# Optimizing Task Scheduling in Heterogeneous Computing Environments: A
Comparative Analysis of CPU, GPU, and ASIC Platforms Using E2C Simulator

1st Ali Mohammadjafari, 1st Poorya Khajouie,

###### Abstract

Efficient task scheduling in heterogeneous computing environments is
imperative for optimizing resource utilization and minimizing task completion
times. In this study, we conducted a comprehensive benchmarking analysis to
evaluate the performance of four scheduling algorithms—First-Come, First-
Served (FCFS), FCFS with No Queuing (FCFS-NQ), Minimum Expected Completion
Time (MECT), and Minimum Expected Execution Time (MEET)—across varying
workload scenarios. We defined three workload scenarios: low, medium, and
high, each representing different levels of computational demands. Through
rigorous experimentation and analysis, we assessed the effectiveness of each
algorithm in terms of total completion percentage, energy consumption, wasted
energy, and energy per completion. Our findings highlight the strengths and
limitations of each algorithm, with MECT and MEET emerging as robust
contenders, dynamically prioritizing tasks based on comprehensive estimates of
completion and execution times. Furthermore, MECT and MEET exhibit superior
energy efficiency compared to FCFS and FCFS-NQ, underscoring their suitability
for resource-constrained environments. This study provides valuable insights
into the efficacy of task scheduling algorithms in heterogeneous computing
environments, enabling informed decision-making to enhance resource
allocation, minimize task completion times, and improve energy efficiency

###### Index Terms:

Task Scheduling, Heterogeneous Computing, Optimization

##  I Introduction

Process scheduling is a fundamental aspect of modern computing systems, that
is playing a critical role in determining the efficiency and responsiveness of
computer operations. At the main part of this mechanism lies the ability to
wisely allocate CPU time to multi-processes. The importance of scheduling
cannot be overstated; it is the arbiter of resource distribution, ensuring
that each process receives a fair share of computing power while minimizing
idle time. This delicate balance is crucial for maintaining system stability
and achieving optimal performance.

The historical development of process scheduling in computer systems is back
to several decades, reflecting the evolution of computing technologies and the
increasing demand for efficient resource utilization in computer technology.
In the early days of computing, scheduling was relatively simplistic, with
batch processing systems executing tasks in a sequential manner. However, as
computing environments became more complex and diverse, the need for more
sophisticated scheduling algorithms became a critical problem. One of the
earliest milestones in process scheduling was the introduction of the first
multitasking operating systems, such as IBM’s OS/360 in the 1960s, which
allowed multiple programs to run concurrently on a single processor. This
innovation built the way for the development of modern scheduling algorithms,
with seminal works like Edsger Dijkstra’s ”The Structure of the
’THE’-Multiprogramming System” in 1968, which introduced concepts such as
process states and priority scheduling. In the recent decades, researchers
have continued to refine and innovate upon scheduling algorithms,
incorporating advancements in computer architecture, real-time computing, and
parallel processing. So, Finding the optimal strategy for workload scheduling
is still a problem[1]. It involves a deep understanding of the intricacies of
both the hardware and the software it supports. By adapting the scheduling
algorithm to the specific characteristics of the workload and the existing
hardware, one can significantly enhance system throughput. This not only
reduces waiting times for processes but also maximizes the overall efficiency
of the system, paving the way for more advanced and responsive operating
systems in our increasingly digital world.

###  I-A Scheduling

Job scheduling in CPU context refers to the process of allocating computing
resources to execute a set of jobs, where each job has specific computational
requirements and resource demands. The goal is to optimize the use of CPU
resources, improve system performance, and ensure that all jobs are completed
in an efficient manner[2]. In Figure 1, we try to show different type of CPU
scheduling algorithms.

Scheduling strategies are categorized in two main groups:

  * •

Non-preemptive Scheduling: Non-preemptive Scheduling: This strategy involves
scheduling jobs so that once a job starts executing, it runs to completion
without being interrupted. This is essential in systems where interrupting a
job can be costly or is not allowed due to system constraints.

  * •

Priority-based Algorithms: These algorithms prioritize jobs based on certain
criteria such as job size, demand, or a combination of both (volume). Examples
include Shortest Job First (SJF), Smallest Demand First (SDF), and Smallest
Volume First (SVF)

Figure 1: Different types of CPU Scheduling Algorithms TABLE I: Comparison of
CPU Scheduling Algorithms

Algorithm | Allocation is | Complexity | AWT | Preemption | Starvation | Performance
---|---|---|---|---|---|---
FCFS | According to arrival time, CPU is allocated. | Simple and easy to implement | Large | No | No | Slow performance
SJF | Based on lowest CPU burst time (BT). | More complex than FCFS | Smaller than FCFS | No | Yes | Minimum AWT
LJFS | Based on highest CPU burst time (BT) | More complex than FCFS | Depending on some measures | No | Yes | Big turn-around time
LRTF | Same as LJFS but preemptive | More complex than FCFS | Depending on some measures | Yes | Yes | Preference to longer jobs
SRTF | Same as SJF but preemptive | More complex than FCFS | Depending on some measures | Yes | Yes | Preference to short jobs
RR | According to order of process with fixed TQ | Complexity depends on TQ size | Large as compared to SJF and Priority | Yes | No | Each process has fixed time
Priority Pre. | According to priority. Bigger priority task executes first | Less complex | Smaller than FCFS | Yes | Yes | Well performance but contains starvation
Priority Non-pre. | According to priority with monitoring incoming higher priority jobs | Less complex than Priority Pre. | Smaller than FCFS | No | Yes | Beneficial with batch systems
MLQ | According to process in bigger queue priority | More complex than priority algorithms | Smaller than FCFS | No | Yes | Good performance but contains starvation
MFLQ | According to process of bigger priority queue | Most complex depends on TQ size | Smaller than other types | No | No | Good performance

As we said before, in the realm of computer systems and operating
environments, optimizing process scheduling is a critical goal to enhance
system performance and resource utilization. The selection of an appropriate
CPU scheduling algorithm profoundly influences various system metrics,
including throughput, response time, and resource utilization efficiency[3].
Consequently, comparing CPU scheduling algorithms serves as a fundamental
aspect of system optimization, enabling researchers and practitioners to
evaluate and understand the strengths and weaknesses of different scheduling
strategies under various workload scenarios. By conducting such comparisons,
insights are gained into how each algorithm performs under different
conditions, aiding in the selection or development of algorithms tailored to
specific system requirements. In table1 we can see a detailed comparison of
CPU scheduling algorithms. This comparative analysis not only facilitates the
identification of optimal scheduling strategies but also contributes to the
advancement of theoretical knowledge and practical implementations in the
field of process scheduling, ultimately leading to more efficient and
responsive computing systems[4].

###  I-B Understanding the Significance of Process Scheduling Optimization

In today’s fast-changing computing landscape, where we can see diverse
environments such as cloud computing, real-time systems, and multi-core
processors, the imperative for efficient process scheduling has never been
more pronounced. The escalating complexity and heterogeneity of these
environments underscore the critical role of scheduling algorithms in
achieving optimal system performance. Efficient process scheduling is not just
merely a technical concern; it directly impacts user experience, application
responsiveness, and overall system efficiency[5]. Delays in process execution
can lead to poor performance and reduced user satisfaction, highlighting the
profound influence of scheduling decisions on the end-user experience.
However, creating effective scheduling algorithms needs to answer to different
challenges and trade-offs. Balancing considerations such as fairness,
throughput, latency, and resource utilization against real-world constraints
and system dynamics is a daunting task faced by system designers and
researchers alike. Despite these challenges, process scheduling remains
integral to a diverse array of applications and domains, spanning batch
processing, scientific computing, multimedia applications, and embedded
systems[6]. The evaluation of scheduling algorithms relies on a variety of
metrics, including average waiting time, turnaround time, response time,
fairness, and scalability, each of which plays a crucial role in assessing
algorithm effectiveness and guiding further optimization efforts.

###  I-C E2C: A Visual Simulator for Heterogeneous Computing Systems

The E2C simulator, introduced in [7], is a cutting-edge visual tool designed
to address the complexities of heterogeneous computing systems. It offers a
cost-effective and time-efficient solution for studying the performance of
diverse system configurations, a challenge that has long plagued researchers
and practitioners in the field of distributed computing. By enabling users to
simulate heterogeneous computing systems, implement scheduling methods,
measure energy consumption, and visualize system performance, E2C opens up a
realm of possibilities for students, researchers, and industry professionals
alike. This simulator not only provides a practical platform for exploring
system heterogeneity but also serves as a valuable resource for understanding
the intricacies of dynamic resource allocation, the concept of dark silicon,
specialized data centers for planet-scale applications, and the multi-tenancy
of latency-sensitive deep learning applications on the edge. With its
comprehensive features, E2C stands as a pivotal tool in advancing the study
and application of heterogeneous computing systems, offering insights and
opportunities for innovation in this rapidly evolving field. So in this paper
we used this simulator to generate workloads in different scenario, then we
used these scenarios to do a computational task scheduling benchmark.

###  I-D Proposed Approach and Research Framework

In this paper, we present a comprehensive analysis of task scheduling across
heterogeneous computing environments, including CPU, GPU, and ASIC. Our
research framework aims to evaluate the efficacy of various scheduling
algorithms in managing workloads of varying volumes, categorized as low,
medium, and high.

