On the K-theory of complete regular local 𝔽_{p}-algebras

Thomas Geisser^{a,1} Lars Hesselholt^{b,2}

aUniversity of Southern California, Los Angeles, California

bMassachusetts Institute of Technology, Cambridge, Massachusetts

Abstract

Let A be a noetherian 𝔽_{p}-algebra that is finitely generated as a module over the subring A^{p} of pth powers. We give an explicit formula for the de Rham-Witt complex of the power series ring A[[t]] in terms of that of the ring A. We use this formula to show that, for every complete regular local 𝔽_{p}-algebra whose residue field is a finite extension of the subfield of pth powers, the canonical map from the algebraic K-theory with ℤ/p^{v}-coefficients to the topological K-theory with ℤ/p^{v}-coefficients is an isomorphism.

Key words: De Rham-Witt complex, K-theory, complete regular local ring.

Introduction

Let (R, 𝔪, k) be a complete regular local 𝔽_{p}-algebra and assume that the residue field k is a finite extension of the subfield k^{p} of pth powers. We show that the canonical map

K_{q}(R, ℤ/p^{v}) → K_{q}^{top}(R, ℤ/p^{v})

from the algebraic K-theory with ℤ/p^{v}-coefficients to the topological K-theory with ℤ/p^{v}-coefficients is an isomorphism. The proof gives an inductive procedure for evaluating the common group in terms of the de Rham-Witt groups of k. This extends previous results of Dundas [2] and the authors [4, Prop. 5.3.1],

1 Supported partly by the National Science Foundation, the Alfred P. Sloan Foundation, and the Japan Society for the Promotion of Science. 2 Supported in part by the National Science Foundation and the Alfred P. Sloan Foundation.

Preprint submitted to Elsevier Science 9 August 2005

where the case of a one-dimensional complete regular local 𝔽_{p}-algebra with perfect residue field was considered. The topological K-groups, which take the 𝔪-adic topology on R into account, are defined to be the homotopy groups with ℤ/p^{v}-coefficients of the homotopy limit spectrum

K^{top}(R) = holim_{n} K(R/𝔪^{n}).

Hence there is natural exact sequence

0 → R^{1} lim_{s} K_{q+1}(R/𝔪^{s}, ℤ/p^{v}) → K_{q}^{top}(R, ℤ/p^{v}) → lim_{s} K_{q}(R/𝔪^{s}, ℤ/p^{v}) → 0

compare Wagoner [19]. By the structure theorem for complete regular local 𝔽_{p}-algebras, the ring R is non-canonically isomorphic to the power series ring k[[t_{1}, …, t_{d}]], where d is the Krull dimension of R. We shall allow for a slightly more general ring of coefficients:

Theorem A Let A be a regular local 𝔽_{p}-algebra and assume that A is finitely generated as a module over the subring A^{p}. Let R = A[[t_{1}, …, t_{d}]] and let I ⊂ R be the ideal generated by t_{1}, …, t_{d}. Then the canonical map

K_{q}(R, ℤ/p^{v}) → K_{q}^{top}(R, ℤ/p^{v})

is an isomorphism, for all integers q and v ⩾ 1.

We note that the Gabber-Suslin rigidity theorem [17,3] implies that for m prime to p and for all s ⩾ 1, the natural projection induces an isomorphism

K_{q}(R, ℤ/m) ⟶^{∼} K_{q}(R/I^{s}, ℤ/m).

Hence, in this case, the limit system is constant. On the other hand, continuity fails rationally, and hence integrally.

We briefly discuss the method of proof. The K-theory spectra of the rings R and R/I^{s} naturally decompose as wedge sums

\begin{aligned}K(A) ∨ K(R, I) & ⟶^{∼} K(R) \\ K(A) ∨ K(R/I^{s}, I) & ⟶^{∼} K(R/I^{s}) \end{aligned}

and the natural projection induces the identity map of the first summand on the left. Thus we may as well show that the induced map of relative theories

K(R, I) → holim_{s} K(R/I^{s}, I)

induces an isomorphism of homotopy groups with ℤ/p^{v}-coefficients. To this end, we compare the K-theory spectra to the corresponding topological cyclic homology spectra via the cyclotomic trace in the following diagram.

\xymatrix{K(R, I) \ar[r]^{tr} \ar[d] & TC(R, I; p) \ar[d] \\ holim_{s} K(R/I^{s}, I) \ar[r]|{tr} & holim_{s} TC(R/I^{s}, I; p).}

By a theorem of McCarthy [15], the lower horizontal map induces an isomorphism of homotopy groups with ℤ/p^{v}-coefficients. A theorem of Popescu states that every regular 𝔽_{p}-algebra is a filtered colimit of smooth 𝔽_{p}-algebras [16]. (See also [18].) This implies that the results of [5] and [6], which were proved originally for smooth 𝔽_{p}-algebras, are valid, more generally, for regular 𝔽_{p}- algebras. Hence, there is a natural long-exact sequence

⋯ → TC_{q}(R, I; p) → W Ω_{(R,I)}^{q} ⟶^{1−F} W Ω_{(R,I)}^{q} → TC_{q−1}(R, I; p) → ⋯

where W Ω_{(R,I)}^{q} is the de Rham-Witt complex of [12]. Moreover, the composite

K_{q}(R, I; ℤ_{p}) → TC_{q}(R, I; p) → W Ω_{R}^{q}

maps the relative K-group with ℤ_{p}-coefficients isomorphically onto the kernel of 1 − F. We prove the following result which is also interesting in its own right. The proof uses the corresponding result, proved in [11], for polynomial algebras, and the extension to power series rings was inspired by Kato’s paper [13].

Theorem B Let A be a noetherian 𝔽_{p}-algebra and suppose that A is finitely generated as an A^{p}-module. Then every element ω^{(n)} ∈ W_{n}Ω_{A[[t]]}^{q} can be written uniquely as an infinite series

\begin{aligned}ω^{(n)} & = ∑_{i∈ℕ_{0}} a_{0,i}^{(n)}[t]_{n}^{i} + ∑_{i∈ℕ}b_{0,i}^{(n)}[t]_{n}^{i−1}d[t]_{n} \\ & + ∑_{s∈ℕ} ∑_{j∈I_{p}} (V^{s}(a_{s,j}^{(n−s)}[t]_{n−s}^{j}) + dV^{s}(b_{s,j}^{(n−s)}[t]_{n−s}^{j})) \end{aligned}

with the components a_{s,i}^{(m)} ∈ W_{m}Ω_{A}^{q} and b_{s,i}^{(m)} ∈ W_{m}Ω_{A}^{q−1}. Here I_{p} is the set of positive integers that are not divisible by p and [t]_{n} is the multiplicative representative of t in W_{n}(A[[t]]).

