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How to Write a Research Paper Introduction (with Examples)

The research paper introduction section, along with the Title and Abstract, can be considered the face of any research paper. The following article is intended to guide you in organizing and writing the research paper introduction for a quality academic article or dissertation.

The research paper introduction aims to present the topic to the reader. A study will only be accepted for publishing if you can ascertain that the available literature cannot answer your research question. So it is important to ensure that you have read important studies on that particular topic, especially those within the last five to ten years, and that they are properly referenced in this section. 1 What should be included in the research paper introduction is decided by what you want to tell readers about the reason behind the research and how you plan to fill the knowledge gap. The best research paper introduction provides a systemic review of existing work and demonstrates additional work that needs to be done. It needs to be brief, captivating, and well-referenced; a well-drafted research paper introduction will help the researcher win half the battle.

The introduction for a research paper is where you set up your topic and approach for the reader. It has several key goals:

• Present your research topic
• Capture reader interest
• Summarize existing research
• Position your own approach
• Define your specific research problem and problem statement
• Highlight the novelty and contributions of the study
• Give an overview of the paper’s structure

The research paper introduction can vary in size and structure depending on whether your paper presents the results of original empirical research or is a review paper. Some research paper introduction examples are only half a page while others are a few pages long. In many cases, the introduction will be shorter than all of the other sections of your paper; its length depends on the size of your paper as a whole.

• Break through writer’s block. Write your research paper introduction with Paperpal Copilot

Table of Contents

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The introduction in a research paper is placed at the beginning to guide the reader from a broad subject area to the specific topic that your research addresses. They present the following information to the reader

• Scope: The topic covered in the research paper
• Context: Background of your topic
• Importance: Why your research matters in that particular area of research and the industry problem that can be targeted

The research paper introduction conveys a lot of information and can be considered an essential roadmap for the rest of your paper. A good introduction for a research paper is important for the following reasons:

• It stimulates your reader’s interest: A good introduction section can make your readers want to read your paper by capturing their interest. It informs the reader what they are going to learn and helps determine if the topic is of interest to them.
• It helps the reader understand the research background: Without a clear introduction, your readers may feel confused and even struggle when reading your paper. A good research paper introduction will prepare them for the in-depth research to come. It provides you the opportunity to engage with the readers and demonstrate your knowledge and authority on the specific topic.
• It explains why your research paper is worth reading: Your introduction can convey a lot of information to your readers. It introduces the topic, why the topic is important, and how you plan to proceed with your research.
• It helps guide the reader through the rest of the paper: The research paper introduction gives the reader a sense of the nature of the information that will support your arguments and the general organization of the paragraphs that will follow. It offers an overview of what to expect when reading the main body of your paper.

What are the parts of introduction in the research?

A good research paper introduction section should comprise three main elements: 2

• What is known: This sets the stage for your research. It informs the readers of what is known on the subject.
• What is lacking: This is aimed at justifying the reason for carrying out your research. This could involve investigating a new concept or method or building upon previous research.
• What you aim to do: This part briefly states the objectives of your research and its major contributions. Your detailed hypothesis will also form a part of this section.

How to write a research paper introduction?

The first step in writing the research paper introduction is to inform the reader what your topic is and why it’s interesting or important. This is generally accomplished with a strong opening statement. The second step involves establishing the kinds of research that have been done and ending with limitations or gaps in the research that you intend to address. Finally, the research paper introduction clarifies how your own research fits in and what problem it addresses. If your research involved testing hypotheses, these should be stated along with your research question. The hypothesis should be presented in the past tense since it will have been tested by the time you are writing the research paper introduction.

The following key points, with examples, can guide you when writing the research paper introduction section:

• Highlight the importance of the research field or topic
• Describe the background of the topic
• Present an overview of current research on the topic

Example: The inclusion of experiential and competency-based learning has benefitted electronics engineering education. Industry partnerships provide an excellent alternative for students wanting to engage in solving real-world challenges. Industry-academia participation has grown in recent years due to the need for skilled engineers with practical training and specialized expertise. However, from the educational perspective, many activities are needed to incorporate sustainable development goals into the university curricula and consolidate learning innovation in universities.

• Reveal a gap in existing research or oppose an existing assumption
• Formulate the research question

Example: There have been plausible efforts to integrate educational activities in higher education electronics engineering programs. However, very few studies have considered using educational research methods for performance evaluation of competency-based higher engineering education, with a focus on technical and or transversal skills. To remedy the current need for evaluating competencies in STEM fields and providing sustainable development goals in engineering education, in this study, a comparison was drawn between study groups without and with industry partners.

• State the purpose of your study
• Highlight the key characteristics of your study
• Describe important results
• Highlight the novelty of the study.
• Offer a brief overview of the structure of the paper.

Example: The study evaluates the main competency needed in the applied electronics course, which is a fundamental core subject for many electronics engineering undergraduate programs. We compared two groups, without and with an industrial partner, that offered real-world projects to solve during the semester. This comparison can help determine significant differences in both groups in terms of developing subject competency and achieving sustainable development goals.

Write a Research Paper Introduction in Minutes with Paperpal

Paperpal Copilot is a generative AI-powered academic writing assistant. It’s trained on millions of published scholarly articles and over 20 years of STM experience. Paperpal Copilot helps authors write better and faster with:

• Real-time writing suggestions
• In-depth checks for language and grammar correction
• Paraphrasing to add variety, ensure academic tone, and trim text to meet journal limits

With Paperpal Copilot, create a research paper introduction effortlessly. In this step-by-step guide, we’ll walk you through how Paperpal transforms your initial ideas into a polished and publication-ready introduction.

How to use Paperpal to write the Introduction section

Step 1: Sign up on Paperpal and click on the Copilot feature, under this choose Outlines > Research Article > Introduction

Step 2: Add your unstructured notes or initial draft, whether in English or another language, to Paperpal, which is to be used as the base for your content.

Step 3: Fill in the specifics, such as your field of study, brief description or details you want to include, which will help the AI generate the outline for your Introduction.

Step 4: Use this outline and sentence suggestions to develop your content, adding citations where needed and modifying it to align with your specific research focus.

Step 5: Turn to Paperpal’s granular language checks to refine your content, tailor it to reflect your personal writing style, and ensure it effectively conveys your message.

You can use the same process to develop each section of your article, and finally your research paper in half the time and without any of the stress.

The purpose of the research paper introduction is to introduce the reader to the problem definition, justify the need for the study, and describe the main theme of the study. The aim is to gain the reader’s attention by providing them with necessary background information and establishing the main purpose and direction of the research.

The length of the research paper introduction can vary across journals and disciplines. While there are no strict word limits for writing the research paper introduction, an ideal length would be one page, with a maximum of 400 words over 1-4 paragraphs. Generally, it is one of the shorter sections of the paper as the reader is assumed to have at least a reasonable knowledge about the topic. 2 For example, for a study evaluating the role of building design in ensuring fire safety, there is no need to discuss definitions and nature of fire in the introduction; you could start by commenting upon the existing practices for fire safety and how your study will add to the existing knowledge and practice.

When deciding what to include in the research paper introduction, the rest of the paper should also be considered. The aim is to introduce the reader smoothly to the topic and facilitate an easy read without much dependency on external sources. 3 Below is a list of elements you can include to prepare a research paper introduction outline and follow it when you are writing the research paper introduction. Topic introduction: This can include key definitions and a brief history of the topic. Research context and background: Offer the readers some general information and then narrow it down to specific aspects. Details of the research you conducted: A brief literature review can be included to support your arguments or line of thought. Rationale for the study: This establishes the relevance of your study and establishes its importance. Importance of your research: The main contributions are highlighted to help establish the novelty of your study Research hypothesis: Introduce your research question and propose an expected outcome. Organization of the paper: Include a short paragraph of 3-4 sentences that highlights your plan for the entire paper

Cite only works that are most relevant to your topic; as a general rule, you can include one to three. Note that readers want to see evidence of original thinking. So it is better to avoid using too many references as it does not leave much room for your personal standpoint to shine through. Citations in your research paper introduction support the key points, and the number of citations depend on the subject matter and the point discussed. If the research paper introduction is too long or overflowing with citations, it is better to cite a few review articles rather than the individual articles summarized in the review. A good point to remember when citing research papers in the introduction section is to include at least one-third of the references in the introduction.

The literature review plays a significant role in the research paper introduction section. A good literature review accomplishes the following: Introduces the topic – Establishes the study’s significance – Provides an overview of the relevant literature – Provides context for the study using literature – Identifies knowledge gaps However, remember to avoid making the following mistakes when writing a research paper introduction: Do not use studies from the literature review to aggressively support your research Avoid direct quoting Do not allow literature review to be the focus of this section. Instead, the literature review should only aid in setting a foundation for the manuscript.

Remember the following key points for writing a good research paper introduction: 4

• Avoid stuffing too much general information: Avoid including what an average reader would know and include only that information related to the problem being addressed in the research paper introduction. For example, when describing a comparative study of non-traditional methods for mechanical design optimization, information related to the traditional methods and differences between traditional and non-traditional methods would not be relevant. In this case, the introduction for the research paper should begin with the state-of-the-art non-traditional methods and methods to evaluate the efficiency of newly developed algorithms.
• Avoid packing too many references: Cite only the required works in your research paper introduction. The other works can be included in the discussion section to strengthen your findings.
• Avoid extensive criticism of previous studies: Avoid being overly critical of earlier studies while setting the rationale for your study. A better place for this would be the Discussion section, where you can highlight the advantages of your method.
• Avoid describing conclusions of the study: When writing a research paper introduction remember not to include the findings of your study. The aim is to let the readers know what question is being answered. The actual answer should only be given in the Results and Discussion section.

To summarize, the research paper introduction section should be brief yet informative. It should convince the reader the need to conduct the study and motivate him to read further. If you’re feeling stuck or unsure, choose trusted AI academic writing assistants like Paperpal to effortlessly craft your research paper introduction and other sections of your research article.

1. Jawaid, S. A., & Jawaid, M. (2019). How to write introduction and discussion. Saudi Journal of Anaesthesia, 13(Suppl 1), S18.

2. Dewan, P., & Gupta, P. (2016). Writing the title, abstract and introduction: Looks matter!. Indian pediatrics, 53, 235-241.

3. Cetin, S., & Hackam, D. J. (2005). An approach to the writing of a scientific Manuscript1. Journal of Surgical Research, 128(2), 165-167.

4. Bavdekar, S. B. (2015). Writing introduction: Laying the foundations of a research paper. Journal of the Association of Physicians of India, 63(7), 44-6.

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Getting started with your research paper outline

Levels of organization for a research paper outline

First level of organization, second level of organization, third level of organization, fourth level of organization, tips for writing a research paper outline, research paper outline template, my research paper outline is complete: what are the next steps, frequently asked questions about a research paper outline, related articles.

The outline is the skeleton of your research paper. Simply start by writing down your thesis and the main ideas you wish to present. This will likely change as your research progresses; therefore, do not worry about being too specific in the early stages of writing your outline.

A research paper outline typically contains between two and four layers of organization. The first two layers are the most generalized. Each layer thereafter will contain the research you complete and presents more and more detailed information.

The levels are typically represented by a combination of Roman numerals, Arabic numerals, uppercase letters, lowercase letters but may include other symbols. Refer to the guidelines provided by your institution, as formatting is not universal and differs between universities, fields, and subjects. If you are writing the outline for yourself, you may choose any combination you prefer.

This is the most generalized level of information. Begin by numbering the introduction, each idea you will present, and the conclusion. The main ideas contain the bulk of your research paper 's information. Depending on your research, it may be chapters of a book for a literature review , a series of dates for a historical research paper, or the methods and results of a scientific paper.

I. Introduction

II. Main idea

III. Main idea

IV. Main idea

V. Conclusion

The second level consists of topics which support the introduction, main ideas, and the conclusion. Each main idea should have at least two supporting topics listed in the outline.

If your main idea does not have enough support, you should consider presenting another main idea in its place. This is where you should stop outlining if this is your first draft. Continue your research before adding to the next levels of organization.

• A. Background information
• B. Hypothesis or thesis
• A. Supporting topic
• B. Supporting topic

The third level of organization contains supporting information for the topics previously listed. By now, you should have completed enough research to add support for your ideas.

The Introduction and Main Ideas may contain information you discovered about the author, timeframe, or contents of a book for a literature review; the historical events leading up to the research topic for a historical research paper, or an explanation of the problem a scientific research paper intends to address.

• 1. Relevant history
• 2. Relevant history
• 1. The hypothesis or thesis clearly stated
• 1. A brief description of supporting information
• 2. A brief description of supporting information

The fourth level of organization contains the most detailed information such as quotes, references, observations, or specific data needed to support the main idea. It is not typical to have further levels of organization because the information contained here is the most specific.

• a) Quotes or references to another piece of literature
• b) Quotes or references to another piece of literature

Tip: The key to creating a useful outline is to be consistent in your headings, organization, and levels of specificity.

• Be Consistent : ensure every heading has a similar tone. State the topic or write short sentences for each heading but avoid doing both.
• Organize Information : Higher levels of organization are more generally stated and each supporting level becomes more specific. The introduction and conclusion will never be lower than the first level of organization.
• Build Support : Each main idea should have two or more supporting topics. If your research does not have enough information to support the main idea you are presenting, you should, in general, complete additional research or revise the outline.

By now, you should know the basic requirements to create an outline for your paper. With a content framework in place, you can now start writing your paper . To help you start right away, you can use one of our templates and adjust it to suit your needs.

After completing your outline, you should:

• Title your research paper . This is an iterative process and may change when you delve deeper into the topic.
• Begin writing your research paper draft . Continue researching to further build your outline and provide more information to support your hypothesis or thesis.
• Format your draft appropriately . MLA 8 and APA 7 formats have differences between their bibliography page, in-text citations, line spacing, and title.
• Finalize your citations and bibliography . Use a reference manager like Paperpile to organize and cite your research.
• Write the abstract, if required . An abstract will briefly state the information contained within the paper, results of the research, and the conclusion.

An outline is used to organize written ideas about a topic into a logical order. Outlines help us organize major topics, subtopics, and supporting details. Researchers benefit greatly from outlines while writing by addressing which topic to cover in what order.

The most basic outline format consists of: an introduction, a minimum of three topic paragraphs, and a conclusion.

You should make an outline before starting to write your research paper. This will help you organize the main ideas and arguments you want to present in your topic.

• Consistency: ensure every heading has a similar tone. State the topic or write short sentences for each heading but avoid doing both.
• Organization : Higher levels of organization are more generally stated and each supporting level becomes more specific. The introduction and conclusion will never be lower than the first level of organization.
• Support : Each main idea should have two or more supporting topics. If your research does not have enough information to support the main idea you are presenting, you should, in general, complete additional research or revise the outline.

How to write an effective introduction for your research paper

Last updated

20 January 2024

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However, the introduction is a vital element of your research paper. It helps the reader decide whether your paper is worth their time. As such, it's worth taking your time to get it right.

In this article, we'll tell you everything you need to know about writing an effective introduction for your research paper.

• The importance of an introduction in research papers

The primary purpose of an introduction is to provide an overview of your paper. This lets readers gauge whether they want to continue reading or not. The introduction should provide a meaningful roadmap of your research to help them make this decision. It should let readers know whether the information they're interested in is likely to be found in the pages that follow.

Aside from providing readers with information about the content of your paper, the introduction also sets the tone. It shows readers the style of language they can expect, which can further help them to decide how far to read.

When you take into account both of these roles that an introduction plays, it becomes clear that crafting an engaging introduction is the best way to get your paper read more widely. First impressions count, and the introduction provides that impression to readers.

• The optimum length for a research paper introduction

While there's no magic formula to determine exactly how long a research paper introduction should be, there are a few guidelines. Some variables that impact the ideal introduction length include:

Field of study

Complexity of the topic

Specific requirements of the course or publication

A commonly recommended length of a research paper introduction is around 10% of the total paper’s length. So, a ten-page paper has a one-page introduction. If the topic is complex, it may require more background to craft a compelling intro. Humanities papers tend to have longer introductions than those of the hard sciences.

The best way to craft an introduction of the right length is to focus on clarity and conciseness. Tell the reader only what is necessary to set up your research. An introduction edited down with this goal in mind should end up at an acceptable length.

• Evaluating successful research paper introductions

A good way to gauge how to create a great introduction is by looking at examples from across your field. The most influential and well-regarded papers should provide some insights into what makes a good introduction.

Dissecting examples: what works and why

We can make some general assumptions by looking at common elements of a good introduction, regardless of the field of research.

A common structure is to start with a broad context, and then narrow that down to specific research questions or hypotheses. This creates a funnel that establishes the scope and relevance.

The most effective introductions are careful about the assumptions they make regarding reader knowledge. By clearly defining key terms and concepts instead of assuming the reader is familiar with them, these introductions set a more solid foundation for understanding.

To pull in the reader and make that all-important good first impression, excellent research paper introductions will often incorporate a compelling narrative or some striking fact that grabs the reader's attention.

Finally, good introductions provide clear citations from past research to back up the claims they're making. In the case of argumentative papers or essays (those that take a stance on a topic or issue), a strong thesis statement compels the reader to continue reading.

Common pitfalls to avoid in research paper introductions

You can also learn what not to do by looking at other research papers. Many authors have made mistakes you can learn from.

We've talked about the need to be clear and concise. Many introductions fail at this; they're verbose, vague, or otherwise fail to convey the research problem or hypothesis efficiently. This often comes in the form of an overemphasis on background information, which obscures the main research focus.

Ensure your introduction provides the proper emphasis and excitement around your research and its significance. Otherwise, fewer people will want to read more about it.

• Crafting a compelling introduction for a research paper

Let’s take a look at the steps required to craft an introduction that pulls readers in and compels them to learn more about your research.

Step 1: Capturing interest and setting the scene

To capture the reader's interest immediately, begin your introduction with a compelling question, a surprising fact, a provocative quote, or some other mechanism that will hook readers and pull them further into the paper.

As they continue reading, the introduction should contextualize your research within the current field, showing readers its relevance and importance. Clarify any essential terms that will help them better understand what you're saying. This keeps the fundamentals of your research accessible to all readers from all backgrounds.

Step 2: Building a solid foundation with background information

Including background information in your introduction serves two major purposes:

It helps to clarify the topic for the reader

It establishes the depth of your research

The approach you take when conveying this information depends on the type of paper.

For argumentative papers, you'll want to develop engaging background narratives. These should provide context for the argument you'll be presenting.

For empirical papers, highlighting past research is the key. Often, there will be some questions that weren't answered in those past papers. If your paper is focused on those areas, those papers make ideal candidates for you to discuss and critique in your introduction.

Step 3: Pinpointing the research challenge

To capture the attention of the reader, you need to explain what research challenges you'll be discussing.

For argumentative papers, this involves articulating why the argument you'll be making is important. What is its relevance to current discussions or problems? What is the potential impact of people accepting or rejecting your argument?

For empirical papers, explain how your research is addressing a gap in existing knowledge. What new insights or contributions will your research bring to your field?

Step 4: Clarifying your research aims and objectives

We mentioned earlier that the introduction to a research paper can serve as a roadmap for what's within. We've also frequently discussed the need for clarity. This step addresses both of these.

When writing an argumentative paper, craft a thesis statement with impact. Clearly articulate what your position is and the main points you intend to present. This will map out for the reader exactly what they'll get from reading the rest.

For empirical papers, focus on formulating precise research questions and hypotheses. Directly link them to the gaps or issues you've identified in existing research to show the reader the precise direction your research paper will take.

Step 5: Sketching the blueprint of your study

Continue building a roadmap for your readers by designing a structured outline for the paper. Guide the reader through your research journey, explaining what the different sections will contain and their relationship to one another.

This outline should flow seamlessly as you move from section to section. Creating this outline early can also help guide the creation of the paper itself, resulting in a final product that's better organized. In doing so, you'll craft a paper where each section flows intuitively from the next.

Step 6: Integrating your research question

To avoid letting your research question get lost in background information or clarifications, craft your introduction in such a way that the research question resonates throughout. The research question should clearly address a gap in existing knowledge or offer a new perspective on an existing problem.

Tell users your research question explicitly but also remember to frequently come back to it. When providing context or clarification, point out how it relates to the research question. This keeps your focus where it needs to be and prevents the topic of the paper from becoming under-emphasized.

Step 7: Establishing the scope and limitations

So far, we've talked mostly about what's in the paper and how to convey that information to readers. The opposite is also important. Information that's outside the scope of your paper should be made clear to the reader in the introduction so their expectations for what is to follow are set appropriately.

Similarly, be honest and upfront about the limitations of the study. Any constraints in methodology, data, or how far your findings can be generalized should be fully communicated in the introduction.

Step 8: Concluding the introduction with a promise

The final few lines of the introduction are your last chance to convince people to continue reading the rest of the paper. Here is where you should make it very clear what benefit they'll get from doing so. What topics will be covered? What questions will be answered? Make it clear what they will get for continuing.

By providing a quick recap of the key points contained in the introduction in its final lines and properly setting the stage for what follows in the rest of the paper, you refocus the reader's attention on the topic of your research and guide them to read more.

• Research paper introduction best practices

Following the steps above will give you a compelling introduction that hits on all the key points an introduction should have. Some more tips and tricks can make an introduction even more polished.

As you follow the steps above, keep the following tips in mind.

Set the right tone and style

Like every piece of writing, a research paper should be written for the audience. That is to say, it should match the tone and style that your academic discipline and target audience expect. This is typically a formal and academic tone, though the degree of formality varies by field.

Kno w the audience

The perfect introduction balances clarity with conciseness. The amount of clarification required for a given topic depends greatly on the target audience. Knowing who will be reading your paper will guide you in determining how much background information is required.

Adopt the CARS (create a research space) model

The CARS model is a helpful tool for structuring introductions. This structure has three parts. The beginning of the introduction establishes the general research area. Next, relevant literature is reviewed and critiqued. The final section outlines the purpose of your study as it relates to the previous parts.

Master the art of funneling

The CARS method is one example of a well-funneled introduction. These start broadly and then slowly narrow down to your specific research problem. It provides a nice narrative flow that provides the right information at the right time. If you stray from the CARS model, try to retain this same type of funneling.

Incorporate narrative element

People read research papers largely to be informed. But to inform the reader, you have to hold their attention. A narrative style, particularly in the introduction, is a great way to do that. This can be a compelling story, an intriguing question, or a description of a real-world problem.

Write the introduction last

By writing the introduction after the rest of the paper, you'll have a better idea of what your research entails and how the paper is structured. This prevents the common problem of writing something in the introduction and then forgetting to include it in the paper. It also means anything particularly exciting in the paper isn’t neglected in the intro.

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How to Write an Introduction for a Research Paper

Table of Contents

Writing an introduction for a research paper is a critical element of your paper, but it can seem challenging to encapsulate enormous amount of information into a concise form. The introduction of your research paper sets the tone for your research and provides the context for your study. In this article, we will guide you through the process of writing an effective introduction that grabs the reader's attention and captures the essence of your research paper.

Understanding the Purpose of a Research Paper Introduction

The introduction acts as a road map for your research paper, guiding the reader through the main ideas and arguments. The purpose of the introduction is to present your research topic to the readers and provide a rationale for why your study is relevant. It helps the reader locate your research and its relevance in the broader field of related scientific explorations. Additionally, the introduction should inform the reader about the objectives and scope of your study, giving them an overview of what to expect in the paper. By including a comprehensive introduction, you establish your credibility as an author and convince the reader that your research is worth their time and attention.

Key Elements to Include in Your Introduction

When writing your research paper introduction, there are several key elements you should include to ensure it is comprehensive and informative.

• A hook or attention-grabbing statement to capture the reader's interest.  It can be a thought-provoking question, a surprising statistic, or a compelling anecdote that relates to your research topic.
• A brief overview of the research topic and its significance. By highlighting the gap in existing knowledge or the problem your research aims to address, you create a compelling case for the relevance of your study.
• A clear research question or problem statement. This serves as the foundation of your research and guides the reader in understanding the unique focus of your study. It should be concise, specific, and clearly articulated.
• An outline of the paper's structure and main arguments, to help the readers navigate through the paper with ease.

Preparing to Write Your Introduction

Before diving into writing your introduction, it is essential to prepare adequately. This involves 3 important steps:

• Conducting Preliminary Research: Immerse yourself in the existing literature to develop a clear research question and position your study within the academic discourse.
• Identifying Your Thesis Statement: Define a specific, focused, and debatable thesis statement, serving as a roadmap for your paper.
• Considering Broader Context: Reflect on the significance of your research within your field, understanding its potential impact and contribution.