We delineate three distinct scenarios based on workload volume to encapsulate
the diverse operational conditions encountered in real-world computing
environments. These scenarios provide a structured framework for assessing the
performance of scheduling algorithms under different levels of computational
demand.

We employ four distinct scheduling algorithms to orchestrate task execution
within each scenario, each tailored to address specific optimization
objectives:

  * •

First-Come, First-Served (FCFS): A rudimentary scheduling approach where tasks
are executed in the order of their arrival, irrespective of their
computational requirements or expected completion times.

  * •

FCFS with No Queuing (FCFS-NQ): A variant of FCFS that eliminates queuing,
thereby prioritizing immediate task execution over waiting, which may be
beneficial in scenarios where latency is critical.

  * •

Minimum Expected Completion Time (MECT): This algorithm prioritizes tasks
based on their expected completion times, aiming to minimize the overall
turnaround time of the task queue.

  * •

Minimum Expected Execution Time (MEET): MEET prioritizes tasks based on their
expected execution times, focusing on minimizing resource utilization and
maximizing throughput efficiency.

Our research framework facilitates a comparative analysis of the performance
of each scheduling algorithm across the defined scenarios. By evaluating
metrics such as throughput, latency, and resource utilization, we aim to
discern the strengths and weaknesses of each algorithm in diverse operational
contexts. Furthermore, we seek to identify the optimal scheduling strategy for
each workload scenario, providing insights for efficient resource management
in heterogeneous computing environments.

This study contributes to the field of task scheduling in heterogeneous
computing environments by offering a systematic evaluation of scheduling
algorithms under various workload scenarios. Our findings shed light on the
performance characteristics of different scheduling strategies, providing
valuable insights for system designers and administrators. Additionally, our
research addresses the practical challenge of optimizing resource utilization
in heterogeneous computing environments, thereby enhancing the overall
efficiency and performance of computational systems. By finding the strengths
and weaknesses of each scheduling algorithm across different workload
scenarios, our work serv‘es as a valuable reference for guiding decision-
making processes in real-world computing environments.

The rest of this paper is organized as follows. Section II discusses
background study and related prior works. We explain our methodology and our
benchmark in detail in Section III. Next, we discuss experimental evaluation
and performance analysis in Section IV. Finally, Section V concludes the paper
and provides a few avenues for the future studies.

##  II Related Works

###  II-A Benchmarking Task Scheduling Algorithms

As we delve into the realm of heterogeneous computing systems, the efficacy of
task scheduling algorithms becomes paramount. Maurya and Tripathi1 in 2018
presents a benchmarking approach to compare the performance of various
scheduling algorithms in a heterogeneous computing environment. The authors
employ a comprehensive set of metrics to assess the efficiency, effectiveness,
and scalability of the algorithms [8]. In this concept, [9] [10] [11] Also
have similar research on doing benchmark on heterogeneous computing systems.
Sirisah and Prasad in 2023 proposed the novel MPEFT (Maximizing Parallelism
for Minimizing Earliest Finish Time) algorithm, which aims to minimize
makespan by enhancing parallelism within workflows. Through rigorous
evaluation against classical scheduling algorithms, the MPEFT algorithm
demonstrates superior performance in terms of makespan, speedup, efficiency,
and frequency of best results. The study contributes valuable insights to the
field of high-performance computing, emphasizing the importance of effective
task scheduling in maximizing system performance [12].

Sirsha in 2023 presented two novel heuristic scheduling algorithms, Global
Highest degree Task First (GHTF) and Critical Path/Earliest Finish Time
(CP/EFT), aimed at minimizing the schedule length in Heterogeneous Computing
Environments. Both algorithms show significant improvements over classical
list scheduling algorithms, with GHTF achieving better schedules by 5–20% and
CP/EFT by 10.76–23.45%3. The paper also includes a comprehensive survey of
existing algorithms, detailed descriptions of the proposed algorithms, and an
evaluation of their performance against widely referred scheduling algorithms
[13].

###  II-B Machine Learning Approaches:

In recent years, researchers have been exploring the integration of machine
learning techniques with process scheduling to enhance system efficiency. By
leveraging approaches like reinforcement learning, supervised learning, and
unsupervised learning, these methods aim to use historical workload data to
make smarter scheduling decisions. This subsection examines the literature on
machine learning-based scheduling algorithms and assesses their effectiveness
in improving system performance. In a recent years a lot of studies provide a
comprehensive overview of machine learning-based scheduling research,
highlighting the growth of this field and the potential for future
advancements. they discuss various machine learning algorithms and scheduling
scenarios, emphasizing the importance of deep learning and reinforcement
learning-based algorithms[14] [15] [16] [17]. In 2023 Mangalampalli et. al.
discusse the use of Deep Reinforcement Learning-based approaches for optimal
task mapping in cloud computing, focusing on improving metrics such as
makespan, energy consumption, and SLA violation. They compare the proposed
algorithm with existing mechanisms and demonstrates superior performance. The
study also emphasizes the importance of dynamic task scheduling based on
priorities to minimize inefficiencies in existing scheduling mechanisms.[18].

In 2023, Hayat et. al. proposed a solution that combines a load-balanced task
scheduler with a machine learning-based device predictor. By forecasting
execution times on CPU and GPU devices and assigning tasks accordingly, the
proposed approach aims to optimize system performance. To counteract the risk
of tasks predominantly mapping to a single device, a work-stealing-based task
scheduler dynamically redistributes workload among devices[19].

###  II-C Adaptive Scheduling Strategies:

In the realm of process scheduling optimization, adaptive strategies take
center stage, dynamically adjusting scheduling decisions to match real-time
workload variations. These strategies embody responsiveness, employing
techniques such as feedback control, dynamic priority adjustment, and online
learning algorithms to efficiently manage fluctuating workloads. As system
dynamics grow more intricate, exploring adaptive scheduling becomes essential,
offering improved performance and efficiency in task management. This
subsection delves into the domain of adaptive scheduling strategies, unveiling
innovative approaches that enhance process scheduling adaptability and
robustness. Mao et. al. in a paper propose a novel approach by decomposing
scheduling strategies into three dimensions and making informed decisions
along each dimension based on workload characteristics. Compared to existing
approaches, MorphStream achieves significantly higher throughput, up to 3.4
times, and reduced processing latency by 69.1% when handling real-world
scenarios characterized by complex and dynamically changing workload
dependencies[20]. Zheng er. al. propose Shockwave, a forward-thinking
scheduler. Drawing upon principles from market theory and stochastic dynamic
programming, Shockwave aims to optimize both efficiency and fairness in
dynamically changing environments. Experimental results indicate that
Shockwave significantly outperforms existing fair scheduling schemes,
enhancing makespan by 1.3× and fairness by 2× for ML job traces characterized
by dynamic adaptation[21]. Li et. al introduce an adaptive batch-stream
scheduling method, a chiplet-based core-cluster binding mechanism, and a
chiplet-based nearest task stealing approach to enhance computing
efficiency[22].

###  II-D Real-Time Scheduling

In the realm of CPU scheduling, real-time scheduling algorithms hold a
critical role in ensuring timely execution of tasks with stringent timing
constraints. These algorithms are specifically designed to meet the
requirements of real-time systems where tasks must complete within
predetermined deadlines to guarantee system stability and reliability. Real-
time scheduling for CPUs encompasses various approaches, such as Rate-
Monotonic Scheduling (RMS), Earliest Deadline First (EDF), and Deadline-
Monotonic Scheduling (DMS), each offering unique strategies for task
prioritization based on deadlines or rates[22]. Saifullah et. al introduce
techniques to minimize CPU energy consumption while ensuring that all tasks
meet their deadlines. They propose a novel integration of a frequency
optimization engine and dynamic voltage and frequency scaling (DVFS) with
classical real-time scheduling policies[23]. In 2021 Kang et. al introduce a
novel approach that combines CPU-friendly quantization with fine-grained
CPU/GPU allocation, aiming to improve the execution of DNN tasks while
preserving timing guarantees and minimizing accuracy loss[24]. Yoon et. al.
introduce a novel swap scheme aimed at reducing memory power consumption by
leveraging high-speed NVM storage and optimizing power savings in both CPU and
memory[25].

##  III Methodology

Before delving into the intricacies of our methodology, it’s essential to
underscore the significance of task scheduling in heterogeneous computing
environments. As computing systems evolve, the integration of diverse
processing units, such as CPUs, GPUs, and ASICs, has become increasingly
prevalent, offering unprecedented computational capabilities[26]. However,
harnessing the full potential of these heterogeneous architectures
necessitates efficient task scheduling strategies tailored to their unique
characteristics and operational requirements.

In this study, we embark on a comprehensive exploration of task scheduling
across CPU, GPU, and ASIC platforms. Our methodology revolves around the
systematic evaluation of scheduling algorithms under varying workload
conditions, categorized into low, medium, and high volumes by using E2C
Simulator [7]. By leveraging a diverse set of scheduling algorithms—First-
Come, First-Served (FCFS), FCFS with No Queuing (FCFS-NQ), Minimum Expected
Completion Time (MECT), and Minimum Expected Execution Time (MEET)—we aim to
discern the optimal scheduling strategy for each workload scenario.

