##  0.1 Data

###  0.1.1 SimpleDC

We introduce the SimpleDC dataset, which is a meticulously curated parallel
corpus, specifically designed for text simplification research within the
medical domain. It comprises educational information for patients from three
reputable institutions: the American Cancer Society (ACS), the Centers for
Disease Control and Prevention (CDC), and the National Cancer Institute (NCI).
These organizations were chosen due to their expertise and comprehensive
coverage of health-related topics. There are many different types of cancer,
each with its own characteristics, and primary sources (ACS, CDC, and NCI)
include comprehensive information on these cancers, far exceeding the
available annotation budget. SimpleDC is focused on the subset of cancers
associated with the digestive system to create a corpus that spans many cancer
types that share similar terminology and topics (e.g. anatomy or symptoms).
The focus on digestive cancers is also motivated by the prevalence and
mortality of these cancers and the role that prevention and screening play in
the outcomes. SimpleDC responds to the urgent need to make medical information
more accessible and comprehensible to a broader audience, especially
traditionally under-resourced populations.

We scraped the ACS, CDC, and NCI web pages containing patient educational
content related to eight digestive cancers: anal, bile duct, colorectal,
gastrointestinal, liver, pancreatic, small intestine, and stomach. For these
cancer types, we collected the text from web pages that present the most
introductory cancer information (e.g. About, Prevention, Screening). The
collected text was cleaned, including the removal of white spaces, adjusting
punctuation, and removing rich-text formatting, and split into sentences. We
developed annotation guidelines for the simplification task and trained a team
of subject matter experts (SMEs) with expertise in both cancer and patient
education, including two nurse oncologists and a nurse practitioner. For each
web page, we created an annotation document with a two-column format, where
the left-hand and right-hand sides contained the original text with each
sentence on its own line. The annotators modified the sentences in the right
column in place based on the annotation guidelines. This approach allowed the
annotators to focus on sentence-level simplification while considering the
entire web page for context. Table LABEL:annotation_example presents
annotation examples for each source, together with the Flesch-Kincaid Grade
Level (FKGL) [kincaid1975derivation] scores.

In the initial training round, two annotators independently generated
simplifications for the same web page. To assess the quality of the
annotation, the third annotator reviewed the simplifications relative to the
original text and indicated their preferred simplification as  A is better; B
is better; Both are good; or  None is good (simplification order shuffled). In
the evaluation phase, the distribution of the third annotator’s preferences
was as follows: 34% of the samples favored the first annotator’s version, 32%
favored the second annotator’s version, 26% were considered equally effective
from both annotators, and 8% were found inadequate by the third annotator,
meaning neither version met the simplification criteria. SimpleDC consists of
29 annotated web pages (14 train, 5 development, and 10 test). It includes
1,183 annotated sentence pairs (361 train, 294 development, and 528 test). The
development and test sets were doubly annotated by two annotators and
adjudicated by the third annotator to create a robust evaluation set. The
training set was singly annotated by the domain experts. Figure
LABEL:fkgl_boxplots presents the distribution of the reading level.

As part of the data collection and exploration process, we collected an
additional 1,395 web pages from ACS, CDC, and NCI related to digestive cancer
that were not annotated/simplified. We randomly sampled 1,000 of these
unannotated web pages for use in the RL procedures described in Section .

###  0.1.2 Med-EASi

Med-EASi dataset [basu2023med] is distinctively crowdsourced and features
fine-grained annotations for supervised text simplification, addressing the
challenge of making complex medical information more accessible to a general
audience. The textual transformations in Med-EASi are classified into four
types: elaboration, replacement, deletion, and insertion. The study employs
the T5-large [raffel2020exploring] model, fine-tuned with different styles of
input-output combinations. They experiment with two control-free and two
controllable versions of the model, enabling varying degrees of text
simplification. A novel aspect of this study is its annotation approach, which
combines the efforts of medical experts, layman crowd-workers, and AI
algorithms, ensuring both the accuracy and readability of simplified texts.
The dataset includes 1,979 expert-simple text pairs, covering a wide range of
medical topics and complexities. The average FKGL of the expert samples is
around 13th grade and the average readability of the layman samples is around
10th grade.

##  0.2 Simplification Models

Our text simplification exploration with SimpleDC and Med-EASi focuses on
state-of-the-art encoder-only LLMs, including Llama 2[touvron2023llama2] and
GPT-4[openai2023gpt4]. Using Llama 2, we explore zero-shot and SFT approaches
to establish performance baselines and introduce a novel RL approach for
simplification. Using GPT-4, we explore different prompt-based strategies and
present a novel self-correction strategy for simplification.

###  0.2.1 Supervised fine-tuning (SFT)

Llama 2 was trained on input-output pairs consisting of the original text and
simplified text using SFT with standard cross-entropy loss. Our SFT
experimentation included both the SimpleDC and Med-EASi data sets.

###  0.2.2 Reinforcement Learning

We explored RL strategies to incorporate unlabeled, in-domain text into
simplification model training. Using Proximal Policy Optimization (PPO)
[schulman2017proximal], we investigated RL reward functions based on
individual metrics related to reading level assessment (FKGL), semantic
similarity (relevance), and likelihood of original vs. simplified. Our
exploration of RL includes Reinforcement Learning with Human Feedback
(RLHF)[christiano2017deep]. Figure LABEL:sft_rlhf presents an overview of the
RL pipeline. Each of these reward metrics is described below. In RL
frameworks, it is common for the reward values to be in the range [0, 1],
where higher values indicate more desirable outcomes and lower values indicate
less desirable outcomes. For each reward metric presented, individual rewards
and composite rewards are scaled to fall in this range.

###  0.2.3 Reward Models

Readability Reward: FKGL is a heuristic function to assess the reading level
of text based on word and sentence length. Although it is an imperfect metric,
FKGL provides an automatic way to assess the reading level of the generated
text. In the TESLEA approach [phatak2022medical], the FKGL score is
essentially normalized to the [-1,1] range based on the target reading level.
To map the FKGL scores to the [0, 1] range for RL, we took inspiration from
the TESLEA approach. The reading level reward, $R_{FKGL}$, was calculated by
normalizing the FKGL score by the reading level target and applying a sigmoid
function:

$R_{FKGL}=sigmoid\left(\frac{6.5-FKGL(generated)}{6.5}\right)$

The normalized reading score (fraction within the sigmoid) generally falls in
the range [-1, 1] for our data, where positive values indicate simpler text
(more desirable) and negative values indicate more complex text (less
desirable). The sigmoid function maps the normalized reading scores to the
range [0, 1], where higher values still indicate lower reading levels.

Relevance Reward: The relevance reward, $R_{Rel}$, is intended to quantify the
semantic relationship between the simplified text and the original text, where
a higher reward indicates greater semantic similarity. We follow the TESLEA
approach [phatak2022medical] for relevance scoring: 1) the original text and
the simplified text are mapped to separate vector representations using an
encoder model and 2) semantic similarity is assessed based on the cosine
similarity between these two vectors. BioSentVec [chen2019biosentvec] was
selected as the encoder, which has been extensively trained in biomedical
literature. Cosine similarity falls in the range [0, 1], where higher values
indicate higher semantic correspondence. The relevance reward incentivizes the
preservation of semantic relatedness between the original and simplified text.
The $R_{Rel}$ reward does not include any information or representation from
the annotated corpus, SimpleDC.

Original vs. Simplified (OvS) Reward: We introduce a new RL reward, $R_{OvS}$,
based on the likelihood that a text sample has been simplified. Using SimpleDC
training data, we trained a binary classifier to label samples as original or
simplified using BioMed-RoBERTa base [domains]. This reward function captures
some aspects of the annotator’s (human) preferences and is considered human
feedback for RL. The OvS classification model achieved an accuracy of 75.1% on
the SimpleDC dev dataset. This accuracy indicates a robust ability to discern
the complexity of the language and the specialized nature of the content. The
reward value, $R_{OvS}$, is the predicted probability that the text is
simplified ($P(simplified)$) providing a continuous reward signal ranging from
[0, 1], where higher values suggest better simplification. Our intuition
behind this reward function is that it provides a mechanism for learning
features of the original and simplified text and propagating this learning to
unlabeled samples during RL.

Reward Aggregation: We calculated aggregated reward values from the individual
metrics using the harmonic mean, $H$, to balance the contribution from each
metric, as

$H(x_{1},x_{2})=\frac{1}{\frac{1}{x_{1}}+\frac{1}{x_{2}}}$

. We developed two aggregated reward functions:

  * •

$\mathbf{R_{FKGL+Rel}}$: We combined the reading level reward, $R_{FKGL}$, and
the relevance reward, $R_{Rel}$, to create an aggregated reward,
$R_{FKGL+Rel}=H(R_{FKGL},R_{Rel})$. This aggregated reward is intended to
capture both text complexity and semantic similarity to yield text that is at
the desired reading level, while preserving the original meaning. This reward
function does not incorporate any learning from SimpleDC and is therefore
considered RL.

  * •

$\mathbf{R_{FKGL+OvS}}$: We also combined the reading level reward,
$R_{FKGL}$, and the Original vs. Simplified reward, $R_{Rel}$, to create an
aggregated reward, $R_{FKGL+OvS}=H(R_{FKGL},R_{OvS})$. This aggregated reward
is intended to assess text complexity ($R_{FKGL}$) and identify linguistic
features that differentiate the original and simplified text ($R_{Rel}$) to
produce text that is at the desired reading level and emulates the
simplification of the SimpleDC. As this reward function incorporates a
training signal derived from the SimpleDC annotations, it is considered RLHF.

###  0.2.4 Fine-tuning With Reward Models

We explored different pipelines for incorporating RL/RLHF, including only
using RL/RLHF, combining SFT and RL/RLHF, and using RL/RLHF for domain
adaptation:

  * •

RL/RLHF only: A pre-trained LLM was trained using RL/RLHF using unlabeled in-
domain data (digestive cancer text) to learn the text simplification task and
text-domain without any SFT of the LLM on in-domain data (SimpleDC). The goal
of this exploration was to understand the achievable performance without any
SFT.

  * •

In-domain SFT + RL/RLHF \- A pre-trained LLM was first trained through SFT on
labeled in-domain data (SimpleDC train set), followed by continued training
through RL/RLHF on unlabeled in-domain data (digestive cancer text). This
experimentation was intended to assess whether RL/RLHF can continue to improve
the model even after in-domain SFT.

  * •

Out-domain SFT + RL/RLHF \- A pre-trained LLM was first trained through SFT on
labeled out-of-domain data (Med-EASi train set), followed by continued
training through RL/RLHF unlabeled in-domain data (digestive cancers text).

In the experimentation described above, the unlabeled in-domain data consisted
of 1,000 web pages from ACS, CDC, and NCI related to digestive cancer that
were not annotated and not incorporated into SimpleDC (no overlap with
SimpleDC).

In all RL experiments, the training used Proximal Policy Optimization (PPO),
where a Kullback-Leibler (KL) divergence constraint
(1975)[csiszar1975divergence] was implemented to limit the excessive deviation
of the LLM trained with RL from the original LLM. This divergence acted as a
regulatory mechanism, ensuring that the LLM maintained linguistic coherence
and avoided overfitting to potentially incorrect or excessive scores from the
reward model. Such a constraint was crucial, considering the reward model’s
inherent limitations in comprehending the extensive range of potential
responses. This approach effectively mitigated bias toward outlier responses
that may not represent genuine quality in text simplification.

###  0.2.5 Prompt-based models

Given our focus on developing simplification models with limited training data
and the success of GPT-4 in many tasks, we present prompt-based approaches for
text simplification with GPT-4 [Achiam2023GPT4TR], including in-context
learning, CoT, and self-correction.

Zero-shot: In the zero-shot approach, GPT-4 was prompted to simplify the text
to an elementary reading level, serving as a baseline to evaluate the efficacy
of more complex prompting strategies in text simplification. The specific
simplification prompt is included in the Baseline module in Figure
LABEL:cot_fig.

In-context Learning: The in-context variant used the same initial prompt as
the zero-shot approach, but was augmented with three representative examples
from the SimpleDC training set. These examples were manually selected to
illustrate the desired simplification style and complexity, providing GPT-4
with contextual clues to guide its simplification process.

FKGL-Enhanced Prompt: Building on the initial prompt, we added a detailed
description of the FKGL heuristic, indicating that lower reading levels are
associated with shorter words and sentences, and provided the FKGL formula for
calculating the grade level from word and sentence lengths.

Chain-of-Thought (CoT): Our CoT approach builds on the FKGL-Enhanced Prompt
approach by providing the model with additional sample-specific context,
including the average word length, average sentence length, and a list of
longer words words ($\geq$ 3 syllables). This model aims to provide explicit
guidance for the simplification process and anchor the simplification in
commonly used readability standards. Figure LABEL:cot_fig illustrates the CoT
approach.

Self-Correction (SC): Inspired by recent work on self-correction [wu2023self,
pan2023automatically], we created a self-correction approach for simple text
identification. Figure LABEL:sc_fig presents the proposed self-correction
approach, which incorporates external assessments of the generated text to
guide the refinement of the simplification. The intent was to both exploit the
reasoning capabilities of the LLM and incorporate additional external training
signals to iteratively refine output. Each generated simplification was scored
for reading level using FKGL and semantic fidelity using BERTScore. Acceptance
criteria include: 1) FKGL score $\leq 6$ or FKGL reduced by 3 grade levels
from the original text and 2) BERTScore between the generated and original
texts $\geq 0.95$. If the simplification met the acceptance criteria, it was
accepted; otherwise, the simplification was fed back into GPT-4 with
instructions for improving the simplification. Each sample was processed
iteratively at most 3 times.

###  0.2.6 Experimental Paradigm

Our experimental pipelines utilized PyTorch [paszke2019pytorch] and Hugging
Face’s Transformers [wolf2019huggingface] package. We used two Llama 2
variants: Llama-2-7b-chat-hf 111https://huggingface.co/meta-
llama/Llama-2-7b-chat-hf and Llama-2-7B-32K-Instruct
222https://huggingface.co/togethercomputer/Llama-2-7B-32K-Instruct. Training
and evaluation utilized A100 GPUs. We also used Weights and Biases [wandb] for
tracking training and performance.