By engaging in these preparatory steps, you can ensure that your introduction is well-informed, focused, and sets the stage for a compelling research paper.

Structuring Your Introduction

Now that you have prepared yourself to tackle the introduction, it's time to structure it effectively. A well-structured introduction will engage the reader from the beginning and provide a logical flow to your research paper.

Starting with a Hook

Begin your introduction with an attention-grabbing hook that captivates the reader's interest. This hook serves as a way to make your introduction more engaging and compelling. For example, if you are writing a research paper on the impact of climate change on biodiversity, you could start your introduction with a statistic about the number of species that have gone extinct due to climate change. This will immediately grab the reader's attention and make them realize the urgency and importance of the topic.

Introducing Your Topic

Provide a brief overview, which should give the reader a general understanding of the subject matter and its significance. Explain the importance of the topic and its relevance to the field. This will help the reader understand why your research is significant and why they should continue reading. Continuing with the example of climate change and biodiversity, you could explain how climate change is one of the greatest threats to global biodiversity, how it affects ecosystems, and the potential consequences for both wildlife and human populations. By providing this context, you are setting the stage for the rest of your research paper and helping the reader understand the importance of your study.

Presenting Your Thesis Statement

The thesis statement should directly address your research question and provide a preview of the main arguments or findings discussed in your paper. Make sure your thesis statement is clear, concise, and well-supported by the evidence you will present in your research paper. By presenting a strong and focused thesis statement, you are providing the reader with the information they could anticipate in your research paper. This will help them understand the purpose and scope of your study and will make them more inclined to continue reading.

Writing Techniques for an Effective Introduction

When crafting an introduction, it is crucial to pay attention to the finer details that can elevate your writing to the next level. By utilizing specific writing techniques, you can captivate your readers and draw them into your research journey.

Using Clear and Concise Language

One of the most important writing techniques to employ in your introduction is the use of clear and concise language. By choosing your words carefully, you can effectively convey your ideas to the reader. It is essential to avoid using jargon or complex terminology that may confuse or alienate your audience. Instead, focus on communicating your research in a straightforward manner to ensure that your introduction is accessible to both experts in your field and those who may be new to the topic. This approach allows you to engage a broader audience and make your research more inclusive.

Establishing the Relevance of Your Research

One way to establish the relevance of your research is by highlighting how it fills a gap in the existing literature. Explain how your study addresses a significant research question that has not been adequately explored. By doing this, you demonstrate that your research is not only unique but also contributes to the broader knowledge in your field. Furthermore, it is important to emphasize the potential impact of your research. Whether it is advancing scientific understanding, informing policy decisions, or improving practical applications, make it clear to the reader how your study can make a difference.

By employing these two writing techniques in your introduction, you can effectively engage your readers. Take your time to craft an introduction that is both informative and captivating, leaving your readers eager to delve deeper into your research.

Revising and Polishing Your Introduction

Once you have written your introduction, it is crucial to revise and polish it to ensure that it effectively sets the stage for your research paper.

Self-Editing Techniques

Review your introduction for clarity, coherence, and logical flow. Ensure each paragraph introduces a new idea or argument with smooth transitions.

Check for grammatical errors, spelling mistakes, and awkward sentence structures.

Ensure that your introduction aligns with the overall tone and style of your research paper.

Seeking Feedback for Improvement

Consider seeking feedback from peers, colleagues, or your instructor. They can provide valuable insights and suggestions for improving your introduction. Be open to constructive criticism and use it to refine your introduction and make it more compelling for the reader.

Writing an introduction for a research paper requires careful thought and planning. By understanding the purpose of the introduction, preparing adequately, structuring effectively, and employing writing techniques, you can create an engaging and informative introduction for your research. Remember to revise and polish your introduction to ensure that it accurately represents the main ideas and arguments in your research paper. With a well-crafted introduction, you will capture the reader's attention and keep them inclined to your paper.

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Research Paper Outline – Types, Example, Template

Table of Contents

By creating a well-structured research paper outline, writers can easily organize their thoughts and ideas and ensure that their final paper is clear, concise, and effective. In this article, we will explore the essential components of a research paper outline and provide some tips and tricks for creating a successful one.

Research Paper Outline

Research paper outline is a plan or a structural framework that organizes the main ideas , arguments, and supporting evidence in a logical sequence. It serves as a blueprint or a roadmap for the writer to follow while drafting the actual research paper .

Typically, an outline consists of the following elements:

• Introduction : This section presents the topic, research question , and thesis statement of the paper. It also provides a brief overview of the literature review and the methodology used.
• Literature Review: This section provides a comprehensive review of the relevant literature, theories, and concepts related to the research topic. It analyzes the existing research and identifies the research gaps and research questions.
• Methodology: This section explains the research design, data collection methods, data analysis, and ethical considerations of the study.
• Results: This section presents the findings of the study, using tables, graphs, and statistics to illustrate the data.
• Discussion : This section interprets the results of the study, and discusses their implications, significance, and limitations. It also suggests future research directions.
• Conclusion : This section summarizes the main findings of the study and restates the thesis statement.
• References: This section lists all the sources cited in the paper using the appropriate citation style.

Research Paper Outline Types

There are several types of outlines that can be used for research papers, including:

Alphanumeric Outline

This is a traditional outline format that uses Roman numerals, capital letters, Arabic numerals, and lowercase letters to organize the main ideas and supporting details of a research paper. It is commonly used for longer, more complex research papers.

I. Introduction

• A. Background information
• B. Thesis statement
• 1 1. Supporting detail
• 1 2. Supporting detail 2
• 2 1. Supporting detail

III. Conclusion

• A. Restate thesis
• B. Summarize main points

Decimal Outline

This outline format uses numbers to organize the main ideas and supporting details of a research paper. It is similar to the alphanumeric outline, but it uses only numbers and decimals to indicate the hierarchy of the ideas.

• 1.1 Background information
• 1.2 Thesis statement
• 1 2.1.1 Supporting detail
• 1 2.1.2 Supporting detail
• 2 2.2.1 Supporting detail
• 1 2.2.2 Supporting detail
• 3.1 Restate thesis
• 3.2 Summarize main points

Full Sentence Outline

This type of outline uses complete sentences to describe the main ideas and supporting details of a research paper. It is useful for those who prefer to see the entire paper outlined in complete sentences.

• Provide background information on the topic
• State the thesis statement
• Explain main idea 1 and provide supporting details
• Discuss main idea 2 and provide supporting details
• Restate the thesis statement
• Summarize the main points of the paper

Topic Outline

This type of outline uses short phrases or words to describe the main ideas and supporting details of a research paper. It is useful for those who prefer to see a more concise overview of the paper.

• Background information
• Thesis statement
• Supporting detail 1
• Supporting detail 2
• Restate thesis
• Summarize main points

Reverse Outline

This is an outline that is created after the paper has been written. It involves going back through the paper and summarizing each paragraph or section in one sentence. This can be useful for identifying gaps in the paper or areas that need further development.

• Introduction : Provides background information and states the thesis statement.
• Paragraph 1: Discusses main idea 1 and provides supporting details.
• Paragraph 2: Discusses main idea 2 and provides supporting details.
• Paragraph 3: Addresses potential counterarguments.
• Conclusion : Restates thesis and summarizes main points.

Mind Map Outline

This type of outline involves creating a visual representation of the main ideas and supporting details of a research paper. It can be useful for those who prefer a more creative and visual approach to outlining.

• Supporting detail 1: Lack of funding for public schools.
• Supporting detail 2: Decrease in government support for education.
• Supporting detail 1: Increase in income inequality.
• Supporting detail 2: Decrease in social mobility.

Research Paper Outline Example

Research Paper Outline Example on Cyber Security:

A. Overview of Cybersecurity

• B. Importance of Cybersecurity
• C. Purpose of the paper

II. Cyber Threats

A. Definition of Cyber Threats

• B. Types of Cyber Threats
• C. Examples of Cyber Threats

III. Cybersecurity Measures

A. Prevention measures

• Anti-virus software
• Encryption B. Detection measures
• Intrusion Detection System (IDS)
• Security Information and Event Management (SIEM)
• Security Operations Center (SOC) C. Response measures
• Incident Response Plan
• Business Continuity Plan
• Disaster Recovery Plan

IV. Cybersecurity in the Business World

A. Overview of Cybersecurity in the Business World

B. Cybersecurity Risk Assessment

C. Best Practices for Cybersecurity in Business

V. Cybersecurity in Government Organizations

A. Overview of Cybersecurity in Government Organizations

C. Best Practices for Cybersecurity in Government Organizations

VI. Cybersecurity Ethics

A. Definition of Cybersecurity Ethics

B. Importance of Cybersecurity Ethics

C. Examples of Cybersecurity Ethics

VII. Future of Cybersecurity

A. Overview of the Future of Cybersecurity

B. Emerging Cybersecurity Threats

C. Advancements in Cybersecurity Technology

VIII. Conclusion

A. Summary of the paper

B. Recommendations for Cybersecurity

• C. Conclusion.

IX. References

A. List of sources cited in the paper

B. Bibliography of additional resources

Introduction

Cybersecurity refers to the protection of computer systems, networks, and sensitive data from unauthorized access, theft, damage, or any other form of cyber attack. B. Importance of Cybersecurity The increasing reliance on technology and the growing number of cyber threats make cybersecurity an essential aspect of modern society. Cybersecurity breaches can result in financial losses, reputational damage, and legal liabilities. C. Purpose of the paper This paper aims to provide an overview of cybersecurity, cyber threats, cybersecurity measures, cybersecurity in the business and government sectors, cybersecurity ethics, and the future of cybersecurity.

A cyber threat is any malicious act or event that attempts to compromise or disrupt computer systems, networks, or sensitive data. B. Types of Cyber Threats Common types of cyber threats include malware, phishing, social engineering, ransomware, DDoS attacks, and advanced persistent threats (APTs). C. Examples of Cyber Threats Recent cyber threats include the SolarWinds supply chain attack, the Colonial Pipeline ransomware attack, and the Microsoft Exchange Server hack.

Prevention measures aim to minimize the risk of cyber attacks by implementing security controls, such as firewalls, anti-virus software, and encryption.

• Firewalls Firewalls act as a barrier between a computer network and the internet, filtering incoming and outgoing traffic to prevent unauthorized access.
• Anti-virus software Anti-virus software detects, prevents, and removes malware from computer systems.
• Encryption Encryption involves the use of mathematical algorithms to transform sensitive data into a code that can only be accessed by authorized individuals. B. Detection measures Detection measures aim to identify and respond to cyber attacks as quickly as possible, such as intrusion detection systems (IDS), security information and event management (SIEM), and security operations centers (SOCs).
• Intrusion Detection System (IDS) IDS monitors network traffic for signs of unauthorized access, such as unusual patterns or anomalies.
• Security Information and Event Management (SIEM) SIEM combines security information management and security event management to provide real-time monitoring and analysis of security alerts.
• Security Operations Center (SOC) SOC is a dedicated team responsible for monitoring, analyzing, and responding to cyber threats. C. Response measures Response measures aim to mitigate the impact of a cyber attack and restore normal operations, such as incident response plans (IRPs), business continuity plans (BCPs), and disaster recovery plans (DRPs).
• Incident Response Plan IRPs outline the procedures and protocols to follow in the event of a cyber attack, including communication protocols, roles and responsibilities, and recovery processes.
• Business Continuity Plan BCPs ensure that critical business functions can continue in the event of a cyber attack or other disruption.
• Disaster Recovery Plan DRPs outline the procedures to recover from a catastrophic event, such as a natural disaster or cyber attack.

Cybersecurity is crucial for businesses of all sizes and industries, as they handle sensitive data, financial transactions, and intellectual property that are attractive targets for cyber criminals.

Risk assessment is a critical step in developing a cybersecurity strategy, which involves identifying potential threats, vulnerabilities, and consequences to determine the level of risk and prioritize security measures.

Best practices for cybersecurity in business include implementing strong passwords and multi-factor authentication, regularly updating software and hardware, training employees on cybersecurity awareness, and regularly backing up data.

Government organizations face unique cybersecurity challenges, as they handle sensitive information related to national security, defense, and critical infrastructure.

Risk assessment in government organizations involves identifying and assessing potential threats and vulnerabilities, conducting regular audits, and complying with relevant regulations and standards.

Best practices for cybersecurity in government organizations include implementing secure communication protocols, regularly updating and patching software, and conducting regular cybersecurity training and awareness programs for employees.

Cybersecurity ethics refers to the ethical considerations involved in cybersecurity, such as privacy, data protection, and the responsible use of technology.

Cybersecurity ethics are crucial for maintaining trust in technology, protecting privacy and data, and promoting responsible behavior in the digital world.

Examples of cybersecurity ethics include protecting the privacy of user data, ensuring data accuracy and integrity, and implementing fair and unbiased algorithms.

The future of cybersecurity will involve a shift towards more advanced technologies, such as artificial intelligence (AI), machine learning, and quantum computing.

Emerging cybersecurity threats include AI-powered cyber attacks, the use of deepfakes and synthetic media, and the potential for quantum computing to break current encryption methods.

Advancements in cybersecurity technology include the development of AI and machine learning-based security tools, the use of blockchain for secure data storage and sharing, and the development of post-quantum encryption methods.

This paper has provided an overview of cybersecurity, cyber threats, cybersecurity measures, cybersecurity in the business and government sectors, cybersecurity ethics, and the future of cybersecurity.

To enhance cybersecurity, organizations should prioritize risk assessment and implement a comprehensive cybersecurity strategy that includes prevention, detection, and response measures. Additionally, organizations should prioritize cybersecurity ethics to promote responsible behavior in the digital world.

C. Conclusion

Cybersecurity is an essential aspect of modern society, and organizations must prioritize cybersecurity to protect sensitive data and maintain trust in technology.

for further reading

X. Appendices

A. Glossary of key terms

B. Cybersecurity checklist for organizations

C. Sample cybersecurity policy for businesses

D. Sample cybersecurity incident response plan

E. Cybersecurity training and awareness resources

Note : The content and organization of the paper may vary depending on the specific requirements of the assignment or target audience. This outline serves as a general guide for writing a research paper on cybersecurity. Do not use this in your assingmets.

Research Paper Outline Template

• Background information and context of the research topic
• Research problem and questions
• Purpose and objectives of the research
• Scope and limitations

II. Literature Review

• Overview of existing research on the topic
• Key concepts and theories related to the research problem
• Identification of gaps in the literature
• Summary of relevant studies and their findings

III. Methodology

• Research design and approach
• Data collection methods and procedures
• Data analysis techniques
• Validity and reliability considerations
• Ethical considerations

IV. Results

• Presentation of research findings
• Analysis and interpretation of data
• Explanation of significant results
• Discussion of unexpected results

V. Discussion

• Comparison of research findings with existing literature
• Implications of results for theory and practice
• Limitations and future directions for research
• Conclusion and recommendations

VI. Conclusion

• Summary of research problem, purpose, and objectives
• Discussion of significant findings
• Contribution to the field of study
• Implications for practice
• Suggestions for future research

VII. References

• List of sources cited in the research paper using appropriate citation style.

Note : This is just an template, and depending on the requirements of your assignment or the specific research topic, you may need to modify or adjust the sections or headings accordingly.

Research Paper Outline Writing Guide

Here’s a guide to help you create an effective research paper outline:

• Choose a topic : Select a topic that is interesting, relevant, and meaningful to you.
• Conduct research: Gather information on the topic from a variety of sources, such as books, articles, journals, and websites.
• Organize your ideas: Organize your ideas and information into logical groups and subgroups. This will help you to create a clear and concise outline.
• Create an outline: Begin your outline with an introduction that includes your thesis statement. Then, organize your ideas into main points and subpoints. Each main point should be supported by evidence and examples.
• Introduction: The introduction of your research paper should include the thesis statement, background information, and the purpose of the research paper.
• Body : The body of your research paper should include the main points and subpoints. Each point should be supported by evidence and examples.
• Conclusion : The conclusion of your research paper should summarize the main points and restate the thesis statement.
• Reference List: Include a reference list at the end of your research paper. Make sure to properly cite all sources used in the paper.
• Proofreading : Proofread your research paper to ensure that it is free of errors and grammatical mistakes.
• Finalizing : Finalize your research paper by reviewing the outline and making any necessary changes.

When to Write Research Paper Outline

It’s a good idea to write a research paper outline before you begin drafting your paper. The outline will help you organize your thoughts and ideas, and it can serve as a roadmap for your writing process.

Here are a few situations when you might want to consider writing an outline:

• When you’re starting a new research project: If you’re beginning a new research project, an outline can help you get organized from the very beginning. You can use your outline to brainstorm ideas, map out your research goals, and identify potential sources of information.
• When you’re struggling to organize your thoughts: If you find yourself struggling to organize your thoughts or make sense of your research, an outline can be a helpful tool. It can help you see the big picture of your project and break it down into manageable parts.
• When you’re working with a tight deadline : If you have a deadline for your research paper, an outline can help you stay on track and ensure that you cover all the necessary points. By mapping out your paper in advance, you can work more efficiently and avoid getting stuck or overwhelmed.

Purpose of Research Paper Outline

The purpose of a research paper outline is to provide a structured and organized plan for the writer to follow while conducting research and writing the paper. An outline is essentially a roadmap that guides the writer through the entire research process, from the initial research and analysis of the topic to the final writing and editing of the paper.

A well-constructed outline can help the writer to:

• Organize their thoughts and ideas on the topic, and ensure that all relevant information is included.
• Identify any gaps in their research or argument, and address them before starting to write the paper.
• Ensure that the paper follows a logical and coherent structure, with clear transitions between different sections.
• Save time and effort by providing a clear plan for the writer to follow, rather than starting from scratch and having to revise the paper multiple times.

Advantages of Research Paper Outline

Some of the key advantages of a research paper outline include:

• Helps to organize thoughts and ideas : An outline helps to organize all the different ideas and information that you want to include in your paper. By creating an outline, you can ensure that all the points you want to make are covered and in a logical order.
• Saves time and effort : An outline saves time and effort because it helps you to focus on the key points of your paper. It also helps you to identify any gaps or areas where more research may be needed.
• Makes the writing process easier : With an outline, you have a clear roadmap of what you want to write, and this makes the writing process much easier. You can simply follow your outline and fill in the details as you go.
• Improves the quality of your paper : By having a clear outline, you can ensure that all the important points are covered and in a logical order. This makes your paper more coherent and easier to read, which ultimately improves its overall quality.
• Facilitates collaboration: If you are working on a research paper with others, an outline can help to facilitate collaboration. By sharing your outline, you can ensure that everyone is on the same page and working towards the same goals.

About the author

Muhammad Hassan

Researcher, Academic Writer, Web developer

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An outline is a formal system used to develop a framework for thinking about what should be the organization and eventual contents of your paper. An outline helps you predict the overall structure and flow of a paper.

Why and How to Create a Useful Outline. The Writing Lab and The OWL. Purdue University.

Importance of...

Writing papers in college requires you to come up with sophisticated, complex, and sometimes very creative ways of structuring your ideas . Taking the time to draft an outline can help you determine if your ideas connect to each other, what order of ideas works best, where gaps in your thinking may exist, or whether you have sufficient evidence to support each of your points. It is also an effective way to think about the time you will need to complete each part of your paper before you begin writing.

A good outline is important because :

• You will be much less likely to get writer's block . An outline will show where you're going and how to get there. Use the outline to set goals for completing each section of your paper.
• It will help you stay organized and focused throughout the writing process and help ensure proper coherence [flow of ideas] in your final paper. However, the outline should be viewed as a guide, not a straitjacket. As you review the literature or gather data, the organization of your paper may change; adjust your outline accordingly.
• A clear, detailed outline ensures that you always have something to help re-calibrate your writing should you feel yourself drifting into subject areas unrelated to the research problem. Use your outline to set boundaries around what you will investigate.
• The outline can be key to staying motivated . You can put together an outline when you're excited about the project and everything is clicking; making an outline is never as overwhelming as sitting down and beginning to write a twenty page paper without any sense of where it is going.
• An outline helps you organize multiple ideas about a topic . Most research problems can be analyzed from a variety of perspectives; an outline can help you sort out which modes of analysis are most appropriate to ensure the most robust findings are discovered.
• An outline not only helps you organize your thoughts, but it can also serve as a schedule for when certain aspects of your writing should be accomplished . Review the assignment and highlight the due dates of specific tasks and integrate these into your outline. If your professor has not created specific deadlines, create your own deadlines by thinking about your own writing style and the need to manage your time around other course assignments.

How to Structure and Organize Your Paper. Odegaard Writing & Research Center. University of Washington; Why and How to Create a Useful Outline. The Writing Lab and The OWL. Purdue University; Lietzau, Kathleen. Creating Outlines. Writing Center, University of Richmond.

Structure and Writing Style

I.   General Approaches

There are two general approaches you can take when writing an outline for your paper:

The topic outline consists of short phrases. This approach is useful when you are dealing with a number of different issues that could be arranged in a variety of different ways in your paper. Due to short phrases having more content than using simple sentences, they create better content from which to build your paper.

The sentence outline is done in full sentences. This approach is useful when your paper focuses on complex issues in detail. The sentence outline is also useful because sentences themselves have many of the details in them needed to build a paper and it allows you to include those details in the sentences instead of having to create an outline of short phrases that goes on page after page.

II.   Steps to Making the Outline

A strong outline details each topic and subtopic in your paper, organizing these points so that they build your argument toward an evidence-based conclusion. Writing an outline will also help you focus on the task at hand and avoid unnecessary tangents, logical fallacies, and underdeveloped paragraphs.

• Identify the research problem . The research problem is the focal point from which the rest of the outline flows. Try to sum up the point of your paper in one sentence or phrase. It also can be key to deciding what the title of your paper should be.
• Identify the main categories . What main points will you analyze? The introduction describes all of your main points; the rest of  your paper can be spent developing those points.
• Create the first category . What is the first point you want to cover? If the paper centers around a complicated term, a definition can be a good place to start. For a paper that concerns the application and testing of a particular theory, giving the general background on the theory can be a good place to begin.
• Create subcategories . After you have followed these steps, create points under it that provide support for the main point. The number of categories that you use depends on the amount of information that you are trying to cover. There is no right or wrong number to use.

Once you have developed the basic outline of the paper, organize the contents to match the standard format of a research paper as described in this guide.

III.   Things to Consider When Writing an Outline

• There is no rule dictating which approach is best . Choose either a topic outline or a sentence outline based on which one you believe will work best for you. However, once you begin developing an outline, it's helpful to stick to only one approach.
• Both topic and sentence outlines use Roman and Arabic numerals along with capital and small letters of the alphabet arranged in a consistent and rigid sequence. A rigid format should be used especially if you are required to hand in your outline.
• Although the format of an outline is rigid, it shouldn't make you inflexible about how to write your paper. Often when you start investigating a research problem [i.e., reviewing the research literature], especially if you are unfamiliar with the topic, you should anticipate the likelihood your analysis could go in different directions. If your paper changes focus, or you need to add new sections, then feel free to reorganize the outline.
• If appropriate, organize the main points of your outline in chronological order . In papers where you need to trace the history or chronology of events or issues, it is important to arrange your outline in the same manner, knowing that it's easier to re-arrange things now than when you've almost finished your paper.
• For a standard research paper of 15-20 pages, your outline should be no more than few pages in length . It may be helpful as you are developing your outline to also write down a tentative list of references.

Muirhead, Brent. “Using Outlines to Improve Online Student Writing Skills.” Journal on School Educational Technology 1, (2005): 17-23; Four Main Components for Effective Outlines. The Writing Lab and The OWL. Purdue University; How to Make an Outline. Psychology Writing Center. University of Washington; Kartawijaya, Sukarta. “Improving Students’ Writing Skill in Writing Paragraph through an Outline Technique.” Curricula: Journal of Teaching and Learning 3 (2018); Organization: Informal Outlines. The Reading/Writing Center. Hunter College; Organization: Standard Outline Form. The Reading/Writing Center. Hunter College; Outlining. Department of English Writing Guide. George Mason University; Plotnic, Jerry. Organizing an Essay. University College Writing Centre. University of Toronto; Reverse Outline. The Writing Center. University of North Carolina; Reverse Outlines: A Writer's Technique for Examining Organization. The Writer’s Handbook. Writing Center. University of Wisconsin, Madison; Using Outlines. Writing Tutorial Services, Center for Innovative Teaching and Learning. Indiana University; Writing: Considering Structure and Organization. Institute for Writing Rhetoric. Dartmouth College.