###  III-A E2C Simulator

In our study, we used E2C Simulator that is published in [7]. E2C developed by
external researchers, to evaluate the performance of various scheduling
algorithms across different workload scenarios. The simulator served as a
crucial tool in our investigation, allowing us to model and analyze task
execution behavior on CPU, GPU, and ASIC platforms. In figure 7, an overview
of the E2C Simulator is shown. E2C core is available for download at the
following address: https://github.com/hpcclab/E2C-Sim The manual document on
how to run E2C and its options and full documentations are available here:
https://hpcclab.github.io/E2C-Sim-docs/

Figure 2: Summary of the E2C Simulator comprising essential elements, namely
the input workload, a queue for incoming tasks, a scheduler (also known as a
load balancer), and a variety of machines depicted in distinct hues [7].

####  III-A1 Simulation Components and Features

TABLE II: Overview of Scenarios  | # of Task1 | # of Task2 | # of Task3 | Start time | End time |
---|---|---|---|---|---|---
Scenario1: Low Workload |  | 700
---
Normal distribution
| 700
---
Exponential distribution
| 700
---
Uniform distribution
0 | 1000 ms |
Scenario2: Medium Workload |  | 1000
---
Normal distribution
| 1000
---
Exponential distribution
| 1000
---
Uniform distribution
0 | 1000ms |
Scenario3: High Workload |  | 1400
---
Normal distribution
| 1400
---
Exponential distribution
| 1400
---
Uniform distribution
0 | 1000ms |

  * •

Task Generation: The simulator enabled us to generate synthetic workloads
representing low, medium, and high task volumes. These workloads consisted of
tasks with varying execution times, resource requirements, and dependencies.
Customization options allowed us to define the number of tasks, their
characteristics, and the inter-task relationships.

  * •

Resource Models: Detailed resource models were incorporated for CPU, GPU, and
ASIC devices. These models considered factors such as processing speed, memory
capacity, and communication latency.

By accurately simulating resource availability and contention during task
execution, we ensured realistic scenarios

  * •

Scheduling Algorithms: Within the E2C Simulator, we implemented four
scheduling algorithms:

    * –

First-Come-First-Serve (FCFS): Tasks executed in the order of arrival.

    * –

FCFS with No Queuing (FCFS-NQ): Similar to FCFS, but tasks could not wait in a
queue

    * –

Minimum Expected Completion Time (MECT): Prioritized tasks with the shortest
expected completion time.

    * –

Minimum Expected Execution Time (MEET): Prioritized tasks with the shortest
expected execution time.

The simulator allowed us to switch between these algorithms dynamically,
observing their impact on overall task completion times.

The E2C Simulator emerged as a valuable asset, enabling us to evaluate
scheduling algorithms across diverse workload scenarios. Its flexibility
allowed us to explore trade-offs and make informed recommendations for
practical task scheduling in heterogeneous computing environments[7].

###  III-B Workload Generation and Task Specifications

In our study, the generation of workloads and the specification of task
characteristics play a crucial role in accurately simulating real-world
scenarios and evaluating the efficacy of different scheduling algorithms
across CPU, GPU, and ASIC platforms. We meticulously design our workload
generation process to encompass a wide range of computational demands and
timing constraints.

Our workload generation process begins with the classification of tasks into
three distinct types, each carefully crafted to represent varying degrees of
computational intensity and timing sensitivity:

  * •

Task1: This task type is characterized by a mean data size of 100 KB and a
slack of 2 milliseconds. Task1 represents computational tasks with moderate
data requirements and relatively lenient timing constraints.

  * •

Task2: With a mean data size of 75 KB and a slack of 1.5 milliseconds, Task2
embodies computational tasks with slightly lower data demands but more
stringent timing requirements compared to Task1.

  * •

Task3: Task3, with a mean data size of 50 KB and a slack of 1 millisecond,
represents computational tasks with relatively low data requirements but tight
timing constraints, requiring swift execution.

These task types are carefully selected to cover a spectrum of computational
requirements commonly encountered in practical applications, ensuring a
comprehensive evaluation of scheduling algorithms.

TABLE III: Machine Types and Task Execution Time (EET) on Respective Platforms | EET( Expected Execution Time) | Power | Idle Power | #Replicas
---|---|---|---|---
| Task1 | Task2 | Task3 |  |  |
CPU | 5ms | 4ms | 3ms | 150 | 15 | 2
GPU | 2ms | 1.5ms | 1ms | 300 | 30 | 4
ASIC | 1ms | 0.8ms | 0.5ms | 50 | 5 | 2

To accurately model task execution across different computing platforms, we
estimate the execution time (EET) for each task type on CPU, GPU, and ASIC
platforms. Table III shows all this information.

In table II, to comprehensively evaluate the performance of scheduling
algorithms under various workload conditions, we define three distinct
scenarios: low, medium, and high, each with specific task counts and arrival
distributions.

TABLE IV: Sample of Low Scenario Workload task_type | data_size | arrival_time | deadline |  |  |
---|---|---|---|---|---|---
Task1 | 110.899 | 692.529 | 694.529 |  |  |
Task1 | 110.663 | 668.942 | 670.942 |  |  |
Task1 | 115.118 | 306.436 | 308.436 |  |  |
Task1 | 100.568 | 617.725 | 619.725 |  |  |
Task2 | 70.814 | 5.612 | 7.112 |  |  |
Task2 | 45.237 | 7.775 | 9.275 |  |  |
Task2 | 77.869 | 10.118 | 11.618 |  |  |
Task2 | 72.863 | 10.903 | 12.403 |  |  |
Task3 | 38.241 | 624.736 | 625.736 |  |  |
Task3 | 43.915 | 493.478 | 494.478 |  |  |
Task3 | 52.198 | 159.911 | 160.911 |  |  |
Task3 | 47.776 | 709.633 | 710.633 |  |  |

This meticulous approach to workload generation and scenario definition
ensures a comprehensive evaluation of scheduling algorithms under diverse
workload conditions, facilitating informed decision-making regarding task
allocation and optimization strategies across different computing platforms.

###  III-C Scheduling Algorithms

In this part, we provide a comprehensive overview of the scheduling algorithms
utilized in the study and elaborate on the methodology employed for workload
simulation and algorithm selection.

  * •

First-Come, First-Served (FCFS): FCFS is a classic scheduling algorithm that
prioritizes tasks based on their arrival time. Tasks are executed in the order
they enter the system, without consideration for their urgency or execution
time.

  * •

FCFS with No Queuing (FCFS-NQ): FCFS-NQ is an extension of FCFS designed to
mitigate queuing delays. It rejects incoming tasks when the system is at full
capacity, thereby preventing queuing and ensuring immediate task rejection if
resources are unavailable.

  * •

Minimum Expected Completion Time (MECT): MECT is a dynamic scheduling
algorithm that estimates the expected completion time for each task based on
factors such as urgency, deadlines, and resource availability. It prioritizes
tasks to minimize the overall completion time of the workload.

  * •

Minimum Expected Execution Time (MEET): MEET predicts the execution time for
each task and schedules them to minimize the total expected execution time. It
considers task characteristics such as complexity and resource requirements to
optimize task execution across heterogeneous architectures.

###  III-D Workload Simulation Methodology

In order to assess the efficacy of scheduling algorithms across diverse
computational architectures, a meticulous methodology was devised to simulate
workloads and evaluate algorithmic performance. This methodology encompasses
several key stages, each contributing to the comprehensive analysis of
scheduling strategies.

####  III-D1 Configuration Setup

A robust configuration setup was paramount to ensuring the fidelity of the
simulation environment. To this end, meticulous attention was paid to
specifying configuration parameters within the _config.json_ file. These
parameters encapsulated crucial aspects such as machine specifications, task
types, and battery capacity. Notably, machine types including CPU, GPU, and
ASIC were defined with their respective power consumption and number of
replicas, thereby laying the foundation for a heterogeneous computational
environment.

####  III-D2 Algorithm Selection and Execution

Workloads are generated using CSV files, where each file contains task arrival
times and associated parameters. The workload generator ensures variability in
workload intensity and composition to simulate real-world scenarios
accurately.

    # Example code for generating workloads
    path_to_arrivals = f’./workloads/{scenario}/
        workload-0.csv’

The simulation framework allows for the selection of scheduling algorithms
through the _config.json_ file. By modifying the _”scheduling-method”_
parameter, different algorithms can be chosen for evaluation.

    #Example code for selecting scheduling
    #algorithm in config.json
    "scheduling_method": "FCFS"

The simulation engine processes the generated workloads using the selected
scheduling algorithm. Performance metrics such as completion percentage,
energy consumption, and wasted energy are recorded and analyzed for each
workload scenario.

    # Example code for running simulation and
    #recording results

    simulation = Simulator(path_to_arrivals,
        path_to_etc, path_to_reports, seed=123)
    simulation.set_scheduling_method
        (config.scheduling_method)
    simulation.run()

By following these steps, researchers can explore the impact of different
scheduling algorithms on workload management across diverse computational
architectures, facilitating informed decision-making in resource allocation
and task prioritization.

##  IV Evaluation of Task Scheduling Algorithms in Heterogeneous Computing
Environments

Task scheduling in heterogeneous computing environments is a multifaceted
endeavor, necessitating the selection of optimal algorithms to ensure
efficient resource utilization and task completion. In this study, we
conducted a comprehensive benchmarking analysis, evaluating four distinct
scheduling algorithms—First-Come, First-Served (FCFS), FCFS with No Queuing
(FCFS-NQ), Minimum Expected Completion Time (MECT), and Minimum Expected
Execution Time (MEET)—across three scenarios of varying workload volumes: low,
medium, and high. Through rigorous experimentation and analysis, we aimed to
discern the strengths and weaknesses of each algorithm and elucidate their
suitability for different workload scenarios.