For SFT, we utilized LORA [hu2021lora] utilities from the Parameter-Efficient
Fine-Tuning (PEFT) package [xu2023parameter, peft], to reduce the time and
space complexity of model training. Hyperparameters were selected based on
computational constraints and performance optimization. We selected a batch
size of 8 to balance memory usage and model updating. The learning rate was
set at 2e-4, following preliminary experiments to find a balance between
catastrophic forgetting and suboptimal convergence. Additional
hyperparameters, such as weight decay (0.001) and maximum gradient norm (0.3),
were included to prevent overfitting and maintain the stability of the
training process. For RLHF, we specified a range of hyperparameters, including
a learning rate of 1.41e-5 and PPO epochs of 2, reflecting a conservative
approach to incremental learning. Our choice of batch and mini-batch sizes,
set to 4, was influenced by the need to manage the computational load
effectively while still capturing the nuances of gradient updates during the
training process.

##  0.3 Evaluation

We utilized widely accepted evaluation metrics, each addressing specific
aspects of text simplification quality. These include System output Against
References and against the Input sentence (SARI) [xu-etal-2016-optimizing],
which evaluates the quality of sentence simplification by considering
additions, deletions, and substitutions; Bilingual Evaluation Understudy
(BLEU) [Papineni02bleu:a, lin-och-2004-orange], evaluating the precision of
the generated text against reference translations; BERTScore [bert-score],
which quantifies the semantic similarity between the simplified text and
reference using contextual embeddings; Recall-Oriented Understudy for Gisting
Evaluation (ROUGE) [lin-2004-rouge], focusing on recall by comparing the
overlap of n-grams between the output and reference texts; and Flesch-Kincaid
Grade Level (FKGL) [kincaid1975derivation], measuring the readability of the
text based on sentence and word length.


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Explain it like I’m 5 years old (5yo) Things are happening awfully fast these
days! I frequently see advances in data processing – text, video, tabular data
– coming at a rate I hadn’t been expecting to see. (5yo) This makes it look
like there’s a superpower out there. Maybe it’s AI? (Me) No, actually it’s
data science. (5yo) What’s the data science superpower? (Me) Okay, on the
fingers of one hand: (1) In the last decade, frictionless services became
available thanks to the modern smartphone information ecosystem (2) Those
frictionless services were applied by Scientists and Technologists to Data
Sharing, Code Sharing and Challenges (3) Some communities of researchers
started frictionlessly sharing research artifacts – code, data, results – and
building on each others’ work. (4) Involved research communities are
progressing much faster. It’s night and day. (5) Soon it will really speed up,
as scientists write computer programs to exploit all the digital artifacts
from previous research. (5yo) But wait, I’m still reading everywhere about AI.
(Me) AI is one of those communities where people are working this way, that’s
why you’re hearing so much news about it. (5yo) But I heard that there’s a
singularity, that AI is going to unemploy everybody and may kill us all. (Me)
There is a singularity (= new superpower) but not in AI. It’s a superpower
allowing us to do research faster and better. It’s not dangerous. You’ll just
be hearing more and more frequently about advances, because research goes
faster and faster.

Abbrevation | Meaning
---|---
AI | Artificial Intelligence
BYODC | Bring-your-own-data Challenge
CASP | Critical Assessment of Protein Structures
CORA | Computation on Research Artifacts
CS | Computer Science
CVPR | Computer Vision and Pattern Recognition
DARPA | Defense Advanced Projects Research Agency
EML | Empirical Machine Learning
FOOM | Hard AI Takeoff
FR | Frictionless Reproducibility
FRX | Frictionless Research Exchange
GPU | Graphics Processing unit
IPR | In-principle reproducibility
ICLR | International Conference on Learning Representations
NASEM | National Academies of Science, Engineering, and Medicine
NeurIPS | Neural Information Processing Systems
NIST | National Institute of Standards and Technology
OSS | Open-Source Software
PMC | Proprietary-Methods Challenge
RCS | Reproducible Computational Science
Table 3: Abbreviations used in this article Label | Meaning
---|---
AI Doom | Artificial Intelligence will wreak terrible destruction
AI Extinction | Artificial Intelligence will extinguish humanity
AI Hegemony | Artificial Intelligence is effectively owned by some few Internet Hegemons
AI Singularity | Artificial Intelligence is soon to disrupt human affairs
Brutal Scaling | Massive data and compute are sufficient and necessary for performance
Emergent Superpowers | Unprecedented powers that emerge under ‘just right’ conditions
Hard Takeoff | Artificial intelligence suddenly improves exponentially and dominates us
Table 4: Narrative Labels


definition[equation]Definition remark[equation]Remark example[equation]Example
19D55 (primary), 55Q91 (secondary) The first and third authors were supported
in part by the National Science Foundation

# On the $K$-theory of truncated polynomial algebras
over the integers

Vigleik Angeltveit  Teena Gerhardt  Lars Hesselholt<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We show that $K_{2i}(\mathbb{Z}[x]/(x^{m}),(x))$ is finite of order
$(mi)!(i!)^{m-2}$ and that $K_{2i+1}(\mathbb{Z}[x]/(x^{m}),(x))$ is free
abelian of rank $m-1$. This is accomplished by showing that the equivariant
homotopy groups $\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};p)$ of the
topological Hochschild $\mathbb{T}$-spectrum $T(\mathbb{Z})$ are free abelian
for $q$ even, and finite for $q$ odd, and by determining their ranks and
orders, respectively.

## Introduction

It was proved by Soulé [13] and Staffeldt [14] that, for every non-negative
integer $q$, the abelian group $K_{q}(\mathbb{Z}[x]/(x^{m}),(x))$ is finitely
generated and that its rank is either $0$ or $m-1$ according as $q$ is even or
odd. In this paper, we prove the following more precise result:

###### Theorem A

Let $m$ be a positive integer and $i$ a non-negative integer. Then:

  1. (i)

The abelian group $K_{2i+1}(\mathbb{Z}[x]/(x^{m}),(x))$ is free of rank $m-1$.

  2. (ii)

The abelian group $K_{2i}(\mathbb{Z}[x]/(x^{m}),(x))$ is finite of order
$(mi)!(i!)^{m-2}$.

In particular, the $p$-primary torsion subgroup of
$K_{2i}(\mathbb{Z}[x]/(x^{m}),(x))$ is zero, for every prime number $p>mi$. At
present, we do not know the group structure of the finite abelian group in
degree $2i$ except for small values of $i$ and $m$. We remark that the result
agrees with the calculation by Geller and Roberts [12] of the group in degree
$2$.

To prove Theorem A, we use the cyclotomic trace map of Bökstedt-Hsiang-Madsen
[4] from the $K$-groups in the statement to the corresponding topological
cyclic homology groups and a theorem of McCarthy [11] which shows that this
map becomes an isomorphism after pro-finite completion. The third author and
Madsen [7, Proposition 4.2.3], in turn, gave a formula for the topological
cyclic homology groups in question in terms of the equivariant homotopy groups

$\operatorname{TR}_{q-\lambda}^{r}(\mathbb{Z})=[S^{q}\wedge(\mathbb{T}/C_{r})_{+},S^{\lambda}\wedge
T(\mathbb{Z})]_{\mathbb{T}}$

of the topological Hochschild $\mathbb{T}$-spectrum $T(\mathbb{Z})$. Here
$\mathbb{T}$ is the multiplicative group of complex numbers of modulus $1$,
$C_{r}\subset\mathbb{T}$ is the finite subgroup of the indicated order,
$\lambda$ is a finite dimensional complex $\mathbb{T}$-representation, and
$S^{\lambda}$ is the one-point compactification of $\lambda$. Since the groups
$K_{q}(\mathbb{Z}[x]/(x^{m}),(x))$ and
$\operatorname{TR}_{q-\lambda}^{r}(\mathbb{Z})$ are finitely generated by [13,
14] and Lemma 1.2, respectively, these earlier results amount to a long exact
sequence

$\textstyle{{\cdots}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\lim_{R}^{\phantom{R}}\operatorname{TR}_{q-1-\lambda_{d}}^{r/m}(\mathbb{Z})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{V_{m}}$$\textstyle{{\lim_{R}^{\phantom{R}}\operatorname{TR}_{q-1-\lambda_{d}}^{r}(\mathbb{Z})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{K_{q}(\mathbb{Z}[x]/(x^{m}),(x))}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\cdots}}$

where $d=d(m,r)$ is the integer part of $(r-1)/m$, and where $\lambda_{d}$ is
the sum

$\lambda_{d}=\mathbb{C}(d)\oplus\mathbb{C}(d-1)\oplus\cdots\oplus\mathbb{C}(1)$

of the one-dimensional complex $\mathbb{T}$-representations defined by
$\mathbb{C}(i)=\mathbb{C}$ with $\mathbb{T}$ acting from the left by $z\cdot
w=z^{i}w$. The two limits range over the positive integers divisible by $m$
and the positive integers, respectively, ordered under division. The structure
maps $R$ and the map $V_{m}$ are explained in Section 1 below. In particular,
we show that for every integer $q$ there exists a positive integer $r=r(m,q)$
divisible by $m$ such that the canonical projections

$\lim_{R}\operatorname{TR}_{q-\lambda_{d}}^{r}(\mathbb{Z})\to\operatorname{TR}_{q-\lambda_{d}}^{r}(\mathbb{Z}),\hskip
28.45274pt\lim_{R}\operatorname{TR}_{q-\lambda_{d}}^{r/m}(\mathbb{Z})\to\operatorname{TR}_{q-\lambda_{d}}^{r/m}(\mathbb{Z})$

are isomorphisms.

After localizing at a prime number $p$, the abelian groups
$\operatorname{TR}_{q-\lambda}^{r}(\mathbb{Z})$ decompose as products of the
$p$-typical equivariant homotopy groups

$\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};p)=\operatorname{TR}_{q-\lambda}^{p^{n-1}}(\mathbb{Z})=[S^{q}\wedge(\mathbb{T}/C_{p^{n-1}})_{+},S^{\lambda}\wedge
T(\mathbb{Z})]_{\mathbb{T}}.$

In addition, the Verschiebung map $V_{m}$ which appears in the long exact
sequence above may be expressed in terms of the $p$-typical Verschiebung map
$V=V_{p}$. The corresponding $p$-typical equivariant homotopy groups with
$\mathbb{Z}/p\mathbb{Z}$-coefficients

$\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};p,\mathbb{Z}/p\mathbb{Z})=[S^{q}\wedge(\mathbb{T}/C_{p^{n-1}})_{+},M_{p}\wedge
S^{\lambda}\wedge T(\mathbb{Z})]_{\mathbb{T}}$

were evaluated by the first and second author [1] and by Tsalidis [16]. More
generally, the first and second author [1] evaluated the
$\operatorname{RO}(\mathbb{T})$-graded equivariant homotopy groups

$\operatorname{TR}_{\alpha}^{n}(\mathbb{Z};p,\mathbb{Z}/p\mathbb{Z})=[S^{\beta}\wedge(\mathbb{T}/C_{p^{n-1}})_{+},M_{p}\wedge
S^{\gamma}\wedge T(\mathbb{Z})]_{\mathbb{T}},$

where $\alpha\in\operatorname{RO}(\mathbb{T})$ is any virtual finite
dimensional orthogonal $\mathbb{T}$-representation, and where $\beta$ and
$\gamma$ are chosen actual representations with $\alpha=[\beta]-[\gamma]$.
Based on these results, we prove:

###### Theorem B

Let $p$ be a prime number, let $n$ be a positive integer, and let $\lambda$ be
a finite dimensional complex $\mathbb{T}$-representation. Then:

  1. (i)

For $q=2i$ even, $\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};p)$ is a free
abelian group whose rank is equal to the number of integers $0\leqslant s<n$
such that $i=\dim_{\mathbb{C}}(\lambda^{C_{p^{s}}})$.

  2. (ii)

For $q=2i-1$ odd, $\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};p)$ is a
finite abelian group whose order is determined, recursively, by

$\lvert\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};p)\rvert=\begin{cases}\lvert\operatorname{TR}_{q-\lambda^{\prime}}^{n-1}(\mathbb{Z};p)\rvert\cdot
p^{n-1}(i-\dim_{\mathbb{C}}(\lambda))&\text{if\,
$i>\dim_{\mathbb{C}}(\lambda)$
}\cr\lvert\operatorname{TR}_{q-\lambda^{\prime}}^{n-1}(\mathbb{Z};p)\rvert&\text{if\,
$i\leqslant\dim_{\mathbb{C}}(\lambda)$, }\cr\end{cases}$

where $\lambda^{\prime}=\rho_{p}^{*}\lambda^{C_{p}}$ is the
$\mathbb{T}/C_{p}$-representation $\lambda^{C_{p}}$ viewed as a
$\mathbb{T}$-representation via the isomorphism
$\rho_{p}\colon\mathbb{T}\to\mathbb{T}/C_{p}$ given by the $p$th root.

  3. (iii)

For every integer $q$, the Verschiebung map

$V\colon\operatorname{TR}_{q-\lambda}^{n-1}(\mathbb{Z};p)\to\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};p)$

is injective, and for $q$ even the cokernel is a free abelian group.

We remark that for $\lambda=0$ the result is that

$\lvert\operatorname{TR}_{2i-1}^{n}(\mathbb{Z};p)\rvert=p^{n(n-1)/2}\,i^{n}$

while the even groups all are zero with the exception of
$\operatorname{TR}_{0}^{n}(\mathbb{Z};p)$ which is a free abelian group of
rank $n$. In the case $n=1$ which was proved by Bökstedt [3], the groups are
all cyclic. For $n>1$, this is not the case. It remains a very interesting
problem to determine the structure of these groups. We refer to [5, Theorem
18] for some partial results.

It follows from Theorem B that with $\mathbb{Z}/p\mathbb{Z}$-coefficients the
Verschiebung map

$V\colon\operatorname{TR}_{q-\lambda}^{n-1}(\mathbb{Z};p,\mathbb{Z}/p\mathbb{Z})\to\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};p,\mathbb{Z}/p\mathbb{Z})$

is injective for $q$ even. We do not know the value of this map for $q$ odd.
The calculation of this map, and hence, the groups
$K_{q}(\mathbb{Z}[x]/(x^{m}),(x);\mathbb{Z}/p\mathbb{Z})$ claimed in [16,
Proposition 7.7] is incorrect. Indeed, in loc. cit., it is only the map
induced by $V$ between the $E^{\infty}$-terms of two spectral sequences which
is evaluated.