Writing Tip

A Disorganized Outline Means a Disorganized Paper!

If, in writing your paper, it begins to diverge from your outline, this is very likely a sign that you've lost your focus. How do you know whether to change the paper to fit the outline, or, that you need to reconsider the outline so that it fits the paper? A good way to check your progress is to use what you have written to recreate the outline. This is an effective strategy for assessing the organization of your paper. If the resulting outline says what you want it to say and it is in an order that is easy to follow, then the organization of your paper has been successful. If you discover that it's difficult to create an outline from what you have written, then you likely need to revise your paper.

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How Can You Create a Well Planned Research Paper Outline

You are staring at the blank document, meaning to start writing your research paper . After months of experiments and procuring results, your PI asked you to write the paper to publish it in a reputed journal. You spoke to your peers and a few seniors and received a few tips on writing a research paper, but you still can’t plan on how to begin!

Writing a research paper is a very common issue among researchers and is often looked upon as a time consuming hurdle. Researchers usually look up to this task as an impending threat, avoiding and procrastinating until they cannot delay it anymore. Seeking advice from internet and seniors they manage to write a paper which goes in for quite a few revisions. Making researchers lose their sense of understanding with respect to their research work and findings. In this article, we would like to discuss how to create a structured research paper outline which will assist a researcher in writing their research paper effectively!

Publication is an important component of research studies in a university for academic promotion and in obtaining funding to support research. However, the primary reason is to provide the data and hypotheses to scientific community to advance the understanding in a specific domain. A scientific paper is a formal record of a research process. It documents research protocols, methods, results, conclusion, and discussion from a research hypothesis .

Table of Contents

What Is a Research Paper Outline?

A research paper outline is a basic format for writing an academic research paper. It follows the IMRAD format (Introduction, Methods, Results, and Discussion). However, this format varies depending on the type of research manuscript. A research paper outline consists of following sections to simplify the paper for readers. These sections help researchers build an effective paper outline.

1. Title Page

The title page provides important information which helps the editors, reviewers, and readers identify the manuscript and the authors at a glance. It also provides an overview of the field of research the research paper belongs to. The title should strike a balance between precise and detailed. Other generic details include author’s given name, affiliation, keywords that will provide indexing, details of the corresponding author etc. are added to the title page.

2. Abstract

Abstract is the most important section of the manuscript and will help the researcher create a detailed research paper outline . To be more precise, an abstract is like an advertisement to the researcher’s work and it influences the editor in deciding whether to submit the manuscript to reviewers or not. Writing an abstract is a challenging task. Researchers can write an exemplary abstract by selecting the content carefully and being concise.

3. Introduction

An introduction is a background statement that provides the context and approach of the research. It describes the problem statement with the assistance of the literature study and elaborates the requirement to update the knowledge gap. It sets the research hypothesis and informs the readers about the big research question.

This section is usually named as “Materials and Methods”, “Experiments” or “Patients and Methods” depending upon the type of journal. This purpose provides complete information on methods used for the research. Researchers should mention clear description of materials and their use in the research work. If the methods used in research are already published, give a brief account and refer to the original publication. However, if the method used is modified from the original method, then researcher should mention the modifications done to the original protocol and validate its accuracy, precision, and repeatability.

It is best to report results as tables and figures wherever possible. Also, avoid duplication of text and ensure that the text summarizes the findings. Report the results with appropriate descriptive statistics. Furthermore, report any unexpected events that could affect the research results, and mention complete account of observations and explanations for missing data (if any).

6. Discussion

The discussion should set the research in context, strengthen its importance and support the research hypothesis. Summarize the main results of the study in one or two paragraphs and show how they logically fit in an overall scheme of studies. Compare the results with other investigations in the field of research and explain the differences.

7. Acknowledgments

Acknowledgements identify and thank the contributors to the study, who are not under the criteria of co-authors. It also includes the recognition of funding agency and universities that award scholarships or fellowships to researchers.

8. Declaration of Competing Interests

Finally, declaring the competing interests is essential to abide by ethical norms of unique research publishing. Competing interests arise when the author has more than one role that may lead to a situation where there is a conflict of interest.

Steps to Write a Research Paper Outline

• Write down all important ideas that occur to you concerning the research paper .
• Answer questions such as – what is the topic of my paper? Why is the topic important? How to formulate the hypothesis? What are the major findings?
• Add context and structure. Group all your ideas into sections – Introduction, Methods, Results, and Discussion/Conclusion.
• Add relevant questions to each section. It is important to note down the questions. This will help you align your thoughts.
• Expand the ideas based on the questions created in the paper outline.
• After creating a detailed outline, discuss it with your mentors and peers.
• Get enough feedback and decide on the journal you will submit to.
• The process of real writing begins.

Benefits of Creating a Research Paper Outline

As discussed, the research paper subheadings create an outline of what different aspects of research needs elaboration. This provides subtopics on which the researchers brainstorm and reach a conclusion to write. A research paper outline organizes the researcher’s thoughts and gives a clear picture of how to formulate the research protocols and results. It not only helps the researcher to understand the flow of information but also provides relation between the ideas.

A research paper outline helps researcher achieve a smooth transition between topics and ensures that no research point is forgotten. Furthermore, it allows the reader to easily navigate through the research paper and provides a better understanding of the research. The paper outline allows the readers to find relevant information and quotes from different part of the paper.

Research Paper Outline Template

A research paper outline template can help you understand the concept of creating a well planned research paper before beginning to write and walk through your journey of research publishing.

1. Research Title

A. Background i. Support with evidence ii. Support with existing literature studies

B. Thesis Statement i. Link literature with hypothesis ii. Support with evidence iii. Explain the knowledge gap and how this research will help build the gap 4. Body

A. Methods i. Mention materials and protocols used in research ii. Support with evidence

B. Results i. Support with tables and figures ii. Mention appropriate descriptive statistics

C. Discussion i. Support the research with context ii. Support the research hypothesis iii. Compare the results with other investigations in field of research

D. Conclusion i. Support the discussion and research investigation ii. Support with literature studies

E. Acknowledgements i. Identify and thank the contributors ii. Include the funding agency, if any

F. Declaration of Competing Interests

5. References

Download the Research Paper Outline Template!

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Writing a scientific paper.

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What is a "good" introduction?

Citing sources in the introduction, "introduction checklist" from: how to write a good scientific paper. chris a. mack. spie. 2018..

• LITERATURE CITED
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This is where you describe briefly and clearly why you are writing the paper. The introduction supplies sufficient background information for the reader to understand and evaluate the experiment you did. It also supplies a rationale for the study.

• Present the problem and the proposed solution
• Presents nature and scope of the problem investigated
• Reviews the pertinent literature to orient the reader
• States the method of the experiment
• State the principle results of the experiment

It is important to cite sources in the introduction section of your paper as evidence of the claims you are making. There are ways of citing sources in the text so that the reader can find the full reference in the literature cited section at the end of the paper, yet the flow of the reading is not badly interrupted. Below are some example of how this can be done:     "Smith (1983) found that N-fixing plants could be infected by several different species of Rhizobium."     "Walnut trees are known to be allelopathic (Smith 1949,  Bond et al. 1955, Jones and Green 1963)."     "Although the presence of Rhizobium normally increases the growth of legumes (Nguyen 1987), the opposite effect has been observed (Washington 1999)." Note that articles by one or two authors are always cited in the text using their last names. However, if there are more than two authors, the last name of the 1st author is given followed by the abbreviation et al. which is Latin for "and others". 

From:  https://writingcenter.gmu.edu/guides/imrad-reports-introductions

• Indicate the field of the work, why this field is important, and what has already been done (with proper citations).
• Indicate a gap, raise a research question, or challenge prior work in this territory.
• Outline the purpose and announce the present research, clearly indicating what is novel and why it is significant.
• Avoid: repeating the abstract; providing unnecessary background information; exaggerating the importance of the work; claiming novelty without a proper literature search. 
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APA Research Paper Outline: Examples and Template

Table of contents

• 1 Why Is Research Paper Format Necessary?
• 2.1 Purpose of research paper outline
• 2.2 APA outline example
• 3.1 APA paper outline example
• 3.2 Introduction:
• 3.4 Conclusion:
• 4 The Basic APA Outline Format
• 5 APA Style Outline Template Breakdown
• 6.1 APA Research Paper Outline Example
• 6.2 APA Paper Outline Format Example
• 7.1 First Paragraph: Hook and Thesis
• 7.2 Main Body
• 7.3 Conclusion
• 7.4 Decimal APA outline format example
• 7.5 Decimal APA outline format layout
• 8.1 A definite goal
• 8.2 Division
• 8.3 Parallelism
• 8.4 Coordination
• 8.5 Subordination
• 8.6 Avoid Redundancy
• 8.7 Wrap it up in a good way
• 8.8 Conclusion

Formatting your paper in APA can be daunting if this is your first time. The American Psychological Association (APA) offers a guide or rules to follow when conducting projects in the social sciences or writing papers. The standard APA fromat a research paper outline includes a proper layout from the title page to the final reference pages. There are formatting samples to create outlines before writing a paper. Amongst other strategies, creating an outline is the easiest way to APA format outline template.

Why Is Research Paper Format Necessary?

Consistency in the sequence, structure, and format when writing a research paper encourages readers to concentrate on the substance of a paper rather than how it is presented. The requirements for paper format apply to student assignments and papers submitted for publication in a peer-reviewed publication. APA paper outline template style may be used to create a website, conference poster, or PowerPoint presentation . If you plan to use the style for other types of work like a website, conference poster, or even PowerPoint presentation, you must format your work accordingly to adjust to requirements. For example, you may need different line spacing and font sizes. Follow the formatting rules provided by your institution or publication to ensure its formatting standards are followed as closely as possible. However, to logically structure your document, you need a research paper outline in APA format. You may ask: why is it necessary to create an outline for an APA research paper?

Concept & Purposes of Research Paper Outline

A path, direction, or action plan! Writing short essays without a layout may seem easy, but not for 10,000 or more words. Yet, confusing a table of contents with an outline is a major issue. The table of contents is an orderly list of all the chapters’ front matter, primary, and back matter. It includes sections and, often, figures in your work, labeled by page number. On the other hand, a research APA-style paper outline is a proper structure to follow.

Purpose of research paper outline

An outline is a formalized essay in which you give your own argument to support your point of view. And when you write your apa outline template, you expand on what you already know about the topic. Academic writing papers examine an area of expertise to get the latest and most accurate information to work on that topic. It serves various purposes, including:

• APA paper outline discusses the study’s core concepts.
• The research paper outlines to define the link between your ideas and the thesis.
• It provides you with manageable portions that you can handle.
• The research paper’s APA outline enables the detection of structural faults or gaps.
• As shown in the example, it must clearly comprehend the subject at hand.

APA outline example

This research paper outline example will guide you in formatting the layout for a clear direction to work on. It eliminates the inconsistency along with lacking proper substance in the paper.

Understanding the APA Outline Format

It would not be wrong to say there is no standard outline format. The official publishing handbook does not give precise guidelines for preparing an outline. But, it requires certain basic guidelines to follow regarding typeface, font size, structure, margins, etc.

APA paper outline example

Moreover, the final shape of your work relies on your instructor’s specifications and your particular preferences for APA citation format. Though, it would be better to follow some standards for formatting your outline, for instance:

Times New Roman is a widely accessible standard typeface for an APA essay format in 12-point font. However, serif and sans serif fonts like Arial and Georgia are acceptable in font size 11pt.

The text of your paper format should be double-spaced.

The primary headlines use Roman and Arabic numerals to write an outline.

Headings & Subheadings

While writing an APA essay, there are particular standards for utilizing headings in your outline: I – Main headings are numbered by Roman numerals like I, II, III, IV A  – Subheadings are numbered with Capital letters (A, B, C, D) 1  – The APA outline uses Arabic numerals (1-9 type numbers) within those subheadings. a  – Below Arabic number subheadings, lower-case letters are used (a, b, a). [1] – Headings below those subheadings use Arabic numbers enclosed in parenthesis.

APA format offers a standard layout for each paper, such as

• 1-inch margins on the top, bottom, left, and right.
• The page number on the upper right corner.

The structure of writing an outline consists of three major sections:

• Introduction

Introduction:

This section highlights crucial background information.

Explain the primary points that support your ideas.

Conclusion:

• Summarize your key arguments.
• Explain how these concepts support your ultimate stance, as shown in APA outline example below.

An outline in APA has three common formats that vary in the numeric sequence of all. To make it easier for you, we have compiled all three templates. You can format your document using these examples for added coherence and structure.

The Basic APA Outline Format

APA Style Outline Template Breakdown

Numbering the APA style format follows five levels of headings that use different alphabets and numbers. For instance, I – Headings use Roman numerals like I, II, and III. A – CAPITAL ALPHABETS”, such as A, B, C, etc. 1 – Headings and subheadings use Arabic numbers (1, 2, 3). a – If there are further headings (the fourth level), use lower-case alphabets. [1] – Headings below that (the fifth level) use Arabic numerals enclosed in parentheses, such as [1], [2], [3].

Full Sentence Outline Format

As the name specifies, the full-sentence style outline format requires every line to be a proper sentence. Full-sentence APA style outline is best recommended for essays and speeches. It gives your writing process an idea or a logical path to follow.

APA Research Paper Outline Example

If you are looking for how to write a research paper outline APA in Full Sentence Format, here is an example:

Full Sentence APA format heading utilizes Roman numerals I, II, and III. Every heading must be a full sentence. Here is an APA style paper outline template for the full-sentence format that will clear all your confusion on how to write an outline in full-sentence format.

APA Paper Outline Format Example

I. Introduction

III. Conclusion

Decimal Outline Format

The decimal outline format for APA research papers differs from other formats. The decimal APA style is simple and uses paragraphs for structure. It contains three main paragraphs, introduction, main body, and conclusion.

First Paragraph: Hook and Thesis

• The first paragraph is a sentence or two that introduces the central concept of your article.
• Introduce your topic or subject of study where your research is applicable as a context for further research.
• Explain why the mentioned issue is essential or relevant to the audience.
• A thesis statement is a claim that you make throughout your whole essay.
• The topic phrase is the first point in any writing to support a thesis statement.
• Give an explanation or provide evidence to support your point.
• Provide verifiable facts, figures, and/or citations from credible sources in your writing. It helps in the substantiating assertion.
• Include as many supporting statements and related evidence in your decimal outline.

Finally, when you write an outline, provide a concluding remark to support your claims.

Decimal APA outline format example

1.0 The main heading 1.1 Subheading under the main heading 1.2 Second digit is represented by subheadings under the main headings 1.2.1 Further division adds another digit in decimal format 1.2.2 You can number them as per the number of paragraphs or points, or lines An easy way to write in decimal APA outline format is to remember the structure, i.e.; 1.1.1 = Heading.Paragraph.Sentence/point under paragraph.”

Decimal APA outline format layout

1.0 Main heading 1.1 First paragraph for first heading. 1.2 Second paragraph for first heading. 1.2.1 First point or sentence for the second paragraph. 2.0 Second heading 2.1 Second heading, first paragraph. 2.2 Second heading, second paragraph. 2.2.1 Second, heading, second paragraph, first sentence, or point. 3.0 Decimal working 3.1 You must remember that each digit represents a segment. 3.2 It is easier to remember the placement of numbers. 3.2.1 First digit represents the heading 3.2.2 Second digit represents the paragraph under the main heading <3.2.3 The third digit represents any point or sentence under the paragraph.

Tips for Writing an Outline: Organize Your Ideas

You may feel it is easier to write without outlines, but once you start writing, organizing your ideas or thoughts becomes hard. Even if you have some fantastic ideas, producing an engaging story is practically hard. If you are not first creating an outline or conceptual guides while writing a research paper, you may lose track. A well-written outline is essential in completing your paper and maintaining quality. Establishing your point in paper writing is easy if you create an outline first. You can find an APA research paper outline template that best suits your requirement. Moreover, these tips can help you polish your writing. These tips and sample papers can help you write outstanding outlines without making any hassle.

A definite goal

For better expression, make a list of primary objectives on a title page in a single phrase or less. Your goal should be specific and measurable. If it is too broad or imprecise, you will not achieve anything. If you are working on a large paper format that covers a variety of themes or topics, you may have a more general purpose in mind. But, if you plan to write an essay, the aim should be as specific and clear as possible to be effective.

Breaking things up rather than allowing them to become verbose is known as the division rule. Make sure that each subsection in the document corresponds to its parent heading. If it doesn’t compare to the section, removing it or moving it to another location is better.

Parallelism

It is mainly related to the consistency and structure of the document. It keeps your paper’s layout tidy and also ensures relevancy. For instance, if you begin one heading with a verb, make sure all other headings and subheadings also start with a verb.

Coordination

Having headings aligned is critical to creating a well-organized outline. This rule also applies to subheadings, which is a good thing. If one title is less important than another, consider changing your layout by incorporating it into a subsection instead.

Subordination

Subordination deals with maintaining a connection between your paper’s headings and subheadings. It helps in the proper sequencing of headings and subheadings. Headings should be broad at the outset. At the same time, the subheadings become more particular as they go further into the document.

Avoid Redundancy

While writing a paper outline, look through it many times and cross out any items that aren’t necessary or have no significance. While outlining, make sure to be specific and concise. It will prevent you from adding information that does not supporting your final essay. Remove all the extra information and points while c that weighs you down while you write.

Wrap it up in a good way

Creating an outline does not only help in writing a coherent term paper, but it also helps in ending with precise understanding. Be considerate of your audience’s time and effort when you write an outline in APA, and ensure it serves its purpose. If you still have any doubts about formatting your paper outline, you can use this APA-style research paper outline template to write your document. We have provided Outline Format Example for every style.

People find it hard to write an outline in APA, but if you are aware of the requirements and structure, it’s no breeze. Sometimes, your instructor may alter your paper format by introducing or removing existing sections. As a result, if you come across any templates for an outline in APA, pay close attention to them. If you are looking for a quick answer to how to outline an APA paper, here’s a standard logical sequence of typical parts to include when writing an outline in APA:

• Thesis statement
• Techniques employed
• Body of paper
• Conclusions section
• List of references

A well-written outline is an excellent tool for presenting an outstanding paper. Including the key components while writing an outline for a research paper is necessary.

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How to Write a Research Methodology for a Research Paper

Have you ever felt overwhelmed by research, not knowing how to start or structure your thoughts effectively?

You're not alone in facing this problem. The challenge of disorganized research is something many students deal with. But don't worry! 

In this blog, we'll not only tell you how to create your research paper outline, but we’ve included some downloadable templates as well.

We'll provide a practical step-by-step solution for organizing your research material so you end with a well-structured research paper. By the end of the blog, you'll understand why outlines matter and how to create compelling research papers.

So, keep reading to learn more!

• 1. What is a Research Paper Outline?
• 2. Different Research Paper Outline Formats
• 3. Steps to Create a Research Paper Outline
• 4. Research Paper Outline Example

What is a Research Paper Outline?

A research paper outline serves as a systematic framework for your paper. It's a structured strategy that assists in the organization of your thoughts and ideas before the writing process begins. 

The outline structures out the main sections, subtopics, and supporting details in your research paper. In essence, it offers a well-structured and coherent roadmap for the entirety of your paper, maintaining academic rigor and clarity.

Different Research Paper Outline Formats

When it comes to creating a research paper outline, you've got options. Let's explore a few different formats that you can choose from:

Numeric Outline

A numeric outline is a structured organizational format for planning a research paper. 

It uses a numerical system to represent the hierarchy of ideas, with each main section or point numbered and subpoints or details indicated by decimal numbers. Numeric outlines are useful for presenting information in a clear and logical sequence.

Here’s a sample research paper outline template for this format:

Alphanumeric Outline

An alphanumeric outline is a hierarchical structure used to outline a research paper, combining numbers and letters to signify the different levels of information. 

Main sections are designated with capital letters (A, B, C), which include major points, while subpoints are indicated by numbers and lowercase letters (1, 2, a, b). Alphanumeric outlines help writers organize complex topics and subtopics effectively.

Here’s a sample sample research paper outline for this format:

Full Sentence Outline

A full-sentence outline is a method of planning a research paper in which each point in the outline is presented as a complete sentence or phrase. 

It provides a detailed overview of the content and structure of the paper. Full-sentence outlines are particularly helpful for writers who prefer thorough planning and want to capture the essence of each section or point.

Here’s a research paper outline format for full sentences:

Steps to Create a Research Paper Outline

Creating a research paper outline doesn't have to be complicated. Follow these simple steps to get started:

Step 1: Choose Your Research Topic

Begin by selecting a research topic that is both interesting to you and relevant to your assignment or academic objectives. Your chosen topic will serve as the foundation for your entire research paper.

Step 2: Identify Your Main Sections

Determine the main sections or chapters your research paper will include. These are the broad thematic areas that will structure your paper, and they provide a high-level overview of the topics you plan to cover. Here are the main sections a typical research paper involves:

• Title Page: This is the first page and includes the paper's title, author's name, institutional affiliation, and often the running head.
• Abstract : A concise summary of the paper, usually around 150-250 words, providing an overview of the research, its key findings, and implications.
• Introduction: Sets the stage for your research, offering background information and a thesis statement , which is a central argument or hypothesis.
• Literature Review : A comprehensive analysis of existing research and literature on your topic, demonstrating your understanding of the subject.
• Methodology: Explain the research methods, data collection techniques, and analytical tools used in your study.
• Findings: Presents the research results in a structured manner, often including data, tables, or charts.
• Discussion: Interpretation of the findings and their implications, offering insights into the research's significance.
• Conclusion: Summarizes the main points, reiterates the thesis, and discusses potential future research directions.
• References: A list of all sources cited in your paper, following a specific citation style (e.g., APA, MLA).

Step 3: Break It Down into Subtopics

Under each main section, further divide your content into smaller subtopics. Subtopics are like the building blocks of your paper; they represent the key points or ideas you intend to explore within each main section.

Step 4: Add Supporting Details

For each subtopic, include supporting details, facts, examples, or arguments that bolster your point. These supporting details form the substance of your paper and provide evidence for your claims or arguments.

Step 5: Organize Your Points

Organize your main sections, subtopics, and supporting details in a logical order that flows smoothly from one point to the next. This step ensures that your research paper maintains coherence and readability.

Step 6: Use Numbers or Letters

To enhance clarity within your outline, use numbering, a lettering system, or Roman numerals. Use numerical sequencing for main sections (e.g., "1.," "2.," "3.") and a combination of numbers and letters for subtopics (e.g., "1.1," "1.2," "2.1").

Step 7: Stay Flexible

Recognize that your outline is not set in stone. As you conduct research and begin writing, your ideas may evolve, and you may discover the need to adjust your outline accordingly. Embrace this flexibility to adapt to new insights and information.

By following these steps, you'll create a well-structured research paper outline that serves as a roadmap for your writing journey. It keeps your research organized and makes writing easier, resulting in a more effective paper.

Research Paper Outline Example

A research paper outline could be created in several different ways. Here is a sample research paper outline for a quick review:

Here are some more examples for different formats and subjects:

APA Research Paper Outline PDF

College Research Paper Outline

Argumentative Research Paper Outline

Sample Research Paper Outline

History Research Paper Outline

Research Paper Outline MLA

Research Paper Outline with Annotated Bibliography

Need to consult more examples? Have a look at these top-quality research paper examples and get inspiration!

In conclusion, with the help of these example templates and our step-by-step guide on creating an outline, you're now well-prepared to create an effective one. 

If you're in a hurry and want to skip the outlining process, our essay writing service is here to help!

You can get our team of expert essay writers to assist you at any stage of your research or to deliver a well-formatted, accurate research paper. 

So, just say, write my research paper or ' write an essay for me ' and we’ll deliver an original, top-quality paper to you!

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Research Paper Outline Examples

Below are examples of research paper outlines. Creating an outline is the first thing you should do before starting on your research paper.