###  IV-A Effectiveness Across Workload Scenarios:

In our evaluation across varied workload scenarios, each scheduling algorithm
exhibited distinct performance attributes, shedding light on their efficacy
under different computational demands. In scenarios characterized by low
workload volumes, FCFS and FCFS-NQ showcased commendable performance,
leveraging their straightforward execution strategy to efficiently manage task
allocation. However, as workload volumes escalated, particularly in scenarios
with stringent task deadlines and varying urgency levels, the limitations of
these algorithms became apparent.As it is shown in figure 3, the total
completion percentage for FCFS and FCFS-NQ dwindled notably as workload
intensity increased, reaching 49.24% and 50.9% respectively in low workload
scenarios, 46.23% and 50.77% in medium workload scenarios, and 41.12% and
47.64% in high workload scenarios.

In stark contrast, MECT and MEET emerged as formidable contenders across all
workload scenarios, adeptly adapting to fluctuating computational demands. By
dynamically prioritizing tasks based on comprehensive completion and execution
time estimates, these algorithms optimized resource utilization and minimized
task completion times. In low workload scenarios, MECT and MEET demonstrated
substantial improvements in total completion percentage, achieving 89.19% and
62.81% respectively, showcasing their ability to efficiently manage tasks even
in scenarios with relatively modest computational demands. As workload volumes
increased to medium and high levels, the resilience of MECT became even more
pronounced, with total completion percentages of 75.57% in medium workload
scenarios, and 54.81% in high workload scenarios, respectively.

In high and medium workload scenarios, the computational resources may become
saturated due to the increased volume of tasks competing for execution. MEET,
which prioritizes tasks based on expected execution time, may struggle to
efficiently allocate resources when the demand exceeds capacity. This can lead
to increased contention and resource bottlenecks, ultimately impacting task
completion rates.

These findings underscore the critical importance of selecting appropriate
scheduling algorithms tailored to the specific characteristics of the
workload. While FCFS-based algorithms offer simplicity and ease of
implementation, MECT emerges as indispensable tools for scenarios
necessitating optimized resource utilization and efficient task completion,
particularly in high-pressure environments with stringent deadlines and
fluctuating computational demands.

Figure 3: Total Completion Percentage.

###  IV-B Resource Utilization and Energy Efficiency:

####  IV-B1 Effectiveness Across Workload Scenarios

In our assessment of scheduling algorithms across diverse workload scenarios,
each algorithm exhibited distinctive performance attributes, showcasing their
inherent strengths and limitations. Particularly in low workload scenarios,
FCFS and FCFS-NQ demonstrated commendable efficiency, leveraging their
simplicity to manage task execution effectively. However, as workload volumes
escalated, particularly in scenarios with stringent task deadlines and varying
urgency levels, the limitations of these algorithms became evident. In
contrast, MECT and MEET emerged as robust contenders, dynamically prioritizing
tasks based on completion and execution time estimates, thereby optimizing
resource utilization and minimizing task completion times.

####  IV-B2 Energy Consumption Analysis

A crucial aspect of algorithm evaluation pertains to energy consumption, a
metric essential for assessing the sustainability and efficiency of task
scheduling strategies. Across all workload scenarios, MECT and MEET exhibited
superior energy efficiency compared to FCFS and FCFS-NQ. In figure 5, in low
workload scenarios, MECT and MEET demonstrated a marked reduction in total
consumed energy, with values of 22.73% and 16.14% respectively, compared to
FCFS and FCFS-NQ. This trend persisted as workload volumes increased,
underscoring the energy-efficient nature of MECT and MEET, which prioritize
tasks judiciously based on completion and execution time estimates.

####  IV-B3 Wasted Energy Consideration

Figure 4: Total Wasted Energy.

Additionally, our analysis encompassed an evaluation of wasted energy,
reflecting inefficiencies in resource utilization. Notably, MECT and MEET
exhibited lower levels of wasted energy across all workload scenarios compared
to FCFS and FCFS-NQ. In low workload scenarios, MECT and MEET demonstrated a
modest increase in wasted energy, attributable to their proactive task
prioritization approach. However, this increase was outweighed by the
significant reduction in total consumed energy, resulting in a net gain in
energy efficiency. The result is shown in figure 4.

####  IV-B4 Energy Per Completion Analysis

The assessment also considered energy per completion, a metric indicative of
the energy expended per task completed. Here again, MECT and MEET outperformed
FCFS and FCFS-NQ across all workload scenarios, exhibiting substantially lower
energy per completion values. Particularly in high workload scenarios, where
computational demands are heightened, MECT and MEET showcased remarkable
efficiency, with energy per completion values of 2.27% and 0.22% respectively,
highlighting their suitability for resource-constrained environments with
demanding task deadlines. The result is shown in figure 6

These findings underscore the significance of energy-efficient task scheduling
algorithms, such as MECT and MEET, in optimizing resource utilization and
mitigating energy consumption across diverse workload scenarios. By
prioritizing tasks based on comprehensive completion and execution time
estimates, these algorithms offer a sustainable and efficient approach to task
scheduling in heterogeneous computing environments.

Figure 5: Total Consumed Energy. Figure 6: Energy Per Completion. Figure 7:
Number of Tasks Assigned to Each Resource.

##  V Conclusion and Future Works

In the realm of heterogeneous computing environments, the selection of
appropriate task scheduling algorithms plays a pivotal role in optimizing
resource utilization, minimizing task completion times, and mitigating energy
consumption. Through our comprehensive benchmarking analysis, we have gained
valuable insights into the performance characteristics of four distinct
scheduling algorithms—First-Come, First-Served (FCFS), FCFS with No Queuing
(FCFS-NQ), Minimum Expected Completion Time (MECT), and Minimum Expected
Execution Time (MEET)—across varying workload scenarios.

Our findings underscore the nuanced interplay between algorithmic
sophistication and workload dynamics. In low workload scenarios, the
simplicity of FCFS and FCFS-NQ facilitates efficient task execution, albeit
with limitations becoming apparent as workload volumes increase. Conversely,
MECT and MEET emerge as robust contenders, dynamically prioritizing tasks
based on comprehensive estimates of completion and execution times, thereby
optimizing resource utilization and minimizing task completion times across
all workload scenarios.

Furthermore, our analysis highlights the critical importance of energy
efficiency in task scheduling algorithms. MECT and MEET consistently
demonstrate superior energy efficiency compared to FCFS and FCFS-NQ, as
evidenced by lower levels of total consumed energy, wasted energy, and energy
per completion across all workload scenarios. This underscores their
suitability for resource-constrained environments and their potential to
significantly reduce operational costs and environmental impact.

In conclusion, our study provides valuable insights into the efficacy of task
scheduling algorithms in heterogeneous computing environments. By
understanding the unique performance characteristics of each algorithm and
their suitability for different workload scenarios, system architects and
developers can make informed decisions to optimize resource allocation,
minimize task completion times, and enhance energy efficiency in heterogeneous
computing environments. Moving forward, further research into algorithm
refinement and adaptation to evolving workload dynamics will be instrumental
in advancing the efficiency and sustainability of task scheduling in
heterogeneous computing environments.

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$X_{k^{\prime}}\to X_{k^{\prime\prime}}\to X$, where ${k^{\prime\prime}}/k$ is
separable and ${k^{\prime}}/{k^{\prime\prime}}$ is purely inseparable. Since
$X_{k^{\prime\prime}}$ is a geometrically connected regular affine curve, the
map
${\operatorname{Br}}(X_{k^{\prime}})\to{\operatorname{Br}}(X_{k^{\prime\prime}})$
is surjective. We can thus assume that ${k^{\prime}}/k$ is separable.

As in the proof of Proposition 11.2, we can find a finite field extension
$l/k$, a geometrically connected smooth projective curve $\overline{Y}$ over
$l$, and a finite surjective morphism of $k$-schemes
$g\colon\overline{Y}\to\overline{X}$ such that $\deg(g)$ and $[l:k]$ are some
powers of $p$. We let $Y=g^{-1}(X)$ and $E=g^{*}(D)$. We let
$A=l\otimes_{k}k^{\prime}$ and $\overline{Y}_{A}=\overline{Y}\times_{{\rm
Spec\,}(l)}{\rm Spec\,}(A)$. Since ${k^{\prime}}/k$ is separable, we see that
$A=\stackrel{{\scriptstyle r}}{{\underset{i=1}{\prod}}}l_{i}$, where each
${l_{i}}/l$ is a finite separable field extension.

We consider the diagrams

(12.5) $\textstyle{\stackrel{{\scriptstyle
r}}{{\underset{i=1}{\prod}}}\overline{Y}_{l_{i}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g^{\prime}}$$\scriptstyle{h}$$\textstyle{\overline{X}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\stackrel{{\scriptstyle
r}}{{\underset{i=1}{\oplus}}}{\operatorname{Br}}({Y}_{l_{i}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g^{\prime}_{*}}$$\scriptstyle{h_{*}}$$\textstyle{{\operatorname{Br}}({X}^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{*}}$$\textstyle{\overline{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{\overline{X}}$$\textstyle{{\operatorname{Br}}({Y})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g_{*}}$$\textstyle{{\operatorname{Br}}(X),}$

where the left square is Cartesian whose all arrows are finite. The right
square is commutative by Proposition 8.11.