The paper is organized as follows. In Section 1, we show that the groups
$\operatorname{TR}_{q-\lambda}^{r}(\mathbb{Z})$ are finitely generated and
determine their ranks. In Section 2, we recall the results of [1] and [16] and
prove Theorem B. In the following Section 3, we evaluate the terms in the long
exact sequence above and prove Theorem A. Finally, in Section 4, we specialize
to the case of the dual numbers and determine the structure of the finite
group $K_{2i}(\mathbb{Z}[x]/(x^{2}),(x))$ of order $(2i)!$ in low degrees.

##  1 The groups $\operatorname{TR}_{q-\lambda}^{r}(\mathbb{Z})$

In this section, we recall the groups
$\operatorname{TR}_{q-\lambda}^{r}(\mathbb{Z})$ and the Frobenius,
Verschiebung, and restriction operators that relate them. We refer to [8,
Section 1] and [5, Section 2] for further details.

Let $A$ be a unital associative ring. The topological Hochschild
$\mathbb{T}$-spectrum $T(A)$ is, in particular, an orthogonal
$\mathbb{T}$-spectrum in the sense of [10, Definition II.2.6]. Therefore, for
every finite dimensional orthogonal $\mathbb{T}$-representation $\lambda$ and
every finite subgroup $C_{r}\subset\mathbb{T}$, we have the equivariant
homotopy group given by the following abelian group of maps in the homotopy
category of orthogonal $\mathbb{T}$-spectra

$\operatorname{TR}_{q-\lambda}^{r}(A)=[S^{q}\wedge(\mathbb{T}/C_{r})_{+},S^{\lambda}\wedge
T(A)]_{\mathbb{T}}.$

For every divisor $s$ of $r$ with quotient $t=r/s$, there are maps

$\displaystyle F_{s}$
$\displaystyle\colon\operatorname{TR}_{q-\lambda}^{r}(A)\to\operatorname{TR}_{q-\lambda}^{t}(A)\hskip
19.91692pt\text{(Frobenius)}$ $\displaystyle V_{s}$
$\displaystyle\colon\operatorname{TR}_{q-\lambda}^{t}(A)\to\operatorname{TR}_{q-\lambda}^{r}(A)\hskip
19.91692pt\text{(Verschiebung)}$

induced by maps $f_{s}\colon(\mathbb{T}/C_{t})_{+}\to(\mathbb{T}/C_{r})_{+}$
and $v_{s}\colon(\mathbb{T}/C_{r})_{+}\to(\mathbb{T}/C_{t})_{+}$ in the
homotopy category of orthogonal $\mathbb{T}$-spectra. The map $f_{s}$ is the
map of suspension $\mathbb{T}$-spectra induced by the canonical projection
$\operatorname{pr}\colon\mathbb{T}/C_{t}\to\mathbb{T}/C_{r}$ and the map
$v_{s}$ is the corresponding transfer map defined as follows. Let
$\iota\colon\mathbb{T}/C_{t}\hookrightarrow\mu$ be an embedding into a finite
dimensional orthogonal $\mathbb{T}$-representation. Then the product embedding
$(\iota,\operatorname{pr})\colon\mathbb{T}/C_{t}\to\mu\times(\mathbb{T}/C_{r})$
has trivial normal bundle, and the linear structure of $\mu$ determines a
canonical trivialization. Therefore, the Pontryagin-Thom construction gives a
map of pointed $\mathbb{T}$-spaces

$S^{\mu}\wedge(\mathbb{T}/C_{r})_{+}\to S^{\mu}\wedge(\mathbb{T}/C_{t})_{+}.$

The induced map of suspension $\mathbb{T}$-spectra determines the homotopy
class of maps of orthogonal $\mathbb{T}$-spectra
$v_{s}\colon(\mathbb{T}/C_{r})_{+}\to(\mathbb{T}/C_{t})_{+}$ and this homotopy
class is independent of the choice of embedding $\iota$ as well as the choices
made in forming the Pontryagin-Thom construction.

The orthogonal $\mathbb{T}$-spectrum $T(A)$ has the additional structure of a
cyclotomic spectrum in the sense of [8, Definition 2.2.]. This implies that,
in the situation above, there is a map

$R_{s}\colon\operatorname{TR}_{q-\lambda}^{r}(A)\to\operatorname{TR}_{q-\lambda^{\prime}}^{t}(A)\hskip
19.91692pt\text{(restriction)},$

where $\lambda^{\prime}=\rho_{s}^{*}(\lambda^{C_{s}})$ is the
$\mathbb{T}/C_{s}$-representation $\lambda^{C_{s}}$ considered as a
$\mathbb{T}$-representation via the isomorphism
$\rho_{s}\colon\mathbb{T}\to\mathbb{T}/C_{s}$ defined by
$\rho_{s}(z)=z^{1/s}C_{s}$. Moreover, the map $R_{s}$ admits a canonical
factorization which we now explain. In general, let $G$ be a compact Lie group
and $\mathscr{F}$ a family of closed subgroups of $G$ stable under conjugation
and passage to subgroups. We recall that a universal $\mathscr{F}$-space is a
$G$-CW-complex $E\mathscr{F}$ with the property that, for every closed
subgroup $H\subset G$, the fixed point set $(E\mathscr{F})^{H}$ is
contractible if $H\in\mathscr{F}$ and empty if $H\notin\mathscr{F}$. It was
proved by tom Dieck [15, Satz 1] that a universal $\mathscr{F}$-space
$E\mathscr{F}$ exists and that, if both $E\mathscr{F}$ and
$E^{\prime}\\!\mathscr{F}$ are universal $\mathscr{F}$-spaces, then there
exists a unique $G$-homotopy class of $G$-homotopy equivalences $f\colon
E\mathscr{F}\to E^{\prime}\\!\mathscr{F}$. Given a universal
$\mathscr{F}$-space $E\mathscr{F}$, the pointed $G$-space
$\tilde{E}\mathscr{F}$ is defined to be the mapping cone of the map $\pi\colon
E\mathscr{F}_{+}\to S^{0}$ that collapses $E\mathscr{F}$ onto the non-base
point such that we have a cofibration sequence of pointed $G$-spaces

$E\mathscr{F}_{+}\xrightarrow{\pi}S^{0}\xrightarrow{\iota}\tilde{E}\mathscr{F}\xrightarrow{\delta}\Sigma
E\mathscr{F}_{+}.$

If $N\subset G$ is a closed normal subgroup, we denote by $\mathscr{F}[N]$ the
family of closed subgroups $H\subset G$ that do not contain $N$ as a subgroup.
Now, the map $R_{s}$ admits a factorization as the composition of the map

$\operatorname{TR}_{q-\lambda}^{r}(A)=[S^{q}\wedge(\mathbb{T}/C_{r})_{+},S^{\lambda}\wedge
T(A)]_{\mathbb{T}}\to[S^{q}\wedge(\mathbb{T}/C_{r})_{+},S^{\lambda}\wedge\tilde{E}\mathscr{F}[C_{s}]\wedge
T(A)]_{\mathbb{T}}$

induced by the map $\iota\colon S^{0}\to\tilde{E}\mathscr{F}[C_{s}]$ and a
canonical isomorphism

$[S^{q}\wedge(\mathbb{T}/C_{r})_{+},S^{\lambda}\wedge\tilde{E}\mathscr{F}[C_{s}]\wedge
T(A)]_{\mathbb{T}}\xrightarrow{\sim}[S^{q}\wedge(\mathbb{T}/C_{t})_{+},S^{\lambda^{\prime}}\wedge
T(A)]_{\mathbb{T}}=\operatorname{TR}_{q-\lambda^{\prime}}^{t}(A)$

induced from the cyclotomic structure of $T(A)$ and [10, Proposition V.4.17].

The group isomorphism $\rho_{s}\colon\mathbb{T}\to\mathbb{T}/C_{s}$ gives rise
to an equivalence of categories $\rho_{s}^{*}$ from the category of orthogonal
$\mathbb{T}/C_{s}$-spectra to the category of orthogonal $\mathbb{T}$-spectra
defined by

$(\rho_{s}^{*}T)({\lambda})=\rho_{s}^{*}(T({(\rho_{s}^{-1})^{*}\lambda})).$

The following result is a generalization of [8, Theorem 2.2].

###### Proposition 1.1

Let $A$ be a unital associative ring, $r$ a positive integer, and $\lambda$ a
finite dimensional orthogonal $\mathbb{T}$-representation. Let $p$ be a prime
number that divides $r$, and $u$ and $v$ positive integers with
$u+v=v_{p}(r)+1$. Then there is a natural long exact sequence

$\cdots\to\mathbb{H}_{q}(C_{p^{u}},\operatorname{TR}_{\,\boldsymbol{\cdot}-\lambda}^{r/p^{u}}(A))\xrightarrow{N_{p^{u}}}\operatorname{TR}_{q-\lambda}^{r}(A)\xrightarrow{R_{p^{v}}}\operatorname{TR}_{q-\lambda^{\prime}}^{r/p^{v}}(A)\xrightarrow{\partial}\mathbb{H}_{q-1}(C_{p^{u}},\operatorname{TR}_{\,\boldsymbol{\cdot}-\lambda}^{r/p^{u}}(A))\to\cdots$

where the left-hand term is the $q$th Borel homology group of the group
$C_{p^{u}}$ with coefficients in the orthogonal $\mathbb{T}$-spectrum defined
by

$\operatorname{TR}_{\,\boldsymbol{\cdot}-\lambda}^{r/p^{u}}(A)=\rho_{r/p^{u}}^{*}(S^{\lambda}\wedge
T(A))^{C_{r/p^{u}}}.$

######  Proof 1.1.

The cofibration sequence of pointed $\mathbb{T}$-spaces

$E\mathscr{F}[C_{p^{v}}]_{+}\xrightarrow{\pi}S^{0}\xrightarrow{\iota}\tilde{E}\mathscr{F}[C_{p^{v}}]\xrightarrow{\delta}\Sigma
E\mathscr{F}[C_{p^{v}}]_{+}$

gives rise to a cofibration sequence of orthogonal $\mathbb{T}$-spectra

$E\mathscr{F}[C_{p^{v}}]_{+}\wedge
T(A)\xrightarrow{\pi}T(A)\xrightarrow{\iota}\tilde{E}\mathscr{F}[C_{p^{v}}]\wedge
T(A)\xrightarrow{\delta}\Sigma E\mathscr{F}[C_{p^{v}}]_{+}\wedge T(A).$

This, in turn, gives rise to a long exact sequence of equivariant homotopy
groups which we now identify with the long exact sequence of the statement. By
definition, we have

$\operatorname{TR}_{q-\lambda}^{r}(A)=[S^{q}\wedge(\mathbb{T}/C_{r})_{+},S^{\lambda}\wedge
T(A)]_{\mathbb{T}},$

and, as recalled above, the restriction map $R_{p^{v}}$ factors through the
canonical isomorphism

$[S^{q}\wedge(\mathbb{T}/C_{r})_{+},S^{\lambda}\wedge\tilde{E}\mathscr{F}[C_{p^{v}}]\wedge
T(A)]_{\mathbb{T}}\xrightarrow{\sim}\operatorname{TR}_{q-\lambda^{\prime}}^{r/p^{v}}(A).$

This identifies the middle and right-hand term of the long exact sequence. To
identify the left-hand term, we recall from [10, Proposition V.2.3] the
change-of-groups isomorphism

$[S^{q}\wedge(C_{r}/C_{r})_{+},S^{\lambda}\wedge
E\mathscr{F}[C_{p^{v}}]_{+}\wedge
T(A)]_{C_{r}}\xrightarrow{\sim}[S^{q}\wedge(\mathbb{T}/C_{r})_{+},S^{\lambda}\wedge
E\mathscr{F}[C_{p^{v}}]_{+}\wedge T(A)]_{\mathbb{T}}.$

On the left-hand side, the family $\mathscr{F}[C_{p^{v}}]$ is equal to the
family of subgroups $C_{s}\subset C_{r}$ for which $v_{p}(s)<v$. Therefore, we
may choose the universal space $E\mathscr{F}[C_{p^{v}}]$ to be a $C_{r}$-CW-
complex that is non-equivariantly contractible and that only has cells of
orbit-type $C_{r}/C_{r/p^{u}}$. Indeed, in this case we have

$(E\mathscr{F}[C_{p^{v}}])^{C_{s}}=\begin{cases}E\mathscr{F}[C_{p^{v}}]&\text{if
$v_{p}(s)<v$}\cr\varnothing&\text{if $v_{p}(s)\geqslant v$}\cr\end{cases}$

as required. We then have canonical isomorphisms

$\displaystyle{[S^{q},S^{\lambda}\wedge E\mathscr{F}[C_{p^{v}}]_{+}\wedge
T(A)]_{C_{r}}}{}$ $\displaystyle\xleftarrow{\sim}[S^{q},(S^{\lambda}\wedge
E\mathscr{F}[C_{p^{v}}]_{+}\wedge T(A))^{C_{r/p^{u}}}]_{C_{r}/C_{r/p^{u}}}$
$\displaystyle\xleftarrow{\sim}[S^{q},E\mathscr{F}[C_{p^{v}}]_{+}\wedge(S^{\lambda}\wedge
T(A))^{C_{r/p^{u}}}]_{C_{r}/C_{r/p^{u}}}$

where for the second isomorphism we use that $E\mathscr{F}[C_{p^{v}}]$ is
chosen to be $C_{r/p^{u}}$-fixed. The group isomorphism $\rho_{r/p^{u}}\colon
C_{p^{u}}\to C_{r}/C_{r/p^{u}}$ induces an isomorphism of categories
$\rho_{r/p^{u}}^{*}$ from the category of orthogonal $C_{p^{u}}$-spectra to
the category of orthogonal $C_{r}/C_{r/p^{u}}$-spectra. In particular, this
gives an isomorphism of the lower group above to the group

$[S^{q},\rho_{r/p^{u}}^{*}E\mathscr{F}[C_{p^{v}}]_{+}\wedge\rho_{r/p^{u}}^{*}(S^{\lambda}\wedge
T(A))^{C_{r/p^{u}}}]_{C_{p^{u}}}=\mathbb{H}_{q}(C_{p^{u}},\operatorname{TR}_{\,\boldsymbol{\cdot}-\lambda}^{r/p^{u}}(A)).$

This is indeed the desired Borel homology group, since
$\rho_{r/p^{u}}^{*}E\mathscr{F}[C_{p^{v}}]$ is a free $C_{p^{u}}$-CW-complex
which is non-equivariantly contractible.