This article is a part of the guide:

• Example of a Paper
• Write a Hypothesis
• Introduction
• Example of a Paper 2

Browse Full Outline

• 1 Write a Research Paper
• 2 Writing a Paper
• 3.1 Write an Outline
• 3.2 Outline Examples
• 4.1 Thesis Statement
• 4.2 Write a Hypothesis
• 5.2 Abstract
• 5.3 Introduction
• 5.4 Methods
• 5.5 Results
• 5.6 Discussion
• 5.7 Conclusion
• 5.8 Bibliography
• 6.1 Table of Contents
• 6.2 Acknowledgements
• 6.3 Appendix
• 7.1 In Text Citations
• 7.2 Footnotes
• 7.3.1 Floating Blocks
• 7.4 Example of a Paper
• 7.5 Example of a Paper 2
• 7.6.1 Citations
• 7.7.1 Writing Style
• 7.7.2 Citations
• 8.1.1 Sham Peer Review
• 8.1.2 Advantages
• 8.1.3 Disadvantages
• 8.2 Publication Bias
• 8.3.1 Journal Rejection
• 9.1 Article Writing
• 9.2 Ideas for Topics

Once you've decided what topic you will be writing about, the next thing you should pay attention to is the scope of your paper or what you will be including in your discussion . The broader your topic is, the more difficult it is to discuss the full details. This is why you should establish early on the scope and limitations of your paper which will provide the foundation for your research paper outline.

Basically, your outline will constitute three main sections: the Introduction, the Body and the Conclusion. But to make sure your paper is complete, consult your instructor for specific parts they wants to be included in your research paper . Sample outlines for research papers will follow. But first, let’s discuss the main sections of your paper and what information each should cover.

The introduction should contain your thesis statement or the topic of your research as well as the purpose of your study. You may include here the reason why you chose that particular topic or simply the significance of your research paper's topic. You may also state what type of approach it is that you'll be using in your paper for the entire discussion of your topic. Generally, your Introduction should orient your readers to the major points the rest of the paper will be covering, and how.

The body of your paper is where you will be presenting all your arguments to support your thesis statement. Remember the “Rule of 3” which states that you should find 3 supporting arguments for each position you take. Start with a strong argument, followed by a stronger one, and end with the strongest argument as your final point.

The conclusion is where you form a summary of all your arguments so you can arrive at your final position. Explain and reiterate why you've ended up with the said conclusion.

As mentioned earlier, here are some sample outlines for research papers:

Thesis Topic: A Study on Factors Affecting the Infant Feeding Practices of Mothers in Las Pinas City

• Statement of the Problem
• Definition of Terms
• Theoretical Framework
• Type of Research
• Respondents
• Questionnaire
• Review of Related Literature
• Scope and Limitations
• Significance of the Study
• Benefits of Breastfeeding
• WHO Recommendations
• The International Code of Marketing of Breast Milk Substitutes
• The Baby-Friendly Hospital Initiative
• The Innocenti Declaration on the Protection, Promotion and Support of Breastfeeding
• National Situationer
• The Milk Code
• BFHI in the Philippines
• Milk Code Violations
• Formula Feeding
• Factors Influencing the Decision Regarding Infant Feeding Method
• Area Situationer
• Socio-economic Demographic Profile of Mothers
• Information Regarding Current (Youngest) Infant
• Exclusive Breastfeeding
• Mixed Feeding
• Previous Infant Feeding Practices
• Maternal Knowledge
• Correlation Tests
• Analytical Summary
• Thesis Reworded
• Recommendations

Topic: Asbestos Poisoning

• Definition of Asbestos Poisoning
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Article Contents

Introduction, 1 review of analytic rings associated to complete affinoid pairs, 2 faithfully flat descent for discrete rings, 3 adic completeness and small complete adic rings, 4 faithfully flat descent over a complete non-archimedean field, 5 descent of pseudo-coherent complexes and perfect complexes, acknowledgments.

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Faithfully Flat Descent of Quasi-Coherent Complexes on Rigid Analytic Varieties via Condensed Mathematics

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Yutaro Mikami, Faithfully Flat Descent of Quasi-Coherent Complexes on Rigid Analytic Varieties via Condensed Mathematics, International Mathematics Research Notices , Volume 2024, Issue 8, April 2024, Pages 7099–7128, https://doi.org/10.1093/imrn/rnad320

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Faithfully flat descent of pseudo-coherent complexes in rigid geometry was proved by Mathew. In this paper, we generalize the result of Mathew to solid quasi-coherent complexes on rigid analytic varieties, which have been introduced by Clausen and Scholze by means of condensed mathematics.

We begin with the statement of faithfully flat descent in commutative algebra. Let |$A\to B$| be a faithfully flat map of (commutative unital) rings. Faithfully flat descent states that an |$A$| -module can be described as a |$B$| -module with a descent datum . Recently, Lurie and Mathew have generalized the theory of faithfully flat descent to complexes of modules in [ 13 ] and [ 15 ] by using higher algebra. This generalization is important even if we are interested only in static objects (i.e., not higher categorical objects). For example, it is used in [ 6 , 8 ] whose main theorems are concerned with static objects.

On the other hand, there is an analogue of the classical theory of faithfully flat descent in rigid geometry. Let |$K$| be a complete non-archimedean field, and let |$A\to B$| be a map of affinoid |$K$| -algebras, which is faithfully flat as a map of ordinary rings. Then a coherent |$A$| -module can be described as a coherent |$B$| -module |$N$| with an isomorphism |$\tau \colon B \otimes _{A} N \cong N \otimes _{A} B$| of |$B\otimes _{A} B$| -modules, which satisfies the cocycle condition (such |$\tau $| is called a descent datum ); see [ 2 , 7 ] for more details. This result has been generalized by Mathew in [ 16 ] as follows:

  Theorem 0.1 ([ 16 , Theorem 1.4]). Let |$K$| and |$A \to B$| be as above. Let |$B^{n/A}$| denote the |$n$| -fold completed tensor product of |$B$| over |$A$|⁠ . Then we have an equivalence of |$\infty $| -categories $$ \begin{align*} &\textrm{PCoh}(A) \overset{\sim}{\longrightarrow} \varprojlim_{[n] \in \Delta} \textrm{PCoh}(B^{(n+1)/A}),\end{align*}$$ where |$\textrm {PCoh}(A)$| is the category of pseudo-coherent complexes of |$A$| -modules and |$\Delta $| is the simplex category.

Unlike commutative algebra, in rigid geometry, we only treat objects that satisfy some finiteness conditions, that is coherent modules and pseudo-coherent complexes. This is because of issues of topological algebraic objects such as topological rings and topological modules.

Recently, Clausen and Scholze have developed a new approach to treat such objects in [ 18 , 19 ], which is called condensed mathematics . In condensed mathematics, topological spaces are generalized to condensed sets , which are sheaves of sets on the site of profinite sets with the topology given by finite jointly surjective families of maps. Similarly, topological groups, rings, etc. are generalized, respectively, to condensed groups, rings, etc . One of the advantages of using condensed objects is that the category of condensed abelian groups becomes an abelian category as opposed to the category of topological abelian groups.

In this context, complete affinoid pairs are generalized to analytic rings . Roughly speaking, an analytic ring |${\mathcal {A}}$| is a pair of a condensed ring |$\underline {{\mathcal {A}}}$| and some additional data |$S\mapsto {\mathcal {A}}[S]$|⁠ , which is called the functor of measures of |${\mathcal {A}}$|⁠ . For an analytic ring |${\mathcal {A}}$|⁠ , we can define a symmetric monoidal stable |$\infty $| -category |${\mathcal {D}}({\mathcal {A}})$|⁠ . For a complete affinoid pair |$(A,A^{+})$|⁠ , we can define an analytic ring |$(A,A^{+})_{\blacksquare }$| associated to |$(A,A^{+})$|⁠ , and the objects in |${\mathcal {D}}((A, A^{+})_{\blacksquare })$| are called solid quasi-coherent complexes over |$(A, A^{+})$|⁠ .

The following is a basic property of solid quasi-coherent complexes.

Let |$X$| be an analytic affinoid adic space and let |$U$| denote an arbitrary open affinoid subspace of |$X$|⁠ . Then the functor |$U \mapsto {\mathcal {D}}(({\mathcal {O}}_{X}(U), {\mathcal {O}}_{X}^{+}(U))_{\blacksquare })$| defines a sheaf of |$\infty $| -categories on |$X$|⁠ .

In [ 16 ], it is expected that faithfully flat descent in rigid geometry can be formulated in this framework. In this paper, we will carry it out. The following is the main theorem in this paper.

  Theorem 0.3. Let |$K$| be a complete non-archimedean field, and let |$A \to B$| be a faithfully flat map of affinoid |$K$| -algebras. Let |$B^{n/A}$| denote the |$n$| -fold completed tensor product of |$B$| over |$A$|⁠ . Then we have an equivalence of |$\infty $| -categories $$ \begin{align*} &{\mathcal{D}}((A,A^{\circ})_{\blacksquare}) \overset{\sim}{\longrightarrow} \varprojlim_{[n] \in \Delta} {\mathcal{D}}((B^{(n+1)/A},(B^{(n+1)/A})^{\circ})_{\blacksquare}). \end{align*}$$

This theorem is a generalization of Theorem 0.1 . In the proof, we will consider formal models and prove faithfully flat descent for them. We explain two difficulties appearing in the proof.

Now, we consider a map of affinoid |$K$| -algebras. Let |$K$| be a complete non-archimedean field and |$\pi $| be a pseudo-uniformizer of |$K$|⁠ . Let |$A \to B$| be a faithfully flat map of admissible |${\mathcal {O}}_{K}$| -algebras (i.e., |$\pi $| -torsion-free topologically finitely presented |${\mathcal {O}}_{K}$| -algebras). In this setting, the functor |$- \otimes _{A_{\blacksquare }}^{{\mathbb {L}}} B_{\blacksquare }$| is not equal to the functor |$-\otimes _{A_{\blacksquare }}^{{\mathbb {L}}} \underline {B}$|⁠ , where |$\underline {B}$| is the object of |${\mathcal {D}}(A_{\blacksquare })$| associated to the topological |$A$| -module |$B$|⁠ . Therefore, we cannot apply the argument in [ 15 ] naively. However, we can prove the following theorem.

The object |$N_{B/A}$| is |$A_{\blacksquare }$| -complete and compact as an object of |${\mathcal {D}}(A_{\blacksquare })$|⁠ .

• We have an equivalence of functors from |${\mathcal {D}}(A_{\blacksquare })$| to |${\mathcal {D}}(B_{\blacksquare })$| $$ \begin{align} -\otimes_{A_{\blacksquare}}^{{\mathbb{L}}} B_{\blacksquare} \overset{\sim}{\longrightarrow} R\operatorname{\underline{Hom}}_{\underline{A}}(N_{B/A}, -).\end{align}$$ (0.1)

By this theorem, we can make use of |$N_{B/A}$| instead of |$\underline {B}$| to prove faithfully flat descent.

Next, we want to reduce faithfully flat descent for admissible |${\mathcal {O}}_{K}$| -algebras to fppf descent for discrete rings by taking a quotient by an ideal of definition. However, for an extremally disconnected set |$S$|⁠ , the map |$A_{\blacksquare }[S]\to \varprojlim _{n \geq 1} (A/\pi ^{n})_{\blacksquare }[S]$| is not necessarily an equivalence, so we cannot reduce the problem to fppf descent for discrete rings naively. To resolve this problem, we will introduce the notion of small complete adic rings. Roughly speaking, a small complete adic ring is a complete adic ring whose reduction is integral over some finitely generated |${\mathbb {Z}}$| -algebra. For a small complete adic ring, the map above becomes an equivalence. By using a limit argument, we can reduce the problem to the case of small complete adic rings. Then we can reduce the problem to the case over discrete rings by taking a quotient by an ideal of definition.

To complete the proof, it is necessary to prove fppf descent for discrete rings in the context of condensed mathematics.

  Theorem 0.5. Let |$A \to B$| be a finitely presented faithfully flat map of discrete rings. Let |$B^{n/A}$| denote the |$n$| -fold tensor product of |$B$| over |$A$|⁠ . Then we have an equivalence of |$\infty $| -categories $$ \begin{align*} &{\mathcal{D}}(A_{\blacksquare}) \overset{\sim}{\longrightarrow} \varprojlim_{[n] \in \Delta} {\mathcal{D}}((B^{(n+1)/A})_{\blacksquare}). \end{align*}$$

In this setting, we can also define |$N_{B/A}$| as in Theorem 0.4 . In order to prove Theorem 0.5 , we will prove a descendable property of |$N_{B/A}$| (Theorem 2.15 ) based on ideas of Mathew in [ 15 ]. It is also necessary to prove uniform boundedness of |$N_{B/A}$| (Proposition 2.20 ) for reducing faithfully flat descent for admissible |${\mathcal {O}}_{K}$| -algebras to fppf descent for discrete rings (see the proof of Theorem 4.13 ).

While writing this paper, the author was informed that Mann proves faithfully flat descent for discrete rings by a slightly different approach in his thesis [ 14 ]. In this paper, it will be proved by using |$N_{B/A}$|⁠ . On the other hand, Mann considers a stable monoidal |$\infty $| -category |${\mathcal {E}}(A_{\blacksquare })$| instead of |${\mathcal {D}}(A_{\blacksquare })$|⁠ . By using it instead of |$N_{B/A}$|⁠ , Mann defines the notion of descendable maps of analytic rings and constructs a general theory of them to prove faithfully flat descent.

Outline of the paper

This paper is organized as follows. In Section 1, we begin with the definition of analytic rings, then recall the construction of analytic rings from affinoid pairs. For more details, see [ 1 , 18 , 19 ]. In Section 2, we prove fppf descent for discrete rings in the context of condensed mathematics. In Section 3, we introduce the notion of adic completeness in the context of condensed mathematics, then introduce the notion of small adic rings. In Section 4, we prove Theorem 0.3 . Finally, in Section 5, we recover Theorem 0.1 from Theorem 0.3 by using the method used in [ 1 , Section 5].

All rings, including condensed ones, are assumed unital and commutative.

For an |$\infty $| -category |${\mathcal {C}}$|⁠ , |$0$| -truncated objects of |${\mathcal {C}}$| are called discrete objects in [ 10 ]. However, this term conflicts with the term “discrete” in the topological sense, so we use the term static object to refer to an |$0$| -truncated object .

In contrast to [ 1 , 14 ], we use the term ring to refer to an ordinary ring (not an animated ring).

For a stable |$\infty $| -category |${\mathcal {D}}$|⁠ , we denote (non |$t$| -exact) limits and colimits in |${\mathcal {D}}$| by |$\mathop {R\varprojlim }$| and |$\mathop {L\varinjlim }$|⁠ , respectively, which correspond to homotopy limits and homotopy colimits in the triangulated category.

All complete adic rings are assumed Hausdorff.

We use the term “extremally disconnected set” to refer to an extremally disconnected compact Hausdorff space .

We use the terms f-adic ring and affinoid pair rather than Huber ring and Huber pair .

We denote the simplex category by |$\Delta $|⁠ , which is the full subcategory of the category of totally ordered sets consisting of the totally ordered sets |$[n]=\{0,\ldots ,n\}$| for all |$n \geq 0$|⁠ . We also denote the subcategory of |$\Delta $| with the same objects but where the morphisms are given by injective maps by |$\Delta _{s}$|⁠ . Moreover, for every |$m \geq 0$|⁠ , we denote the full subcategory of |$\Delta _{s}$| consisting of |$[n]$| for all |$0 \leq n \leq m$| by |$\Delta _{s,\leq m}$|⁠ .

For an f-adic ring |$A$|⁠ , we denote the ring of power-bounded elements of |$A$| by |$A^{\circ }$|⁠ .

For a topological ring |$A$|⁠ , we denote the discrete ring whose underlying ring is the underlying ring of |$A$| by |$A_{\textrm {disc}}$|⁠ .

In this section, we recall some results of [ 1 , 14 , 18 , 19 ], which will be needed later. Most of the propositions in this section are stated without proofs. For complete proofs, see [ 1 , 14 , 18 , 19 ].

1.1 Analytic rings

We begin with the definition of condensed sets, condensed abelian groups, condensed rings, etc.

Let * proét denote the pro-étale site of the point |$\ast $|⁠ , that is, the category of profinite sets with covers given by finite families of jointly surjective maps. A condensed set is a sheaf of sets on * proét . Similarly, a condensed ring, group, etc. is a sheaf of rings, groups, etc. on * proét , respectively. We denote the category of condensed sets by |$\textrm {Cond}$|⁠ .

This definition has minor set-theoretic issues. For the correct definition, see [ 18 , Appendix to Lecture II] or [ 14 , §2.1]. Clausen-Scholze’s solution to these issues is different from the one adopted in [ 10 ]. We often use the results of [ 10 ], and they are justified by the methods used in [ 14 , §2.9]. In this paper, we will ignore this kind of problems.

Let |$X$| be a |$T_{1}$| space (i.e., a topological space all of whose points are closed). Then the presheaf |$\underline {X}$| defined by sending a profinite set |$S$| to the set of continuous maps from |$S$| to |$X$| becomes a condensed set. Here, the |$T_{1}$| condition is related to set-theoretical issues. Similarly, for a Hausdorff topological abelian group or ring |$A$|⁠ , we can define a condensed abelian group or ring |$\underline {A}$|⁠ .

  Remark 1.4. We can identify the category of condensed sets, rings, groups, etc. with the category of functors $$ \begin{align*} &\{\mbox{extremally disconnected sets}\}^{\textrm{op}} \to \{\mbox{sets, rings, groups, etc.}\}\end{align*}$$ sending finite disjoint unions to finite products, respectively (see [ 18 , Lecture II]). However, it is not yet known that the |$\infty $| -category of sheaves of spaces on |$ \ast _{\textrm {pro\'{e}t}}$| is equivalent to the |$\infty $| -category of functors $$ \begin{align*} &\{\mbox{extremally disconnected sets}\}^{\textrm{op}} \to {\mathcal{S}}\end{align*}$$ sending finite disjoint unions to finite products, where |${\mathcal {S}}$| is the |$\infty $| -category of spaces. It is claimed in [ 3 , Warning 2.2.2] that the former category is not hypercomplete, but the latter category is hypercomplete. In such a case, we will use the latter as condensed objects.

For a condensed ring |${\mathcal {A}}$|⁠ , we denote the category of condensed |${\mathcal {A}}$| -modules by |$\textrm {Mod}_{{\mathcal {A}}}^{\textrm {cond}}$|⁠ , and the derived |$\infty $| -category of condensed |${\mathcal {A}}$| -modules by |${\mathcal {D}}({\mathcal {A}})$|⁠ .

The category |$\textrm {Mod}_{{\mathcal {A}}}^{\textrm {cond}}$| is an abelian category that satisfies Grothendieck’s axioms (AB3), (AB4), (AB5), (AB6), (AB3*), and (AB4*). Furthermore, it is generated by compact projective objects.

Note that, for an extremally disconnected set |$S$|⁠ , the functor |$\textrm {Mod}_{{\mathcal {A}}}^{\textrm {cond}} \to \textrm {Ab} ;\; M \mapsto M(S)$| is exact. Let |${\mathcal {A}}[S]$| denote the sheafification of |$T \mapsto {\mathcal {A}}(T)^{\oplus \underline {S}(T)}$|⁠ . Then |$\operatorname {Hom}_{{\mathcal {A}}}({\mathcal {A}}[S], M)$| is isomorphic to |$M(S)$| for every condensed |${\mathcal {A}}$| -module |$M$|⁠ . Therefore, |${\mathcal {A}}[S]$| is a compact projective condensed |${\mathcal {A}}$| -module for any extremally disconnected set |$S$|⁠ , and such modules form a family of compact projective generators (see [ 18 , Lecture II]). Moreover, |$\textrm {Mod}_{{\mathcal {A}}}^{\textrm {cond}}$| has a tensor product and an internal Hom as usual (for details, see [ 14 , Proposition 2.1.11]). We denote the latter by |$\underline {\operatorname {Hom}}_{{\mathcal {A}}}(-,-)$|⁠ . Note that |$\underline {\operatorname {Hom}}_{{\mathcal {A}}}(M,N)(S)$| is equal to |$\operatorname {Hom}_{{\mathcal {A}}}(M \otimes _{{\mathcal {A}}} {\mathcal {A}}[S], N)$| for each extremally disconnected set |$S$|⁠ . In addition, for an object |$M \in {\mathcal {D}}({\mathcal {A}})$|⁠ , we denote |$R\Gamma (S,M) \in {\mathcal {D}}(\textrm {Ab})$| by |$M(S)$|⁠ .

Next, we define (static) analytic rings.

  Definition 1.6 ([ 18 , Definitions 7.1, 7.4]). An uncompleted pre-analytic ring |${\mathcal {A}}$| is a condensed ring |$\underline {{\mathcal {A}}}$| equipped with a functor $$ \begin{align*} &\{\textrm{extremally disconnected sets}\}\to \textrm{Mod}^{\textrm{cond}}_{{\mathcal{A}}};\; S\mapsto {\mathcal{A}}[S]\end{align*}$$ that sends finite disjoint unions to finite products, and a natural transformation |$\Phi _{{\mathcal {A}}}\colon \underline {S} \to {\mathcal {A}}[S]$| of functors from the category of extremally disconnected sets to |$\textrm {Cond}.$| For an uncompleted pre-analytic ring |${\mathcal {A}}$|⁠ , we call |$\underline {{\mathcal {A}}}$| the underlying condensed ring of |${\mathcal {A}}$| and the functor |$S \mapsto {\mathcal {A}}[S]$| the functor of measures of |${\mathcal {A}}$|⁠ . An uncompleted pre-analytic ring |${\mathcal {A}}$| is called an uncompleted analytic ring if, for every complex $$ \begin{align*} &C\colon \cdots \rightarrow C_i\rightarrow \cdots \rightarrow C_1\rightarrow C_0\rightarrow 0 \end{align*}$$ of |$\underline {{\mathcal {A}}}$| -modules such that each |$C_{i}$| is a direct sum of objects of the form |${\mathcal {A}}[T]$| for various extremally disconnected sets |$T$|⁠ , the map $$ \begin{align*} &R\operatorname{\underline{Hom}}_{\underline{{\mathcal{A}}}}({\mathcal{A}}[S], C) \to R\operatorname{\underline{Hom}}_{\underline{{\mathcal{A}}}}(\underline{{\mathcal{A}}}[S], C)\end{align*}$$ is an equivalence for all extremally disconnected sets |$S$|⁠ . An uncompleted analytic ring |${\mathcal {A}}$| is called an analytic ring if the map |$\underline {{\mathcal {A}}} \to {\mathcal {A}}[\ast ]$| is an isomorphism.

Our terminology follows [ 14 ]. In [ 18 ], uncompleted pre-analytic rings, uncompleted analytic rings, and analytic rings are called pre-analytic rings, analytic rings, and normalized analytic rings, respectively.

• We define |$\textrm {Mod}_{{\mathcal {A}}}^{\textrm {cond}} \subset \textrm {Mod}_{\underline {{\mathcal {A}}}}^{\textrm {cond}}$| to be the full subcategory of all |$\underline {{\mathcal {A}}}$| -modules |$M$| such that for all extremally disconnected sets |$S$|⁠ , the map $$ \begin{align*} &\operatorname{Hom}_{\underline{{\mathcal{A}}}}({\mathcal{A}}[S], M) \to \operatorname{Hom}_{\underline{{\mathcal{A}}}}(\underline{{\mathcal{A}}}[S], M) \cong M(S)\end{align*}$$ is an isomorphism. An object of |$\textrm {Mod}_{\underline {{\mathcal {A}}}}^{\textrm {cond}}$| is said to be |${\mathcal {A}}$| -complete if it lies in |$\textrm {Mod}_{{\mathcal {A}}}^{\textrm {cond}}$|⁠ .
• We define |${\mathcal {D}}({\mathcal {A}}) \subset {\mathcal {D}}(\underline {{\mathcal {A}}})$| to be the full |$\infty $| -subcategory of all complexes of |$\underline {{\mathcal {A}}}$| -modules |$M$| such that for all extremally disconnected sets |$S$|⁠ , the map $$ \begin{align*} &R\operatorname{Hom}_{\underline{{\mathcal{A}}}}({\mathcal{A}}[S], M) \to R\operatorname{Hom}_{\underline{{\mathcal{A}}}}(\underline{{\mathcal{A}}}[S], M)\end{align*}$$ is an equivalence. An object of |${\mathcal {D}}(\underline {{\mathcal {A}}})$| is said to be |${\mathcal {A}}$| -complete if it lies in |${\mathcal {D}}({\mathcal {A}})$|⁠ .