Using this result we prove, by induction on the number of variables, that the map 1 − F in the sequence above is surjective. It follows that the cyclotomic trace induces an isomorphism

K_{q}(R, I; ℤ_{p}) ⟶^{∼} TC_{q}(R, I; p).

Finally, we show that the canonical map

TC(R, I; p) → holim_{s} TC(R/I^{s}, I; p)

is a weak equivalence. This completes the outline of the proof of Thm. A.

We mention that Thm. B gives a canonical isomorphism

W Ω_{A,log}^{q} ⊕ WΩ_{A}^{q−1} ⟶^{∼} W Ω_{A[[t]],log}^{q}

where the second summand on the left is the big de Rham-Witt group considered in [9]. Previously, the graded pieces of a complete and separated filtration of the left-hand side were known [13].

In this paper, A will always denote an 𝔽_{p}-algebra, R the power series algebra A[[t_{1}, …, t_{d}]], and I ⊂ R the ideal generated by t_{1}, …, t_{d}. A pro-object of a category C is a functor to C from the set of positive integers viewed as a category with one morphism from m to n, if m ⩾ n. A strict map from a pro-object X to a pro-object Y is a natural transformation and a general map from X to Y is an element of the set

Hom_{pro −C}(X, Y) = lim_{n} colim_{m} Hom_{C}(X_{m}, Y_{n}).

1 Continuity and topological cyclic homology

Let A be a regular 𝔽_{p}-algebra, finitely generated as a module over the subring A^{p}, and let R and I ⊂ R be as in the introduction. In this paragraph, we show that the canonical map

TC(R; p) → holim_{s} TC(R/I^{s}; p)

is a weak equivalence. We briefly recall the definition of TC(R; p), refering the reader to [10, Sect. 1] for a fuller discussion.

For any ring S, the spectrum TC(S; p) is defined as the homotopy fixed points of an operator called Frobenius on another spectrum TR(S; p). Hence, there is a natural cofibration sequence

TC(S; p) → TR(S; p) ⟶^{1−F} TR(S; p) ⟶^{∂} Σ TC(S; p).

The spectrum TR(S; p), in turn, is the homotopy limit of a pro-spectrum TR^{·}(S; p) and there are strict maps of pro-spectra

F: TR^{n}(S; p) → TR^{n−1}(S; p)
V : TR^{n−1}(S; p) → TR^{n}(S; p).

The spectrum TR^{1}(S; p) is the topological Hochschild spectrum TH(S). It has an action by the circle group 𝕋 and the higher levels in the pro-system by definition are the fixed sets of the cyclic subgroups of 𝕋 of p-power order

TR^{n}(S; p) = TH(S)^{C_{p^{n−1}}}.

The map F is the obvious inclusion and V is the accompanying transfer. The structure map R in the pro-system is harder to define and uses the so-called cyclotomic structure of TH(S). There is a natural cofibration sequence

ℍ_{·}(C_{p^{n−1}}, TH(S)) ⟶^{N} TR^{n}(S; p) ⟶^{R} TR^{n−1}(S; p) ⟶^{∂} Σℍ_{·}(C_{p^{n−1}}, TH(S))

which gives the “layers” in the tower TR^{·}(S; p). Here the left-hand term is the group homology spectrum (or borel spectrum) of the group C_{p^{n−1}} acting on TH(S). Its homotopy groups are given by a strongly convergent first quadrant homology type spectral sequence

E_{s,t}^{2} = H_{s}(C_{p^{v}}, TH_{t}(S)) ⇒ ℍ_{s+t}(C_{p^{v}}, TH(S)).

We first show that, quite generally, continuity for topological Hochschild homology implies continuity for topological cyclic homology.

Lemma 1.1 Let G be a finite group and let X_{n}, n ⩾ 1, be a limit system of G-spectra. Suppose that there exists m ∈ ℤ such that π_{t}X_{n} vanishes for all n ⩾ 1 if t < m. Then the canonical map

ℍ_{·}(G, holim_{n} X_{n}) → holim_{n} ℍ_{·}(G, X_{n})

is a weak equivalence.

Proof. Let EG be a free contractible G-CW complex. Then by definition

ℍ_{·}(G, X) = (X ∧ EG_{+})^{G}.

The filtration of EG by the skeleta defines an exact couple

\begin{aligned}D_{s,t} & = π_{s+t}((X ∧ E_{s}G_{+})^{G}) \\ E_{s,t} & = π_{s+t}((X ∧ (E_{s}G/E_{s−1}G))^{G}) \end{aligned}

and this in turn gives rise to the spectral sequence

E_{s,t}^{2} = H_{s}(G, π_{t}X) ⇒ ℍ_{s+t}(G, X).

In particular, if π_{t}X vanishes for t < m, then the map induced from the canonical inclusion

π_{n}((X ∧ E_{s}G_{+})^{G}) → π_{n}((X ∧ EG_{+})^{G})

is an isomorphism for n ⩽ s + m. Now consider the following diagram

\xymatrix{((holim_{n} X_{n}) ∧ E_{s}G_{+})^{G} \ar[r] \ar[d] & ((holim_{n} X_{n}) ∧ EG_{+})^{G} \ar[d] \\ holim_{n} ((X_{n} ∧ E_{s}G_{+})^{G}) \ar[r] & holim_{n} ((X_{n} ∧ EG_{+})^{G}).}

Since G is finite, we can choose EG such that E_{s}G is a finite CW complex, for all s ⩾ 0. This implies that the left-hand vertical map is a weak equivalence. Indeed, E_{s}G_{+} has a dual. Moreover, the horizontal maps induce isomorphism of homotopy groups in degrees less than m. (The assumption that π_{t}X_{n} vanishes for for all n ⩾ 1 if t < m implies that π_{t} holim_{n} X_{n} vanishes for t < m − 1.) Hence the right-hand vertical map will induce an isomorphism of homotopy groups in degrees less than s + m. But this is true for all s ⩾ 0, so the right-hand vertical map is a weak equivalence. □

Proposition 1.2 Let S be a ring and let J ⊂ S be an ideal. If the canonical map TH(S) → holim_{s} TH(S/J^{s}) is a weak equivalence, then so is

TC(S; p) → holim_{s} TC(S/J^{s}; p).

Proof. We show inductively that the canonical map

TR^{n}(S; p) → holim_{s} TR^{n}(S/J^{s}; p)

is a weak equivalence. The case n = 1 is the assumption. In the induction step, we must show that the canonical map

ℍ_{·}(C_{p^{v}}, TH(S)) → holim_{s} ℍ_{·}(C_{p^{v}}, TH(S/J^{s}))

is a weak equivalence. This map is equal to the composite map

ℍ_{·}(C_{p^{v}}, TH(S)) → ℍ_{·}(C_{p^{v}}, holim_{s} TH(S/J^{s})) → holim_{s} ℍ_{·}(C_{p^{v}}, TH(S/J^{s})).