We recall that the Borel homology groups that appear in the statement of
Proposition 1.1 are the abutment of the first quadrant skeleton spectral
sequence

$E_{s,t}^{2}=H_{s}(C_{p^{u}},\operatorname{TR}_{t-\lambda}^{r/p^{u}}(A))\Rightarrow\mathbb{H}_{s+t}(C_{p^{u}},\operatorname{TR}_{\,\boldsymbol{\cdot}-\lambda}^{r/p^{u}}(A))$
(1.2)

from the group homology of $C_{p^{u}}$ with coefficients in the trivial
$C_{p^{u}}$-module $\operatorname{TR}_{t-\lambda}^{r/p^{u}}(A)$; see for
instance [5, Section 4].

We now specialize to the case $A=\mathbb{Z}$ and recall from Bökstedt [3] that
$\operatorname{TR}_{q}^{1}(\mathbb{Z})$ is zero if either $q$ is negative or
$q$ is positive and even, a free abelian group of rank one if $q=0$, and a
finite cyclic group of order $i$ if $q=2i-1$ is positive and odd; see also
[9].

######  Lemma 1.2.

Let $r$ be a positive integer, let $q$ be an integer, and let $\lambda$ be a
finite dimensional complex $\mathbb{T}$-representation. Then
$\operatorname{TR}_{q-\lambda}^{r}(\mathbb{Z})$ is a finitely generated
abelian group whose rank is equal to the number of positive divisors $e$ of
$r$ for which $q=2\dim_{\mathbb{C}}(\lambda^{C_{e}})$. The group is zero for
$q<2\dim_{\mathbb{C}}(\lambda^{C_{r}})$.

######  Proof 1.3.

Let $\ell(r,q,\lambda)$ denote the number of positive divisors $e$ of $r$ with
$q=2\dim_{\mathbb{C}}(\lambda^{C_{e}})$ and note that $\ell(r,q,\lambda)$ is
zero, for $q$ odd. We prove by induction on the number $k$ of prime divisors
in $r$ that $\operatorname{TR}_{q-\lambda}^{r}(\mathbb{Z})$ is a finitely
generated abelian group of rank $\ell(r,q,\lambda)$. If $k=0$, or
equivalently, if $r=1$ the statement follows from the result of Bökstedt which
we recalled above. Indeed, it follows from [10, Proposition V.2.3] that, up to
isomorphism,

$\operatorname{TR}_{q-\lambda}^{1}(\mathbb{Z})=\operatorname{TR}_{q-2\dim_{\mathbb{C}}(\lambda)}^{1}(\mathbb{Z}).$

So we let $k\geqslant 1$ and assume that the lemma has been proved, for all
$q$ and $\lambda$ as in the statement if $r$ has $k-1$ prime divisors. Let $p$
be a prime divisor of $r$ and write $r=p^{n}r^{\prime}$ with $r^{\prime}$ not
divisible by $p$. We consider the long exact sequence of Proposition 1.1 with
$u=n$ and $v=1$,

${\cdots}\to{\mathbb{H}_{q}(C_{p^{n}},\operatorname{TR}_{\,\boldsymbol{\cdot}-\lambda}^{r^{\prime}}(\mathbb{Z}))}\xrightarrow{N_{p^{n}}}{\operatorname{TR}_{q-\lambda}^{r}(\mathbb{Z})}\xrightarrow{R_{p}}{\operatorname{TR}_{q-\lambda^{\prime}}^{r/p}(\mathbb{Z})}\xrightarrow{\partial}{\mathbb{H}_{q-1}(C_{p^{n}},\operatorname{TR}_{\,\boldsymbol{\cdot}-\lambda}^{r^{\prime}}(\mathbb{Z}))}\to{\cdots}$

Since $r^{\prime}$ has only $k-1$ prime divisors, the inductive hypothesis
implies that in the skeleton spectral sequence (1.2), $E_{0,q}^{2}$ is a
finitely generated abelian group of rank $\ell(q,r^{\prime},\lambda)$ and that
the groups $E_{s,t}^{2}$ with $s>0$ are finite. Hence, the left-hand group in
the long exact sequence is finitely generated of rank
$\ell(q,r^{\prime},\lambda)$. By further induction on $n\geqslant 0$, we may
assume that the group
$\operatorname{TR}_{q-\lambda^{\prime}}^{r/p}(\mathbb{Z})$ is finitely
generated of rank $\ell(q,r/p,\lambda^{\prime})$. The first part of the lemma
now follows from the formula

$\ell(q,r^{\prime},\lambda)+\ell(q,r/p,\lambda^{\prime})=\ell(q,r,\lambda)$

which holds since the two summands on the left-hand side count the number of
positive divisors $e$ of $r$ with $q=2\dim_{\mathbb{C}}(\lambda^{C_{e}})$ for
which $e$ is, respectively, prime to $p$ and divisible by $p$. The second part
of the lemma is proved in a similar manner.

###### Addendum 1.3

(i) Let $m,r\geqslant 1$, $0\leqslant\epsilon\leqslant 1$, and $i$ be
integers, and let $d=d(m,r)$ be the integer part of $(r-1)/m$. Then the
canonical projection induces an isomorphism

$\lim_{R}\operatorname{TR}_{2i+\epsilon-\lambda_{d}}^{r}(\mathbb{Z})\xrightarrow{\sim}\operatorname{TR}_{2i+\epsilon-\lambda_{d}}^{r}(\mathbb{Z}),$

provided that $m(i+1)<p^{v_{p}(r)+1}$ for every prime number $p$.

(ii) Let $m\geqslant 1$, $0\leqslant\epsilon\leqslant 1$, and $i$ be integers,
let $r\geqslant 1$ be an integer divisible by $m$, and let $d=d(m,r)$. Then
the canonical projection induces an isomorphism

$\lim_{R}\operatorname{TR}_{2i+\epsilon-\lambda_{d}}^{r/m}(\mathbb{Z})\xrightarrow{\sim}\operatorname{TR}_{2i+\epsilon-\lambda_{d}}^{r/m}(\mathbb{Z})$

provided that $i+1<p^{v_{p}(r/m)+1}$ for every prime number $p$.

######  Proof 1.4.

We prove statement (i); the proof of statement (ii) is similar. It suffices to
show that for every prime number $p$ the restriction map

$R_{p}\colon\operatorname{TR}_{q-\lambda_{d(m,pr)}}^{pr}(\mathbb{Z})\to\operatorname{TR}_{q-\lambda_{d(m,r)}}^{r}(\mathbb{Z})$

is an isomorphism if $q=2i+\epsilon$ with $m(i+1)<p^{v_{p}(r)+1}$. We write
$r=p^{n-1}r^{\prime}$ with $r^{\prime}$ not divisible by $p$ and consider the
long exact sequence of Proposition 1.1 with $u=n$ and $v=1$,

$\cdots\to\mathbb{H}_{q}(C_{p^{n}},\operatorname{TR}_{\,\boldsymbol{\cdot}-\lambda_{d(m,pr)}}^{r^{\prime}}(\mathbb{Z}))\xrightarrow{N_{p^{n}}}\operatorname{TR}_{q-\lambda_{d(m,pr)}}^{pr}(\mathbb{Z})\xrightarrow{R_{p}}\operatorname{TR}_{q-\lambda_{d(m,r)}}^{r}(\mathbb{Z})\to\cdots$

The skeleton spectral sequence (1.2) and Lemma 1.2 show that the left-hand
group vanishes, provided that
$q<2\dim_{\mathbb{C}}(\lambda_{d(m,pr)}^{C_{r^{\prime}}})$. Therefore, the map
$R_{p}$ is an isomorphism for

$i<\dim_{\mathbb{C}}(\lambda_{d(m,pr)}^{C_{r^{\prime}}})=\lfloor
d(m,pr)/r^{\prime}\rfloor.$

We claim that $d(m,p^{n})\leqslant\lfloor d(m,pr)/r^{\prime}\rfloor$. Indeed,
this inequality is equivalent to the inequality $d(m,p^{n})\leqslant
d(m,pr)/r^{\prime}$ which is equivalent to the inequality
$r^{\prime}d(m,p^{n})\leqslant d(m,pr)$ which, in turn, is equivalent to the
inequality $r^{\prime}d(m,p^{n})\leqslant(p^{n}r^{\prime}-1)/m$. We may
rewrite this inequality as $mr^{\prime}d(m,p^{n})\leqslant p^{n}r^{\prime}-1$
or $mr^{\prime}d(m,p^{n})<p^{n}r^{\prime}$ or $md(m,p^{n})<p^{n}$. But this
inequality is equivalent to the inequality $md(m,p^{n})\leqslant p^{n}-1$
which, in turn, is equivalent to the inequality
$d(m,p^{n})\leqslant(p^{n}-1)/m$ which holds. The claim follows. Finally, a
similar argument shows that the inequalities $i<d(m,p^{n})$ and $m(i+1)<p^{n}$
are equivalent.

##  2 The $p$-typical groups
$\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z}_{;}p)$

In this section, we prove Theorem B of the introduction. We first show that
after localization at the prime number $p$, the groups
$\operatorname{TR}_{q-\lambda}^{r}(A)$ decompose as products of the
$p$-typical groups

$\operatorname{TR}_{q-\lambda}^{n}(A;p)=\operatorname{TR}_{q-\lambda}^{p^{n-1}}(A)=[S^{q}\wedge(\mathbb{T}/C_{p^{n-1}})_{+},S^{\lambda}\wedge
T(A)]_{\mathbb{T}}.$

######  Proposition 1.

Let $A$ be a unital associative ring, $r\geqslant 1$ and $q$ integers, and
$\lambda$ a finite dimensional orthogonal $\mathbb{T}$-representation. Let $p$
be a prime number and write $r=p^{n-1}r^{\prime}$ with $r^{\prime}$ not
divisible by $p$. Then the map

$\gamma\colon\operatorname{TR}_{q-\lambda}^{r}(A)\to\textstyle{\prod_{j\mid
r^{\prime}}}\operatorname{TR}_{q-\lambda^{\prime}}^{n}(A;p)$

whose $j$th component is the composite map

$\textstyle{{\operatorname{TR}_{q-\lambda}^{r}(A)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{j}}$$\textstyle{{\operatorname{TR}_{q-\lambda}^{r/j}(A)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{R_{r^{\prime}/j}}$$\textstyle{{\operatorname{TR}_{q-\lambda^{\prime}}^{p^{n-1}}(A)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\operatorname{TR}_{q-\lambda^{\prime}}^{n^{\phantom{n}}}(A;p)}}$

becomes an isomorphism after localization at $p$.

######  Proof 2.1.

The proof is by induction on the number $k$ of positive divisors of
$r^{\prime}$. If $k=1$, or equivalently, if $r^{\prime}=1$ then $\gamma$ is
the identity map and the statement holds trivially. So we let $k\geqslant 2$
and assume that the statement holds whenever $r^{\prime}$ has at most $k-1$
divisors. Let $\ell$ be a prime divisor of $r^{\prime}$, and let
$v=v_{\ell}(r^{\prime})=v_{\ell}(r)$. We show that the map

$(R_{\ell},F_{\ell^{v}})\colon\operatorname{TR}_{q-\lambda}^{r}(A)\to\operatorname{TR}_{q-\lambda^{\prime}}^{r/\ell}(A)\times\operatorname{TR}_{q-\lambda}^{r/\ell^{v}}(A)$

becomes an isomorphism after localization at $p$. This will prove the
induction step, since $r/\ell$ and $r/\ell^{v}$ have at most $k-1$ divisors
and $\gamma=(\gamma\times\gamma)\circ(R_{\ell},F_{\ell^{v}})$. Now, by
Proposition 1.1, we have the long exact sequence

$\cdots\to\mathbb{H}_{q}(C_{\ell^{v}},\operatorname{TR}_{\,\boldsymbol{\cdot}-\lambda}^{r/\ell^{v}}(A))\xrightarrow{N_{\ell^{v}}}\operatorname{TR}_{q-\lambda}^{r}(A)\xrightarrow{R_{\ell}}\operatorname{TR}_{q-\lambda^{\prime}}^{r/\ell}(A)\xrightarrow{\partial}\mathbb{H}_{q-1}(C_{\ell^{v}},\operatorname{TR}_{\,\boldsymbol{\cdot}-\lambda}^{r/\ell^{v}}(A))\to\cdots$

Moreover, one readily shows that the composite map

$\operatorname{TR}_{q-\lambda}^{r/\ell^{v}}(A)\xrightarrow{\epsilon}\mathbb{H}_{q}(C_{\ell^{v}},\operatorname{TR}_{\,\boldsymbol{\cdot}-\lambda}^{r/\ell^{v}}(A))\xrightarrow{N_{\ell^{v}}}\operatorname{TR}_{q-\lambda}^{r}(A)\xrightarrow{F_{\ell^{v}}}\operatorname{TR}_{q-\lambda}^{r/\ell^{v}}(A),$

where $\epsilon$ is the edge homomorphism of the skeleton spectral sequence
(1.2), is equal to the composition $F_{\ell^{v}}V_{\ell^{v}}$ of the Frobenius
and Verschiebung maps which, in turn, is equal to the map given by
multiplication by $\ell^{v}$. Hence, after localization at $p$, $F_{\ell^{v}}$
is the projection onto a direct summand of the group
$\operatorname{TR}_{q-\lambda}^{r}(Z)$. The long exact sequence shows that the
map $(R_{\ell},F_{\ell^{v}})$ becomes an isomorphism after localization at $p$
as desired.

