅ W_{j+1}(k).

Finally, we recall from [HM], proposition 9.1, that under the isomorphism of (4.2.11) and (4.2.12) the map

V_{p^{i}}: holim_{\overleftarrow{R}} T (k)_{V_{[ \frac{p^{r} d−1}{n}]}}^{C_{p^{r−i}}} → holim_{\overleftarrow{R}} T (k)_{V_{[ \frac{p^{r} d−1}{n}]}}^{C_{p^{r−i}}}

induces the Verschiebung V_{p^{i}}: W_{s−i}(k) → W_{s}(k) of Witt vectors. Now the theorem follows from (4.1.11) and (4.2.9). □

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Massachusetts Institute of Technology  Aarhus University
Cambridge, Massachusetts  Aarhus, Denmark