The right-hand map is an equivalence by Lemma 1.1, and the left-hand map is an equivalence since forming the group homology spectrum preserves weak equivalences. Finally, the result for topological cyclic homology follows since homotopy limits commute. □

We will show in Prop. 1.7 below that if A is a regular 𝔽_{p}-algebra which is finitely generated as an A^{p}-module, then the canonical map

TH(R) → holim_{s} TH(R/I^{s})

is a weak equivalence. The proof is based on a series of lemmas.

Lemma 1.3 Let S be a ring and J ⊂ S an ideal. Then the canonical map

S/J^{·} ⊗_{S} Ω_{S}^{q} → Ω_{S/J^{·}}^{q}

is an isomorphism of pro-abelian groups.

Proof. Let Ω_{(S,J)}^{∗} ⊂ Ω_{S}^{∗} be the differential graded ideal generated by J. Then the projection induces an isomorphism Ω_{S}^{∗}/Ω_{(S,J^{·})}^{∗} ⟶^{∼} Ω_{S/J}^{∗}, and by the Leibniz rule, Ω_{(S,J^{2s})}^{∗} ⊂ J^{s} ⊗_{S} Ω_{(S,J^{s})}^{∗}. It follows that in the diagram

\xymatrix{S/J^{2s} ⊗_{S} Ω_{(S,J^{2s})}^{q} \ar[r] \ar[d]^{0} & S/J^{2s} ⊗_{S} Ω_{S}^{q} \ar[r] \ar[d] & Ω_{S/J^{2s}}^{q} \ar[r] \ar[d] & 0 \\ S/J^{s} ⊗_{S} Ω_{(S,J^{s})}^{q} \ar[r] & S/J^{s} ⊗_{S} Ω_{S}^{q} \ar[r] & Ω_{S/J^{s}}^{q} \ar[r] & 0}

the left-hand vertical map is zero, and this, in turn, shows that the the map of the statement is an isomorphism of pro-abelian groups. □

Lemma 1.4 Let A be a ring, let S = A[t_{1}, …, t_{d}], and let J ⊂ S be the ideal generated by t_{1}, …, t_{d}. Then the canonical map

S/J^{·} ⊗_{S} TH_{q}(S) → TH_{q}(S/J^{·})

is an isomorphism of pro-abelian groups.

Proof. Let J_{s} ⊂ S be the ideal generated by t_{1}^{s}, …, t_{d}^{s}. We note that

J_{s} ⊂ J^{s} ⊂ J_{[s/d]}

where [m] is the largest integer less than or equal to m. This shows that in the diagram

\xymatrix{S/J_{·} ⊗_{S} TH_{q}(S) \ar[r] \ar[d] & TH_{q}(S/J_{·}) \ar[d] \\ S/J^{·} ⊗_{S} TH_{q}(S) \ar[r] & TH_{q}(S/J^{·})}

the vertical maps are isomorphisms of pro-abelian groups. Hence, we may instead show that the lower horizontal map is an isomorphism of pro-abelian groups. The rings S and S/J_{s} both are pointed monoid algebras in the sense of [8, Sect. 7] and their topological Hochschild homology groups therefore are given by loc. cit. Thm. 7.1. We use this to show that in the diagram

\xymatrix{S/J_{·} ⊗_{S} TH_{∗}(S) \ar[r] & TH_{∗}(S/J_{·}) \\ S/J_{·} ⊗_{S} Ω_{S}^{∗} ⊗_{Ω_{A}^{∗}} TH_{∗}(A) \ar[r] \ar[u] & Ω_{S/J_{·}}^{∗} ⊗_{Ω_{A}^{∗}} TH_{∗}(A) \ar[u]}

the left-hand vertical map is an isomorphism, and the right-hand vertical map an isomorphism of pro-abelian groups. Since Lemma 1.3 implies that also the lower horizontal map is an isomorphism of pro-abelian groups, the lemma will follow.

For the ring S in question, we get a weak equivalence

TH(A) ∧ N^{cy}(Π_{∞}^{∧d}) → TH(S)

where the second factor on the left is the cyclic bar-construction of the d-fold smash product of the pointed monoid Π_{∞} = {0, 1, t, t^{2}, … }. Moreover, the cyclic bar-construction decomposes as wedge sum

⋁_{i_{1},...,i_{d}∈ℕ_{0}}(N^{cy}(Π_{∞}, i_{1}) ∧ ⋯ ∧ N^{cy}(Π_{∞}; i_{d})) ⟶^{∼} N^{cy}(Π_{∞}^{∧d}).

It is proved in [6, Lemma 3.1.6] that N^{cy}(Π_{∞}, 0) = S^{0} and that for i > 0, there is a 𝕋-equivariant deformation retract

𝕋/C_{i+} ⟶^{∼} N^{cy}(Π_{∞}; i)

where 𝕋 and C_{i} denote the circle group and the cyclic subgroup of order i. On homotopy groups in degree q, this shows that for i > 0,

π_{q}(TH(A) ∧ N^{cy}(Π_{∞}, i)) = TH_{q}(A) · t^{i} ⊕ TH_{q−1}(A) · t^{i−1}dt,

and hence the canonical map

Ω_{S}^{∗} ⊗_{Ω_{A}^{∗}} TH_{∗}(A) → TH_{∗}(S)

is an isomorphism.

The description of the topological Hochschild homology of the ring S/J_{s} is completely parallel. There is a natural equivalence

⋁_{i_{1},...,i_{d}∈ℕ_{0}} TH(A) ∧ (N^{cy}(Π_{s}, i_{1}) ∧ ⋯ ∧ N^{cy}(Π_{s}; i_{d})) ⟶^{∼} TH(S/J_{s})

where Π_{s} is the pointed monoid {0, 1, t, …, t^{s−1}} with base-point 0 and with t^{s} = 0. The 𝕋-equivariant homotopy type of the spaces N^{cy}(Π_{s}, i) was determined in [7, Thm. B]. In particular, the natural projection

N^{cy}(Π_{∞}, i) → N^{cy}(Π_{s}, i)

is a homeomorphism, for i < s, and for i = s, there is a cofibration sequence

N^{cy}(Π_{∞}, 1) → N^{cy}(Π_{∞}, s) → N^{cy}(Π_{s}, s) → ΣN^{cy}(Π_{∞}, 1).

On homotopy groups in degree q, the latter gives rise to a short exact sequence

0 → (TH_{q−1}(A)/s) · t^{s−1}dt → π_{q}(TH(A) ∧ N^{cy}(Π_{s}, s)) → TH_{q−2}(A)[s] · t → 0.