The full subcategory |$\textrm {Mod}_{{\mathcal {A}}}^{\textrm {cond}} \subset \textrm {Mod}_{\underline {{\mathcal {A}}}}^{\textrm {cond}}$| is an abelian category stable under limits, colimits, and extensions. The objects of the form |${\mathcal {A}}[S]$|⁠ , where |$S$| is an extremally disconnected set, form a family of compact projective generators.

• The inclusion functor |$\textrm {Mod}_{{\mathcal {A}}}^{\textrm {cond}} \to \textrm {Mod}_{\underline {{\mathcal {A}}}}^{\textrm {cond}} $| admits a left adjoint $$ \begin{align*} &\textrm{Mod}_{\underline{{\mathcal{A}}}}^{\textrm{cond}} \to \textrm{Mod}_{{\mathcal{A}}}^{\textrm{cond}} ;\; M \mapsto M\otimes_{\underline{{\mathcal{A}}}} {\mathcal{A}},\end{align*}$$ which is the unique colimit-preserving extension of |$\underline {{\mathcal {A}}}[S] \mapsto {\mathcal {A}}[S]$|⁠ . There is a unique symmetric monoidal tensor product |$-\otimes _{{\mathcal {A}}}-$| making the functor above symmetric monoidal.

For |$M,N \in \textrm {Mod}_{{\mathcal {A}}}^{\textrm {cond}}$|⁠ , |$\underline {\operatorname {Hom}}_{\underline {{\mathcal {A}}}}(M,N)$| is also |${\mathcal {A}}$| -complete, and |$\underline {\operatorname {Hom}}_{\underline {{\mathcal {A}}}}(-,-)$| becomes an internal Hom of |$\textrm {Mod}_{{\mathcal {A}}}^{\textrm {cond}}$|⁠ .

• The full |$\infty $| -subcategory |${\mathcal {D}}({\mathcal {A}}) \subset {\mathcal {D}}(\underline {{\mathcal {A}}})$| is a stable subcategory stable under limits and colimits. The image of the functor |${\mathcal {D}}(\textrm {Mod}_{{\mathcal {A}}}^{\textrm {cond}}) \to {\mathcal {D}}(\underline {{\mathcal {A}}})$| is included in |${\mathcal {D}}({\mathcal {A}})$|⁠ , and |${\mathcal {D}}(\textrm {Mod}_{{\mathcal {A}}}^{\textrm {cond}}) \to {\mathcal {D}}({\mathcal {A}})$| is an equivalence. For |$M \in {\mathcal {D}}({\mathcal {A}})$|⁠ , $$ \begin{align*} &R\operatorname{\underline{Hom}}_{\underline{{\mathcal{A}}}}({\mathcal{A}}[S], M) \to R\operatorname{\underline{Hom}}_{\underline{{\mathcal{A}}}}(\underline{{\mathcal{A}}}[S], M)\end{align*}$$ is also an equivalence. An object |$M \in {\mathcal {D}}(\underline {{\mathcal {A}}})$| is |${\mathcal {A}}$| -complete if and only if |$H^{i}(M)$| is |${\mathcal {A}}$| -complete for all |$i\in {\mathbb {Z}}$|⁠ .
• The inclusion functor |${\mathcal {D}}({\mathcal {A}}) \to {\mathcal {D}}(\underline {{\mathcal {A}}})$| admits a left adjoint $$ \begin{align*} &{\mathcal{D}}(\underline{{\mathcal{A}}}) \to {\mathcal{D}}({\mathcal{A}}) ;\; M \mapsto M\otimes_{\underline{{\mathcal{A}}}}^{{\mathbb{L}}} {\mathcal{A}},\end{align*}$$ which is a left derived functor of |$M \mapsto M\otimes _{\underline {{\mathcal {A}}}} {\mathcal {A}}$|⁠ . There is a unique symmetric monoidal tensor product |$-\otimes _{{\mathcal {A}}}^{{\mathbb {L}}}-$| making the functor above symmetric monoidal.

For |$M,N \in {\mathcal {D}}({\mathcal {A}})$|⁠ , |$R\operatorname {\underline {Hom}}_{\underline {{\mathcal {A}}}}(M,N)$| is also |${\mathcal {A}}$| -complete, and |$R\operatorname {\underline {Hom}}_{\underline {{\mathcal {A}}}}(-,-)$| becomes an internal Hom of |${\mathcal {D}}({\mathcal {A}})$|⁠ .

Let |${\mathcal {A}}$| and |${\mathcal {B}}$| be uncompleted analytic rings. A map of uncompleted analytic rings from |${\mathcal {A}}$| to |${\mathcal {B}}$| is a map |$\underline {{\mathcal {A}}} \to \underline {{\mathcal {B}}}$| of underlying condensed rings such that for all extremally disconnected sets |$S$|⁠ , the |$\underline {{\mathcal {A}}}$| -module |${\mathcal {B}}[S]$| is |${\mathcal {A}}$| -complete.

For a map |${\mathcal {A}} \to {\mathcal {B}}$| of uncompleted analytic rings, every object |$M\in {\mathcal {D}}({\mathcal {B}})$| is |${\mathcal {A}}$| -complete, since |${\mathcal {D}}({\mathcal {B}})$| is generated by |${\mathcal {B}}[S]$| for various extremally disconnected sets |$S$| under shifts and sifted colimits and |${\mathcal {B}}[S]$| is |${\mathcal {A}}$| -complete.

• The composite functor $$ \begin{align*} &\textrm{Mod}_{\underline{{\mathcal{A}}}}^{\textrm{cond}} \xrightarrow{- \otimes_{\underline{{\mathcal{A}}}} \underline{{\mathcal{B}}}}\textrm{Mod}_{\underline{{\mathcal{B}}}}^{\textrm{cond}} \xrightarrow{-\otimes_{\underline{{\mathcal{B}}}} {\mathcal{B}}} \textrm{Mod}_{{\mathcal{B}}}^{\textrm{cond}}\end{align*}$$ factors over |$\textrm {Mod}_{{\mathcal {A}}}^{\textrm {cond}}$|⁠ , via a functor denoted by $$ \begin{align*} &\textrm{Mod}_{{\mathcal{A}}}^{\textrm{cond}} \to \textrm{Mod}_{{\mathcal{B}}}^{\textrm{cond}} ;\; M \to M\otimes_{{\mathcal{A}}}{\mathcal{B}}.\end{align*}$$ This is the left adjoint of the forgetful functor |$\textrm {Mod}_{{\mathcal {B}}}^{\textrm {cond}} \to \textrm {Mod}_{{\mathcal {A}}}^{\textrm {cond}}$|⁠ .
• The composite functor $$ \begin{align*} &{\mathcal{D}}(\underline{{\mathcal{A}}}) \xrightarrow{- \otimes_{\underline{{\mathcal{A}}}}^{{\mathbb{L}}} \underline{{\mathcal{B}}}}{\mathcal{D}}(\underline{{\mathcal{B}}}) \xrightarrow{-\otimes_{\underline{{\mathcal{B}}}}^{{\mathbb{L}}} {\mathcal{B}}} {\mathcal{D}}({\mathcal{B}})\end{align*}$$ factors over |${\mathcal {D}}({\mathcal {A}})$|⁠ , via a functor denoted by $$ \begin{align*} &{\mathcal{D}}({\mathcal{A}}) \to {\mathcal{D}}({\mathcal{B}}) ;\; M \to M\otimes_{{\mathcal{A}}}^{{\mathbb{L}}}{\mathcal{B}}.\end{align*}$$ This is the left adjoint functor of the forgetful functor |${\mathcal {D}}({\mathcal {B}}) \to {\mathcal {D}}({\mathcal {A}})$|⁠ .

Similarly, we can also make a definition of a (complete commutative) analytic animated ring |${\mathcal {A}}$|⁠ , and similar propositions hold true for it. In essence, however, we treat only (0-truncated) analytic rings in this paper, so we do not give the precise definition. See [ 19 , Lectures XI, XII], [ 14 , §2.3] for more details. We denote the |$\infty $| -category of (complete commutative) analytic animated rings by |$\textrm {AnRing}$|⁠ . Then the category of (complete) analytic rings becomes a full |$\infty $| -subcategory of |$\textrm {AnRing}$| in a natural way.

The |$\infty $| -category |$\textrm {AnRing}$| admits all small colimits. In particular, |$\textrm {AnRing}$| admits pushouts. Moreover, sifted colimits in |$\textrm {AnRing}$| commute with the functor |${\mathcal {A}} \mapsto {\mathcal {A}}[S]$| to |${\mathcal {D}}^{\leq 0}(\textrm {Cond}(\textrm {Ab}))$| for every extremally disconnected set |$S$|⁠ , where |$\textrm {Cond}(\textrm {Ab})$| is the category of condensed abelian groups.

The following lemma is useful for showing that a given diagram is a pushout diagram.

The object |$\underline {{\mathcal {E}}}$| is |${\mathcal {B}}$| -complete, and the natural map |$\underline {{\mathcal {C}}} \otimes _{{\mathcal {A}}}^{{\mathbb {L}}} {\mathcal {B}} \to \underline {{\mathcal {E}}}$| is an equivalence.

For any object |$M\in {\mathcal {D}}(\underline {{\mathcal {E}}})$|⁠ , |$M$| is |${\mathcal {E}}$| -complete if and only if |$M$| is |${\mathcal {B}}$| -complete and |${\mathcal {C}}$| -complete.

A map |$f \colon {\mathcal {A}} \to {\mathcal {B}}$| in |$\textrm {AnRing}$| is steady if for all maps |$g \colon {\mathcal {A}} \to {\mathcal {C}}$| in |$\textrm {AnRing}$| and all |$M \in {\mathcal {D}}({\mathcal {C}})$|⁠ , the object |$M \otimes _{{\mathcal {A}}}^{{\mathbb {L}}} {\mathcal {B}}$|⁠ , which is a priori an object of |${\mathcal {D}}(\underline {{\mathcal {C}}} \otimes _{{\mathcal {A}}}^{{\mathbb {L}}} {\mathcal {B}})$|⁠ , is |${\mathcal {C}}$| -complete.

By definition, the class of steady maps is stable under base change and composition.

The original definition [ 19 , Definition 12.13] of steady maps is different from our definition. It is claimed in [ 19 , page 83] that the original definition is equivalent to our definition, but this is not true. However, this causes no issues in the result of [ 19 ] because only our definition is used there.

1.2 Analytic rings associated to complete affinoid pairs

We begin with the construction of (complete) analytic rings associated to complete affinoid pairs.

The underlying ring is |$\underline {A}$|⁠ .

For an extremally disconnected set |$S= \varprojlim _{i} S_{i}$| (each |$S_{i}$| is a finite set), we define |$A_{\blacksquare }[S]= \varprojlim _{i} \underline {A}[S_{i}]$|⁠ .

The natural transform |$\underline {S} \to A_{\blacksquare }[S]$| is defined in the obvious way.

For an extremally disconnected set |$S$|⁠ , we define |$(A, A^{+})_{\blacksquare }[S]= \varinjlim _{B \subset A^{+}} \underline {A} \otimes _{\underline {B}} B_{\blacksquare }[S]$|⁠ , where the colimit is taken over all finitely generated |${\mathbb {Z}}$| -subalgebras |$B$| of |$A^{+}$|⁠ .

The natural transform |$\underline {S} \to (A,A^{+})_{\blacksquare }[S]$| is defined in the obvious way.

For an extremally disconnected set |$S$|⁠ , we define |$(A, A^{+})_{\blacksquare }[S]= \underline {A} \otimes _{(A_{\textrm {disc}}, A^{+}_{\textrm {disc}})_{\blacksquare }} (A_{\textrm {disc}}, A^{+}_{\textrm {disc}})_{\blacksquare }[S]$|⁠ .

We recall some results about such analytic rings.

Let |$f \colon A \to B$| be an integral map of discrete rings. Then |$M \in {\mathcal {D}}(\underline {B})$| is |$B_{\blacksquare }$| -complete if and only if |$M$| is |$A_{\blacksquare }$| -complete.

Let |$A$| be an f-adic ring, and |$f_{I}=\{f_{i} \mid i \in I \}$| be a collection of elements of |$A^{\circ }$|⁠ . Let |$A^{+}$| be the minimal ring of integral elements containing |$f_{I}$|⁠ . Then |$M \in {\mathcal {D}}(\underline {A})$| is |$(A, A^{+})_{\blacksquare }$| -complete if and only if |$M$| is |${\mathbb {Z}}[f_{i}]_{\blacksquare }$| -complete for all |$i \in I$|⁠ , where we regard |${\mathbb {Z}}[f_{i}]$| as a discrete ring.

It follows from Proposition 1.22 and Proposition 1.23 that for a map |$(A,A^{+})\to (B,B^{+})$| of complete affinoid pairs, |$(B,B^{+})_{\blacksquare }[S]$| is |$(A,A^{+})_{\blacksquare }$| -complete for any extremally disconnected set |$S$|⁠ . In particular, we get a map |$(A,A^{+})_{\blacksquare }\to (B,B^{+})_{\blacksquare }$| of analytic rings.

  Proposition 1.25 ([ 1 , Proposition 3.34]). Let |$\textrm {cAff}$| denote the category of complete affinoid pairs. Then the following functor is fully faithful: $$ \begin{align*} & \textrm{cAff} \to \textrm{AnRing} ;\; (A, A^+) \mapsto (A, A^+)_{\blacksquare}.\end{align*}$$ Moreover, the image of this functor is contained in the full |$\infty $| -subcategory consisting of ( ⁠|$0$| -truncated) analytic rings.

  Proposition 1.26 ([ 1 , Proposition 3.33]). Let |$A$| be a discrete ring, and |$B,C$| be discrete |$A$| -algebras such that |$B \otimes _{A} C \simeq B \otimes _{A}^{{\mathbb {L}}} C$|⁠ . Then we have the following equivalence: $$ \begin{align*} &B_{\blacksquare} \otimes_{A_{\blacksquare}}^{{\mathbb{L}}} C_{\blacksquare} \simeq (B \otimes_{A} C)_{\blacksquare}.\end{align*}$$

The natural map |${\mathbb {Z}}_{\blacksquare } \to {\mathbb {Z}}[T]_{\blacksquare }$| in |$\textrm {AnRing}$| is steady.

  Lemma 1.28 ([ 1 , Lemma 3.14]). Let |$M$| be a prodiscrete topological abelian group. Let |$M\langle T_{1},\ldots ,T_{n} \rangle $| denote the topological abelian group of convergent formal power series with |$n$| variables. Then we have the following equivalence: $$ \begin{align*} &M \otimes_{{\mathbb{Z}}_{\blacksquare}}^{{\mathbb{L}}} {\mathbb{Z}}[T_1,\ldots,T_n]_{\blacksquare} \simeq \underline{M\langle T_1,\ldots,T_n \rangle}.\end{align*}$$

  Lemma 1.29 ([ 1 , Lemma 4.7]). Let |$(A,A^{+})$| be a complete affinoid pair. Then we have the following equivalence: $$ \begin{align*} &(A\langle T\rangle,A^+\langle T\rangle)_{\blacksquare} \simeq (A, A^+)_{\blacksquare} \otimes_{{\mathbb{Z}}_{\blacksquare}}^{{\mathbb{L}}} {\mathbb{Z}}[T]_{\blacksquare}.\end{align*}$$

Finally, we compute |$(A,A^{+})_{\blacksquare }[S]$| for an extremally disconnected set |$S$|⁠ .

  Lemma 1.31 ([ 18 , Corollary 5.5]). There exists a set |$J$| and an isomorphism of condensed abelian groups $$ \begin{align*} &{\mathbb{Z}}_{\blacksquare}[S] \cong \prod_{J} \underline{{\mathbb{Z}}}.\end{align*}$$

Let |$A$| be an f-adic ring, and |$B$| be a finitely generated |${\mathbb {Z}}$| -subalgebra of |$A^{\circ }$|⁠ . Then a |$B$| -submodule |$M$| of |$A$| is said to be quasi-finitely generated over |$B$| if |$M$| is closed in |$A$| and for some pair of definition |$(A_{0}, I)$| such that |$B \subset A_{0}$|⁠ , the image of |$M \to A/(I^{n}A_{0})$| is a finitely generated |$B$| -module for any positive integer |$n$|⁠ .

If |$M$| is quasi-finitely generated over |$B$|⁠ , then for any pair of definition |$(A_{0}, I)$| such that |$B \subset A_{0}$|⁠ , the image of |$M \to A/(I^{n}A_{0})$| is a finitely generated |$B$| -module for any positive integer |$n$|⁠ .

When |$A$| is discrete, a |$B$| -submodule |$M$| of |$A$| is quasi-finitely generated over |$B$| if and only if |$M$| is finitely generated over |$B$|⁠ .

  Theorem 1.35 ([ 1 , Theorem 3.27]). Let |$(A,A^{+})$| be a complete affinoid pair, and let |$J$| be a set such that |${\mathbb {Z}}_{\blacksquare }[S] \cong \prod _{J} {\mathbb {Z}}$|⁠ . Then we have an isomorphism $$ \begin{align*} &(A, A^+)_{\blacksquare}[S] \cong \varinjlim_{B \subset A^+, M} \prod_J \underline{M},\end{align*}$$ where the colimit is taken over all the finitely generated |${\mathbb {Z}}$| -subalgebras |$B \subset A^{+}$| and all the quasi-finitely generated |$B$| -submodules |$M$| of |$A$|⁠ .

  Remark 1.36. We assume that |$A$| is discrete. Then by Remark 1.34 , we get an isomorphism $$ \begin{align*} &(A, A^+)_{\blacksquare}[S] \cong \varinjlim_{B \subset A^+} \prod_J \underline{M},\end{align*}$$ where the colimit is taken over all the finitely generated |${\mathbb {Z}}$| -subalgebras |$B \subset A^{+}$| and all the finitely generated |$B$| -submodules |$M$| of |$A$|⁠ .

Let |$(A,A^{+})$| be a complete affinoid pair, and let |$J$| be a set. Then |$\varinjlim _{B \subset A^{+}, M} \prod _{J} \underline {M}$| is a compact projective object of |$\textrm {Mod}_{(A,A^{+})_{\blacksquare }}^{\textrm {cond}}$|⁠ .

In what follows, we prove a version of fppf descent for discrete rings in condensed mathematics. In this section, all rings are discrete (static) topological rings unless otherwise stated.

Let |${\mathcal {A}}$| be a condensed ring. Then a finite projective |${\mathcal {A}}$| -module is a condensed |${\mathcal {A}}$| -module that is a direct summand of the finite free |${\mathcal {A}}$| -module |${\mathcal {A}}^{n}$| for some |$n$|⁠ .

Let |$A$| be a ring and endow every |$A$| -module with the discrete topology. Then the functor |$\textrm {Mod}_{A} \to \textrm {Mod}_{\underline {A}}^{\textrm {cond}} ;\; M \mapsto \underline {M}$| extends to a fully faithful exact functor |$\textrm {dCond}_{A} \colon {\mathcal {D}}(A) \to {\mathcal {D}}(A_{\blacksquare })$|⁠ , which is called the discrete condensification functor . For an object |$M\in {\mathcal {D}}(A)$|⁠ , |$\textrm {dCond}_{A}(M)$| is given by |$\underline {N^{\bullet }}$|⁠ , where |$N^{\bullet }$| is any complex representing |$M$|⁠ .

From this, we find that for a perfect |$A$| -module |$M$| (i.e., |$M[0]$| is a perfect complex), |$\underline {M}$| is quasi-isomorphic to a bounded complex of finite projective |$\underline {A}$| -modules.

Let |$A$| be a ring and |$M$| be a pseudo-coherent complex of |$A$| -modules. Let |$a,b\in {\mathbb {Z}}$| such that |$a \leq b$|⁠ . If for every maximal ideal |${\mathfrak {m}} \subset A$|⁠ , we have |$H^{i}(M \otimes ^{{\mathbb {L}}}_{A} A/{\mathfrak {m}})=0$| for all |$i\notin [a,b]$|⁠ , then |$M$| is a perfect complex with Tor-amplitude in |$[a,b]$|⁠ .

It easily follows from [ 9 , Corollary 8.3.6.5] or [ 20 , Tag 068V ].

By using the above lemma, we get the following decomposition (cf. [ 20 , Tag 068Y ]).

  Lemma 2.5. Let |$A$| be a finitely generated |${\mathbb {Z}}$| -algebra, and |$B$| be a finitely presented flat |$A$| -algebra. Then there exists a decomposition $$ \begin{align*} &A \to A[T_1,\ldots,T_n] \to B\end{align*}$$ such that |$B$| is a perfect |$A[T_{1},\ldots ,T_{n}]$| -module.

  Proof. We take a surjection |$A[T_{1},\ldots ,T_{n}] \to B$|⁠ , and we will show that |$B$| is a perfect |$A[T_{1},\ldots ,T_{n}]$| -module. We take any maximal ideal |${\mathfrak {m}}$| of |$A[T_{1},\ldots ,T_{n}]$|⁠ . Since |$A$| is a Jacobson ring, the prime ideal |${\mathfrak {n}} = {\mathfrak {m}} \cap A$| of |$A$| is a maximal ideal. Therefore, the global dimension of |$A/{\mathfrak {n}}[T_{1},\ldots ,T_{n}]$| is equal to |$n$|⁠ . We have the following equivalence: $$ \begin{align*} &B \otimes_{A[T_{1},\ldots,T_{n}]}^{{\mathbb{L}}} A[T_{1},\ldots,T_{n}]/{\mathfrak{m}} \\ \simeq{} & (B \otimes_{A[T_{1},\ldots,T_{n}]}^{{\mathbb{L}}} A/{\mathfrak{n}}[T_{1},\ldots,T_{n}]) \otimes_{A/{\mathfrak{n}}[T_{1},\ldots,T_{n}]}^{{\mathbb{L}}} A[T_{1},\ldots,T_{n}]/{\mathfrak{m}} \\ \simeq{} & (B \otimes_{A[ T_{1},\ldots,T_{n}]}^{{\mathbb{L}}} (A[T_{1},\ldots,T_{n}]\otimes_{A}^{{\mathbb{L}}} A/{\mathfrak{n}})) \otimes_{A/{\mathfrak{n}}[T_{1},\ldots,T_{n}]}^{{\mathbb{L}}} A[T_{1},\ldots,T_{n}]/{\mathfrak{m}} \\ \simeq{} & (B \otimes_{A}^{{\mathbb{L}}} A/{\mathfrak{n}}) \otimes_{A/{\mathfrak{n}}[T_{1},\ldots,T_{n}]}^{{\mathbb{L}}} A[T_{1},\ldots,T_{n}]/{\mathfrak{m}} \\ \simeq{} &B/{{\mathfrak{n}} B} \otimes_{A/{\mathfrak{n}}[T_{1},\ldots,T_{n}]}^{{\mathbb{L}}} A[T_{1},\ldots,T_{n}]/{\mathfrak{m}}, \end{align*}$$ where the flatness of |$B$| over |$A$| is used in the fourth equivalence. Therefore, we get |$H^{i}(B \otimes _{A[T_{1},\ldots ,T_{n}]}^{{\mathbb {L}}} A[T_{1},\ldots ,T_{n}]/{\mathfrak {m}})=0$| for all |$i \notin [-n, 0]$|⁠ . Since |$A[T_{1},\ldots ,T_{n}]$| is noetherian, |$B$| is a pseudo-coherent |$A[T_{1},\ldots ,T_{n}]$| -module, so we get the claim by Lemma 2.4 .

Let |$A$| be a ring, and |$B$| be a finitely presented flat |$A$| -algebra. Then |$A_{\blacksquare } \to B_{\blacksquare }$| is steady.