We now proceed as follows. Since $\overline{Y}$ is a geometrically connected
smooth projective curve over $l$, we have argued previously that
$h_{*}\colon{\operatorname{Br}}(Y_{l_{i}})\to{\operatorname{Br}}(Y)$ is
surjective for each $i$. Since $\deg(g)=p^{n}$ for some $n\geq 1$, we have
also seen previously that
$g_{*}\colon{\operatorname{Br}}(Y)\to{\operatorname{Br}}(X)$ is surjective. In
particular, $f_{*}\circ g^{\prime}_{*}=g_{*}\circ h_{*}$ is surjective. It
follows that $f_{*}$ is surjective. This concludes the proof. ∎

###  12.2. Proof of Theorem 1.1

We let the notations and assumptions be as stated in the beginning of § 12. We
shall divide the proof of Theorem 1.1 into several cases. We begin with the
following case. Recall that
$H^{q}_{cc}(X^{o},{\mathbb{G}}_{m})={\varprojlim}_{n}H^{q}(X,{\mathbb{G}}_{m,(X,nD)})\cong{\varprojlim}_{n}H^{q}(X,{\mathcal{K}}^{M}_{1,(X,nD)})$.

######  Lemma 12.4.

One has
$H^{0}_{cc}(X^{o},{\mathcal{O}}^{\times}_{X^{o}})=H^{i}(X^{o},{\mathcal{O}}^{\times}_{X^{o}})=0$
for all $i\geq 3$.

###### Proof.

It is easy to see that $H^{0}(X,{\mathcal{K}}^{M}_{1,(X,nD)})=0$ for every
$n\geq 1$. In particular,
$H^{0}_{cc}(X^{o},{\mathcal{O}}^{\times}_{X^{o}})=0$. To show that
$H^{i}(X^{o},{\mathcal{O}}^{\times}_{X^{o}})=0$ for $i\geq 3$ is equivalent to
show that $H^{i}(X^{o},{\mathcal{O}}^{\times}_{X^{o}})_{\rm tor}=0$ for all
$i\geq 3$ (e.g., see [4, Lem. 3.5.3]). It already follows from Remark 10.9
that $H^{i}(X^{o},{\mathcal{O}}^{\times}_{X^{o}})\\{p\\}=0$ for all $i\geq 3$.

We now show the prime-to-$p$ case. For $i\geq 4$, we have a surjection
$H^{i}(X^{o},{{\mathbb{Q}}}/{{\mathbb{Z}}}\\{p^{\prime}\\}(1))\twoheadrightarrow
H^{i}(X^{o},{\mathcal{O}}^{\times}_{X^{o}})\\{p^{\prime}\\}$. On the other
hand, the term on the left of this surjection is zero because $cd(X^{o})\leq
3$. For $i=3$, note that there is an inclusion
$H^{3}(X^{o},{\mathcal{O}}^{\times}_{X^{o}})\\{p^{\prime}\\}\hookrightarrow
H^{4}_{D}(X,{\mathcal{O}}^{\times}_{X})\\{p^{\prime}\\}$ by Corollary 10.10.
On the other hand, there is a canonical surjection
$H^{4}_{D}(X,{{\mathbb{Q}}}/{{\mathbb{Z}}}\\{p^{\prime}\\}(1))\twoheadrightarrow
H^{4}_{D}(X,{\mathcal{O}}^{\times}_{X})\\{p^{\prime}\\}$. Using Gabber’s
purity, the term on the left of this surjection is isomorphic to
${\underset{x\in
D}{\bigoplus}}H^{2}(k(x),{{\mathbb{Q}}}/{{\mathbb{Z}}}\\{p^{\prime}\\})$.

We are thus reduced to showing that
$H^{2}(k^{\prime},{{\mathbb{Q}}}/{{\mathbb{Z}}}\\{p^{\prime}\\})=0$ for $i\geq
2$ if ${k^{\prime}}/k$ is any finite field extension. By [15, Thm. 9.9], we
have an isomorphism
$H^{2}(k^{\prime},{{\mathbb{Q}}}/{{\mathbb{Z}}}\\{p^{\prime}\\})\cong{\varinjlim}_{p\nmid
n}{\rm
Hom}_{{\mathbf{Ab}}}(\mu_{n}(k^{\prime}),{{\mathbb{Q}}}/{{\mathbb{Z}}})$,
where the limit is taken with respect to the maps
$\mu_{mn}(k^{\prime})\xrightarrow{m}\mu_{n}(k^{\prime})$. But this limit is
zero since $({k^{\prime}}^{\times})\\{p^{\prime}\\}$ is finite. This concludes
the proof. ∎

Recall that there is an exact sequence

(12.6) ${{\mathbb{Z}}}^{r}\to{\rm CH}_{0}(X)\xrightarrow{j^{*}}{\rm
CH}_{0}(X^{o})\to 0,$

where the first arrow is the sum of cycle class maps for points in $D$. We
endow ${\rm CH}_{0}(X^{o})$ with the quotient of the adic topology of ${\rm
CH}_{0}(X)$ via this exact sequence. Note that this topology of ${\rm
CH}_{0}(X^{o})$ is independent of any choice of a regular compactification of
$X^{o}$. We shall consider $H^{2}_{cc}(X^{o},{\mathcal{O}}^{\times}_{X^{o}})$
to be a discrete abelian group.

######  Lemma 12.5.

There is a perfect pairing of topological abelian groups

${\rm CH}_{0}(X^{o})^{\rm pf}\times
H^{2}_{cc}(X^{o},{\mathcal{O}}^{\times}_{X^{o}})\to{{\mathbb{Q}}}/{{\mathbb{Z}}}.$

###### Proof.

We look at the diagram

(12.7)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{2}_{cc}(X,{\mathcal{O}}^{\times}_{X^{o}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\operatorname{Br}}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{X}}$$\textstyle{{\operatorname{Br}}(D^{\dagger})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{D^{\dagger}}}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
CH}_{0}(X^{o})^{\star}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(j^{*})^{\star}}$$\textstyle{{\rm
CH}_{0}(X)^{\star}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{({{\mathbb{Q}}}/{{\mathbb{Z}}})^{r},}$

where the right vertical arrow is the sum (over $x\in D^{\dagger}$) of the
maps ${\rm inv}_{k(x)}$. It follows from the exactness of the top row that the
canonical map $H^{2}_{cc}(X,{\mathcal{O}}^{\times}_{X^{o}})\to
H^{2}(X,{\mathcal{K}}^{M}_{1,(X,D^{\dagger})})$ is an isomorphism.

It is immediate from the construction of the Brauer-Manin pairing (9.5) (with
$D=\emptyset$) that the right square in (12.7) is commutative. The middle and
the right vertical arrows are isomorphisms. It follows that there is a unique
isomorphism $\beta^{1}_{X^{o}}\colon
H^{2}_{cc}(X^{o},{\mathcal{O}}^{\times}_{X^{o}})\xrightarrow{\cong}{\rm
CH}_{0}(X^{o})^{\star}$ such that (12.7) is commutative. Since
$\beta^{1}_{X^{o}}$ is automatically continuous, we get a continuous pairing
${\rm CH}_{0}(X^{o})^{\rm pf}\times
H^{2}_{cc}(X^{o},{\mathcal{O}}^{\times}_{X^{o}})\to{{\mathbb{Q}}}/{{\mathbb{Z}}}$.
Since ${\rm CH}_{0}(X)^{\star}$ is a torsion group, ${\rm
CH}_{0}(X^{o})^{\star}$ must also be torsion. We can now apply Lemma 2.5 and
the Pontryagin duality between profinite and discrete torsion groups to
conclude the proof. ∎

End of the proof of Theorem 1.1: Combine Theorem 1.6 with Lemmas 2.5, 8.8,
12.4 and Lemma 12.5. $\hfill\qed$

######  Corollary 12.6.

There is a perfect pairing of topological abelian groups

${\operatorname{Br}}(X^{o})\times C(X^{o})^{{\rm
pf}}\to{{\mathbb{Q}}}/{{\mathbb{Z}}}.$

###### Proof.

Combine Theorem 1.1 and Lemma 11.1. ∎

###  12.3. Duality for the cohomology of ${\mathbb{G}}_{m,K}$

Let the notations and assumptions be as stated in the beginning of § 12.
Recall that $K$ denotes the function field of $X$. We let
$H^{q}_{cc}(K,{\mathbb{G}}_{m}):={\underset{D\in{\rm
Div}(X)}{\varprojlim}}H^{q}(X,{\mathbb{G}}_{m,(X,D)})$. We let $C(K)$ be the
cokernel of the canonical map
$K^{\times}\xrightarrow{\alpha}\widehat{I}(X):={\underset{x\in
X_{(0)}}{\prod^{\prime}}}\widehat{K}^{\times}_{x}$, where $\alpha$ is the
canonical inclusion (cf. (9.9)). We endow $H^{1}_{cc}(K,{\mathbb{G}}_{m})$
with the inverse limit of the adic topologies of ${\rm Pic}(X|D)$ for
$D\in{\rm Div}(X)$, and ${\operatorname{Br}}(K)$ with the discrete topology.

To prove the duality theorem for the étale cohomology of ${\mathbb{G}}_{m,K}$,
we need the following.

######  Lemma 12.7.

For every effective divisor $D\subset X$ with complement $X^{o}$, the
canonical maps $H^{1}_{cc}(K,{\mathbb{G}}_{m})\to
H^{1}_{cc}(X^{o},{\mathbb{G}}_{m})\to H^{1}(X,{\mathbb{G}}_{m,(X,D)})$ are
surjective.