It follows that the map

Ω_{S/J_{s}}^{∗} ⊗_{Ω_{A}^{∗}} TH_{∗}(A) → TH_{∗}(S/J_{s})

is injective and that the cokernel C_{s}^{∗} only involves the summands (i_{1}, …, i_{d}) where i_{j} ⩾ s for some j = 1, …, d. Finally, since N^{cy}(Π_{s}, i) is [(i − 1)/s]- connected, the canonical inclusion

⋁_{0⩽i_{1},...,i_{d}<qs+1}TH(A) ∧ (N^{cy}(Π_{s}, i_{1}) ∧ ⋯ ∧ N^{cy}(Π_{s}; i_{d})) → TH(S/J_{s})

induces an isomorphism on homotopy groups in degrees ⩽ q. It follows that in degree q, the map of cokernels C_{qs+1}^{q} → C_{s}^{q} is zero. □

Remark 1.5 It would be interesting to determine if, generally, given a ring S and an ideal J ⊂ S, the canonical map

S/J^{·} ⊗_{S} TH_{q}(S)) → TH_{q}(S/J^{·})

is an isomorphism of pro-abelian groups. As far as we know the analogous question for ordinary Hochschild homology also has not been settled.

Lemma 1.6 Let A be a regular 𝔽_{p}-algebra. Then the canonical map

Ω_{R/I^{·}}^{∗} ⊗ TH_{∗}(𝔽_{p}) → TH_{∗}(R/I^{·})

is an isomorphism of pro-abelian groups.

Proof. Let S be the polynomial algebra A[t_{1}, …, t_{d}] and let J be the ideal generated by t_{1}, …, t_{d}. Then, since S/J^{s} ⟶^{∼} R/I^{s}, the map of the statement may be identified with the lower horizontal map in the diagram

\xymatrix{S/J^{·} ⊗_{S} Ω_{S}^{∗} ⊗ TH_{∗}(𝔽_{p}) \ar[r] \ar[d] & S/J^{·} ⊗_{S} TH_{∗}(S) \ar[d] \\ Ω_{S/J^{·}}^{∗} ⊗ TH_{∗}(𝔽_{p}) \ar[r] & TH_{∗}(S/J^{·}).}

In this diagram, the left-hand vertical map is an isomorphism of pro-abelian groups by Lemma 1.3 and the right-hand vertical map is an isomorphism of pro-abelian groups by Lemma 1.4. Finally, the upper horizontal map is an isomorphism by [6, Thm. B]. □

Proposition 1.7 Suppose that A is regular and finitely generated as a module over the subring A^{p}. Then the canonical map

TH_{q}(R) → lim_{s} TH_{q}(R/I^{s})

is an isomorphism and the derived limit R^{1} lim_{s} TH_{q}(R/I^{s}) vanishes.

Proof. We consider the following diagram

\xymatrix{Ω_{R}^{∗} ⊗ TH_{∗}(𝔽_{p}) \ar[r] \ar[d] & TH_{∗}(R) \ar[d] \\ lim_{s} Ω_{R/I^{s}}^{∗} ⊗ TH_{∗}(𝔽_{p}) \ar[r] & lim_{s} TH_{∗}(R/I^{s})}

where the upper and lower horizontal maps are isomorphisms by [6, Thm. B] and Lemma 1.6. We recall that TH_{q}(𝔽_{p}) is isomorphic to 𝔽_{p}, if q is a nonnegative even integer, and is zero, otherwise. We show that the canonical map

Ω_{R}^{q} → lim_{s} Ω_{R/I^{s}}^{q}

is an isomorphism and that the derived limit R^{1} lim_{s} Ω_{R/I^{s}}^{q} vanishes. The canonical map is equal to the composite map

Ω_{R}^{q} → lim_{s}(R/I^{s} ⊗_{R} Ω_{R}^{q}) → lim_{s} Ω_{R/I^{s}}^{q}.

It follows from Lemma 1.3 that the right-hand map and the corresponding map of derived limits are isomorphisms. Hence, it suffices to show that the R-module Ω_{R}^{q} is I-adically complete. We recall from [14, Thm. 8.7] that since R is noetherian and I-adically complete, every finitely generated R-module, too, is I-adically complete. So it will suffice to prove that the R-module Ω_{R}^{q} is finitely generated. But if x_{r}^{s}, r = 1, …, m, 0 ⩽ s < p, generate A as an A^{p}-module, then dx_{1}, …, dx_{m}, dt_{1}, …, dt_{d} generates Ω_{R}^{1} as an R-module. □

2 De Rham-Witt complexes

In this section, we show that if the 𝔽_{p}-algebra A is finitely generated as a module over the subring A^{p}, then the map

1 − F: W Ω_{(R,I)}^{q} → W Ω_{(R,I)}^{q}

is surjective. We begin with some lemmas on Witt vectors and completion. We note that the definition of the ring W_{n}(J) of Witt vectors associated with a ring J does not require the ring to be unital.

Lemma 2.1 Let S be a ring and J ⊂ S an ideal. Then W_{n}(J) ⊂ W_{n}(S) is an ideal and the canonical projection induces an isomorphism

W_{n}(S)/W_{n}(J) ⟶^{∼} W_{n}(S/J).

Proof. It is clear that W_{n}(J) ⊂ W_{n}(S) is an ideal and that the natural projection factors as stated. Indeed, as a set W_{n}(A) is the n-fold product of copies of A. We show inductively that the sequence

0 → W_{n}(J) → W_{n}(S) → W_{n}(S/J) → 0

is exact. The case n = 1 is trivial. It is also clear that the left-hand map is injective, that the right-hand map is surjective, and that the composite of the two maps is zero. The proof of the induction step now follows from the natural exact sequence

0 → A ⟶^{V^{n−1}} W_{n}(A) ⟶^{R} W_{n−1}(A) → 0

by a diagram chase. □

Lemma 2.2 Let S be a ring and let x_{1}, …, x_{d} ∈ S. Then

([x_{1}]_{n}^{p^{n−1}ds}, …, [x_{d}]_{n}^{p^{n−1}ds}) ⊂ W_{n}((x_{1}, …, x_{d})^{p^{n−1}ds}) ⊂ ([x_{1}]_{n}^{s}, …, [x_{d}]_{n}^{s}),

where [x]_{n} ∈ W_{n}(S) is the Teichmüller representative.

Proof. The inclusion on the left is trivial. To prove the right-hand inclusion, we first note that since

(x_{1}, …, x_{d})^{p^{n−1}ds} ⊂ (x_{1}^{p^{n−1}s}, …, x_{d}^{p^{n−1}s}),

it will suffice to prove that for all y_{1}, …, y_{d} ∈ S,

W_{n}((y_{1}^{p^{n−1}}, …, y_{d}^{p^{n−1}})) ⊂ ([y_{1}]_{n}, …, [y_{d}]_{n}).