The maps $F_{s}$, $V_{s}$, and $R_{s}$ may also be expressed as products of
their $p$-typical analogs

$\displaystyle F=F_{p}\colon$
$\displaystyle\operatorname{TR}_{q-\lambda}^{n}(A;p)\to\operatorname{TR}_{q-\lambda}^{n-1}(A;p)\hskip
19.91692pt\text{(Frobenius)}\hfill$ $\displaystyle V=V_{p}\colon$
$\displaystyle\operatorname{TR}_{q-\lambda}^{n-1}(A;p)\to\operatorname{TR}_{q-\lambda}^{n}(A;p)\hskip
19.91692pt\text{(Verschiebung)}\hfill$ $\displaystyle R=R_{p}\colon$
$\displaystyle\operatorname{TR}_{q-\lambda}^{n}(A;p)\to\operatorname{TR}_{q-\lambda^{\prime}}^{n-1}(A;p)\hskip
17.92523pt\text{(restriction)}\hfill$

Suppose that $r=st$ and write $s=p^{v}s^{\prime}$ and $t=p^{n-v-1}t^{\prime}$
with $s^{\prime}$ and $t^{\prime}$ not divisible by $p$. Then there are three
commutative square diagrams

$\textstyle{{\operatorname{TR}_{q-\lambda}^{r}(A)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\scriptstyle{F_{s}}$$\textstyle{{\prod_{j\mid
r^{\prime}}\operatorname{TR}_{q-\lambda^{\prime}}^{n}(A;p)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F_{s}^{\gamma}}$$\textstyle{{\operatorname{TR}_{q-\lambda}^{r}(A)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\scriptstyle{R_{s}}$$\textstyle{{\prod_{j\mid
r^{\prime}}\operatorname{TR}_{q-\lambda^{\prime}}^{n}(A;p)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{R_{s}^{\gamma}}$$\textstyle{{\operatorname{TR}_{q-\lambda}^{t}(A)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\scriptstyle{V_{s}}$$\textstyle{{\prod_{j\mid
t^{\prime}}\operatorname{TR}_{q-\lambda^{\prime}}^{n-v}(A;p)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{V_{s}^{\gamma}}$$\textstyle{{\operatorname{TR}_{q-\lambda^{\prime}}^{t}(A)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\textstyle{{\prod_{j\mid
t^{\prime}}\operatorname{TR}_{q-\lambda^{\prime\prime}}^{n-v}(A;p),}}$ (2.1)

where the maps $F_{s}^{\gamma}$, $V_{s}^{\gamma}$, and $R_{s}^{\gamma}$ are
defined as follows: The map $F_{s}^{\gamma}$ takes the factor indexed by a
divisor $j$ of $r^{\prime}$ that is divisible by $s^{\prime}$ to the factor
indexed by the divisor $j/s^{\prime}$ of $t^{\prime}$ by the map $F^{v}$ and
annihilates the remaining factors. The map $V_{s}^{\gamma}$ takes the factor
indexed by the divisor $j$ of $t^{\prime}$ to the factor indexed by the
divisor $s^{\prime}j$ of $r^{\prime}$ by the map $s^{\prime}V^{v}$. Finally,
the map $R_{s}^{\gamma}$ takes the factor indexed by a divisor $j$ of
$t^{\prime}$ to the factor indexed by the same divisor $j$ of $t^{\prime}$ by
the map $R^{v}$ and annihilates the factors indexed by divisors $j$ of
$r^{\prime}$ that do not divide $t^{\prime}$.

Let $M_{p}$ be the equivariant Moore spectrum defined by the mapping cone of
the multiplication by $p$ map on the sphere $\mathbb{T}$-spectrum. The
equivariant homotopy groups with $\mathbb{Z}/p\mathbb{Z}$-coefficients

$\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};p,\mathbb{Z}/p\mathbb{Z})=[S^{q}\wedge(\mathbb{T}/C_{p^{n-1}})_{+},M_{p}\wedge
S^{\lambda}\wedge T(\mathbb{Z})]_{\mathbb{T}}$

were evaluated for $p$ odd by Tsalidis [16], and for all $p$ by the first and
second authors [1]. We recall the result.

######  Theorem 2.1 ((Angeltveit-Gerhardt, Tsalidis)).

Let $p$ be a prime number, let $n$ be a positive integer, let $\lambda$ be a
finite dimensional complex $\mathbb{T}$-representation, and define

$\delta_{p}(\lambda)=(1-p)\sum_{s\geqslant
0}\dim_{\mathbb{C}}(\lambda^{C_{p^{s}}})p^{s}.$

Then the finite $\mathbb{Z}_{(p)}$-modules
$\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};p,\mathbb{Z}/p\mathbb{Z})$ have
the following structure:

  1. (i)

For $q\geqslant 2\dim_{\mathbb{C}}(\lambda)$,
$\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};p,\mathbb{Z}/p\mathbb{Z})$ has
length $n$, if $q$ is congruent to $2\delta_{p}(\lambda)$ or
$2\delta_{p}(\lambda)-1$ modulo $2p^{n}$, and $n-1$, otherwise.

  2. (ii)

For $2\dim_{\mathbb{C}}(\lambda^{C_{p^{s}}})\leqslant
q<2\dim_{\mathbb{C}}(\lambda^{C_{p^{s-1}}})$ with $1\leqslant s<n$,
$\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};p,\mathbb{Z}/p\mathbb{Z})$ has
length $n-s$, if $q$ is congruent to $2\delta_{p}(\lambda^{C_{p^{s}}})$ or
$2\delta_{p}(\lambda^{C_{p^{s}}})-1$ modulo $2p^{n-s}$, and $n-s-1$,
otherwise.

  3. (iii)

For $q<2\dim_{\mathbb{C}}(\lambda^{C_{p^{n-1}}})$,
$\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};p,\mathbb{Z}/p\mathbb{Z})$ is
zero.

We show in Corollary 3 below that the groups
$\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};p,\mathbb{Z}/p\mathbb{Z})$ have
exponent $p$ for all prime numbers $p$.

######  Proof 2.2 (of Theorem B (i)).

By Lemma 1.2, $\operatorname{TR}_{2i-\lambda}^{n}(\mathbb{Z};p)$ is a finitely
generated abelian group, and hence, it suffices to show that it is torsion
free. We first show that the $p$-torsion subgroup is trivial. Comparing
Theorem 2.2 and Lemma 1.2, we find that for all integers $i$

$\operatorname{length}_{\mathbb{Z}_{(p)}}\\!\\!\operatorname{TR}_{2i-\lambda}^{n}(\mathbb{Z};p,\mathbb{Z}/p\mathbb{Z})-\operatorname{length}_{\mathbb{Z}_{(p)}}\\!\\!\operatorname{TR}_{2i-1-\lambda}^{n}(\mathbb{Z};p,\mathbb{Z}/p\mathbb{Z})=\operatorname{rk}_{\mathbb{Z}}\operatorname{TR}_{2i-\lambda}^{n}(\mathbb{Z};p).$

Moreover, Lemma 1.2 shows that for every integer $i$,
$\operatorname{TR}_{2i-1-\lambda}^{n}(\mathbb{Z};p)_{(p)}$ is a finite
$p$-primary torsion group. By a Bockstein spectral sequence argument, we
conclude that $\operatorname{TR}_{2i-\lambda}^{n}(\mathbb{Z};p)_{(p)}$ is
torsion free; compare [5, Proposition 13]. This shows that the group
$\operatorname{TR}_{2i-\lambda}^{n}(\mathbb{Z};p)$ has no $p$-torsion. To see
that it has no prime to $p$ torsion, we use that, by Proposition 1, the map

$(R^{n-1-s}F^{s})\colon\operatorname{TR}_{2i-\lambda}^{n}(\mathbb{Z};p)\to\prod_{0\leqslant
s<n}\operatorname{TR}_{2i-\lambda^{(n-1-s)}}^{1}(\mathbb{Z};p)$

becomes an isomorphism after inverting $p$. Therefore, Bökstedt’s result
recalled earlier shows that also
$\operatorname{TR}_{2i-\lambda}^{n}(\mathbb{Z};p)[1/p]$ is torsion free. This
completes the proof.

######  Lemma 2.3.

Let $p$ be a prime number, let $n$ be a positive integer, and let $\lambda$ be
a finite dimensional complex $\mathbb{T}$-representation. Then the restriction
map

$R\colon\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};p)\to\operatorname{TR}_{q-\lambda^{\prime}}^{n-1}(\mathbb{Z};p)$

is surjective for every even integer $q$.

######  Proof 2.4.

We see as in the proof of Addendum 1.3 that the map of the statement is an
epimorphism, for $q\leqslant 2\dim_{\mathbb{C}}(\lambda)$. Moreover, Lemma 1.2
and Theorem B(i) show that
$\operatorname{TR}_{q-\lambda^{\prime}}^{n-1}(\mathbb{Z};p)$ is zero, for
$q>2\dim_{\mathbb{C}}(\lambda^{\prime})$ and even. The lemma follows, since
$\dim_{\mathbb{C}}(\lambda^{\prime})\leqslant\dim_{\mathbb{C}}(\lambda)$.

######  Proposition 2.

Let $p$ be a prime number, let $n$ be a positive integer, and let $\lambda$ be
a finite dimensional complex $\mathbb{T}$-representation. Then in the skeleton
spectral sequence

$E_{s,t}^{2}=H_{s}(C_{p^{n-1}},\operatorname{TR}_{t-\lambda}^{1}(\mathbb{Z};p))\Rightarrow\mathbb{H}_{s+t}(C_{p^{n-1}},\operatorname{TR}_{\,\boldsymbol{\cdot}-\lambda}^{1}(\mathbb{Z};p)),$

every non-zero differential $d^{r}\colon E_{s,t}^{r}\to E_{s-r,t+r-1}^{r}$ is
supported in odd total degree.

######  Proof 2.5.

We must show that if $s+t$ is even then $d^{r}\colon E_{s,t}^{r}\to
E_{s-r,t+r-1}^{r}$ is zero. Suppose first that $s$ and $t$ are both even. Then
$E_{s,t}^{2}$ is zero unless $s=0$ and $t=2\dim_{\mathbb{C}}(\lambda)$.
Therefore, in this case, $d^{r}\colon E_{s,t}^{r}\to E_{s-r,t+r-1}^{r}$ is
zero. Suppose next that $s$ and $t$ are both odd and that $r$ is even. Then
$E_{s,t}^{2}$ is zero unless $t>2\dim_{\mathbb{C}}(\lambda)$, and
$E_{s-r,t+r-1}^{2}$ is zero unless $t+r-1=2\dim_{\mathbb{C}}(\lambda)$. It
follows that, also in this case, $d^{r}\colon E_{s,t}^{r}\to
E_{s-r,t+r-1}^{r}$ is zero. It remains to prove that if $r$, $s$, and $t$ are
all odd then $d^{r}\colon E_{s,t}^{r}\to E_{s-r,t+r-1}^{r}$ is zero.

To this end, we use that, for all integers $n^{\prime}\geqslant n\geqslant 1$,
the iterated Frobenius map

$F^{n^{\prime}-n}\colon\mathbb{H}_{q}(C_{p^{n^{\prime}-1}},\operatorname{TR}_{\,\boldsymbol{\cdot}-\lambda}^{1}(\mathbb{Z};p))\to\mathbb{H}_{q}(C_{p^{n-1}},\operatorname{TR}_{\,\boldsymbol{\cdot}-\lambda}^{1}(\mathbb{Z};p))$

induces a map of skeleton spectral sequences that we write

$F^{n^{\prime}-n}\colon E_{s,t}^{r}(n^{\prime},\lambda)\to
E_{s,t}^{r}(n,\lambda).$

The map of $E^{2}$-terms is given by the transfer map in group homology and is
readily evaluated; see for instance [5, Lemma 6]. It is surjective if $s$ is
odd, and zero if is $s$ even, $t$ is odd, and $n^{\prime}-n$ is sufficiently
large. We prove by induction on $r\geqslant 2$ that, for all odd integers $s$
and $t$ and for all $n$ and $\lambda$ as in the statement, the differential
$d^{r}\colon E_{s,t}^{r}(n,\lambda)\to E_{s-r,t+r-1}^{r}(n,\lambda)$ is zero
and the Frobenius map $F\colon E_{s,t}^{r}(n+1,\lambda)\to
E_{s,t}^{r}(n,\lambda)$ surjective. The case $r=2$ was proved above. So we let
$r\geqslant 3$ and assume, inductively, that the statement has been proved for
$r-1$. Since the differential $d^{r-1}\colon E_{s,t}^{r-1}(n,\lambda)\to
E_{s-(r-1),t+r-2}^{r-1}(n,\lambda)$ is zero by the inductive hypothesis, the
canonical isomorphism

$H(E_{s+r-1,t-(r-2)}^{r-1}(n,\lambda)\xrightarrow{d^{r-1}}E_{s,t}^{r-1}(n,\lambda)\xrightarrow{d^{r-1}}E_{s-(r-1),t+r-2}^{r-1}(n,\lambda))\xrightarrow{\sim}E_{s,t}^{r}(n,\lambda)$

gives rise to a canonical surjection $\pi\colon
E_{s,t}^{r-1}(n,\lambda)\twoheadrightarrow E_{s,t}^{r}(n,\lambda)$. Moreover,
by naturality of the skeleton spectral sequence, the diagram

$\textstyle{{E_{s,t}^{r-1}(n+1,\lambda)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{F}$$\textstyle{{E_{s,t}^{r}(n+1,\lambda)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F}$$\textstyle{{E_{s,t}^{r-1}(n,\lambda)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{{E_{s,t}^{r}(n,\lambda)}}$

commutes. Since the left-hand vertical map $F$ is surjective by the inductive
hypothesis, we conclude that the right-hand vertical map $F$ is surjective as
desired. It remains to prove that the differential $d^{r}\colon
E_{s,t}^{r}(n,\lambda)\to E_{s-r,t+r-1}^{r}(n,\lambda)$ is zero. The case
where $r$ is even was proved above, and in the case where $r$ is odd, we
consider the following commutative diagram:

$\textstyle{{E_{s,t}^{r}(n^{\prime},\lambda)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d^{r}}$$\scriptstyle{F^{n^{\prime}-n}}$$\textstyle{{E_{s-r,t+r-1}^{r}(n^{\prime},\lambda)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F^{n^{\prime}-n}}$$\textstyle{{E_{s,t}^{r}(n,\lambda)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d^{r}}$$\textstyle{{E_{s-r,t+r-1}^{r}(n,\lambda)}}$

We have just proved that the left-hand vertical map is surjective. Moreover,
as recalled above, the right-hand vertical map is zero if $n^{\prime}-n$ is
sufficiently large. Indeed, $s-r$ is even and $t-r+1$ is odd. Hence, the lower
horizontal map $d^{r}$ is zero as desired. This proves the induction step and
the proposition.