By a limit argument, we can take a finitely generated |${\mathbb {Z}}$| -subalgebra |$A_{0}$| of |$A$| and a finitely presented flat |$A_{0}$| -algebra |$B_{0}$| such that |$A \otimes _{A_{0}} B_{0} \cong B$|⁠ . By Proposition 1.26 , we have an equivalence |$A_{\blacksquare } \otimes _{(A_{0})_{\blacksquare }}^{{\mathbb {L}}} (B_{0})_{\blacksquare } \simeq B_{\blacksquare }$|⁠ . Since the class of steady maps is stable under base change, we may assume that |$A$| is a finitely generated |${\mathbb {Z}}$| -algebra. Since the class of steady maps is stable under composition, we may assume that |$B$| is equal to |$A[T]$| or |$B$| is a perfect |$A$| -module by Lemma 2.5 . If |$B$| is equal to |$A[T]$|⁠ , then |$B_{\blacksquare }$| is equivalent to |$A_{\blacksquare } \otimes _{{\mathbb {Z}}_{\blacksquare }}^{{\mathbb {L}}} {\mathbb {Z}}[T]_{\blacksquare }$|⁠ . Therefore, |$A_{\blacksquare } \to B_{\blacksquare }$| is steady, since |${\mathbb {Z}}_{\blacksquare } \to {\mathbb {Z}}[T]_{\blacksquare }$| is steady by Lemma 1.27 . Next, we show the claim when |$B$| is a perfect |$A$| -module. Since |$\underline {B}$| is quasi-isomorphic to a bounded complex of finite projective |$\underline {A}$| -modules, for an object |$M \in {\mathcal {D}}(A_{\blacksquare })$|⁠ , |$M \otimes _{\underline {A}}^{{\mathbb {L}}} \underline {B}$| is |$A_{\blacksquare }$| -complete and also |$B_{\blacksquare }$| -complete by Proposition 1.22 . This implies that |$M \otimes _{A_{\blacksquare }}^{{\mathbb {L}}} B_{\blacksquare }$| is equivalent to |$M\otimes _{\underline {A}}^{{\mathbb {L}}} \underline {B}$|⁠ . Therefore, for all maps |$A_{\blacksquare }\to {\mathcal {C}}$| in |$\textrm {AnRing}$| and all |$M \in {\mathcal {D}}({\mathcal {C}})$|⁠ , |$M \otimes _{A_{\blacksquare }}^{{\mathbb {L}}} B_{\blacksquare } \simeq M\otimes _{\underline {A}}^{{\mathbb {L}}} \underline {B}$| is |${\mathcal {C}}$| -complete since |$\underline {B}$| is quasi-isomorphic to a bounded complex of finite projective |$\underline {A}$| -modules. From the above, we get that |$A_{\blacksquare } \to B_{\blacksquare }$| is steady.

By the proof of this proposition, we can also show that |$A_{\blacksquare } \to B_{\blacksquare }$| is steady if |$A \to B$| is a perfect ring map, that is, there exists a decomposition as in Lemma 2.5 . More generally, it is proved in [ 14 , Proposition 2.9.7] that for every ring map |$A\to B$| of discrete rings, |$A_{\blacksquare } \to B_{\blacksquare }$| is steady.

|$R\operatorname {\underline {Hom}}_{\underline {{\mathbb {Z}}}}(\bigoplus _{J} \underline {{\mathbb {Z}}}, \underline {{\mathbb {Z}}}) \simeq \prod _{J} \underline {{\mathbb {Z}}}$|⁠ .

|$R\operatorname {\underline {Hom}}_{\underline {{\mathbb {Z}}}}(\prod _{J} \underline {{\mathbb {Z}}}, \underline {{\mathbb {Z}}}) \simeq \bigoplus _{J} \underline {{\mathbb {Z}}}$|⁠ .

|$(\prod _{J} \underline {{\mathbb {Z}}}) \otimes _{{\mathbb {Z}}_{\blacksquare }}^{{\mathbb {L}}} A_{\blacksquare } \simeq \prod _{J} \underline {A}$|⁠ .

The first equivalence follows from Theorem 1.5 . The second equivalence follows from [ 18 , Theorem 4.3] and its proof. For the third equivalence, we can take an extremally disconnected set |$S$| and a set |$I$| containing |$J$| as a subset such that |${\mathbb {Z}}_{\blacksquare }[S] \cong \prod _{I} \underline {{\mathbb {Z}}}$|⁠ . Then the claim follows from the equivalence |${\mathbb {Z}}_{\blacksquare }[S] \otimes _{{\mathbb {Z}}_{\blacksquare }}^{{\mathbb {L}}} A_{\blacksquare } \simeq A_{\blacksquare }[S]$| and |$A_{\blacksquare }[S] \cong \prod _{I} \underline {A}$| by Theorem 1.35 .

The following lemma is key in this paper.

  Lemma 2.9. We have an equivalence of functors from |${\mathcal {D}}({\mathbb {Z}}_{\blacksquare })$| to |${\mathcal {D}}({\mathbb {Z}}[T_{1},\ldots ,T_{n}]_{\blacksquare })$| $$ \begin{align*} &-\otimes_{{\mathbb{Z}}_{\blacksquare}}^{{\mathbb{L}}} {\mathbb{Z}}[T_1,\ldots,T_n]_{\blacksquare} \overset{\sim}{\longrightarrow} R\operatorname{\underline{Hom}}_{\underline{{\mathbb{Z}}}}(R\operatorname{\underline{Hom}}_{\underline{{\mathbb{Z}}}}(\underline{{\mathbb{Z}}[T_1,\ldots,T_n]}, \underline{{\mathbb{Z}}}), -).\end{align*}$$

  Proof. We begin with the outline of the proof. We have a natural map of functors from |${\mathcal {D}}({\mathbb {Z}}_{\blacksquare })$| to |${\mathcal {D}}({\mathbb {Z}}_{\blacksquare })$| $$ \begin{align*} &\textrm{id}=R\operatorname{\underline{Hom}}_{\underline{{\mathbb{Z}}}}(R\operatorname{\underline{Hom}}_{\underline{{\mathbb{Z}}}}(\underline{{\mathbb{Z}}},\underline{{\mathbb{Z}}}), -) \to R\operatorname{\underline{Hom}}_{\underline{{\mathbb{Z}}}}(R\operatorname{\underline{Hom}}_{\underline{{\mathbb{Z}}}}(\underline{{\mathbb{Z}}[T_1,\ldots,T_n]}, \underline{{\mathbb{Z}}}), -),\end{align*}$$ which is induced by the map |$\underline {{\mathbb {Z}}} \to \underline {{\mathbb {Z}}[T_{1},\ldots ,T_{n}]}$|⁠ . First, we will show that for any object |$M \in {\mathcal {D}}({\mathbb {Z}}_{\blacksquare })$|⁠ , |$R\operatorname {\underline {Hom}}_{\underline {{\mathbb {Z}}}}(R\operatorname {\underline {Hom}}_{\underline {{\mathbb {Z}}}}(\underline {{\mathbb {Z}}[T_{1},\ldots ,T_{n}]}, \underline {{\mathbb {Z}}}), M)$| is |${\mathbb {Z}}[T_{1},\ldots ,T_{n}]_{\blacksquare }$| -complete. Then we get a map of functors from |${\mathcal {D}}({\mathbb {Z}}_{\blacksquare })$| to |${\mathcal {D}}({\mathbb {Z}}[T_{1}, \ldots , T_{n}]_{\blacksquare })$| $$ \begin{align*} &-\otimes_{{\mathbb{Z}}_{\blacksquare}}^{{\mathbb{L}}} {\mathbb{Z}}[T_1,\ldots,T_n]_{\blacksquare} \to R\operatorname{\underline{Hom}}_{\underline{{\mathbb{Z}}}}(R\operatorname{\underline{Hom}}_{\underline{{\mathbb{Z}}}}(\underline{{\mathbb{Z}}[T_1,\ldots,T_n]}, \underline{{\mathbb{Z}}}), -),\end{align*}$$ and we will show that this is an equivalence. We take an object |$M \in {\mathcal {D}}({\mathbb {Z}}_{\blacksquare })$|⁠ , and we will show that $$ \begin{align*} &R\operatorname{\underline{Hom}}_{\underline{{\mathbb{Z}}}}(R\operatorname{\underline{Hom}}_{\underline{{\mathbb{Z}}}}(\underline{{\mathbb{Z}}[T_1,\ldots,T_n]}, \underline{{\mathbb{Z}}}), M)\end{align*}$$ is |${\mathbb {Z}}[T_{1},\ldots ,T_{n}]_{\blacksquare }$| -complete. The object |$R\operatorname {\underline {Hom}}_{\underline {{\mathbb {Z}}}}(\underline {{\mathbb {Z}}[T_{1},\ldots ,T_{n}]}, \underline {{\mathbb {Z}}}) \in {\mathcal {D}}({{\mathbb {Z}}_{\blacksquare }})$| is equivalent to |$\prod \underline {{\mathbb {Z}}}$| by Lemma 2.8 , so it is compact as an object of |${\mathcal {D}}({{\mathbb {Z}}_{\blacksquare }})$|⁠ . Therefore, the functor $$ \begin{align} R\operatorname{\underline{Hom}}_{\underline{{\mathbb{Z}}}}(R\operatorname{\underline{Hom}}_{\underline{{\mathbb{Z}}}}(\underline{{\mathbb{Z}}[T_{1},\ldots,T_{n}]}, \underline{{\mathbb{Z}}}), -) \colon {\mathcal{D}}({\mathbb{Z}}_{\blacksquare}) \to {\mathcal{D}}(\underline{{\mathbb{Z}}[T_{1},\ldots,T_{n}]})\end{align}$$ (2.1) commutes with small colimits. Since the objects of the form |${\mathbb {Z}}_{\blacksquare }[S]$| for an extremally disconnected set |$S$| form a family of compact generators of |${\mathcal {D}}({\mathbb {Z}}_{\blacksquare })$| and |${\mathcal {D}}({\mathbb {Z}}[T_{1},\ldots ,T_{n}]_{\blacksquare }) \subset {\mathcal {D}}(\underline {{\mathbb {Z}}[T_{1},\ldots ,T_{n}]})$| is stable under small colimits, we may assume that |$M$| is equal to |${\mathbb {Z}}_{\blacksquare }[S]$| for an extremally disconnected set |$S$|⁠ . By Lemma 1.31 , we have an isomorphism |${\mathbb {Z}}_{\blacksquare }[S] \cong \prod _{J} \underline {{\mathbb {Z}}}$|⁠ . Since the functor ( 2.1 ) commutes with small limits and |${\mathcal {D}}({\mathbb {Z}}[T_{1},\ldots ,T_{n}]_{\blacksquare }) \subset {\mathcal {D}}(\underline {{\mathbb {Z}}[T_{1},\ldots ,T_{n}]})$| is stable under small limits, we may assume that |$M$| is equal to |$\underline {{\mathbb {Z}}}$|⁠ . By Lemma 2.8 (1), (2), the natural map $$ \begin{align*} &\underline{{\mathbb{Z}}[T_1,\ldots,T_n]} \to R\operatorname{\underline{Hom}}_{\underline{{\mathbb{Z}}}}(R\operatorname{\underline{Hom}}_{\underline{{\mathbb{Z}}}}(\underline{{\mathbb{Z}}[T_1,\ldots,T_n]}, \underline{{\mathbb{Z}}}), \underline{{\mathbb{Z}}})\end{align*}$$ is an equivalence. In particular, $$ \begin{align*} &R\operatorname{\underline{Hom}}_{\underline{{\mathbb{Z}}}}(R\operatorname{\underline{Hom}}_{\underline{{\mathbb{Z}}}}(\underline{{\mathbb{Z}}[T_1,\ldots,T_n]}, \underline{{\mathbb{Z}}}), \underline{{\mathbb{Z}}})\end{align*}$$ is |${\mathbb {Z}}[T_{1},\ldots ,T_{n}]_{\blacksquare }$| -complete. Finally, we prove that the map of functors from |${\mathcal {D}}({\mathbb {Z}}_{\blacksquare })$| to |${\mathcal {D}}({\mathbb {Z}}[T_{1}, \ldots , T_{n}]_{\blacksquare })$| $$ \begin{align*} &-\otimes_{{\mathbb{Z}}_{\blacksquare}}^{{\mathbb{L}}} {\mathbb{Z}}[T_1,\ldots,T_n]_{\blacksquare} \to R\operatorname{\underline{Hom}}_{\underline{{\mathbb{Z}}}}(R\operatorname{\underline{Hom}}_{\underline{{\mathbb{Z}}}}(\underline{{\mathbb{Z}}[T_1,\ldots,T_n]}, \underline{{\mathbb{Z}}}), -)\end{align*}$$ is an equivalence. Since both functors commute with small colimits, it is enough to show that $$ \begin{align*} &\prod_{J} \underline{{\mathbb{Z}}} \otimes_{{\mathbb{Z}}_{\blacksquare}}^{{\mathbb{L}}} {\mathbb{Z}}[T_1,\ldots,T_n]_{\blacksquare} \to R\operatorname{\underline{Hom}}_{\underline{{\mathbb{Z}}}}(R\operatorname{\underline{Hom}}_{\underline{{\mathbb{Z}}}}(\underline{{\mathbb{Z}}[T_1,\ldots,T_n]}, \underline{{\mathbb{Z}}}), \prod_{J} \underline{{\mathbb{Z}}})\end{align*}$$ is an equivalence for any set |$J$|⁠ . We may also assume that |$J$| is a singleton by Lemma 2.8 (3), and in this case the claim is already shown.

  Theorem 2.10. Let |$A$| be a finitely generated |${\mathbb {Z}}$| -algebra and |$B$| be a finitely presented flat |$A$| -algebra. Then we have an equivalence of functors from |${\mathcal {D}}(A_{\blacksquare })$| to |${\mathcal {D}}(B_{\blacksquare })$| $$ \begin{align*} &-\otimes_{A_{\blacksquare}}^{{\mathbb{L}}} B_{\blacksquare} \overset{\sim}{\longrightarrow} R\operatorname{\underline{Hom}}_{\underline{A}}(R\operatorname{\underline{Hom}}_{\underline{A}}(\underline{B}, \underline{A}), -).\end{align*}$$

|$R\operatorname {\underline {Hom}}_{\underline {A}}(\underline {B}, \underline {A})$| is compact as an object of |${\mathcal {D}}(A_{\blacksquare })$|⁠ .

The map |$\underline {B} \to R\operatorname {\underline {Hom}}_{\underline {A}}(R\operatorname {\underline {Hom}}_{\underline {A}}(\underline {B}, \underline {A}), \underline {A})$| is an equivalence.

The functor |$- \otimes _{A_{\blacksquare }}^{{\mathbb {L}}} B_{\blacksquare }$| commutes with small limits.

Next, we check the conditions (1), (2), and (3) when |$B$| is a perfect |$A$| -module. The complex |$\underline {B}$| is quasi-isomorphic to a bounded complex of finite projective |$\underline {A}$| -modules, and the functor |$- \otimes _{A_{\blacksquare }}^{{\mathbb {L}}} B_{\blacksquare }$| is equivalent to |$- \otimes _{\underline {A}}^{{\mathbb {L}}} \underline {B}$| by the proof of Proposition 2.6 , so the conditions are independent of the ring structure of |$B$| (i.e., they depend only on the |$A$| -module structure on |$B$|⁠ ). Therefore, we may assume that |$B$| is a finitely generated projective |$A$| -module, and then the claim is clear.

• We have an equivalence of functors from |${\mathcal {D}}(A_{\blacksquare })$| to |${\mathcal {D}}(B_{\blacksquare })$| $$ \begin{align} -\otimes_{A_{\blacksquare}}^{{\mathbb{L}}} B_{\blacksquare} \overset{\sim}{\longrightarrow} R\operatorname{\underline{Hom}}_{\underline{A}}(N_{B/A}, -).\end{align}$$ (2.2)

  Proof. By a limit argument, we can take a finitely generated |${\mathbb {Z}}$| -subalgebra |$A_{0}$| of |$A$| and a finitely presented flat |$A_{0}$| -algebra |$B_{0}$| such that |$A \otimes _{A_{0}} B_{0} \cong B$|⁠ . We define |$N_{B/A}=R\operatorname {\underline {Hom}}_{\underline {A_{0}}}(\underline {B_{0}}, \underline {A_{0}})\otimes _{(A_{0})_{\blacksquare }}^{{\mathbb {L}}} A_{\blacksquare }$|⁠ , which is an object of |${\mathcal {D}}(\underline {B})$|⁠ . By Theorem 2.10 , we have the following equivalence for every object |$M\in {\mathcal {D}}(A_{\blacksquare })$|⁠ : $$ \begin{align*} &R\operatorname{\underline{Hom}}_{\underline{A}}(N_{B/A}, M) \simeq R\operatorname{\underline{Hom}}_{\underline{A_0}}(R\operatorname{\underline{Hom}}_{\underline{A_0}}(\underline{B_0}, \underline{A_0}), M) \simeq M\otimes_{(A_0)_{\blacksquare}}^{{\mathbb{L}}} (B_0)_{\blacksquare}.\end{align*}$$ Since |$R\operatorname {\underline {Hom}}_{\underline {A}}(N_{B/A}, M)$| is |$A_{\blacksquare }$| -complete and |$M\otimes _{(A_{0})_{\blacksquare }}^{{\mathbb {L}}} (B_{0})_{\blacksquare }$| is |$(B_{0})_{\blacksquare }$| -complete, we find that |$R\operatorname {\underline {Hom}}_{\underline {A}}(N_{B/A}, M)$| is also |$B_{\blacksquare }$| -complete, where we use Proposition 1.14 and Proposition 1.26 . We have a natural map |$N_{B/A} \to \underline {A}$| induced by the natural map |$R\operatorname {\underline {Hom}}_{\underline {A_{0}}}(\underline {B_{0}}, \underline {A_{0}})\to \underline {A_{0}}$|⁠ . Therefore, we get a natural map of functors from |${\mathcal {D}}(A_{\blacksquare })$| to |${\mathcal {D}}(B_{\blacksquare })$| $$ \begin{align*} &-\otimes_{A_{\blacksquare}} B_{\blacksquare} \to R\operatorname{\underline{Hom}}_{\underline{A}}(N_{B/A}, -)\end{align*}$$ induced by the natural map |$N_{B/A} \to \underline {A}$|⁠ , and it is an equivalence by Proposition 2.6 and Proposition 1.18 .

  Remark 2.12. The equivalence ( 2.2 ) is functorial in the following sense: let |$\textrm {Alg}_{A}^{\textrm {fpf}}$| be the category of finitely presented flat |$A$| -algebras endowed with the co-Cartesian monoidal structure (i.e., the monoidal structure given by tensor products over |$A$|⁠ ), and let |$\operatorname {End}_{{\mathcal {D}}(A_{\blacksquare })}({\mathcal {D}}(A_{\blacksquare }))$| be the |$\infty $| -category of |${\mathcal {D}}(A_{\blacksquare })$| -enriched endofunctors (for the definition, see [ 14 , Definition A.4.4]). Then there exists a monoidal functor $$ \begin{align*} &N \colon \textrm{Alg}_{A}^{\textrm{fpf}} \to {\mathcal{D}}(A_{\blacksquare})^{\textrm{op}};\; B \mapsto N_{B/A},\end{align*}$$ and an equivalence of functors from |$\textrm {Alg}_{A}^{\textrm {fpf}} \to \operatorname {End}_{{\mathcal {D}}(A_{\blacksquare })}({\mathcal {D}}(A_{\blacksquare }))$| $$ \begin{align*} &B \mapsto -\otimes_{A_{\blacksquare}}^{{\mathbb{L}}} B_{\blacksquare},\end{align*}$$ and $$ \begin{align*} &B \mapsto R\operatorname{\underline{Hom}}_{\underline{A}}(N_{B/A}, -).\end{align*}$$ To see this, first observe that we have monoidal functors $$ \begin{align*} &F\colon\textrm{Alg}_{A}^{\textrm{fpf}} \to \operatorname{End}_{{\mathcal{D}}(A_{\blacksquare})}({\mathcal{D}}(A_{\blacksquare})) ;\; B \mapsto -\otimes_{A_{\blacksquare}}^{{\mathbb{L}}} B_{\blacksquare}\end{align*}$$ and $$ \begin{align*} &G\colon{\mathcal{D}}(A_{\blacksquare})^{\textrm{op}} \to \operatorname{End}_{{\mathcal{D}}(A_{\blacksquare})}({\mathcal{D}}(A_{\blacksquare}));\; M\mapsto R\operatorname{\underline{Hom}}_{\underline{A}}(M, -),\end{align*}$$ where the monoidal structure on |$\operatorname {End}_{{\mathcal {D}}(A_{\blacksquare })}({\mathcal {D}}(A_{\blacksquare }))$| is defined by compositions and the monoidal structure on |$F$| is defined by Proposition 1.18 and Proposition 2.6 . By [ 14 , Corollary A.4.9] |$G$| is fully faithful, and the image of |$F$| is included in the essential image of |$G$| by Corollary 2.11 , so we get the desired functoriality result.

From the corollary above, we find that it is enough to examine properties of |$N_{B/A}$| to prove faithfully flat descent. To examine them, we will use a property of maps of rings introduced by Mathew in [ 15 ].

Let |${\mathcal {C}}$| be a presentable symmetric monoidal stable |$\infty $| -category where the |$\otimes $| -product commutes with small colimits in each variable, which is called a stable homotopy theory in [ 15 ]. A map |$A \to B$| of |${\mathbb {E}}_{\infty }$| -algebras in |${\mathcal {C}}$| is descendable if the pro-object |$\{\textrm {Tot}_{n}(B^{\bullet +1/A})\}_{n}$| of |${\mathcal {D}}(A)$| is a constant pro-object which converges to |$A$|⁠ , where |$B^{\bullet +1/A}$| is the Čech nerve of |$A \to B$| and |$\textrm {Tot}_{n}(B^{\bullet +1/A})$| is the limit of the |$n$| -truncation of |$B^{\bullet +1/A}$|⁠ .

A finitely presented faithfully flat map |$A \to B$| of rings is descendable.

Let |$A \to B$| be a finitely presented faithfully flat map of rings. Let |$N_{B/A} \in {\mathcal {D}}(\underline {B})$| be as in Corollary 2.11 , and let |$N_{B/A} \to \underline {A}$| be the natural map. Then the induced augmented semisimplicial diagram |$\{N_{B/A}^{\otimes (m+1)}\}_{m\geq -1}$| is a colimit diagram (i.e., |$\mathop {L\varinjlim }_{[m] \in \Delta _{s}^{\textrm {op}}}N_{B/A}^{\otimes (m+1)} \overset {\sim }{\to } A$|⁠ ), where |$N_{B/A}^{\otimes (m+1)}$| is the |$(m+1)$| -fold derived tensor product of |$N_{B/A}$| over |$A_{\blacksquare }$|⁠ .

  Proof. We may assume that |$A$| is a finitely generated |${\mathbb {Z}}$| -algebra by a limit argument. Since |$A_{\blacksquare } \to B_{\blacksquare }$| is steady, we have the following equivalence of functors from |${\mathcal {D}}(A_{\blacksquare })$| to |${\mathcal {D}}(A_{\blacksquare })$|⁠ : $$ \begin{align*} &- \otimes_{A_{\blacksquare}}^{{\mathbb{L}}} (B\otimes_{A} B)_{\blacksquare} \\ \simeq{} &(-\otimes_{A_{\blacksquare}}^{{\mathbb{L}}} B_{\blacksquare}) \otimes_{A_{\blacksquare}}^{{\mathbb{L}}} B_{\blacksquare} \\ \simeq{} &R\operatorname{\underline{Hom}}_{\underline{A}}(N_{B/A}, R\operatorname{\underline{Hom}}_{\underline{A}}(N_{B/A}, -))\\ \simeq{} &R\operatorname{\underline{Hom}}_{\underline{A}}(N_{B/A}\otimes_{A_{\blacksquare}}^{{\mathbb{L}}} N_{B/A},-). \end{align*}$$ By repeating the same argument, we get the following equivalence of functors from |${\mathcal {D}}(A_{\blacksquare })$| to |${\mathcal {D}}(A_{\blacksquare })$|⁠ : $$ \begin{align*} &- \otimes_{A_{\blacksquare}}^{{\mathbb{L}}} (B^{m/A})_{\blacksquare} \simeq R\operatorname{\underline{Hom}}_{\underline{A}}(N_{B/A}^{\otimes m},-),\end{align*}$$ where |$B^{m/A}$| is the |$m$| -fold tensor product of |$B$| over |$A$|⁠ . Therefore, the induced augmented semisimplicial diagram |$\{N_{B/A}^{\otimes (m+1)}\}_{m\geq -1}$| is equivalent to the underlying augmented semisimplicial diagram of the augmented simplicial diagram $$ \begin{align*} &\{R\operatorname{\underline{Hom}}_{\underline{A}}(\underline{B^{(m+1)/A}}, \underline{A})\}_{m \geq -1}.\end{align*}$$ Since the inclusion functor |$N(\Delta _{s}^{\textrm {op}}) \subset N(\Delta ^{\textrm {op}})$| is cofinal by [ 10 , Lemma 6.5.3.7], the theorem follows from Proposition 2.14 .