To this end, we show by descending induction on 0 ⩽ s ⩽ n that

V^{s}W_{n}((y_{1}^{p^{n−1}}, …, y_{d}^{p^{n−1}})) ⊂ ([y_{1}]_{n}, …, [y_{d}]_{n}).

The case s = n is trivial. We write J = (y_{1}^{p^{n−1}}, …, y_{d}^{p^{n−1}}) and assume the statement is true for s + 1. The elements V^{s}([x]_{n−s}) with x ∈ J form a set of coset representatives of V^{s+1}W_{n}(J) as a subgroup of V^{s}W_{n}(J). Now if

x = a_{1}y_{1}^{p^{n−1}} + … a_{d}y_{d}^{p^{n−1}}

then, modulo V W_{n−s}(J),

[x]_{n−s} ≡ [a_{1}]_{n−s}[y_{1}]_{n−s}^{p^{n−1}} + ⋯ + [a_{d}]_{n−s}[y_{d}]_{n−s}^{p^{n−1}}

which shows that, modulo V^{s+1}W_{n}(J),

V^{s}([x]_{n−s}) ≡ V^{s}([a_{1}]_{n−s})[y_{1}]_{n}^{p^{n−1−s}} + ⋯ + V^{s}([a_{d}]_{n−s})[y_{d}]_{n}^{p^{n−1−s}}.

This proves the induction step. □

Corollary 2.3 Let S be a ring and J ⊂ S a finitely generated ideal. Then for all s ⩾ 1, there exists r ⩾ s such that

W_{n}(J^{r}) ⊂ W_{n}(J)^{s} ⊂ W_{n}(J^{s}).

Proof. Indeed, if x_{1}, …, x_{d} generates J, then by Lemma 2.2,

W_{n}(J^{p^{n−1}ds}) ⊂ ([x_{1}]_{n}^{s}, …, [x_{d}]_{n}^{s}) ⊂ W_{n}(J)^{s},

which proves the left-hand inclusion of the statement with r = p^{n−1}ds. The right-hand inclusion follows from the more general fact that if I, J ⊂ S are two ideals, then W_{n}(I)W_{n}(J) ⊂ W_{n}(IJ). □

The de Rham-Witt complex W_{·}Ω_{S}^{∗} is defined, for any ring S, to be the initial example of an algebraic structure called a Witt complex over S [12,11]. By definition a Witt complex over S consists of the following data (i)–(iii).

(i) A pro-differential graded ring E_{·}^{∗} and a strict map of pro-rings

λ: W_{·}(S) → E_{·}^{0}.

(ii) A strict map of pro-graded rings

F: E_{n}^{∗} → E_{n−1}^{∗}

such that F λ = λF and such that

F dλ([a]_{n}) = λ([a]_{n−1})^{p−1}dλ([a]_{n−1}), for all a ∈ S.

(iii) A strict map of pro-graded modules over the pro-graded ring E_{·}^{∗},

V : F_{∗}E_{n−1}^{∗} → E_{n}^{∗},

such that V λ = λV and such that F V = p and F dV = d.

A map of Witt functors is a strict map of pro-differential graded rings that commutes with the maps λ, F and V. The structure map in the de Rham-Witt complex W_{·}Ω_{S}^{∗} is called the restriction and written R.

Lemma 2.4 Let S be a ring, let J ⊂ S be an ideal, and let W_{n}Ω_{(S,J)}^{∗} be the differential graded ideal in W_{n}Ω_{S}^{∗} generated by W_{n}(J). Then the canonical projection induces an isomorphism

W_{n}Ω_{S}^{q}/W_{n}Ω_{(S,J)}^{q} ⟶^{∼} W_{n}Ω_{S/J}^{q}.

Proof. We claim that W_{·}Ω_{S}^{∗}/W_{·}Ω_{(S,J)}^{∗} is a Witt complex over S/J. Indeed, by definition and by Lemma 2.1, it is a pro-differential graded ring with underlying pro-ring W_{·}(S/J). Hence, we need only show that the operators F, R and V on W_{·}Ω_{S}^{∗} descend to operators on W_{·}Ω_{S}^{∗}/W_{·}Ω_{(S,J)}^{∗}, or equivalently, that

\begin{aligned}RW_{n}Ω_{(S,J)}^{q} & ⊂ W_{n−1}Ω_{(S,J)}^{q} \\ F W_{n}Ω_{(S,J)}^{q} & ⊂ W_{n−1}Ω_{(S,J)}^{q} \\ V W_{n}Ω_{(S,J)}^{q} & ⊂ W_{n+1}Ω_{(S,J)}^{q}. \end{aligned}

The elements of W_{n}Ω_{(S,J)}^{q} are sums of elements of the form ω = a_{0}da_{1} … da_{q} with a_{i} ∈ W_{n}(S), for all i, and a_{i} ∈ W_{n}(J), for some i. We find that

V (ω) = V (a_{0} F dV (a_{1}) … F dV (a_{q})) = V (a_{0})dV (a_{1}) … dV (a_{q}),

which is therefore in W_{n+1}Ω_{(S,J)}^{q}. Since the Frobenius is multiplicative,

F (ω) = F a_{0} · F da_{1} · . . . · F da_{q}.

If a_{0} ∈ W_{n}(J) then F (a_{0}) ∈ W_{n−1}(J), and hence, F (ω) ∈ W_{n−1}Ω_{(S,J)}^{q}. Suppose that a_{i} ∈ W_{n}(J), for some 1 ⩽ i ⩽ q. We write out a_{i} in Witt coordinates

a_{i} = [a_{i,0}]_{n} + V [a_{i,1}]_{n−1} + ⋯ + V^{n−1}[a_{i,n−1}]_{1}

and find that

F da_{i} = [a_{i,0}]_{n−1}^{p−1}d[a_{i,0}]_{n−1} + d[a_{i,1}]_{n−1} + dV [a_{i,2}]_{n−2} + ⋯ + dV^{n−2}[a_{i,n−1}]_{1}.

This shows that F da_{i} ∈ W_{n−1}Ω_{(S,J)}^{1} and hence F (ω) ∈ W_{n−1}Ω_{(S,J)}^{q}. Finally, the statement for R is clear.