###### Proof.

Using Lemma 11.1, it suffices to show that for every reduced effective divisor
$D\subset X$ with $X^{o}=X\setminus D$, the map $C(K)\to C(X^{o})$ is
surjective, and the map $C(K)\to{\varprojlim}_{D}C(X^{o})$ is bijective, where
the limit is over all reduced effective divisors $D\subset X$.

We have a commutative diagram of exact sequences

(12.8)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K^{\times}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{\widehat{I}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{D}}$$\textstyle{C(K)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma_{D}}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K^{\times}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{D}}$$\textstyle{{\mathcal{Z}}_{0}(X^{o})\oplus
B(D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C(X^{o})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

where we let $B(D):=\widehat{K}^{\times}_{\infty}={\prod}_{x\in
D}\widehat{K}^{\times}_{x}$. We let $A(D)={\mathcal{Z}}_{0}(X^{o})\oplus
B(D)$. It is straightforward to check that each $\beta_{D}$ is surjective. In
particular, each transition map of the cofiltered system $\\{A(D)\\}$ is
surjective. It remains to show that the map
$\beta:={\varprojlim}_{D}\beta_{D}\colon\widehat{I}(X)\to{\varprojlim}_{D}A(D)$
is bijective.

It is easy to see that ${\prod}_{x\in
X_{(0)}}\widehat{K}^{\times}_{x}\xrightarrow{\cong}{\varprojlim}_{D}B(D)$. We
let $\pi_{D}\colon A(D)\to B(D)$ denote the projection. For $D^{\prime}>D$,
the transition map $\Phi_{D^{\prime}>D}\colon A(D^{\prime})\to A(D)$ has the
property that if $(a_{1},a_{2},a_{3})\in A(D^{\prime})$ with
$a_{1}\in{\mathcal{Z}}_{0}(X\setminus D^{\prime}),\ a_{2}\in{\prod}_{x\in
D}\widehat{K}^{\times}_{x}$ and $a_{3}\in{\prod}_{x\in D^{\prime}\setminus
D}\widehat{K}^{\times}_{x}$, then

(12.9) $\Phi_{D^{\prime}>D}(a_{1})=a_{1},\ \Phi_{D^{\prime}>D}(a_{2})=a_{2}\
\mbox{and}\ \Phi_{D^{\prime}>D}(a_{3})={\prod}_{x\in D^{\prime}\setminus
D}v_{x}(a_{3}).$

It follows that
$\pi_{D}\circ\Phi_{D^{\prime}>D}=\Phi_{D^{\prime}>D}\circ\pi_{D^{\prime}}$.
That is, the projection $\\{\pi_{D}\\}\colon\\{A(D)\\}\to\\{B(D)\\}$ is a
strict morphism of pro-abelian groups. Taking the limits, we get a commutative
diagram

(12.10)
$\textstyle{{\varprojlim}_{D}A(D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{{\prod}_{x\in
X_{(0)}}\widehat{K}^{\times}_{x}}$$\textstyle{\widehat{I}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces,}$$\scriptstyle{\beta}$$\scriptstyle{\beta^{\prime}}$

where $\beta^{\prime}$ is the canonical inclusion
$\widehat{I}(X)\hookrightarrow{\prod}_{x\in X_{(0)}}\widehat{K}^{\times}_{x}$.
In particular, $\beta$ is injective.

We claim that $\pi$ is injective. Indeed, if $(a_{D})\in{\varprojlim}_{D}A(D)$
is an element indexed by reduced divisors $D\subset X$ with
$(a_{D})=(a_{0},a_{1})$ (where $a_{0}\in{\mathcal{Z}}_{0}(X\setminus D)$ and
$a_{1}\in B(D)$) such that $\pi((a_{D}))=0$, then we must have $a_{1}=0$ for
each reduced divisor $D\subset X$. That is,
$(a_{D})\in{\varprojlim}_{D}{\mathcal{Z}}_{0}(X\setminus
D)={\bigcap}_{D}{\mathcal{Z}}_{0}(X\setminus D)$. The claim now follows
because it is not hard to see that ${\bigcap}_{D}{\mathcal{Z}}_{0}(X\setminus
D)=0$.

To prove that $\beta$ is surjective, let
$(a_{D})\in{\varprojlim}_{D}A(D)\subset{\prod}_{x\in
X_{(0)}}\widehat{K}^{\times}_{x}$. Let $b=\pi((a_{D}))$ and write
$b=(b_{x})_{x\in X_{(0)}}$. If $v_{x}(b_{x})\neq 0$ for infinitely many $x\in
X_{(0)}$, then it follows from (12.9) that the projection map
${\varprojlim}_{D}A(D)\to A(D)$ is not defined at $b$ for any reduced divisor
$D\subset X$. But this is absurd. We conclude that
$b=\pi((a_{D}))\in\widehat{I}(X)$. This shows that $\beta$ is surjective, and
concludes the proof. ∎

Using Lemma 12.7, we endow $C(K)$ with the inverse limit of the adic
topologies of ${\rm Pic}(X|D)$ for $D\in{\rm Div}(X)$. We can now prove the
duality theorem for the cohomology of ${\mathbb{G}}_{m,K}$.

######  Theorem 12.8.

For every integer $q\neq 0$, there is a bilinear pairing

$H^{q}(K,{\mathbb{G}}_{m})\times
H^{3-q}_{cc}(K,{\mathbb{G}}_{m})\to{{\mathbb{Q}}}/{{\mathbb{Z}}}$

which induces perfect pairings of topological abelian groups

$H^{1}(K,{\mathbb{G}}_{m})^{{\rm pf}}\times
H^{2}_{cc}(K,{\mathbb{G}}_{m})\to{{\mathbb{Q}}}/{{\mathbb{Z}}},$
$H^{2}(K,{\mathbb{G}}_{m})\times H^{1}_{cc}(K,{\mathbb{G}}_{m})^{{\rm
pf}}\to{{\mathbb{Q}}}/{{\mathbb{Z}}}$

and

$H^{3}(K,{\mathbb{G}}_{m})\times H^{0}_{cc}(K,{\mathbb{G}}_{m})^{{\rm
pf}}\to{{\mathbb{Q}}}/{{\mathbb{Z}}}.$

###### Proof.

The existence of the pairing is an immediate consequence of Theorem 1.1 and a
limit argument (over $D\in{\rm Div}(X)$). The perfectness of the third pairing
follows from Lemma 12.4 using a limit argument. The perfectness of the second
pairing follows by applying Theorem 1.6, Lemmas 2.5, 2.6, 8.8 and 12.7, and
taking limit over ${\rm Div}(X)$. To prove the perfectness of the first
pairing, we only need to show that $H^{2}_{cc}(K,{\mathbb{G}}_{m})=0$. But
this follows by taking limit over ${\rm Div}(X)$ of the top exact sequence in
(12.7) and observing that the map ${\operatorname{Br}}(X)\to{\prod}_{x\in
X_{(0)}}{\operatorname{Br}}(k(x))\cong{\varprojlim}_{D}{\operatorname{Br}}(D_{\rm
red})$ is injective by Lemma 12.1 (with $D=\emptyset$). ∎

By (9.9), we have a bilinear pairing
${\operatorname{Br}}(K)\times\widehat{I}(X)\to{{\mathbb{Q}}}/{{\mathbb{Z}}}$,
and the proof of Proposition 9.4 shows that this factors through
${\operatorname{Br}}(K)\times C(K)\to{{\mathbb{Q}}}/{{\mathbb{Z}}}$. The
following result is the global version of [46, Thm. 2.10, Cor. 2.12] and
provides an explicit description of ${\operatorname{Br}}(K)$.

######  Corollary 12.9.

The above pairing induces a perfect pairing of topological abelian groups

${\operatorname{Br}}(K)\times C(K)^{{\rm
pf}}\to{{\mathbb{Q}}}/{{\mathbb{Z}}}.$

In particular, ${\operatorname{Br}}(K)\xrightarrow{\cong}(C(K)^{\star})_{\rm
tor}$.

##  13\. Brauer-Manin pairing for singular curves

In this section, we shall apply Theorem 1.6 to extend the Brauer-Manin pairing
for smooth projective curves à la Lichtenbaum-Saito to singular curves over
local fields. To achieve this, we shall introduce a refined version of the
Brauer group for singular varieties. We shall then show that this refined
group is directly related to the Picard group for singular curves. We fix a
local field $k$ of exponential characteristic $p\geq 1$.

###  13.1. Levine-Weibel Brauer group of singular varieties

Before we introduce the refined Brauer group for singular quasi-projective
varieties, we recall some known facts about the completions of local rings.
For any semilocal ring $R$, let $\widehat{R}$ denote the completion of $R$
with respect to its Jacobson radical. Let $A$ be a local integral domain which
is essentially of finite type over a field and let $B$ denote the integral
closure of $A$. We let ${\mathfrak{m}}_{A}$ (resp. ${\mathfrak{m}}_{B}$)
denote the maximal ideal of $A$ (resp. the Jacobson radical of $B$). It is an
elementary fact (e.g., see [1, Chap. 10]) that
$\widehat{A}\hookrightarrow\widehat{B}\cong
B\otimes_{A}\widehat{A}\cong\prod_{i}\widehat{B_{{\mathfrak{m}}_{i}}}$, where
the product is over all maximal ideals of $B$. Furthermore, $\widehat{B}$
coincides with the normalization of $\widehat{A}$ if $\text{\rm dim}(A)=1$. In
particular, it is a product of discrete valuation rings
$\widehat{B_{{\mathfrak{m}}_{i}}}$. In the latter case, it is also easy to see
using the faithfully flatness property of $A\hookrightarrow\widehat{A}$ that
$\widehat{A}\cap K=A$, where $K$ is the quotient field of $A$.