  Corollary 2.16. Let |$A\to B$| be a finitely presented faithfully flat map of rings. Then the functor $$ \begin{align*} &- \otimes_{A_{\blacksquare}}^{{\mathbb{L}}} B_{\blacksquare} \colon {\mathcal{D}}(A_{\blacksquare}) \to {\mathcal{D}}(B_{\blacksquare})\end{align*}$$ is conservative.

  Proof. We take an object |$M \in {\mathcal {D}}(A_{\blacksquare })$| such that |$M \otimes _{A_{\blacksquare }}^{{\mathbb {L}}} B_{\blacksquare } \simeq 0$|⁠ . We want to show that |$M$| is equivalent to |$0$|⁠ . By Theorem 2.15 , we get the following equivalence: $$ \begin{align*} M \simeq{} & R\operatorname{\underline{Hom}}_{\underline{A}}(\underline{A},M)\\ \simeq{} &R\operatorname{\underline{Hom}}_{\underline{A}}(\mathop{L\varinjlim}_{[m] \in \Delta_{s}^{\textrm{op}}} N_{B/A}^{\otimes (m+1)}, M)\\ \simeq{} &\mathop{R\varprojlim}_{[m] \in \Delta_{s}}R\operatorname{\underline{Hom}}_{\underline{A}}(N_{B/A}^{\otimes (m+1)}, M). \end{align*}$$ On the other hand, we have $$ \begin{align*} &R\operatorname{\underline{Hom}}_{\underline{A}}(N_{B/A}^{\otimes (m+1)}, M)\\ \simeq{} &R\operatorname{\underline{Hom}}_{\underline{A}}(N_{B/A}^{\otimes m}, R\operatorname{\underline{Hom}}_{\underline{A}}(N_{B/A},M))\\ \simeq{} &R\operatorname{\underline{Hom}}_{\underline{A}}(N_{B/A}^{\otimes m}, M \otimes_{A_{\blacksquare}}^{{\mathbb{L}}} B_{\blacksquare})\\ \simeq{} &0 \end{align*}$$ for every |$m \geq 0$|⁠ . Therefore, we find |$M \simeq 0$|⁠ .

Faithfully flat descent follows from the following general result.

The map |${\mathcal {A}} \to {\mathcal {B}}$| is steady.

The functor |$- \otimes _{{\mathcal {A}}}^{{\mathbb {L}}} {\mathcal {B}} \colon {\mathcal {D}}({\mathcal {A}}) \to {\mathcal {D}}({\mathcal {B}})$| commutes with small limits.

The functor |$- \otimes _{{\mathcal {A}}}^{{\mathbb {L}}} {\mathcal {B}} \colon {\mathcal {D}}({\mathcal {A}}) \to {\mathcal {D}}({\mathcal {B}})$| is conservative.

  Proof. We will apply [ 12 , Proposition 5.2.2.36] to the functor $$ \begin{align*} &\chi \colon\Delta^{\triangleleft} \to {\mathcal{C}}at_{\infty};\; [n] \mapsto {\mathcal{D}}({\mathcal{B}}^{(n+1)/{\mathcal{A}}}),\end{align*}$$ where we denote the cone point of |$\Delta ^{\triangleleft }$| by |$[-1]$|⁠ , and we will check the conditions (a) and (b) of loc. cit. The condition (a) follows from (3). Let us check the condition (b). We note that the functor $$ \begin{align*} &\chi \colon\Delta^{\triangleleft} \to {\mathcal{C}}at_{\infty};\; [n] \mapsto {\mathcal{D}}({\mathcal{B}}^{(n+1)/{\mathcal{A}}})\end{align*}$$ classifies a bi-Cartesian fibration |$q\colon {\mathcal {C}} \to \Delta ^{\triangleleft }$| since for every |$\alpha \colon [m] \to [n]$|⁠ , $$ \begin{align*} &\chi(\alpha) \colon {\mathcal{D}}({\mathcal{B}}^{(m+1)/{\mathcal{A}}}) \to {\mathcal{D}}({\mathcal{B}}^{(n+1)/{\mathcal{A}}})\end{align*}$$ has a right adjoint functor ([ 12 , Proposition 4.7.4.17]). We take |$M \in \operatorname {Fun}_{\Delta ^{\triangleleft }}(\Delta , {\mathcal {C}})$|⁠ , which carries each morphism in |$\Delta $| to a |$q$| -co-Cartesian morphism in |${\mathcal {C}}$|⁠ . We put $$ \begin{align*} &M^n =M([n]) \in q^{-1}([n]) \simeq {\mathcal{D}}({\mathcal{B}}^{(n+1)/{\mathcal{A}}}).\end{align*}$$ By [ 10 , Corollary 4.3.1.11] and its proof, we can extend |$M$| to a |$q$| -limit diagram |$\overline {M} \in \operatorname {Fun}_{\Delta ^{\triangleleft }}(\Delta ^{\triangleleft }, {\mathcal {C}})$|⁠ , and |$M([-1]) \in q^{-1}([-1]) \simeq {\mathcal {D}}({\mathcal {A}})$| is equivalent to |$\mathop {R\varprojlim }_{[n] \in \Delta } M^{n}$|⁠ . We will show that |$\overline {M}$| carries each map |$[-1] \to [m]$| to a |$q$| -co-Cartesian morphism in |${\mathcal {C}}$|⁠ . It is equivalent to show $$ \begin{align*} &(\mathop{R\varprojlim}_{[n] \in \Delta} M^n)\otimes_{{\mathcal{A}}}^{{\mathbb{L}}}{\mathcal{B}}^{(m+1)/{\mathcal{A}}}\simeq M^m.\end{align*}$$ We define a functor |$F \colon \Delta ^{\triangleleft } \to \Delta ;\; [n] \mapsto [n+1]$| by $$ \begin{align*} & F(\alpha)(i)= \begin{cases} \alpha(i) & (i\in [n]), \\ m+1 & (i=n+1) \end{cases} \end{align*}$$ for |$\alpha \colon [n] \to [m]$| in |$\Delta ^{\triangleleft }$|⁠ . Let |$\overline {M}^{\prime }\in \operatorname {Fun}(\Delta ^{\triangleleft }, {\mathcal {C}})$| be the augmented simplicial object given by composing |$M$| with the functor |$F$|⁠ , and |$M^{\prime }$| be the underling simplicial object of |$\overline {M}^{\prime }$|⁠ . We regard |$\overline {M}$|⁠ , |$\overline {M}^{\prime }$| as augmented simplicial objects of |${\mathcal {D}}({\mathcal {A}})$|⁠ , |${\mathcal {D}}({\mathcal {B}})$|⁠ . Then by (1), we have an equivalence |$M \otimes _{{\mathcal {A}}}^{{\mathbb {L}}}{\mathcal {B}} \simeq M^{\prime }$|⁠ . By [ 10 , Lemma 6.1.3.16]), |$\overline {M}^{\prime }$| is a limit diagram, that is, |$\mathop {R\varprojlim }_{[n] \in \Delta } M^{(n+1)} \simeq M^{0}$|⁠ . Therefore, we get |$(\mathop {R\varprojlim }_{[n] \in \Delta } M^{n})\otimes _{{\mathcal {A}}}^{{\mathbb {L}}}{\mathcal {B}}\simeq M^{0}$| from (2). We let |$f_{m} \colon {\mathcal {B}} \to {\mathcal {B}}^{(m+1)/{\mathcal {A}}}$| denote the morphism of analytic rings induced from the map |$[0] \to [m] ;\; 0 \mapsto 0$|⁠ . Then we get the following equivalence: $$ \begin{align*} &(\mathop{R\varprojlim}_{[n] \in \Delta} M^{n})\otimes_{{\mathcal{A}}}^{{\mathbb{L}}}{\mathcal{B}}^{(m+1)/{\mathcal{A}}}\\ \simeq &((\mathop{R\varprojlim}_{[n] \in \Delta} M^{n})\otimes_{{\mathcal{A}}}^{{\mathbb{L}}}{\mathcal{B}}) \otimes_{{\mathcal{B}}, f_{m}}^{{\mathbb{L}}} {\mathcal{B}}^{(m+1)/{\mathcal{A}}}\\ \simeq &M^{0} \otimes_{{\mathcal{B}}, f_{m}}^{{\mathbb{L}}} {\mathcal{B}}^{(m+1)/{\mathcal{A}}}\\ \simeq & M^{m}. \end{align*}$$

Since the codiagonal map |${\mathcal {B}} \otimes _{{\mathcal {A}}}^{{\mathbb {L}}} {\mathcal {B}} \to {\mathcal {B}}$| in |$\textrm {AnRing}$| is steady for a steady map |${\mathcal {A}} \to {\mathcal {B}}$| (cf. [ 14 , Proposition 2.3.19 (2)]), the condition (1) of Lemma 2.17 implies that for every map |$[m] \to [n]$| in |$\Delta $|⁠ , the map |${\mathcal {B}}^{(m+1)/{\mathcal {A}}} \to {\mathcal {B}}^{(n+1)/{\mathcal {A}}}$| is steady. Therefore, Lemma 2.17 follows also from the Barr–Beck–Lurie monadicity theorem [ 12 , 4.7.5], and we can weaken the condition (2) of Lemma 2.17 as stated in loc. cit.

  Theorem 2.19. Let |$A \to B$| be a finitely presented faithfully flat map of rings. Let |$B^{n/A}$| denote the |$n$| -fold tensor product of |$B$| over |$A$|⁠ . Then we have an equivalence of |$\infty $| -categories $$ \begin{align*} &{\mathcal{D}}(A_{\blacksquare}) \overset{\sim}{\longrightarrow} \varprojlim_{[n] \in \Delta} {\mathcal{D}}((B^{(n+1)/A})_{\blacksquare}). \end{align*}$$

It follows from Proposition 1.26 , Proposition 2.6 , Corollary 2.11 , Corollary 2.16 , and Lemma 2.17 .

Finally, we prove the uniform boundedness of |$N_{B/A}$|⁠ , which is essential for the proof of the main theorem.

Let |$A \to B$| be a finitely presented flat map of rings. We take |$N_{B/A} \in {\mathcal {D}}(\underline {B})$| as in Corollary 2.11 . Then for every |$n \geq 1$|⁠ , |$N_{B/A}^{\otimes n}$| is quasi-isomorphic to a complex of the form |$0 \to M^{0} \to M^{1} \to 0$| where |$M^{0}$| is placed in cohomological degree |$0$| and |$M^{0}, M^{1}$| are projective objects of |$\textrm {Mod}_{A_{\blacksquare }}^{\textrm {cond}}$|⁠ .

By the proof of Theorem 2.15 , it is enough to show the claim in the case where |$n=1$|⁠ . We take a finitely generated |${\mathbb {Z}}$| -subalgebra |$A_{0}$| of |$A$| and a finitely presented flat |$A_{0}$| -algebra |$B_{0}$| such that |$A \otimes _{A_{0}} B_{0} \cong B$|⁠ . Then |$N_{B/A}$| is quasi-isomorphic to |$R\operatorname {\underline {Hom}}_{\underline {A_{0}}}(\underline {B_{0}}, \underline {A_{0}})\otimes _{(A_{0})_{\blacksquare }}^{{\mathbb {L}}} A_{\blacksquare }$|⁠ , so it is enough to show that |$R\operatorname {\underline {Hom}}_{\underline {A_{0}}}(\underline {B_{0}}, \underline {A_{0}})$| is quasi-isomorphic to a complex of the form |$0 \to \prod _{I} \underline {A_{0}} \to \prod _{J} \underline {A_{0}} \to 0$|⁠ , where we note that |$\prod _{I} \underline {A_{0}}, \prod _{J} \underline {A_{0}}$| are projective objects of |$\textrm {Mod}_{(A_{0})_{\blacksquare }}^{\textrm {cond}}$|⁠ . By [ 17 , Seconde partie: Corollary 3.3.2], |$B_{0}$| has a free resolution of length |$1$| over |$A_{0}$|⁠ , and the claim follows from taking |$A_{0}$| -duals of the resolution of |$B_{0}$|⁠ .

In this section, we will introduce some notions that are necessary for proving the main theorem.

3.1 Adic completeness

First, we will introduce the notion of adic completeness in the context of analytic rings. It is an analogue of derived adic completeness of ordinary rings (cf. [ 20 , Tag 091N ]). For our need, it is sufficient to consider analytic rings associated to complete |$\pi $| -adic rings where |$\pi $| is a non-zero-divisor. For more general theory, see [ 14 , §2.12].

Let |$A$| be a complete |$\pi $| -adic ring where |$\pi \in A$| is a non-zero-divisor.

An object |$M \in {\mathcal {D}}(A_{\blacksquare })$| is |$\pi $| -adically complete if |$R\varprojlim (\cdots \overset {\pi }{\to }M \overset {\pi }{\to }M\overset {\pi }{\to }M)$| is equivalent to |$0$|⁠ . We denote the full |$\infty $| -subcategory of |$\pi $| -adically complete objects of |${\mathcal {D}}(A_{\blacksquare })$| by |$\widehat {{\mathcal {D}}}_{\pi }(A_{\blacksquare })$|⁠ .

The object |$M$| is |$\pi $| -adically complete.

For every integer |$i \in {\mathbb {Z}}$|⁠ , |$H^{i}(M)$| is |$\pi $| -adically complete (as an object of |${\mathcal {D}}(A_{\blacksquare })$|⁠ ).

The natural map |$M \to R\varprojlim _{n} (M\otimes _{A_{\blacksquare }}^{{\mathbb {L}}} \underline {A/\pi ^{n}})$| is an equivalence.

Let |$M$| be a |$\pi $| -adically complete object of |${\mathcal {D}}(A_{\blacksquare })$|⁠ . If |$M \otimes _{A_{\blacksquare }}^{{\mathbb {L}}} \underline {A/\pi }$| belongs to |${\mathcal {D}}^{\leq -2}(A_{\blacksquare })$|⁠ , then we have |$M \in {\mathcal {D}}^{\leq -1}(A_{\blacksquare })$|⁠ .

We can check |$M \otimes _{A}^{{\mathbb {L}}} A/\pi ^{n}\in {\mathcal {D}}^{\leq -2}(A)$| for |$n \geq 1$| by induction, so the lemma follows from the fact that the cohomological dimension of |$R\varprojlim _{n \in {\mathbb {N}}}$| in |$\textrm {Mod}_{A_{\blacksquare }}^{\textrm {cond}}$| is 1.

3.2 Small complete adic rings

Next, we define small complete adic rings.

Let |$A$| be a complete adic ring. The topological ring |$A$| is said to be small if there exists an ideal of definition |$I \subset A$|⁠ , a finitely generated |${\mathbb {Z}}$| -algebra |$R$| and an integral ring map |$R \to A/I$|⁠ .

The ring |$A$| is small.

There exists a finitely generated |${\mathbb {Z}}$| -algebra |$R$| and a ring map |$R \to A$| such that, for every ideal of definition |$I\subset A$|⁠ , |$R\to A/I$| is integral.

It follows from a straightforward argument.

The ring of integers of a non-archimedean local field is small. Moreover, |${\mathcal {O}}_{{\mathbb {C}}_{p}}$| is small.

Let |$A$| be a complete adic ring and |$I \subset A$| be an ideal of definition, and |$p \colon A \to A/I$| be the projection map. Let |$\overline {R} \subset A/I$| be a finitely generated |${\mathbb {Z}}$| -subalgebra. Then |$R=p^{-1}(\overline {R})$| is a small complete |$IR$| -adic ring.

The following proposition is the reason why we introduce small complete adic rings.

Let |$A$| be a small complete |$\pi $| -adic ring where |$\pi \in A$| is a non-zero-divisor. Then for every extremally disconnected set |$S$|⁠ , |$A_{\blacksquare }[S]$| is |$\pi $| -adically complete. In particular, every compact object of |${\mathcal {D}}(A_{\blacksquare })$| is |$\pi $| -adically complete.

The latter part follows from [ 14 , Lemma A.2.1] and the fact that |$\widehat {{\mathcal {D}}}_{\pi }(A_{\blacksquare })$| is a full |$\infty $| -subcategory of |${\mathcal {D}}(A_{\blacksquare })$| stable under finite colimits and retracts.

  Lemma 3.9. Let |$A$| be a complete |$I$| -adic ring, where |$I \subset A$| is a finitely generated ideal. We write |$A/I$| as a filtered colimit |$\varinjlim _{\lambda \in \Lambda } \overline {A_{\lambda }}$| where each |$\overline {A_{\lambda }}$| is a finitely generated |${\mathbb {Z}}$| -subalgebra of |$A/I$|⁠ . Let |$A_{\lambda }$| be the preimage of |$\overline {A_{\lambda }}$| under |$A\to A/I$|⁠ , which is a small complete |$I$| -adic ring. Then in |$\textrm {AnRing}$|⁠ , we have the following equivalence: $$ \begin{align*} &A_{\blacksquare} \simeq \varinjlim_{\lambda \in \Lambda} (A_{\lambda})_{\blacksquare}.\end{align*}$$

It follows from a straightforward computation by using Theorem 1.35 .

This lemma is useful because of the following. For an |$\infty $| -category |${\mathcal {C}}$|⁠ , we denote the full |$\infty $| -subcategory of compact objects of |${\mathcal {C}}$| by |${\mathcal {C}}_{\textrm {cpt}}$|⁠ .

  Lemma 3.10 ([ 14 , Lemma 2.7.4]). Let |${\mathcal {A}} = \varinjlim _{\lambda \in \Lambda } {\mathcal {A}}_{\lambda }$| be a filtered colimit in |$\textrm {AnRing}$|⁠ . Then the following map induced by the scalar extension functors is an equivalence: $$ \begin{align*} & \varinjlim_{\lambda \in \Lambda} {\mathcal{D}}({\mathcal{A}}_{\lambda})_{\textrm{cpt}} \to {\mathcal{D}}({\mathcal{A}})_{\textrm{cpt}}.\end{align*}$$

In this section, we will prove faithfully flat descent for affinoid pairs over a complete non-archimedean field.

Let |$K$| be a complete non-archimedean field, and let |${\mathcal {O}}_{K}$| denote the ring of integers of |$K$|⁠ , and |$\pi $| denote a pseudo-uniformizer of |$K$|⁠ . We begin with a review of classical results.

A topologically finitely presented |${\mathcal {O}}_{K}$| -algebra is a topological |${\mathcal {O}}_{K}$| -algebra that is a topological quotient of |${\mathcal {O}}_{K}\langle T_{1}, \ldots , T_{n} \rangle $| for some |$n$| by a finitely generated ideal, where |${\mathcal {O}}_{K}\langle T_{1}, \ldots , T_{n} \rangle $| is equipped with the |$\pi $| -adic topology. An admissible |${\mathcal {O}}_{K}$| -algebra is a |$\pi $| -torsion-free topologically finitely presented |${\mathcal {O}}_{K}$| -algebra.

An affinoid |$K$| -algebra is a topological |$K$| -algebra that is a quotient topological ring of |$K\langle T_{1}, \ldots , T_{n} \rangle $| for some |$n$| by an ideal.

For a topologically finitely presented |${\mathcal {O}}_{K}$| -algebra |$A$|⁠ , the topology of |$A$| coincides with the |$\pi $| -adic topology of |$A$|⁠ .

For a topologically finitely presented |${\mathcal {O}}_{K}$| -algebra |$A$|⁠ , |$A[1/\pi ]$| naturally becomes an affinoid |$K$| -algebra.

Affinoid |$K$| -algebras are noetherian.

Topologically finitely presented |${\mathcal {O}}_{K}$| -algebras are not necessarily noetherian but coherent (see [ 4 , Proposition 1.3]).

A map |$A\to B$| of topologically finitely presented |${\mathcal {O}}_{K}$| -algebras (resp. affinoid |$K$| -algebras) is flat (resp. faithfully flat) if it is flat (resp. faithfully flat) as a map of ordinary rings.

Let |$A\to B$| be a faithfully flat map of affinoid |$K$| -algebras. Then there exists a rational covering |$\{\operatorname {Spa}(A_{i}, (A_{i})^{\circ }) \to \operatorname {Spa}(A,A^{\circ }) \}$|⁠ , which satisfies that for each |$i$| there exists a faithfully flat map |$A_{i}^{\prime } \to B_{i}^{\prime }$| of admissible |${\mathcal {O}}_{K}$| -algebras such that |$A_{i}^{\prime }[1/\pi ] \to B_{i}^{\prime }[1/\pi ]$| is isomorphic to |$A_{i} \to B \mathbin {\widehat {\otimes }}_{A} A_{i}$|⁠ , where |$ \mathbin {\widehat {\otimes }}$| is the completed tensor product.

  Lemma 4.5. Let |$A$| be a topologically finitely presented |${\mathcal {O}}_{K}$| -algebra. Then we have the following equivalence in |$\textrm {AnRing}$|⁠ : $$ \begin{align*} &(A[1/\pi], A[1/\pi]^{\circ})_{\blacksquare} \simeq A_{\blacksquare} \otimes_{{{\mathcal{O}}_K}_{\blacksquare}}^{{\mathbb{L}}} (K, {\mathcal{O}}_K)_{\blacksquare}.\end{align*}$$

  Proof. We check the conditions (1) and (2) of Lemma 1.15 . As |$\underline {A[1/\pi ]}$| is |$(K, {\mathcal {O}}_{K})_{\blacksquare }$| -complete, we get the natural map |$\underline {A} \otimes _{{{\mathcal {O}}_{K}}_{\blacksquare }}^{{\mathbb {L}}} (K, {\mathcal {O}}_{K})_{\blacksquare } \to \underline {A[1/\pi ]}$|⁠ , and we want to prove that it is an equivalence. Since |$\underline {A} \otimes _{\underline {{\mathcal {O}}_{K}}}^{{\mathbb {L}}} \underline {K}$| is equivalent to the colimit of the system in |${\mathcal {D}}(\underline {{\mathcal {O}}_{K}})$| $$ \begin{align*} &\underline{A} \overset{\pi}{\to}\underline{A} \overset{\pi}{\to}\underline{A} \to \cdots,\end{align*}$$ |$\underline {A} \otimes _{\underline {{\mathcal {O}}_{K}}}^{{\mathbb {L}}} \underline {K}$| is equivalent to |$\underline {A[1/\pi ]}$|⁠ , and it is |$(K, {\mathcal {O}}_{K})_{\blacksquare }$| -complete. Therefore, we have |$\underline {A[1/\pi ]} \simeq \underline {A} \otimes _{\underline {{\mathcal {O}}_{K}}}^{{\mathbb {L}}} \underline {K} \simeq \underline {A} \otimes _{{{\mathcal {O}}_{K}}_{\blacksquare }}^{{\mathbb {L}}} (K, {\mathcal {O}}_{K})_{\blacksquare }$|⁠ , which proves (1). Since |$A[1/\pi ]^{\circ }$| is the integral closure of |$A$| in |$A[1/\pi ]$|⁠ , (2) follows from Proposition 1.23 .

Let |$A$| be a topologically finitely presented |${\mathcal {O}}_{K}$| -algebra, and |$M$| be a perfect |$A$| -module with the |$\pi $| -adic topology. Then |$\underline {M}$| is quasi-isomorphic to a bounded complex of finite projective |$\underline {A}$| -modules.

It easily follows from [ 1 , Lemma 3.1].

The following is an analogue of Lemma 2.5 .

  Lemma 4.7. Let |$A \to B$| be a flat map of admissible |${\mathcal {O}}_{K}$| -algebras or a flat map of affinoid |$K$| -algebras. Then there exists a decomposition $$ \begin{align*} &A \to A\langle T_1,\ldots,T_n\rangle \to B\end{align*}$$ such that |$B$| is a perfect |$A\langle T_{1},\ldots ,T_{n}\rangle $| -module.