It remains to show that the quotient W_{·}Ω_{S}^{∗}/W_{·}Ω_{(S,J)}^{∗} is the universal Witt complex over S/J. Let E_{·}^{∗} be a Witt complex over S/J. By regarding E_{·}^{∗} as a Witt complex over S, we have the unique map W_{·}Ω_{S}^{∗} → E_{·}^{∗} of Witt complexes of S. But this factors through the canonical projection to give a map

W_{·}Ω_{S}^{∗}/W_{·}Ω_{(S,J)}^{∗} → E_{·}^{∗}

of Witt complexes over S/J. Finally, this map is unique because the canonical projection is surjective. This completes the proof. □

Proposition 2.5 Let S be a ring and let J ⊂ S be a finitely generated ideal. Then for all n ⩾ 1 and q ⩾ 0, the natural map

W_{n}(S/J^{·}) ⊗_{W_{n}(S)} W_{n}Ω_{S}^{q} → W_{n}Ω_{S/J^{·}}^{q}

is an isomorphism of pro-abelian groups.

Proof. It follows from Lemma 2.4 that we have a natural exact sequence of limit systems

W_{n}(S/J^{·}) ⊗_{W_{n}(S)} W_{n}Ω_{(S,J^{·})}^{q} → W_{n}(S/J^{·}) ⊗_{W_{n}(S)} W_{n}Ω_{S}^{q} → W_{n}Ω_{S/J^{·}}^{q} → 0.

Hence, it suffices to show that the limit system on the left-hand side is zero as a pro-abelian group. This means that given s ⩾ 1, we must find r ⩾ s such that the map

W_{n}(S/J^{r}) ⊗_{W_{n}(S)} W_{n}Ω_{(S,J^{r})}^{q} → W_{n}(S/J^{s}) ⊗_{W_{n}(S)} W_{n}Ω_{(S,J^{s})}^{q}

is zero, or equivalently, such that

W_{n}Ω_{(S,J^{r})}^{∗} ⊂ W_{n}(J^{s}) ⊗_{W_{n}(S)} W_{n}Ω_{S}^{∗}.

To this end, we choose r ⩾ s such that W_{n}(J^{r}) ⊂ W_{n}(J^{s})^{2}. This is possible by Cor. 2.3 since the ideal J and hence also J^{s} is finitely generated. The desired inclusion follows from the Leibniz rule. □

Lemma 2.6 Let A be a noetherian 𝔽_{p}-algebra and suppose that A is finitely generated as a module over A^{p}. Then W_{n}(A) is a noetherian ℤ/p^{n}-algebra.

Proof. It is enough to show that every prime ideal of W_{n}(A) is finitely generated [14, Thm. 3.4]. We prove this by induction on n. The case n = 1 is by assumption. In the induction step, we use that V^{n−1}(A) ⊂ W_{n}(A) is a nilpotent ideal and that the restriction map induces an isomorphism

R: W_{n}(A)/V^{n−1}(A) ⟶^{∼} W_{n−1}(A).

It follows that every prime ideal 𝔭 ⊂ W_{n}(A) is of the form 𝔭 = R^{−1}(𝔭̄) for some prime ideal 𝔭̄ ⊂ W_{n−1}(A). Hence, we have a short-exact sequence of W_{n}(A)-modules

0 → V^{n−1}(A) → 𝔭 → R_{∗}𝔭̄ → 0

and it will suffice to show that the right and left-hand modules are finitely generated. By the induction hypothesis, W_{n−1}(A) is noetherian, and hence 𝔭̄ is a finitely generated over W_{n−1}(A). This implies, since the restriction is surjective, that R_{∗}𝔭̄ is a finitely generated W_{n}(A)-module. Finally, the iterated Verschiebung gives an isomorphism of W_{n}(A)-modules

V^{n−1} : F_{∗}^{n−1}A ⟶^{∼} V^{n−1}(A),

and the iterated Frobenius factors as the composite map

F^{n−1} : W_{n}(A) ⟶^{R^{n−1}} A ⟶^{φ^{n−1}} A

where φ is the Frobenius endomorphism of A. The left-hand map is always finite, and the right-hand map is finite by assumption. Hence V^{n−1}(A) is a finitely generated W_{n}(A)-module. □

Lemma 2.7 Suppose that A is finitely generated as an module over the subring A^{p}. Then W_{n}Ω_{R}^{q} is a finitely generated W_{n}(R)-module.

Proof. The proof is by induction on n. The case n = 1 was treated in the proof of Prop. 1.7. In the induction step we use the following exact sequence of W_{n}(R)-module from [10, Prop. 3.2.6]

hW_{n}Ω_{R}^{q} ⟶^{N} W_{n}Ω_{R}^{q} ⟶^{R} W_{n−1}Ω_{R}^{q} → 0.

The left-hand term, as an abelian group, is the quotient of Ω_{R}^{q−1} ⊕ Ω_{R}^{q} by the subgroup consisting of the elements (p^{n−1}ω, −dω) with ω ∈ Ω_{R}^{q−1}. We recall from op. cit., Lemma 3.2.5, that it has a natural W_{n}(R)-module structure such that there is an exact sequence of W_{n}(R)-modules

F_{∗}^{n−1}Ω_{R}^{q} → _{h}W_{n}Ω_{R}^{q} → F_{∗}^{n−1}(Ω_{R}^{q−1}/p^{n−1}Ω_{R}^{q−1}) → 0.

Since A/A^{p} is finitely generated, Ω_{R}^{q} is a finitely generated R-module, and F_{∗}^{n−1}R is a finitely generated W_{n}(R)-module. This completes the proof of the induction step. □

Proof of Thm. B Let R = A[[t]], let S = A[t], and let I ⊂ R and J ⊂ S be the ideals generated by t. Then in the diagram

\xymatrix{lim_{s}(W_{n}(S/J^{s}) ⊗_{W_{n}(S)} W_{n}Ω_{S}^{q}) \ar[r] \ar[d] & lim_{s}(W_{n}(R/I^{s}) ⊗_{W_{n}(R)} W_{n}Ω_{R}^{q}) \ar[d] \\ lim_{s} W_{n}Ω_{S/J^{s}}^{q} \ar[r] & lim_{s} W_{n}Ω_{R/I^{s}}^{q}}

the vertical maps are isomorphisms by Prop. 2.5 and the lower horizontal map is an isomorphism since S/J^{s} ⟶^{∼} R/I^{s}. Hence the top horizontal map is an isomorphism. Moreover, we claim that the map

W_{n}Ω_{R}^{q} → lim_{s}(W_{n}(R/I^{s}) ⊗_{W_{n}(R)} W_{n}Ω_{R}^{q})

is an isomorphism. According to Cor. 2.3 this is equivalent to the statement that the W_{n}(R)-module W_{n}Ω_{R}^{q} is W_{n}(I)-adically complete. We know from Lemma 2.7 that W_{n}Ω_{R}^{q} is a finitely generated W_{n}(R)-module, from Lemma 2.6 that W_{n}(R) is a noetherian ring, and from Cor. 2.3 that the ring W_{n}(R) is W_{n}(I)-adically complete. The claim then follows from [14, Thm. 8.7]. Concluding, we have isomorphisms