We let $X$ be an integral quasi-projective curve over $k$. Let
$\pi\colon{X^{\prime}}\to X$ be the normalization of $X$ and let $K$ denote
the function field of $X$. For every closed point $x\in X$ and every
$y\in\pi^{-1}(x)$, let $U^{0}_{x,y}\subset\widehat{K}_{y}$ be the image of
$(\widehat{{\mathcal{O}}_{X,x}})^{\times}$ under the composite map
$\widehat{{\mathcal{O}}_{X,x}}\hookrightarrow\widehat{{\mathcal{O}}_{X^{\prime},y}}\hookrightarrow\widehat{K}_{y}$.

######  Definition 13.1.

We let ${\operatorname{Br}}^{{\rm lw}}(X)$ be the subgroup of
${\operatorname{Br}}(X_{\rm reg})$ consisting of elements $w$ such that for
every singular point $x\in X$ and every $y\in\pi^{-1}(x)$, the image of $w$
under the canonical map ${\operatorname{Br}}(X_{\rm
reg})\hookrightarrow{\operatorname{Br}}(K)\to{\operatorname{Br}}(\widehat{K}_{y})$
has the property that the associated character (under Kato’s pairing)
$\chi_{w}\colon\widehat{K}^{\times}_{y}\to{{\mathbb{Q}}}/{{\mathbb{Z}}}$
annihilates $U^{0}_{x,y}$.

Let $X$ be an integral quasi-projective variety over $k$. Let
${\mathcal{C}}^{\rm l.c.i.\\!}(X)$ be the set of all integral curves $C\subset
X$ such that $C$ is not contained in $X_{\rm sing}$ and the inclusion
$C\hookrightarrow X$ is a local complete intersection at every point of $C\cap
X_{\rm sing}$.

######  Definition 13.2.

We let ${\operatorname{Br}}^{{\rm lw}}(X)$ be the subgroup of
${\operatorname{Br}}(X_{\rm reg})$ consisting of elements $w$ such that for
every $C\in{\mathcal{C}}^{\rm l.c.i.\\!}(X),\ x\in X_{\rm sing}\cap C$ and
$y\in\nu^{-1}(x)$, the image of $w$ under the canonical composite map
${\operatorname{Br}}(X_{\rm reg})\to{\operatorname{Br}}(\nu^{-1}(X_{\rm
reg}))\to{\operatorname{Br}}(\widehat{k(C)}_{y})$ has the property that the
associated character (under Kato’s pairing)
$\chi_{w}\colon\widehat{k(C)}^{\times}_{y}\to{{\mathbb{Q}}}/{{\mathbb{Z}}}$
annihilates $U^{0}_{x,y}$, where $\nu\colon C_{n}\to X$ is the canonical map.

It easily follows from the above definitions that ${\operatorname{Br}}^{{\rm
lw}}(X)\subset{\operatorname{Br}}^{{\rm lw}}(U)$ if $U\subset X$ is open. This
is in contrast with the classical Brauer group of singular varieties. It is
also clear that ${\operatorname{Br}}^{{\rm lw}}(X)={\operatorname{Br}}(X)$ if
$X$ is regular. We shall refer to ${\operatorname{Br}}^{{\rm lw}}(X)$ as the
‘Levine-Weibel Brauer group’ of $X$.

###  13.2. The Brauer-Manin pairing

For the rest of § 13, we shall work with the following set-up. Let $X$ be a
geometrically integral projective curve over $k$ and $\pi\colon X_{n}\to X$
the normalization of $X$. It is clear that ${X}_{n}$ is geometrically
integral. We shall assume that ${X}_{n}$ is smooth over $k$ (this is automatic
if $p=1$). We let $K$ denote the function field of $X$. We let $S$ be the
singular locus of $X$ with the reduced closed subscheme structure and let $E$
be the scheme theoretic inverse image of $S$ in ${X}_{n}$. We let
$X^{o}=X_{\rm reg}$ be the regular locus of $X$. We represent this datum in
the Cartesian squares

(13.1)
$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota^{\prime}}$$\scriptstyle{\pi^{\prime}}$$\textstyle{{X}_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{X^{o}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j^{\prime}}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\textstyle{X}$$\textstyle{X^{o}\ignorespaces\ignorespaces\ignorespaces\ignorespaces.}$$\scriptstyle{j}$

We have recalled the definition of the Levine-Weibel Chow group ${\rm
CH}^{{\rm lw}}_{0}(X)$ and the isomorphism ${\operatorname{\rm
cyc}}_{X}\colon{\rm CH}^{{\rm lw}}_{0}(X)\xrightarrow{\cong}{\rm Pic}(X)$
after Lemma 5.4. We let ${\rm CH}^{{\rm lw}}_{0}(X)_{0}={\rm
Ker}(\deg\colon{\rm CH}^{{\rm lw}}_{0}(X)\to{\mathbb{Z}})\cong{\rm
Pic}^{0}(X)$. We shall use ${\rm Pic}(X)$ and ${\rm CH}^{{\rm lw}}_{0}(X)$
interchangeably. In order to construct a duality between the Levine-Weibel
Brauer group and the Picard group of $X$, we need to give a new description of
the latter group.

For a point $x\in S$, we let $E_{x}=\pi^{-1}(x)$ and
$\widehat{K}_{x,\infty}={\underset{y\in E_{x}}{\prod}}\widehat{K}_{y}$. We let
$U^{0}_{x,\infty}={\underset{y\in E_{x}}{\prod}}U^{0}_{x,y}$. We have the
canonical maps
$K^{\times}\hookrightarrow\widehat{K}^{\times}_{x,\infty}\twoheadrightarrow{\widehat{K}^{\times}_{x,\infty}}/{U^{0}_{x,\infty}}$.
We let $\delta_{x}$ be the composite map. Let $\delta={\underset{y\in
E_{x}}{\prod}}\delta_{x}$,

######  Lemma 13.3.

There is a canonical exact sequence

(13.2) $0\to k^{\times}\to K^{\times}\xrightarrow{({\rm
div},\delta)}{\mathcal{Z}}_{0}(X^{o})\bigoplus\left({\underset{x\in
S}{\prod}}\frac{\widehat{K}^{\times}_{x,\infty}}{U^{0}_{x,\infty}}\right)\to{\rm
CH}^{{\rm lw}}_{0}(X)\to 0.$

###### Proof.

We consider the diagram

(13.3)
$\textstyle{{\mathcal{O}}^{\times}_{X,S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
div}}$$\textstyle{{\mathcal{Z}}_{0}(X^{o})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
CH}^{{\rm
lw}}_{0}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{K^{\times}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{({\rm
div},\delta)}$$\textstyle{{{\mathcal{Z}}_{0}(X^{o})\bigoplus\left({\underset{x\in
S}{\prod}}\frac{\widehat{K}^{\times}_{x,\infty}}{U^{0}_{x,\infty}}\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{{\rm
CH}^{{\rm
lw}}_{0}(X)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

where the vertical arrows are the canonical inclusions and $\overline{{\rm
CH}^{LW}_{0}(X)}$ is defined to make the bottom row exact. The top row is
exact by definition of ${\rm CH}^{{\rm lw}}_{0}(X)$. It is clear that the left
square is commutative. This yields a unique homomorphism $\psi_{X}\colon{\rm
CH}^{{\rm lw}}_{0}(X)\to\overline{{\rm CH}^{{\rm lw}}_{0}(X)}$ such that the
right square also commutes.

To prove that $\psi_{X}$ is bijective, it is equivalent to show that

(13.4) $0\to{\mathcal{O}}^{\times}_{X,S}\to
K^{\times}\xrightarrow{\delta}{\underset{x\in
S}{\prod}}\frac{\widehat{K}^{\times}_{x,\infty}}{U^{0}_{x,\infty}}\to 0$

is exact. Now, it is clear that if $f\in K^{\times}$ is such that
$\delta_{x}(f)=0$ for every $x\in S$, then $f\in{\underset{x\in
S}{\bigcap}}{\mathcal{O}}^{\times}_{X,x}$ (see the first paragraph of § 13.1).
But it is not hard to see that ${\mathcal{O}}^{\times}_{X,S}={\underset{x\in
S}{\bigcap}}{\mathcal{O}}^{\times}_{X,x}$, using the fact that
${\mathcal{O}}^{\times}_{X,x}$ is the set of rational functions on $X$ which
are regular without a zero or a pole in a neighborhood of $x$. It remains to
show that $\delta$ is surjective.