######  Remark 2.6.

The proof of Proposition 2 also shows that in the Tate spectral sequence

$\hat{E}_{s,t}^{2}=\hat{H}^{-s}(C_{p^{n-1}},\operatorname{TR}_{t-\lambda}^{1}(\mathbb{Z};p))\Rightarrow\hat{\mathbb{H}}^{-s-t}(C_{p^{n-1}},\operatorname{TR}_{\,\boldsymbol{\cdot}-\lambda}^{1}(\mathbb{Z};p)),$

every non-zero differential is supported in even total degree. It remains an
important problem to determine the differential structure of the two spectral
sequences.

######  Proof 2.7 (of Theorem B (ii)).

First, for $i\leqslant\dim_{\mathbb{C}}(\lambda)$, we see as in the proof of
Lemma 2.3 that the restriction map

$R\colon\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};p)\to\operatorname{TR}_{q-\lambda^{\prime}}^{n-1}(\mathbb{Z};p)$

is an isomorphism, so the statement holds in this case. Next, for
$i=\dim_{\mathbb{C}}(\lambda)+1$, Proposition 1.1 and Lemmas 1.2 and 2.3 give
a short exact sequence

$0\to\mathbb{H}_{q}(C_{p^{n-1}},\operatorname{TR}_{\,\boldsymbol{\cdot}-\lambda}^{1}(\mathbb{Z};p))\to\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};p)\to\operatorname{TR}_{q-\lambda^{\prime}}^{n-1}(\mathbb{Z};p)\to
0$

and the skeleton spectral sequence (1.2) shows that the left-hand group has
order $p^{n-1}$. So the statement also holds in this case. Finally, for
$i>\dim_{\mathbb{C}}(\lambda)+1$, Proposition 1.1 and Lemma 1.2 give a four-
term exact sequence

$\displaystyle 0$
$\displaystyle\to\mathbb{H}_{q}(C_{p^{n-1}},\operatorname{TR}_{\,\boldsymbol{\cdot}-\lambda}^{1}(\mathbb{Z};p))\to\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};p)\to\operatorname{TR}_{q-\lambda^{\prime}}^{n-1}(\mathbb{Z};p)$
$\displaystyle\to\mathbb{H}_{q-1}(C_{p^{n-1}},\operatorname{TR}_{\,\boldsymbol{\cdot}-\lambda}^{1}(\mathbb{Z};p))\to
0$

which shows that the orders of the four groups satisfy the equality

$\lvert\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};p)\rvert\big{/}\lvert\operatorname{TR}_{q-1-\lambda^{\prime}}^{n-1}(\mathbb{Z};p)\rvert=\lvert\mathbb{H}_{q}(C_{p^{n-1}},\operatorname{TR}_{\,\boldsymbol{\cdot}-\lambda}^{1}(\mathbb{Z};p))\rvert\big{/}\lvert\mathbb{H}_{q-1}(C_{p^{n-1}},\operatorname{TR}_{\,\boldsymbol{\cdot}-\lambda}^{1}(\mathbb{Z};p))\rvert.$

To evaluate the ratio on the right-hand side, we consider the skeleton
spectral sequence (1.2). We may write the ratio in question as

$\lvert\mathbb{H}_{q}(C_{p^{n-1}},\operatorname{TR}_{\,\boldsymbol{\cdot}-\lambda}^{1}(\mathbb{Z};p))\rvert\big{/}\lvert\mathbb{H}_{q-1}(C_{p^{n-1}},\operatorname{TR}_{\,\boldsymbol{\cdot}-\lambda}^{1}(\mathbb{Z};p))\rvert=\big{(}\\!\prod_{s+t=q}\lvert
E_{s,t}^{\infty}\rvert\big{)}\big{/}\big{(}\\!\\!\prod_{s+t=q-1}\lvert
E_{s,t}^{\infty}\rvert\big{)}.$

By Proposition 2, for all $r\geqslant 2$ and all $s$ and $t$ with $s+t$ is odd
we have an exact sequence

$0\to E_{s,t}^{r+1}\to E_{s,t}^{r}\xrightarrow{d^{r}}E_{s-r,t+r-1}^{r}\to
E_{s-r,t+r-1}^{r+1}\to 0.$

Hence, by induction on $r$, we find that

$\big{(}\\!\prod_{s+t=q}\lvert
E_{s,t}^{\infty}\rvert\big{)}\big{/}\big{(}\\!\\!\prod_{s+t=q-1}\lvert
E_{s,t}^{\infty}\rvert\big{)}=\big{(}\\!\prod_{s+t=q}\lvert
E_{s,t}^{2}\rvert\big{)}\big{/}\big{(}\\!\\!\prod_{s+t=q-1}\lvert
E_{s,t}^{2}\rvert\big{)}$

and the ratio on the right-hand side is readily seen to be equal to

$\lvert E_{0,q}^{2}\rvert\cdot\lvert
E_{q-2\dim_{\mathbb{C}}(\lambda),2\dim_{\mathbb{C}}(\lambda)}^{2}\rvert=(i-\dim_{\mathbb{C}}(\lambda))\cdot
p^{n-1}.$

This completes the proof.

######  Proof 2.8 (of Theorem B (iii)).

First, for $q$ odd, we use that the Verschiebung map in question is equal to
the composition of the edge homomorphism

$\epsilon\colon\operatorname{TR}_{q-\lambda}^{n-1}(\mathbb{Z};p)\to\mathbb{H}_{q}(C_{p},\operatorname{TR}_{\,\boldsymbol{\cdot}-\lambda}^{n-1}(\mathbb{Z};p))$

of the skeleton spectral sequence (1.2) and the norm map $N_{n-1}$ in the long
exact sequence

$\textstyle{{\cdots}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{H}_{q}(C_{p},\operatorname{TR}_{\boldsymbol{\cdot}-\lambda}^{n-1}(\mathbb{Z};p))}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{N_{n-1}}$$\textstyle{{\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};p)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{R^{n-1}}$$\textstyle{{\operatorname{TR}_{q-\lambda^{(n-1)}}^{1}(\mathbb{Z};p)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\cdots}}$

from Proposition 1.1. Since $q$ is odd, Lemma 2.3 shows that the latter map is
injective. Hence, it will suffice to show that also the edge homomorphism is
injective, or equivalently, that in the skeleton spectral sequence, all
differentials of the form

$d^{r}\colon E_{r,q-r+1}^{r}\to E_{0,q}^{r}$

are zero. If $r$ is even, then $q-r+1$ is even and Theorem B (i) shows that
the group $E_{r,q-r+1}^{r}$ is zero. Hence, $d^{r}$ is zero in this case. If
$r$ is odd, we consider the iterated Frobenius map

$F^{v^{\prime}-v}\colon\mathbb{H}_{q}(C_{p^{v^{\prime}}},\operatorname{TR}_{\boldsymbol{\cdot}-\lambda}^{n-1}(\mathbb{Z};p))\to\mathbb{H}_{q}(C_{p^{v}},\operatorname{TR}_{\boldsymbol{\cdot}-\lambda}^{n-1}(\mathbb{Z};p)).$

It induces a map of spectral sequences that we write

$F^{v^{\prime}-v}\colon E_{s,t}^{r}(v^{\prime},\lambda)\to
E_{s,t}^{r}(v,\lambda).$

As in the proof of Theorem B (ii), an induction on $r\geqslant 2$ shows that,
for all $v^{\prime}\geqslant v\geqslant 1$, the left-hand vertical map and the
horizontal maps in the diagram

$\textstyle{{E_{r,q-r+1}^{r}(v^{\prime},\lambda)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d^{r}}$$\scriptstyle{F^{v^{\prime}-v}}$$\textstyle{{E_{0,q}^{r}(v^{\prime},\lambda)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F^{v^{\prime}-v}}$$\textstyle{{E_{r,q-r+1}^{r}(v,\lambda)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d^{r}}$$\textstyle{{E_{0,q}^{r}(v,\lambda)}}$

are surjective and zero, respectively. The proof of the induction step uses
that, for $v^{\prime}-v$ sufficiently large, the right-hand vertical map is
zero. This proves the statement for $q$ odd.

Finally, suppose that $q$ is even. We let $\mu$ be the direct sum of $\lambda$
and the one dimensional complex representation $\mathbb{C}(p^{n-2})$. Then [6,
Proposition 4.2] gives a long exact sequence

$\cdots\to\operatorname{TR}_{q+1-\mu}^{n}(\mathbb{Z};p)\xrightarrow{(F,-Fd)}\overset{\displaystyle{\operatorname{TR}_{q-1-\lambda}^{n-1}(\mathbb{Z};p)}}{\underset{\displaystyle{\operatorname{TR}_{q-\lambda}^{n-1}(\mathbb{Z};p)}}{\oplus}}\xrightarrow{dV+V}\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};p)\xrightarrow{\iota_{*}}\operatorname{TR}_{q-\mu}^{n}(\mathbb{Z};p)\to\cdots$

Now, we proved in Thm B (i) that
$\operatorname{TR}_{q+1-\mu}^{n}(\mathbb{Z};p)$ and
$\operatorname{TR}_{q-1-\lambda}^{n-1}(\mathbb{Z};p)$ are finite abelian
groups while $\operatorname{TR}_{q-\lambda}^{n-1}(\mathbb{Z};p)$,
$\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};p)$, and
$\operatorname{TR}_{q-\mu}^{n}(\mathbb{Z};p)$ are free abelian groups. Hence,
we obtain the exact sequence of free abelian groups

$\textstyle{{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\operatorname{TR}_{q-\lambda}^{n-1}(\mathbb{Z};p)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{V}$$\textstyle{{\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};p)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{*}}$$\textstyle{{\operatorname{TR}_{q-\mu}^{n}(\mathbb{Z};p)}}$

which shows that for $q$ even, the Verschiebung map is the inclusion of a
direct summand as stated. This concludes the proof of Theorem B.

######  Corollary 3.

Let $n$ be a positive integer, $p$ a prime number, and $\lambda$ a finite
dimensional complex $\mathbb{T}$-representation. Then
$\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};p,\mathbb{Z}/p\mathbb{Z})$ has
exponent $p$ for every integer $q$.

######  Proof 2.9.

We first let $p=2$ and consider the coefficient long exact sequence

$\cdots\to\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};2)\xrightarrow{2}\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};2)\xrightarrow{\iota}\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};2,\mathbb{Z}/2\mathbb{Z})\xrightarrow{\beta}\operatorname{TR}_{q-1-\lambda}^{n}(\mathbb{Z};2)\to\cdots.$

It is proved in [2, Theorem 1.1] that the composition

$\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};2,\mathbb{Z}/2\mathbb{Z})\xrightarrow{\beta}\operatorname{TR}_{q-1-\lambda}^{n}(\mathbb{Z};2)\xrightarrow{\eta}\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};2)\xrightarrow{\iota}\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};2,\mathbb{Z}/2\mathbb{Z})$

is equal to multiplication by $2$. Now, Theorem B shows that for $q$ odd the
map $\beta$ is zero, and that for $q$ even the map $\eta$ is zero. Hence, the
group $\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};2,\mathbb{Z}/2\mathbb{Z})$
is annihilated by multiplication by $2$ as stated. Finally, for $p$ odd, [2,
Theorem 1.1] shows that the multiplication by $p$ map on
$\operatorname{TR}_{q-\lambda}^{n}(\mathbb{Z};p,\mathbb{Z}/p\mathbb{Z})$ is
equal to zero.

##  3 The groups $K_{q}(\mathbb{Z}[x]/(x^{m}),(x))$

In this section, we prove Theorem A of the introduction.

######  Proposition 4.

Let $m$ and $r$ be positive integers, let $i$ be a non-negative integer, and
let $d=d(m,r)$ be the integer part of $(r-1)/m$. Then:

  1. (i)

The abelian group $\lim_{R}\operatorname{TR}_{2i-\lambda_{d}}^{r}(\mathbb{Z})$
is free of rank $m$.

  2. (ii)

The abelian group
$\lim_{R}\operatorname{TR}_{2i-1-\lambda_{d}}^{r}(\mathbb{Z})$ is finite of
order $(mi)!(i!)^{m}$.