W_{n}Ω_{R}^{q} ⟶^{∼} lim_{s}(W_{n}(R/I^{s}) ⊗_{W_{n}(R)} W_{n}Ω_{R}^{q}) ⟵^{∼} lim_{s}(W_{n}(S/J^{s}) ⊗_{W_{n}(S)} W_{n}Ω_{S}^{q})

and the right-hand term is equal to the completion of the W_{n}(S)-module W_{n}Ω_{S}^{q} with respect to the topology given by the ideals W_{n}(J^{s}), s ⩾ 0. We recall from [11, Thm. B] that every element ω^{(n)} ∈ W_{n}Ω_{S}^{q} can be written uniquely

\begin{aligned}ω^{(n)} & = ∑_{i∈ℕ_{0}} a_{0,i}^{(n)}[t]_{n}^{i} + ∑_{i∈ℕ}b_{0,i}^{(n)}[t]_{n}^{i−1}d[t]_{n} \\ & + ∑_{s∈ℕ} ∑_{j∈I_{p}} (V^{s}(a_{s,j}^{(n−s)}[t]_{n−s}^{j}) + dV^{s}(b_{s,j}^{(n−s)}[t]_{n−s}^{j})) \end{aligned}

where a_{s,i}^{(m)} ∈ W_{m}Ω_{A}^{q}, b_{s,i}^{(m)} ∈ W_{m}Ω_{A}^{q−1}, and where only finitely many a_{s,i}^{(r)} and b_{s,i}^{(r)} are non-zero. The effect of the completion is to remove the latter requirement. Indeed, according to Cor. 2.3, the ideals W_{n}(J^{s}) and ([t]_{n}^{s}), where s ⩾ 0, define the same topology on W_{n}Ω_{S}^{q}. □

Proposition 2.8 Let A be a noetherian 𝔽_{p}-algebra and suppose that A is a finitely generated A^{p}-module. Then the map

1 − F: W Ω_{(R,I)}^{q} → W Ω_{(R,I)}^{q}

is surjective.

Proof. We first reduce to the one variable case. Let R = A[[t_{1}, …, t_{d}]] and let I and J be the ideals generated by t_{1}, …, t_{d} and by t_{d}, respectively. Then Lemma 2.4 and the snake lemma give rise to an exact sequence

0 → W Ω_{(R,J)}^{q} → W Ω_{(R,I)}^{q} → W Ω_{(R/J,I/J)}^{q} → 0,

and by induction, the endomorphisms 1 − F of the right and left-hand terms are surjective. Here, for the left-hand term, we have used that if A is a finitely generated A^{p}-module, then R is a finitely generated R^{p}-module.

So suppose that R = A[[t]] and I = (t). Then W_{n}Ω_{(R,I)}^{q} is given by Thm. B, and the value of the Frobenius endomorphism is given by

\begin{aligned}F (w^{(n)}) & = ∑_{i∈ℕ}(F a_{0,i}^{(n)}[t]_{n−1}^{pi} + F b_{0,i}^{(n)}[t]_{n−1}^{pi−1}d[t]_{n−1}) \\ & + ∑_{s∈ℕ} ∑_{j∈I_{p}} (V^{s−1}(pa_{s,j}^{(n−s)}[t]_{n−s}^{j}) + dV^{s−1}(b_{s,j}^{(n−s)}[t]_{n−s}^{j})). \end{aligned}

To see that 1 − F is onto, let ω = (ω^{(n)})_{n∈ℕ} be an element of

W Ω_{(R,I)}^{q} = lim_{R} W_{n}Ω_{(R,I)}^{q}

and write

\begin{aligned}ω^{(n)} & = ∑_{i∈ℕ}(a_{0,i}^{(n)}[t]_{n}^{i} + b_{0,i}^{(n)}[t]_{n}^{i−1}d[t]_{n}) \\ & + ∑_{s∈ℕ} ∑_{j∈I_{p}} (V^{s}(a_{s,j}^{(n−s)}[t]_{n−s}^{j}) + dV^{s}(b_{s,j}^{(n−s)}[t]_{n−s}^{j})) \end{aligned}

with a_{s,i}^{(m)} ∈ W_{m}Ω_{A}^{d} and b_{s,i}^{(m)} ∈ W_{m}Ω_{A}^{q−1}. We consider the case where b_{s,j}^{(n)} = 0, for all n, s ∈ ℕ and j ∈ I_{p}, and the case where a_{0,i}^{(n)} = b_{0,i}^{(n)} = 0, for all n, i ∈ ℕ, and a_{s,j}^{(n)} = 0, for all s, n ∈ ℕ and j ∈ I_{p}, separately. In the first case, the geometric series

ψ^{(n)} = ∑_{m⩾0}F^{m}ω^{(m+n)} ∈ W_{n}Ω_{(R,I)}^{q}

converges and defines an element ψ = (ψ^{(n)})_{n∈ℕ} with (1 − F)ψ = ω. In the second case, we define

φ^{(n)} = − ∑_{m∈ℕ} ∑_{s∈ℕ} ∑_{j∈I_{p}} dV^{m+s}(b_{s,j}^{(n−m−s)}[t]_{n−m−s}^{j}).

This makes sense since, for each n ∈ ℕ, the sum over m ∈ ℕ is finite. The resulting element φ = (φ^{(n)})_{n∈ℕ} satisfies (1−F)φ = ω. We note that, formally, we may write φ = ∑_{s⩾1} F^{−s}ω. This completes the proof. □

Remark 2.9 It is interesting to use Thm. B to evaluate the kernel of 1 − F. Let ω = (ω^{(n)})_{n∈ℕ} be an element of W Ω_{(A[[t]],(t))}^{q}. Then

\begin{aligned}F (w^{(n)}) & = ∑_{i∈ℕ}(F a_{0,i}^{(n)}[t]_{n−1}^{pi} + F b_{0,i}^{(n)}[t]_{n−1}^{pi−1}d[t]_{n−1}) \\ & + ∑_{s∈ℕ} ∑_{j∈I_{p}} (V^{s−1}(pa_{s,j}^{(n−s)}[t]_{n−s}^{j}) + dV^{s−1}(b_{s,j}^{(n−s)}[t]_{n−s}^{j})), \end{aligned}

which we compare to

\begin{aligned}ω^{(n−1)} & = ∑_{i∈ℕ}(a_{0,i}^{(n−1)}[t]_{n−1}^{i} + b_{0,i}^{(n−1)}[t]_{n−1}^{i−1}d[t]_{n−1}) \\ & + ∑_{s∈ℕ} ∑_{j∈I_{p}} (V^{s}(a_{s,j}^{(n−1−s)}[t]_{n−1−s}^{j}) + dV^{s}(b_{s,j}^{(n−1−s)}[t]_{n−1−s}^{j})). \end{aligned}