We let $I_{x}\subset\widehat{{\mathcal{O}}_{X,x}}$ be a conductor ideal for
the normalization
$\widehat{{\mathcal{O}}_{X,x}}\hookrightarrow\widehat{{\mathcal{O}}_{X_{n},E_{x}}}={\underset{y\in
E_{x}}{\prod}}\widehat{{\mathcal{O}}_{X_{n},y}}$. Then
$I_{x}\twoheadrightarrow
I_{x}\widehat{{\mathcal{O}}_{X_{n},y}}\subset\widehat{{\mathfrak{m}}_{y}}$ for
every $y\in E_{x}$, where the latter is the maximal ideal of
$\widehat{{\mathcal{O}}_{X_{n},y}}$. It is also clear that
$\sqrt{I_{x}\widehat{{\mathcal{O}}_{X_{n},y}}}=\widehat{{\mathfrak{m}}_{y}}$.
As $I_{x}\xrightarrow{\cong}{\underset{y\in
E_{x}}{\prod}}I_{x}\widehat{{\mathcal{O}}_{X_{n},y}}$, it follows that
$(1+I_{x})\xrightarrow{\cong}{\underset{y\in
E_{x}}{\prod}}(1+I_{x}\widehat{{\mathcal{O}}_{X_{n},y}})$. In particular,
${\underset{y\in E_{x}}{\prod}}(1+\widehat{{\mathfrak{m}}_{y}}^{n})\subset
U^{0}_{x,\infty}$ for all $n\gg 0$. In other words, we have a factorization

(13.5) $K^{\times}\to{\underset{x\in
S}{\prod}}\frac{\widehat{K}^{\times}_{x,\infty}}{U^{n}_{x,\infty}}\twoheadrightarrow{\underset{x\in
S}{\prod}}\frac{\widehat{K}^{\times}_{x,\infty}}{U^{0}_{x,\infty}}$

for all $n\gg 0$, where we let $U^{n}_{x,\infty}={\underset{y\in
E_{x}}{\prod}}(1+\widehat{{\mathfrak{m}}_{y}}^{n})$. We are now done because
the left arrow is well known to be surjective as an application of the
approximation lemma in number theory (e.g., see [28, Lem. 6.3]). This proves
the exactness of (13.4).

To conclude the proof of the lemma, it remains to show that $k^{\times}={\rm
Ker}({\rm div})\bigcap{\rm Ker}(\delta)$. But this is easy because we have
shown that ${\rm Ker}({\rm div})\bigcap{\rm Ker}(\delta)={\rm Ker}({\rm
div})\bigcap{\mathcal{O}}^{\times}_{X,S}$ and the latter is $k^{\times}$
because $X$ is geometrically integral. ∎

We now define the Brauer-Manin pairing for $X$ as follows. Let
$w\in{\operatorname{Br}}^{{\rm lw}}(X)$. We define the associated character
$\chi_{w}\colon{\mathcal{Z}}_{0}(X^{o})\bigoplus({\underset{x\in
S}{\prod}}\widehat{K}^{\times}_{x,\infty})\to{{\mathbb{Q}}}/{{\mathbb{Z}}}$
exactly as we did to define the pairing (9.4) in § 9.2. It follows from the
definition of ${\operatorname{Br}}^{{\rm lw}}(X)$ that $\chi_{w}$ factors
through ${\mathcal{Z}}_{0}(X^{o})\bigoplus{\underset{x\in
S}{\prod}}\frac{\widehat{K}^{\times}_{x,\infty}}{U^{0}_{x,\infty}}$. The proof
of the claim that $\chi_{w}\circ({\rm div},\delta)=0$ is identical to the
proof of Proposition 9.4, mutatis mutandis. We have thus shown the following.

######  Proposition 13.4.

There exists a canonical bilinear pairing

(13.6) ${\operatorname{Br}}^{{\rm lw}}(X)\times{\rm CH}^{{\rm
lw}}_{0}(X)\to{{\mathbb{Q}}}/{{\mathbb{Z}}}.$

###  13.3. Perfectness of the pairing

We now state and prove the main result of § 13. We shall continue to work with
the set-up described in § 13.2. Since $X$ is geometrically integral, it
follows from [29, Thm. 9.2.5, 9.4.8] that there is a group scheme
${\mathbf{Pic}}(X)$ over $k$ which is locally of finite type and whose
identity component ${\mathbf{Pic}}^{0}(X)$ is a smooth quasi-projective group
scheme over $k$, and ${\mathbf{Pic}}(X)(k)\cong{\rm Pic}(X)$. One also has
that ${\mathbf{Pic}}(X_{k^{\prime}})\cong{\mathbf{Pic}}(X)_{k^{\prime}}$ for
every field extension ${k^{\prime}}/k$. It follows that ${\rm Pic}(X)$ is
equipped with its canonical adic topology and ${\rm Pic}^{0}(X)$ is its open
subgroup. We shall let ${\operatorname{Br}}^{{\rm lw}}(X)$ have the discrete
topology.

######  Lemma 13.5.

The pairing (13.6) is continuous.

###### Proof.

It suffices to show that the image of the induced homomorphism
$\beta_{X}\colon{\operatorname{Br}}^{{\rm lw}}(X)\to{\rm Pic}(X)^{\vee}$ lies
in ${\rm Pic}(X)^{\star}$ (cf. § 4.2). To that end, we choose a conductor
ideal sheaf ${\mathcal{I}}_{Z}\subset{\mathcal{O}}_{X}$ associated to $\pi$ so
that we have an isomorphism
$(1+{\mathcal{I}}_{nZ})\xrightarrow{\cong}\pi_{*}((1+{\mathcal{I}}_{nD}))$ for
every $n\geq 1$, where we let $Z\subset X$ be the closed subscheme defined by
${\mathcal{I}}_{Z}$ and $D$ the scheme theoretic inverse image of $Z$ under
$\pi$. It follows from Corollary 6.10 that the canonical map ${\rm
Pic}(X_{n}|D)\twoheadrightarrow{\rm Pic}(X_{n})$ has a factorization ${\rm
Pic}(X_{n}|D)\twoheadrightarrow{\rm Pic}(X)\twoheadrightarrow{\rm Pic}(X_{n})$
in which all maps are topological quotient maps between adic spaces.

We now let $w\in{\operatorname{Br}}^{{\rm lw}}(X)$ and let $\chi_{w}\colon{\rm
Pic}(X)\to{{\mathbb{Q}}}/{{\mathbb{Z}}}$ be the associated character. Since
$w\in{\operatorname{Br}}(X^{o})$, it lies in ${\operatorname{Br}}(X_{n}|D)$
for some conductor closed subscheme $Z\subset X$ and $D=Z\times_{X}X_{n}$ as
above. It is easily seen that the composite map $\chi_{w}\colon{\rm
Pic}(X_{n}|D)\twoheadrightarrow{\rm Pic}(X)\to{{\mathbb{Q}}}/{{\mathbb{Z}}}$
is same as the one in (9.8) (see Remark 9.1). Proposition 11.2 then implies
that this composite map is continuous. Since ${\rm
Pic}(X_{n}|D)\twoheadrightarrow{\rm Pic}(X)$ is a topological quotient, it
follows that $\chi_{w}$ is continuous on ${\rm Pic}(X)$. This concludes the
proof. ∎

Let the notations be as in the proof of Lemma 13.5. From (13.5), one also
deduces the following.

######  Lemma 13.6.

For all $m\gg 0$, we have ${\operatorname{Br}}^{\rm
lw}(X)\subseteq{\operatorname{Br}}(X_{n}|mD)$.

######  Theorem 13.7.

The pairing (13.6) induces a perfect pairing of topological abelian groups

${\operatorname{Br}}^{{\rm lw}}(X)\times{\rm CH}^{{\rm lw}}_{0}(X)^{\rm
pf}\to{{\mathbb{Q}}}/{{\mathbb{Z}}}.$

###### Proof.

The continuity of the pairing follows from Lemmas 2.5 and 13.5. As in the
proof of Theorem 1.6, it suffices to show that the map
$\beta_{X}\colon{\operatorname{Br}}^{{\rm lw}}(X)\to F$ is bijective if we let
$F={\rm CH}^{{\rm lw}}_{0}(X)^{\star}$ when $p>1$ and $F=({\rm CH}^{{\rm
lw}}_{0}(X)^{\star})_{\rm tor}$ when $p=1$.

Using Lemma 13.6, we choose conductor subschemes $Z\subset X$ and
$D=Z\times_{X}X_{n}\subset X_{n}$ such that ${\operatorname{Br}}^{\rm
lw}(X)\subseteq{\operatorname{Br}}(X_{n}|D)$. We let $G={\rm
CH}_{0}(X_{n}|D)^{\star}$ (resp. $({\rm CH}_{0}(X_{n}|D)^{\star})_{\rm tor}$)
if $p>1$ (resp. $p=1$). We consider the commutative diagram

(13.7) $\textstyle{{\operatorname{Br}}^{{\rm
lw}}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{X}}$$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\operatorname{Br}}(X_{n}|D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{X_{n}|D}}$$\textstyle{G.}$

The right vertical arrow is injective by Corollary 6.10. Since
$\beta_{X_{n}|D}$ is injective by Lemma 12.1, it follows that $\beta_{X}$ is
injective. To prove its surjectivity, we let $\chi\in F$. If we consider
$\chi$ as an element of $G$ , then Lemma 12.2 says that
$\chi=\beta_{X_{n}|D}(w)$ for some $w\in{\operatorname{Br}}(X_{n}|D)$. In this
case, it follows directly from the definition of ${\operatorname{Br}}^{{\rm
lw}}(X)$ and Lemma 13.3 that $w\in{\operatorname{Br}}^{{\rm lw}}(X)$. This
implies subsequently that $\beta_{X}(w)=\chi$. This concludes the proof of the
theorem. ∎

_Acknowledgements._ JR and SS would like to thank IISc, Bangalore for hosting
them during the work on this paper.

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