######  Proof 3.1.

It follows from Lemma 1.2 and Addendum 1.3 that the groups
$\lim_{R}\operatorname{TR}_{q-\lambda_{d}}^{r}(\mathbb{Z})$ are finitely
generated. Hence, it suffices to show that for every prime number $p$, the
$\mathbb{Z}_{(p)}$-module
$\lim_{R}\operatorname{TR}_{2i-\lambda_{d}}^{r}(\mathbb{Z})_{(p)}$ has finite
rank $m$ and the $\mathbb{Z}_{(p)}$-module
$\lim_{R}\operatorname{TR}_{2i-1-\lambda_{d}}^{r}(\mathbb{Z})_{(p)}$ has
finite length $v_{p}((mi)!(i!)^{m})$. We fix a prime number $p$ and let
$I_{p}$ be the set of positive integers not divisible by $p$. It follows from
Proposition 1 that there is a canonical isomorphism

$\lim_{R}\operatorname{TR}_{q-\lambda_{d}}^{r}(\mathbb{Z})_{(p)}\xrightarrow{\sim}\prod_{j\in
I_{p}}\lim_{R}\operatorname{TR}_{q-\lambda_{d}}^{n}(\mathbb{Z};p)_{(p)},$

where, on the left-hand side, the limit ranges over the set of positive
integers ordered under division and $d=d(m,r)$, and where, on the right-hand
side, the limits range over the set of non-negative integers ordered
additively and $d=d(m,p^{n-1}j)$. Moreover, on the $j$th factor of the
product, the canonical projection

$\lim_{R}\operatorname{TR}_{q-\lambda_{d}}^{n}(\mathbb{Z};p)_{(p)}\to\operatorname{TR}_{q-\lambda_{d}}^{s}(\mathbb{Z};p)_{(p)}$

is an isomorphism for $q<2d(m,p^{s}j)$; see [6, Lemma 2.6]. The requirement
that $2i-2$ and $2i-1$ be strictly smaller than $2d(m,p^{s}j)$ is equivalent
to the requirement that $mi<p^{s}j$. Hence, for $q=2i-2$ or $q=2i-1$, we have
a canonical isomorphism

$\lim_{R}\operatorname{TR}_{q-\lambda_{d}}^{r}(\mathbb{Z})_{(p)}\xrightarrow{\sim}\prod_{\begin{subarray}{c}1\leqslant
j\leqslant mi\\\ j\in
I_{p}\end{subarray}}\operatorname{TR}_{q-\lambda_{d}}^{s}(\mathbb{Z};p)_{(p)},$

where $s=s_{p}(m,i,j)$ is the unique integer such that

$p^{s-1}j\leqslant mi<p^{s}j.$

Now, from Theorem B (i) we find that

$\displaystyle\operatorname{rk}_{\mathbb{Z}_{(p)}}\lim_{R}\operatorname{TR}_{2i-\lambda_{d}}^{r}(\mathbb{Z})_{(p)}=\lvert\\{j\in
I_{p}\mid\text{$i=d(m,p^{n-1}j)$, for some $n\geqslant 1$}\\}\rvert$
$\displaystyle=\lvert\\{r\in\mathbb{N}\mid
i=d(m,r)\\}\rvert=\lvert\\{mi+1,mi+2,\dots,mi+m\\}\rvert=m$

which proves statement (i). Similarly, we have

$\operatorname{length}_{\mathbb{Z}_{(p)}}\lim_{R}\operatorname{TR}_{2i-1-\lambda_{d}}^{r}(\mathbb{Z})_{(p)}=\sum_{\begin{subarray}{c}1\leqslant
j\leqslant mi\\\ j\in
I_{p}\end{subarray}}\operatorname{length}_{\mathbb{Z}_{(p)}}\operatorname{TR}_{2i-1-\lambda_{d}}^{s}(\mathbb{Z};p)_{(p)},$

where $s=s_{p}(m,i,j)$, and Theorem B (ii) shows that the right-hand side is
equal to

$\displaystyle\sum_{\begin{subarray}{c}1\leqslant j\leqslant mi\\\ j\in
I_{p}\end{subarray}}\sum_{1\leqslant t\leqslant
s}\big{(}v_{p}(i-d(m,p^{t-1}j))+t-1\big{)}=\sum_{1\leqslant k\leqslant
mi}\big{(}v_{p}(i-d(m,k))+v_{p}(k)\big{)}$ $\displaystyle=m\sum_{0\leqslant
l<i}v_{p}(i-l)+\\!\\!\sum_{1\leqslant k\leqslant mi}v_{p}(k)=m\sum_{1\leqslant
k\leqslant i}v_{p}(k)+\\!\\!\sum_{1\leqslant k\leqslant
mi}v_{p}(k)=v_{p}((i!)^{m}(mi)!)$

which proves statement (ii).

######  Proposition 5.

Let $m$ and $r$ be positive integers, let $i$ be a non-negative integer, and
let $d=d(m,r)$ be the integer part of $(r-1)/m$. Then:

  1. (i)

The abelian group
$\lim_{R}\operatorname{TR}_{2i-\lambda_{d}}^{r/m}(\mathbb{Z})$ is free of rank
$1$.

  2. (ii)

The abelian group
$\lim_{R}\operatorname{TR}_{2i-1-\lambda_{d}}^{r/m}(\mathbb{Z})$ is finite of
order $(i!)^{2}$.

######  Proof 3.2.

It follows from Lemma 1.2 and Addendum 1.3 that the groups
$\lim_{R}\operatorname{TR}_{q-\lambda_{d}}^{r/m}(\mathbb{Z})$ are finitely
generated. We fix a prime number $p$ and write $m=p^{v}m^{\prime}$ with
$m^{\prime}$ not divisible by $p$. Then for $q=2i-2$ and $q=2i-1$, there is a
canonical isomorphism

$\lim_{R}\operatorname{TR}_{q-\lambda_{d}}^{r/m}(\mathbb{Z})_{(p)}\xrightarrow{\sim}\prod_{\begin{subarray}{c}1\leqslant
j\leqslant mi\\\ j\in
m^{\prime}I_{p}\end{subarray}}\operatorname{TR}_{q-\lambda_{d}}^{s-v}(\mathbb{Z};p)_{(p)},$

where $s=s_{p}(m,i,j)$. From Theorem B (i) we find

$\displaystyle\operatorname{rk}_{\mathbb{Z}_{(p)}}\lim_{R}\operatorname{TR}_{2i-\lambda_{d}}^{r/m}(\mathbb{Z})_{(p)}=\lvert\\{j\in
m^{\prime}I_{p}\mid\text{$i=d(m,p^{n+v-1}j)$, for some $n\geqslant
1$}\\}\rvert$ $\displaystyle=\lvert\\{r\in m\mathbb{N}\mid
i=d(m,r)\\}\rvert=\lvert\\{mi+m\\}\rvert=1$

which proves statement (i). Similarly, from Theorem B (ii), we have

$\displaystyle\operatorname{length}_{\mathbb{Z}_{(p)}}\lim_{R}\operatorname{TR}_{2i-1-\lambda_{d}}^{r}(\mathbb{Z})_{(p)}=\sum_{\begin{subarray}{c}1\leqslant
j\leqslant mi\\\ j\in
m^{\prime}I_{p}\end{subarray}}\operatorname{length}_{\mathbb{Z}_{(p)}}\operatorname{TR}_{2i-1-\lambda_{d}}^{s-v}(\mathbb{Z};p)_{(p)}$
$\displaystyle=\sum_{\begin{subarray}{c}1\leqslant j\leqslant mi\\\ j\in
m^{\prime}I_{p}\end{subarray}}\sum_{1\leqslant t\leqslant
s-v}\big{(}v_{p}(i-d(m,p^{t+v-1}j))+t-1\big{)}=\sum_{1\leqslant k\leqslant
i}\big{(}v_{p}(i-d(m,km))+v_{p}(k)\big{)}$ $\displaystyle=\sum_{0\leqslant
l<i}v_{p}(i-l)+\\!\\!\sum_{1\leqslant k\leqslant i}v_{p}(k)=2\sum_{1\leqslant
k\leqslant i}v_{p}(k)=v_{p}((i!)^{2})$

which proves statement (ii).

######  Proposition 6.

Let $m$ and $r$ be positive integers, and let $d=d(m,r)$ be the integer part
of $(r-1)/m$. Then the Verschiebung map

$V_{m}\colon\lim_{R}\operatorname{TR}_{q-\lambda_{d}}^{r/m}(\mathbb{Z})\to\lim_{R}\operatorname{TR}_{q-\lambda_{d}}^{r}(\mathbb{Z})$

is injective for all integers $q$, and has free abelian cokernel for all even
integers $q$.

######  Proof 3.3.

We fix a prime number $p$ and show that the Verschiebung map

$V_{m}\colon\lim_{R}\operatorname{TR}_{q-\lambda_{d}}^{r/m}(\mathbb{Z})_{(p)}\to\lim_{R}\operatorname{TR}_{q-\lambda_{d}}^{r}(\mathbb{Z})_{(p)}$

is injective for all integers $q$, and has cokernel a free
$\mathbb{Z}_{(p)}$-module for all even integers $q$. We write
$m=p^{v}m^{\prime}$ with $m^{\prime}$ not divisible by $p$. Then for $q=2i-2$
and $q=2i-1$ the map $V_{m}$ is canonically isomorphic to the map

$m^{\prime}V^{v}\colon\prod_{\begin{subarray}{c}1\leqslant j\leqslant mi\\\
j\in
m^{\prime}I_{p}\end{subarray}}\operatorname{TR}_{q-\lambda_{d}}^{s-v}(\mathbb{Z};p)_{(p)}\to\prod_{\begin{subarray}{c}1\leqslant
j\leqslant mi\\\ j\in
I_{p}\end{subarray}}\operatorname{TR}_{q-\lambda_{d}}^{s}(\mathbb{Z};p)_{(p)},$

where $s=s_{p}(m,i,j)$. The statement now follows from Theorem B (iii).

######  Proof 3.4 (of Theorem A).

The statement follows immediately from the long exact sequence recalled in the
introduction together with Propositions 4, 5, and 6.

##  4 The dual numbers

It follows from Theorem B that $K_{2i}(\mathbb{Z}[x]/(x^{2}),(x))$ is a finite
abelian group of order $(2i)!$. In this section, we investigate the structure
of these groups in low degrees.

######  Theorem 4.0.

There are isomorphisms

$\displaystyle K_{2}(\mathbb{Z}[x]/(x^{2}),(x))$
$\displaystyle\approx\mathbb{Z}/2\mathbb{Z}$ $\displaystyle
K_{4}(\mathbb{Z}[x]/(x^{2}),(x))$
$\displaystyle\approx\mathbb{Z}/8\mathbb{Z}\oplus\mathbb{Z}/3\mathbb{Z}$
$\displaystyle K_{6}(\mathbb{Z}[x]/(x^{2}),(x))$
$\displaystyle\approx\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/4\mathbb{Z}\oplus\mathbb{Z}/9\mathbb{Z}\oplus\mathbb{Z}/5\mathbb{Z}$

######  Proof 4.1.

We know from Theorem B that the orders of these three groups are as stated.
Hence, it suffices to show that the $2$-primary torsion subgroup of the groups
in degree $4$ and $6$ and the $3$-primary torsion subgroup of the group in
degree $6$ are as stated.

We first consider the group in degree $4$ which is given by the short exact
sequence

$0\to\lim_{R}\operatorname{TR}_{3-\lambda_{d}}^{r/2}(\mathbb{Z})\xrightarrow{V_{2}}\lim_{R}\operatorname{TR}_{3-\lambda_{d}}^{r}(\mathbb{Z})\to
K_{4}(\mathbb{Z}[x]/(x^{2}),(x))\to 0.$

The middle term in the short exact sequence decomposes $2$-locally as the
direct sum

$\lim_{R}\operatorname{TR}_{3-\lambda_{d}}^{r}(\mathbb{Z})_{(2)}\xrightarrow{\sim}\operatorname{TR}_{3-\lambda_{1}}^{3}(\mathbb{Z};2)_{(2)}\oplus\operatorname{TR}_{3-\lambda_{1}}^{1}(\mathbb{Z};2)_{(2)},$

where the first and second summands on the right-hand side correspond to $j=1$
and $j=3$, respectively. Similarly, the left-hand term in the short exact
sequence decomposes $2$-locally as

$\lim_{R}\operatorname{TR}_{3-\lambda_{d}}^{r/2}(\mathbb{Z})_{(2)}\xrightarrow{\sim}\operatorname{TR}_{3-\lambda_{1}}^{2}(\mathbb{Z};2)_{(2)}\oplus\operatorname{TR}_{3-\lambda_{1}}^{0}(\mathbb{Z};2)_{(2)}.$

The summands corresponding to $j=3$ are both zero. Hence, the $2$-primary
torsion subgroup of $K_{4}(\mathbb{Z}[x]/(x^{2}),(x))$ is canonically
isomorphic to the cokernel of the Verschiebung map

$V\colon\operatorname{TR}_{3-\lambda_{1}}^{2}(\mathbb{Z};2)_{(2)}\to\operatorname{TR}_{3-\lambda_{1}}^{3}(\mathbb{Z};2)_{(2)}.$

To evaluate this cokernel, we consider the following diagram

$\textstyle{{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\operatorname{TR}_{3}^{1}(\mathbb{Z};2)_{(2)}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{V}$$\textstyle{{\operatorname{TR}_{3}^{2}(\mathbb{Z};2)_{(2)}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{*}}$$\scriptstyle{V}$$\textstyle{{\operatorname{TR}_{3-\lambda_{1}}^{2}(\mathbb{Z};2)_{(2)}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{V}$$\textstyle{{0}}$$\textstyle{{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\operatorname{TR}_{3}^{1}(\mathbb{Z};2)_{(2)}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{V^{2}}$$\textstyle{{\operatorname{TR}_{3}^{3}(\mathbb{Z};2)_{(2)}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{*}}$$\textstyle{{\operatorname{TR}_{3-\lambda_{1}}^{3}(\mathbb{Z};2)_{(2)}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{0}}$

where the rows are exact by [6, Proposition 4.2]. It follows from [5, Theorem
18] that this diagram is isomorphic to the diagram

$\textstyle{{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{Z}/2\mathbb{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{4}$$\textstyle{{\mathbb{Z}/8\mathbb{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1}$$\scriptstyle{(a,b)}$$\textstyle{{\mathbb{Z}/4\mathbb{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(a,b)}$$\textstyle{{0}}$$\textstyle{{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{Z}/2\mathbb{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(0,4)}$$\textstyle{{\mathbb{Z}/8\mathbb{Z}\oplus\mathbb{Z}/8\mathbb{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1\oplus
1}$$\textstyle{{\mathbb{Z}/8\mathbb{Z}\oplus\mathbb{Z}/4\mathbb{Z}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{0}}$

where $a\in 2\mathbb{Z}/8\mathbb{Z}$ and $b\in(\mathbb{Z}/8\mathbb{Z})^{*}$.
Now, the cokernels of the middle and right-hand vertical maps are isomorphic
to $\mathbb{Z}/8\mathbb{Z}$. This shows that
$K_{4}(\mathbb{Z}[x]/(x^{2}),(x))_{(2)}$ is as stated.