We find that for all n, i ∈ ℕ, s ∈ ℕ_{0} and j ∈ I_{p}, the coefficients a_{s,j}^{(m)} and b_{s,j}^{(m)} must satisfy the equations

\begin{aligned}a_{0,pi}^{(n−1)} & = F a_{0,i}^{(n)}, a_{s−1,j}^{(n−s)} = pa_{s,j}^{(n−s)}, \\ b_{0,pi}^{(n−1)} & = F b_{0,i}^{(n)}, b_{s−1,j}^{(n−s)} = b_{s,j}^{(n−s)}. \end{aligned}

It follows that for all j ∈ I_{p}, there exist unique elements

a_{j} = (a_{j}^{(n)}) ∈ lim_{R} Hom(ℤ[\frac{1}{p}], W_{n}Ω_{A}^{q}),
b_{j} = (b_{j}^{(n)}) ∈ W Ω_{A}^{q−1} = lim_{R} W_{n}Ω_{A}^{q−1},

such that for all s ∈ ℕ_{0} and all i ∈ ℕ,

\begin{aligned}a_{s,j}^{(n)} & = a_{j}^{(n)}(p^{−s}), a_{0,i}^{(n)} = F^{v}a_{j}^{(n+v)}(1), \\ b_{s,j}^{(n)} & = b_{j}^{n}, b_{0,i}^{(n)} = F^{v}b_{j}^{(n+v)}, \end{aligned}

where we write i = p^{v}d with j ∈ I_{p}. But W_{n}Ω_{A}^{q} is p^{n}-torsion, and hence, the coefficients a_{j} must all be zero. It follows that the kernel of 1 − F is isomorphic to a product indexed by I_{p} of copies of W Ω_{A}^{q−1}. This group is canonically isomorphic to the big de Rham Witt group WΩ_{A}^{q−1} introduced in [9]. Hence, we can write our findings as a short exact sequence

0 → WΩ_{A}^{q−1} → W Ω_{(A[[t]],(t))}^{q} ⟶^{1−F} W Ω_{(A[[t]],(t))}^{q} → 0

which is valid whenever A is a noetherian 𝔽_{p}-algebra that is finitely generated as a module over A^{p}. This also implies a canonical isomorphism

K_{q}(A[[t]], (t), ℤ_{p}) ⟶^{∼} WΩ_{A}^{q−1}

with the relative p-adic K-group on the left.

3 Proof of Theorem A

Let A be a regular 𝔽_{p}-algebra. It follows from [6, Thm. B] and [16] that the canonical map

W_{·}Ω_{A}^{∗} → TR_{∗}^{·}(A; p)

is an isomorphism of pro-abelian groups, and hence, there is a natural longexact sequence

⋯ → TC_{q}(A; p) → W Ω_{A}^{q} ⟶^{1−F} W Ω_{A}^{q} → TC_{q−1}(A; p) → ⋯

The cyclotomic trace induces a map

K_{q}(A; ℤ_{p}) → TC_{q}(A; p)

that is defined to be the composite

π_{q}(K(A), ℤ_{p}) → π_{q}(TC(A; p), ℤ_{p}) ⟵^{∼} π_{q}(TC(A; p)).

The right-hand map is an isomorphism since the topological cyclic homology spectrum of an 𝔽_{p}-algebra is p-complete. We recall the following result from [5] and [4].

Theorem 3.1 Let A be a regular local 𝔽_{p}-algebra. Then the composite map

K_{q}(A, ℤ_{p}) → TC_{q}(A; p) → W Ω_{A}^{q}

is an isomorphism onto the kernel of 1 − F.

Proof. Suppose first that A is an essentially smooth local 𝔽_{p}-algebra. Then it was proved in [5] that K_{q}(A) is p-torsion free and that K_{q}(A)/p^{n} is generated by symbols. It follows that the composite

K_{q}(A) → TC_{q}^{n}(A; p) → TR_{q}^{n}(A; p)

factors through the canonical map from W_{n}Ω_{A}^{q} to TR_{q}^{n}(A; p). Indeed, in the following diagram, the upper horizontal map is an isomorphism.

\xymatrix{(W_{n}Ω_{A}^{1})^{⊗q} \ar[r] \ar[d] & TR_{1}^{n}(A; p)^{⊗q} \ar[d] \\ W_{n}Ω_{A}^{q} \ar[r] & TR_{q}^{n}(A; p)}

It follows further from [5], [1], and [4] that the induced map

K_{q}(A)/p^{n} → W_{n}Ω_{A}^{q}

is injective. Moreover, according to [12, I.5.7.4], its image W_{n}Ω_{A,log}^{q} is related to the kernel K_{n}^{q} of

R − F: W_{n}Ω_{A}^{q} → W_{n−1}Ω_{A}^{q−1}

by

W_{n}Ω_{A,log}^{q} ⊂ K_{n}^{q} ⊂ W_{n}Ω_{A,log}^{q} + Fil^{n−1} W_{n}Ω_{A}^{q}.

The above statements are all stable under filtered colimits, and hence they remain valid for any regular local 𝔽_{p}-algebra. Indeed, by [16] a regular local 𝔽_{p}-algebra is a filtered colimit of essentially smooth local 𝔽_{p}-algebras. The result follows by forming the limit over n. □

Proof of Thm. A Let A be a regular local 𝔽_{p}-algebra which is finitely generated as a module over A^{p}. We consider the square from the introduction

\xymatrix{K(R, I) \ar[r] \ar[d] & TC(R, I; p) \ar[d] \\ holim_{s} K(R/I^{s}, I) \ar[r] & holim_{s} TC(R/I^{s}, I; p).}

Since R is regular, the topological cyclic group in the upper right-hand corner is given by the long exact sequence

⋯ → TC_{q}(R, I; p) → W Ω_{(R,I)}^{q} ⟶^{1−F} W Ω_{(R,I)}^{q} ⟶^{∂} TC_{q−1}(R, I; p) → …

and we proved in Prop. 2.8 above that the map 1 − F is surjective. Hence Thm. 3.1 shows that the top horizontal map in the diagram above becomes a weak equivalence after p-completion. Moreover, Props. 1.2 and 1.7 show that the right-hand vertical map is weak equivalence. Finally, the lower horizontal map becomes a weak equivalence after p-completion by the main theorem of [15]. Hence the left-hand vertical map becomes a weak equivalence after p-completion. This is the statement of Thm. A. □

Acknowledgements

The majority of the work reported here was done while the authors were visiting the University of Tokyo. It is a great pleasure to thank the university, and in particular, Kazuya Kato and Takeshi Saito for their hospitality and for providing a stimulating environment.

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