We next consider the group in degree $6$ which is given by the short exact
sequence

$0\to\lim_{R}\operatorname{TR}_{5-\lambda_{d}}^{r/2}(\mathbb{Z})\xrightarrow{V_{2}}\lim_{R}\operatorname{TR}_{5-\lambda_{d}}^{r}(\mathbb{Z})\to
K_{6}(\mathbb{Z}[x]/(x^{2}),(x))\to 0$

and begin by evaluating the $2$-primary torsion subgroup. The middle term in
the short exact sequence decomposes $2$-locally as the direct sum

$\lim_{R}\operatorname{TR}_{5-\lambda_{d}}^{r}(\mathbb{Z})_{(2)}\xrightarrow{\sim}\operatorname{TR}_{5-\lambda_{1}}^{3}(\mathbb{Z};2)_{(2)}\oplus\operatorname{TR}_{5-\lambda_{2}}^{2}(\mathbb{Z};2)_{(2)}\oplus\operatorname{TR}_{5-\lambda_{2}}^{1}(\mathbb{Z};2)_{(2)},$

where the three summands on the right-hand side correspond to $j=1$, $j=3$,
and $j=5$, respectively. Similarly, the left-hand term in the short exact
sequence decomposes $2$-locally as

$\lim_{R}\operatorname{TR}_{5-\lambda_{d}}^{r/2}(\mathbb{Z})_{(2)}\xrightarrow{\sim}\operatorname{TR}_{5-\lambda_{1}}^{2}(\mathbb{Z};2)_{(2)}\oplus\operatorname{TR}_{5-\lambda_{2}}^{1}(\mathbb{Z};2)_{(2)}\oplus\operatorname{TR}_{5-\lambda_{2}}^{0}(\mathbb{Z};2)_{(2)}.$

The summands corresponding to $j=5$ are both zero. Hence, the $2$-primary
torsion subgroup of $K_{6}(\mathbb{Z}[x]/(x^{2}),(x))$ is canonically
isomorphic to the direct sum of the cokernels of

$\displaystyle V$
$\displaystyle\colon\operatorname{TR}_{5-\lambda_{1}}^{2}(\mathbb{Z};2)_{(2)}\to\operatorname{TR}_{5-\lambda_{1}}^{3}(\mathbb{Z};2)_{(2)}$
$\displaystyle V$
$\displaystyle\colon\operatorname{TR}_{5-\lambda_{2}}^{1}(\mathbb{Z};2)_{(2)}\to\operatorname{TR}_{5-\lambda_{2}}^{2}(\mathbb{Z};2)_{(2)}.$

We show that these are isomorphic to
$\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$ and
$\mathbb{Z}/4\mathbb{Z}$, respectively. The statement for the latter cokernel
follows directly from Theorems B and 2.2. The two theorems also show that the
group $\operatorname{TR}_{5-\lambda_{1}}^{2}(\mathbb{Z};2)_{(2)}$ is
isomorphic to $\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$ and that
the group $\operatorname{TR}_{5-\lambda_{1}}^{3}(\mathbb{Z};2)_{(2)}$ is
isomorphic to either $\mathbb{Z}/4\mathbb{Z}\oplus\mathbb{Z}/4\mathbb{Z}$ or
$\mathbb{Z}/8\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$. We will prove that the
latter group is isomorphic to
$\mathbb{Z}/4\mathbb{Z}\oplus\mathbb{Z}/4\mathbb{Z}$ by showing that it
contains $\mathbb{Z}/4\mathbb{Z}$ as a direct summand. To this end, we
consider the following diagram

$\textstyle{{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{H}_{5}(C_{2},\operatorname{TR}_{\,\boldsymbol{\cdot}-\lambda_{1}}^{1}(\mathbb{Z};2))_{(2)}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{V}$$\textstyle{{\operatorname{TR}_{5-\lambda_{1}}^{2}(\mathbb{Z};2)_{(2)}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{V}$$\textstyle{{\operatorname{TR}_{5}^{1}(\mathbb{Z};2)_{(2)}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{V}$$\textstyle{{0}}$$\textstyle{{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{H}_{5}(C_{4},\operatorname{TR}_{\,\boldsymbol{\cdot}-\lambda_{1}}^{1}(\mathbb{Z};2))_{(2)}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F^{2}}$$\textstyle{{\operatorname{TR}_{5-\lambda_{1}}^{3}(\mathbb{Z};2)_{(2)}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F^{2}}$$\textstyle{{\operatorname{TR}_{5}^{2}(\mathbb{Z};2)_{(2)}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{0}}$$\textstyle{{\operatorname{TR}_{3}^{1}(\mathbb{Z};2)_{(2)}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\operatorname{TR}_{3}^{1}(\mathbb{Z};2)_{(2)}}}$

where the rows, but not the columns, are exact. It follows from Theorem B that
the top middle and right-hand vertical maps $V$ are injective. Hence, also the
top left-hand vertical map $V$ is injective. Moreover, [6, Proposition 4.2]
and [5, Proposition 15] show that the bottom left-hand vertical map $F^{2}$ is
surjective. Hence, also the bottom right-hand vertical map $F^{2}$ is
surjective. The skeleton spectral sequence

$E_{s,t}^{2}=H_{s}(C_{4},\operatorname{TR}_{t-\lambda_{1}}^{1}(\mathbb{Z};2))_{(2)}\Rightarrow\mathbb{H}_{s+t}(C_{4},\operatorname{TR}_{\,\boldsymbol{\cdot}-\lambda_{1}}^{1}(\mathbb{Z};2))_{(2)}$

shows that the middle left-hand group is an extension of
$E_{2,3}^{\infty}=\mathbb{Z}/4\mathbb{Z}$ by
$E_{0,5}^{\infty}=\mathbb{Z}/2\mathbb{Z}$ and the diagram above shows that the
extension is split. It follows from [5, Lemma 6] that

$F\colon\mathbb{H}_{5}(C_{4},\operatorname{TR}_{\,\boldsymbol{\cdot}-\lambda_{1}}^{1}(\mathbb{Z};2))_{(2)}\to\mathbb{H}_{5}(C_{2},\operatorname{TR}_{\,\boldsymbol{\cdot}-\lambda_{1}}^{1}(\mathbb{Z};2))_{(2)}$

maps the generator of the summand $E_{0,5}^{\infty}=\mathbb{Z}/2\mathbb{Z}$ to
zero. Hence, the lower left-hand vertical map $F^{2}$ in the diagram above
maps the generator of the summand $E_{0,5}^{\infty}=\mathbb{Z}/2\mathbb{Z}$ to
zero. But the map $F^{2}$ is surjective, and therefore, maps a generator of
the summand $E_{3,2}^{\infty}=\mathbb{Z}/4\mathbb{Z}$ non-trivially. It
follows that $\operatorname{TR}_{5-\lambda_{1}}^{3}(\mathbb{Z};2)_{(2)}$
contains direct summand isomorphic to $\mathbb{Z}/4\mathbb{Z}$, and hence, is
isomorphic to $\mathbb{Z}/4\mathbb{Z}\oplus\mathbb{Z}/4\mathbb{Z}$. This shows
that the cokernel of the upper middle vertical map $V$ is isomorphic to
$\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$, and hence, that
$K_{6}(\mathbb{Z}[x]/(x^{2}),(x))_{(2)}$ is as stated.

It remains to evaluate $K_{6}(\mathbb{Z}[x]/(x^{2}),(x))_{(3)}$. This group is
canonically isomorphic to the direct sum of
$\operatorname{TR}_{5-\lambda_{1}}^{2}(\mathbb{Z};3)_{(3)}$ and
$\operatorname{TR}_{5-\lambda_{2}}^{1}(\mathbb{Z};3)_{(3)}$. It follows from
Theorem B that the former group has order $9$ and that the latter group is
zero and from Theorem 2.2 that the former group is cyclic. This completes the
proof.

######  Theorem 4.1.

Let $p$ be an odd prime number. Then for $2i<p^{2}$ the $p$-primary torsion
subgroup of $K_{2i}(\mathbb{Z}[x]/(x^{2}),(x))$ is isomorphic to
$(\mathbb{Z}/p\mathbb{Z})^{r_{1}}\oplus(\mathbb{Z}/p^{2}\mathbb{Z})^{r_{2}}$,
where

$(r_{1},r_{2})=\begin{cases}(0,\lfloor i/p\rfloor)&\text{if $2i+1\equiv 0$
modulo $p$}\cr(\lfloor 2i/p\rfloor-2,1)&\text{if $2i+1\equiv j$ modulo $p$
with $1\leqslant j\leqslant 2i/p$ odd}\cr(\lfloor
2i/p\rfloor,0)&\text{otherwise.}\cr\end{cases}$

Here $\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$.

######  Proof 4.2.

After localizing at the odd prime number $p$, the short exact sequence

$0\to\lim_{R}\operatorname{TR}_{2i-1-\lambda_{d}}^{r/2}(\mathbb{Z})\xrightarrow{V_{2}}\lim_{R}\operatorname{TR}_{2i-1-\lambda_{d}}^{r}(\mathbb{Z})\to
K_{2i}(\mathbb{Z}[x]/(x^{2}),(x))\to 0$

induces a canonical isomorphism

$\bigoplus_{j}\operatorname{TR}_{2i-1-\lambda_{d}}^{s}(\mathbb{Z};p)_{(p)}\xrightarrow{\sim}K_{2i}(\mathbb{Z}[x]/(x^{2}),(x))_{(p)},$

where the sum runs over integers $1\leqslant j\leqslant 2i$ coprime to both
$2$ and $p$ and $s=s_{p}(2,i,j)$ is the unique integer that satisfies
$p^{s-1}j\leqslant 2i<p^{s}j$. Since $2i<p^{2}$ we find

$s_{p}(2,i,j)=\begin{cases}2&\text{if $1\leqslant j\leqslant 2i/p$}\cr
1&\text{if $2i/p<j\leqslant 2i$.}\cr\end{cases}$

If $2i/p<j\leqslant 2i$ we have $d=(j-1)/2$, and hence

$\operatorname{length}_{\mathbb{Z}_{(p)}}\operatorname{TR}_{2i-1-\lambda_{d}}^{1}(\mathbb{Z};p)_{(p)}=v_{p}(i-d)=v_{p}(2i+1-j).$

The length is at most $1$ since $2i<p^{2}$. If $1\leqslant j\leqslant 2i/p$ we
have $d=(pj-1)/2$, and in this case Theorem B shows that

$\displaystyle\operatorname{length}_{\mathbb{Z}_{(p)}}\operatorname{TR}_{2i-1-\lambda_{d}}^{2}(\mathbb{Z};p)_{(p)}$
$\displaystyle=\operatorname{length}_{\mathbb{Z}_{(p)}}\operatorname{TR}_{2i-1-\lambda_{d}^{\prime}}^{1}(\mathbb{Z};p)_{(p)}+v_{p}(i-d)+1$
$\displaystyle=v_{p}(2i+1-j)+v_{p}(2i+1)+1,$

where we have used that $\lambda_{d}^{\prime}=\lambda_{d^{\prime}}$ with
$d^{\prime}=(j-1)/2$. The length is at most $2$ since $2i<p^{2}$ and since $j$
is coprime to $p$. We claim that the group
$\operatorname{TR}_{2i-1-\lambda_{d}}^{2}(\mathbb{Z};p)_{(p)}$ is always
cyclic. By Theorem 2.2, the claim is equivalent to the congruence

$2i-1\not\equiv 2\delta_{p}(\lambda_{d})-1\mod 2p^{2}.$

We compute that modulo $p^{2}$

$\delta_{p}(\lambda_{d})\equiv(1-p)((pj-1)/2+(j-1)/2\cdot p)$

Hence, we have $2i-1\equiv 2\delta_{p}(\lambda_{d})-1$ modulo $2p^{2}$ if and
only if $2i+1\equiv 2pj+p^{2}$ modulo $2p^{2}$. This is possible only if
$2i+1$ is congruent to $0$ modulo $p$. If we write $2i+1=ap$ then $1\leqslant
a\leqslant p$ and $1\leqslant j<a$. Hence, $p\leqslant ap\leqslant p^{2}$ and
$2p+p^{2}\leqslant 2pj+p^{2}<2ap+p^{2}\leqslant 3p^{2}$ which implies that the
congruence $ap\equiv 2pj+p^{2}$ modulo $2p^{2}$ is equivalent to the equality
$ap+p^{2}=2pj$. But then $a+p=2j$ which contradicts that $j<a$. The claim
follows.

We now show that the integers $(r_{1},r_{2})$ are as stated. It follows from
Theorem A that

$r_{1}+2r_{2}=v_{p}((2i)!)=\lfloor 2i/p\rfloor.$

In the case where $2i+1$ is congruent to $0$ modulo $p$, we proved above that
$r_{1}=0$. If we write $2i+1=ap$ then $a$ is odd and

$r_{2}=\lfloor 2i/p\rfloor/2=(a-1)/2=\lfloor(a-1)/2+(p-1)/2p\rfloor=\lfloor
i/p\rfloor$

as stated. In the case where $2i+1\equiv j$ modulo $p$ with $1\leqslant
j\leqslant 2i/p$ odd, we proved above that $r_{2}=1$. Hence, $r_{1}=\lfloor
2i/p\rfloor-2$. Finally, in the remaining case, $r_{2}=0$, and hence,
$r_{1}=\lfloor 2i/p\rfloor$. This completes the proof.

######  Example 4.3.

Let $p$ be an odd prime number. We spell out the statement of Theorem 4.2, for
$i\leqslant p+1$. The $p$-primary torsion subgroup of
$K_{2i}(\mathbb{Z}[x]/(x^{2}),(x))$ is zero for $i\leqslant(p-1)/2$, is cyclic
of order $p$ for $(p+1)/2\leqslant i<p$, and is cyclic of order $p^{2}$ for
$i=p$. The structure of the $p$-primary torsion subgroup of
$K_{2p+2}(\mathbb{Z}[x]/(x^{2}),(x))$ depends on the odd prime $p$. It is
cyclic of order $9$ if $p=3$, and the direct sum of two cyclic groups of order
$p$ if $p\geqslant 5$.

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###### Acknowledgements.

This paper was written in part while the third author visited the Hausdorff
Center for Mathematics at the University of Bonn. He would like to thank the
center and in particular Stefan Schwede and Christian Ausoni for their
hospitality and support. Finally, we thank an anonymous referee for helpful
comments.

Vigleik Angeltveit
Department of Mathematics
The University of Chicago
Chicago, Illinois
USA
Teena Gerhardt
Department of Mathematics
Indiana University
Bloomington, Indiana
USA
Lars Hesselholt
Graduate School of Mathematics
Nagoya University
Chikusa-ku
Nagoya, Japan Japan