  Proof. We will prove it when |$A \to B$| is a flat map of topologically finitely presented |${\mathcal {O}}_{K}$| -algebras. We can also prove the other case by almost the same way. We take a surjection |$A\langle T_{1},\ldots ,T_{n}\rangle \to B$|⁠ , and we will show that |$B$| is a perfect |$A\langle T_{1},\ldots ,T_{n}\rangle $| -module. We take any maximal ideal |${\mathfrak {m}}$| of |$A\langle T_{1},\ldots ,T_{n}\rangle $|⁠ . Since |$\pi $| is topologically nilpotent, we have |$\pi \in {\mathfrak {m}}$|⁠ . Since the ring |$(A/\pi )/{\sqrt {(0)}}$| is finitely generated over the residue field of |${\mathcal {O}}_{K}$|⁠ , |$A/\pi $| is a Jacobson ring. Therefore, the prime ideal |${\mathfrak {n}} = {\mathfrak {m}} \cap A$| of |$A$| is a maximal ideal, and the global dimension of |$A/{\mathfrak {n}}[T_{1},\ldots ,T_{n}]$| is equal to |$n$|⁠ . Since |$A,B$| are |$\pi $| -torsion-free, we have the following equivalence: $$ \begin{align*} & B \otimes_{A\langle T_{1},\ldots,T_{n}\rangle}^{{\mathbb{L}}} A/{\mathfrak{n}}[T_{1},\ldots,T_{n}]\\ \simeq{} &(B \otimes_{A\langle T_{1},\ldots,T_{n}\rangle}^{{\mathbb{L}}} A/\pi[T_{1},\ldots,T_{n}])\otimes_{A/\pi[T_{1},\ldots,T_{n}]}^{{\mathbb{L}}} A/{\mathfrak{n}}[T_{1},\ldots,T_{n}]\\ \simeq{} &B/\pi \otimes_{A/\pi[T_{1},\ldots,T_{n}]}^{{\mathbb{L}}} (A/\pi[T_{1},\ldots,T_{n}]\otimes_{A/\pi}^{{\mathbb{L}}} A/{\mathfrak{n}}) \\ \simeq{} &B/\pi \otimes_{A/\pi}^{{\mathbb{L}}} A/{\mathfrak{n}} \\ \simeq{} &B/{{\mathfrak{n}} B}. \end{align*}$$ Therefore, we get the following equivalence: $$ \begin{align*} &B \otimes_{A\langle T_{1},\ldots,T_{n}\rangle}^{{\mathbb{L}}} A\langle T_{1},\ldots,T_{n}\rangle/{\mathfrak{m}} \\ \simeq{} & (B \otimes_{A\langle T_{1},\ldots,T_{n}\rangle}^{{\mathbb{L}}} A/{\mathfrak{n}}[T_{1},\ldots,T_{n}]) \otimes_{A/{\mathfrak{n}}[T_{1},\ldots,T_{n}]}^{{\mathbb{L}}} A\langle T_{1},\ldots,T_{n}\rangle/{\mathfrak{m}} \\ \simeq{} &B/{{\mathfrak{n}} B} \otimes_{A/{\mathfrak{n}}[T_{1},\ldots,T_{n}]}^{{\mathbb{L}}} A\langle T_{1},\ldots,T_{n}\rangle/{\mathfrak{m}}. \end{align*}$$ Therefore, we have |$H^{i}(B \otimes _{A\langle T_{1},\ldots ,T_{n}\rangle }^{{\mathbb {L}}} A\langle T_{1},\ldots ,T_{n}\rangle /{\mathfrak {m}})=0$| for all |$i \notin [-n, 0]$|⁠ . Since |$A\langle T_{1},\ldots ,T_{n}\rangle $| is coherent, |$B$| is a pseudo-coherent |$A\langle T_{1},\ldots ,T_{n}\rangle $| -module, we get the claim by Lemma 2.4 .

  Corollary 4.8. Let |$A\to B$| and |$A \to C$| be flat maps of admissible |${\mathcal {O}}_{K}$| -algebras. Then we have the following equivalence: $$ \begin{align*} &(B \mathbin{\widehat{\otimes}}_A C)_{\blacksquare} \simeq B_{\blacksquare} \otimes_{A_{\blacksquare}}^{{\mathbb{L}}} C_{\blacksquare}.\end{align*}$$ Moreover, a similar statement holds true for affinoid |$K$| -algebras.

By Lemma 4.7 , we may assume that |$B$| is equal to |$A\langle T_{1},\ldots , T_{n}\rangle $| or |$B$| is a perfect |$A$| -module. If |$B$| is equal to |$A\langle T_{1},\ldots , T_{n}\rangle $|⁠ , then the claim follows from Lemma 1.28 and Lemma 1.29 . If |$B$| is a perfect |$A$| -module, then the claim follows from Lemma 1.15 .

By using Lemma 4.7 , we can prove the following analogue of Proposition 2.6 .

Let |$A \to B$| be a flat map of admissible |${\mathcal {O}}_{K}$| -algebras. Then |$A_{\blacksquare } \to B_{\blacksquare }$| is steady. Moreover, a similar statement holds true for affinoid |$K$| -algebras.

The same argument as in the proof of Proposition 2.6 works well.

We can prove the following analogue of Corollary 2.11 .

• We have an equivalence of functors from |${\mathcal {D}}(A_{\blacksquare })$| to |${\mathcal {D}}(B_{\blacksquare })$| $$ \begin{align*} &-\otimes_{A_{\blacksquare}}^{{\mathbb{L}}} B_{\blacksquare} \overset{\sim}{\longrightarrow} R\operatorname{\underline{Hom}}_{\underline{A}}(N_{B/A}, -).\end{align*}$$

By the same argument as in the proof of Theorem 2.10 , we may assume that |$B$| is equal to |$A\langle T_{1},\ldots ,T_{n} \rangle $| or |$B$| is a perfect |$A$| -module. If |$B$| is a perfect |$A$| -module, then the same argument as in Theorem 2.10 works well. We assume |$B=A\langle T_{1},\ldots ,T_{n} \rangle $|⁠ . We define |$N_{A\langle T_{1},\ldots ,T_{n} \rangle /A} = R\operatorname {\underline {Hom}}_{\underline {{\mathbb {Z}}}}({\mathbb {Z}}[T_{1},\ldots ,T_{n}],{\mathbb {Z}}) \otimes _{{\mathbb {Z}}_{\blacksquare }}^{{\mathbb {L}}} A_{\blacksquare }$|⁠ , which is an object of |${\mathcal {D}}(\underline {A\langle T_{1},\ldots ,T_{n} \rangle })$|⁠ . Then we can show that it satisfies the conditions by the same argument as in Corollary 2.11 .

The functoriality result as in Remark 2.12 holds true. It can be proved by the same way.

  Lemma 4.12. Let |${\mathcal {D}}$| be a stable |$\infty $| -category, and |$X_{\bullet }$| be a semisimplicial object of |${\mathcal {D}}$|⁠ . Then for every |$n \geq 1$|⁠ , we have the following fiber sequence: $$ \begin{align*} &\mathop{L\varinjlim}_{[m] \in \Delta_{s,\leq n-1}^{\textrm{op}}}X_m \to \mathop{L\varinjlim}_{[m] \in \Delta_{s,\leq n}^{\textrm{op}}}X_m \to X_n[n].\end{align*}$$

  Proof. By [ 11 , Remark 4.3.4], we have the following fiber sequence: $$ \begin{align*} &\mathop{L\varinjlim}_{[m] \in \Delta_{s,\leq n-1}^{\textrm{op}}}X_m \to \mathop{L\varinjlim}_{[m] \in \Delta_{s,\leq n}^{\textrm{op}}}X_m \to \operatorname{cofib}(X_n\otimes \partial \Delta^n \to X_n).\end{align*}$$ Therefore, the claim follows from the equivalence |$\operatorname {cofib}(X_{n}\otimes \partial \Delta ^{n} \to X_{n}) \simeq X_{n}[n].$|

Let |$f \colon A \to B$| be a faithfully flat map of admissible |${\mathcal {O}}_{K}$| -algebras. Let |$N_{B/A} \in {\mathcal {D}}(\underline {B})$| be as in Theorem 4.10 and |$N_{B/A} \to \underline {A}$| be the natural map in |${\mathcal {D}}(A_{\blacksquare })$|⁠ . Then the map |$\mathop {L\varinjlim }_{[m] \in \Delta _{s,\leq 2}^{\textrm {op}}} N_{B/A}^{\otimes (m+1)} \to \underline {A}$| has a section, where |$N_{B/A}^{\otimes (m+1)}$| is the |$(m+1)$| -fold derived tensor product of |$N_{B/A}$| over |$A_{\blacksquare }$|⁠ .

  Proof. We take a directed system |$(A_{\lambda })_{\lambda \in \Lambda }$| of small complete |$\pi $| -adic subrings of |$A$| as in Lemma 3.9 . Since |$N_{B/A}$| is compact as an object of |${\mathcal {D}}(A_{\blacksquare })$|⁠ , we can take |$\lambda \in \Lambda $| and |$N^{\lambda }_{B/A} \to \underline {A_{\lambda }}$| in |${\mathcal {D}}((A_{\lambda })_{\blacksquare })_{\textrm {cpt}}$| such that $$ \begin{align*} &(N^{\lambda}_{B/A} \to \underline{A_{\lambda}}) \otimes_{(A_{\lambda})_{\blacksquare}}^{{\mathbb{L}}} A_{\blacksquare} \simeq (N_{B/A} \to \underline{A})\end{align*}$$ in |${\mathcal {D}}(A_{\blacksquare })$| by Lemma 3.10 . By enlarging |$\lambda $|⁠ , if necessary, we may assume that there exists a finitely presented faithfully flat |$A_{\lambda }/\pi $| -algebra |$S_{\lambda }$| such that |$S_{\lambda } \otimes _{A_{\lambda }/\pi } A/\pi \cong B/\pi $|⁠ . Then, the two maps $$ \begin{align}& (N^{\lambda}_{B/A} \to \underline{A_{\lambda}}) \otimes_{(A_{\lambda})_{\blacksquare}}^{{\mathbb{L}}} (A_{\lambda}/\pi)_{\blacksquare}\end{align}$$ (4.1) and $$ \begin{align}& N_{S_{\lambda}/(A_{\lambda}/\pi)} \to \underline{A_{\lambda}/\pi}\end{align}$$ (4.2) in |${\mathcal {D}}((A_{\lambda }/\pi )_{\blacksquare })_{\textrm {cpt}}$| become equivalent after applying |$-\otimes _{(A_{\lambda }/\pi )_{\blacksquare }}^{{\mathbb {L}}} (A/\pi )_{\blacksquare }$|⁠ . Therefore, we may assume that two maps ( 4.1 ) and ( 4.2 ) are equivalent by enlarging |$\lambda $|⁠ , if necessary. We put $$ \begin{align*} &N_n:= \mathop{L\varinjlim}_{[m] \in \Delta_{s,\leq n}^{\textrm{op}}}(N^{\lambda}_{B/A})^{\otimes (m+1)} \otimes_{(A_{\lambda})_{\blacksquare}}^{{\mathbb{L}}} (A_{\lambda}/\pi)_{\blacksquare}.\end{align*}$$ By Lemma 4.12 , we have a fiber sequence $$ \begin{align*} &N_n \to N_{n+1} \to ((N^{\lambda}_{B/A})^{\otimes (n+2)} \otimes_{(A_{\lambda})_{\blacksquare}}^{{\mathbb{L}}} (A_{\lambda}/\pi)_{\blacksquare})[n+1].\end{align*}$$ Since |$(N^{\lambda }_{B/A})^{\otimes (n+2)} \otimes _{(A_{\lambda })_{\blacksquare }}^{{\mathbb {L}}} (A_{\lambda }/\pi )_{\blacksquare }$| lies in |${\mathcal {D}}^{[0,1]}((A_{\lambda }/\pi )_{\blacksquare })$| by Proposition 2.20 , we get that |$\tau ^{\geq 2-n}(N_{n}) \to \tau ^{\geq 2-n}(N_{n+1})$| is an equivalence. By Theorem 2.15 , we have |$\varinjlim _{n} N_{n} \simeq \underline {A_{\lambda }/\pi }$|⁠ , so we get an equivalence |$\tau ^{\geq 0} N_{2} \simeq \underline {A_{\lambda }/\pi }$|⁠ . Since |$H^{-1}(\underline {A_{\lambda }/\pi })=0$|⁠ , we get |$\operatorname {cofib}(N_{2} \to \underline {A_{\lambda }/\pi }) \in {\mathcal {D}}^{\leq -2}((A_{\lambda }/\pi )_{\blacksquare })$|⁠ . We put |$N_{2}^{\prime } := \mathop {L\varinjlim }_{[m] \in \Delta _{s,\leq 2}^{\textrm {op}}}(N^{\lambda }_{B/A})^{\otimes (m+1)}$|⁠ , which is a compact object of |${\mathcal {D}}((A_{\lambda })_{\blacksquare })$| and therefore |$\pi $| -adically complete by Proposition 3.8 . We have $$ \begin{align*} &\operatorname{cofib}(N_2^{\prime} \to \underline{A_{\lambda}})\otimes_{(A_{\lambda})_{\blacksquare}}^{{\mathbb{L}}} (A_{\lambda}/\pi)_{\blacksquare} \simeq \operatorname{cofib}(N_2 \to \underline{A_{\lambda}/\pi}),\end{align*}$$ so we get |$\operatorname {cofib}(N_{2}^{\prime } \to \underline {A_{\lambda }}))\in {\mathcal {D}}^{\leq -1}((A_{\lambda })_{\blacksquare })$| by Lemma 3.3 . We have the following fiber sequence: $$ \begin{align*} & N_2^{\prime} \to \underline{A_{\lambda}} \overset{\alpha}{\longrightarrow} \mathrm{cofib}\big(N_2^{\prime} \to \underline{A_{\lambda}}\big). \end{align*}$$ By |$\operatorname {cofib}(N_{2}^{\prime } \to \underline {A_{\lambda }}))\in {\mathcal {D}}^{\leq -1}((A_{\lambda })_{\blacksquare })$|⁠ , |$\alpha$| is zero, where we note that |$\underline {A_{\lambda }}$| is a projective object in |$\textrm {Mod}_{(A_{\lambda })_{\blacksquare }}^{\textrm {cond}}$|⁠ , so |$N_{2}^{\prime } \to \underline {A_{\lambda }}$| has a section. Therefore, $$ \begin{align*} &(\mathop{L\varinjlim}_{[m] \in \Delta_{s,\leq 2}^{\textrm{op}}} N_{B/A}^{\otimes (m+1)} \to \underline{A}) \simeq (N_2^{\prime} \to \underline{A_{\lambda}}) \otimes_{(A_{\lambda})_{\blacksquare}}^{{\mathbb{L}}} A_{\blacksquare}\end{align*}$$ also has a section.

By the similar argument as in [ 15 , Proposition 3.20], we can deduce from Theorem 4.13 that the ind-object |$\{\mathop {L\varinjlim }_{[m] \in \Delta _{s,\leq n}^{\textrm {op}}} N_{B/A}^{\otimes (m+1)}\}_{n}$| of |${\mathcal {D}}(A_{\blacksquare })$| is a constant ind-object that converges to |$\underline {A}$|⁠ , but we will not use it, so we omit the details.

  Corollary 4.15. Let |$f \colon A \to B$| be a faithfully flat map of admissible |${\mathcal {O}}_{K}$| -algebras. Then, the functor $$ \begin{align*} &- \otimes_{A_{\blacksquare}}^{{\mathbb{L}}} B_{\blacksquare} \colon {\mathcal{D}}(A_{\blacksquare}) \to {\mathcal{D}}(B_{\blacksquare})\end{align*}$$ is conservative.

  Proof. We take an object |$M \in {\mathcal {D}}(A_{\blacksquare })$| such that |$M \otimes _{A_{\blacksquare }}^{{\mathbb {L}}} B_{\blacksquare } \simeq 0$|⁠ . We want to show that |$M$| is equivalent to |$0$|⁠ . Since |$M \simeq R\operatorname {\underline {Hom}}_{\underline {A}}(\underline {A},M)$| is a retract of $$ \begin{align*} &R\operatorname{\underline{Hom}}_{\underline{A}}(\mathop{L\varinjlim}_{[m] \in \Delta_{s,\leq 2}^{\textrm{op}}} N_{B/A}^{\otimes (m+1)}, M)\simeq \mathop{R\varprojlim}_{[m] \in \Delta_{s,\leq 2}}R\operatorname{\underline{Hom}}_{\underline{A}}(N_{B/A}^{\otimes (m+1)}, M),\end{align*}$$ it is enough to show that |$R\operatorname {\underline {Hom}}_{\underline {A}}(N_{B/A}^{\otimes (m+1)}, M)\simeq 0$|⁠ , which can be shown by the same way as in Corollary 2.16 .

If |$K$| is a non-archimedean local field, then we can prove the equivalence |$N_{B/A} \simeq R\operatorname {\underline {Hom}}_{\underline {A}}(\underline {B}, \underline {A})$| by the same argument as in the proof of Theorem 2.10 . Therefore, the proofs become easier by using the results in [ 16 ].

From Theorem 4.13 , we get our main theorems.

  Theorem 4.17. Let |$f \colon A \to B$| be a faithfully flat map of admissible |${\mathcal {O}}_{K}$| -algebras. Let |$B^{n/A}$| denote the |$n$| -fold completed tensor product of |$B$| over |$A$|⁠ . Then we have an equivalence of |$\infty $| -categories $$ \begin{align*} &{\mathcal{D}}(A_{\blacksquare}) \overset{\sim}{\longrightarrow} \varprojlim_{[n] \in \Delta} {\mathcal{D}}((B^{(n+1)/A})_{\blacksquare}). \end{align*}$$

It follows from Lemma 2.17 , Corollary 4.8 , Proposition 4.9 , Theorem 4.10 , and Corollary 4.15 .

  Theorem 4.18. Let |$f \colon A \to B$| be a faithfully flat map of affinoid |$K$| -algebras. Let |$B^{n/A}$| denote the |$n$| -fold completed tensor product of |$B$| over |$A$|⁠ . Then we have an equivalence of |$\infty $| -categories $$ \begin{align*} &{\mathcal{D}}((A,A^{\circ})_{\blacksquare}) \overset{\sim}{\longrightarrow} \varprojlim_{[n] \in \Delta} {\mathcal{D}}((B^{(n+1)/A}, (B^{(n+1)/A})^{\circ})_{\blacksquare}). \end{align*}$$

By Theorem 1.30 and Lemma 4.4 , we may assume that there exists a faithfully flat map |$A^{\prime } \to B^{\prime }$| of admissible |${\mathcal {O}}_{K}$| -algebras such that |$A^{\prime }[1/\pi ] \to B^{\prime }[1/\pi ]$| is isomorphic to |$A \to B$|⁠ . Noting that there is an equivalence |$(B, B^{\circ })_{\blacksquare } \simeq (A, A^{\circ })_{\blacksquare } \otimes _{A^{\prime }_{\blacksquare }}^{{\mathbb {L}}} B^{\prime }_{\blacksquare }$| by Lemma 4.5 , we can easily check the conditions of Lemma 2.17 as in Theorem 4.17 .

In this section, we will recover descent for pseudo-coherent complexes and perfect complexes from the main theorem by the method used in [ 1 , Section 5]. Let us begin by briefly recalling the notion used in [ 1 , Section 5].

An object of |${\mathcal {D}}(A)$| is called a pseudo-coherent complex if the following are satisfied:

|$M$| is bounded above.

For each integer |$i$|⁠ , the functor |$\operatorname {Ext}_{A}^{0}(M,-) \colon {\mathcal {D}}^{\geq i}(A) \to \textrm {Ab}$| commutes with small filtered colimits.

We denote the full |$\infty $| -subcategory of pseudo-coherent complexes in |${\mathcal {D}}(A)$| by |$\textrm {PCoh}(A)$|⁠ . In addition, for every integer |$n$|⁠ , we denote the full |$\infty $| -subcategory of pseudo-coherent complexes in |${\mathcal {D}}^{\leq n}(A)$| by |$\textrm {PCoh}^{\leq n}(A)$|⁠ .

• An object of |${\mathcal {D}}(A)$| is called a perfect complex if it is quasi-isomorphic to a complex of the form $$ \begin{align*} & 0 \to P_{n+m} \to \cdots \to P_{n+1} \to P_{n} \to 0,\end{align*}$$ where |$P_{i}$| are finite projective |$A$| -modules. We denote the full |$\infty $| -subcategory of perfect complexes in |${\mathcal {D}}(A)$| by |$\textrm {Perf}(A)$|⁠ . In addition, for every pair of integers |$a \leq b$|⁠ , we denote the full |$\infty $| -subcategory of perfect complexes with Tor-amplitude in |$[a,b]$| by |$\textrm {Perf}^{[a,b]}(A)$|⁠ .

The category |$\textrm {FP}(A)$| of finite projective |$A$| -modules is equivalent to |$\textrm {Perf}^{[0,0]}(A)$|⁠ .

  Remark 5.3 ([ 1 , Proposition 5.14]). An object of |${\mathcal {D}}(A)$| is a pseudo-coherent complex if and only if it is quasi-isomorphic to a complex of the form $$ \begin{align*} & \cdots \to P_{n+m} \to \cdots \to P_{n+1} \to P_{n} \to 0,\end{align*}$$ where |$P_{i}$| are finite projective |$A$| -modules.

Let |${\mathcal {A}}$| be a (0-truncated) analytic ring.

For each integer |$i$|⁠ , the functor |$\operatorname {Ext}_{\underline {{\mathcal {A}}}}^{0}(M,-) \colon {\mathcal {D}}^{\geq i}({\mathcal {A}}) \to \textrm {Ab}$| commutes with small filtered colimits.

Let |$M$| be an object of |$\textrm {PCoh}({\mathcal {A}})$|⁠ . Then for each integer |$i$|⁠ , the functor |$R\operatorname {Hom}_{\underline {{\mathcal {A}}}}(M,-) \colon {\mathcal {D}}^{\geq i}({\mathcal {A}}) \to {\mathcal {D}}(\textrm {Ab})$| commutes with small filtered colimits.

It easily follows from the definition.

An object |$M \in {\mathcal {D}}({\mathcal {A}})$| is a pseudo-coherent complex if and only if |$M$| is quasi-isomorphic to a bounded above complex with terms of the form |${\mathcal {A}}[S]$| for various extremally disconnected sets |$S$|⁠ .

We note that |$\bigoplus _{i=1}^{n} {\mathcal {A}}[S_{i}]$| is isomorphic to |${\mathcal {A}}[\coprod _{i=1}^{n} S_{i}]$| for extremally disconnected sets |$S_{i}$|⁠ , and that |$\coprod _{i=1}^{n} S_{i}$| is an extremally disconnected set. Then the lemma follows from [ 1 , Proposition 5.14].

Let |$M$| be an object of |${\mathcal {D}}({\mathcal {A}})$|⁠ . Then the object |$R\operatorname {\underline {Hom}}_{\underline {{\mathcal {A}}}}(M, {\mathcal {A}}) \in {\mathcal {D}}({\mathcal {A}})$| is called the dual of |$M$| and denoted by |$M^{\lor }$|⁠ .

  Definition 5.8 ([ 1 , Definition 5.30]). An object |$M\in {\mathcal {D}}({\mathcal {A}})$| is said to be nuclear if for any extremally disconnected set |$S$|⁠ , the natural map $$ \begin{align*} &({\mathcal{A}}[S]^{\lor}\otimes_{{\mathcal{A}}}^{{\mathbb{L}}} M)(\ast) \to M(S)\end{align*}$$ is an equivalence in |${\mathcal {D}}(\textrm {Ab})$|⁠ . We denote the full |$\infty $| -subcategory of nuclear objects in |${\mathcal {D}}({\mathcal {A}})$| by |${\mathcal {D}}({\mathcal {A}})_{\textrm {nc}}$|⁠ .

Next, we define a functor that associates |$\underline {A}$| -modules to |$A$| -modules.

  Definition 5.9 ([ 1 , Definition 5.8, Theorem 5.9]). Let |$(A,A^{+})$| be an analytic complete affinoid pair. Then the condensification functor is the fully faithful functor |$\textrm {Cond}_{(A,A^{+})} \colon {\mathcal {D}}(A) \to {\mathcal {D}}((A,A^{+})_{\blacksquare })$| given by $$ \begin{align*} &M \mapsto \textrm{dCond}_{A_{\textrm{disc}}}(M) \otimes_{(A_{\textrm{disc}}, A^+_{\textrm{disc}})_{\blacksquare}}^{{\mathbb{L}}} (A,A^+)_{\blacksquare}.\end{align*}$$ An object of |${\mathcal {D}}((A,A^{+})_{\blacksquare })$| is said to be discrete if it lies in the essential image of |$\textrm {Cond}_{(A,A^{+})}$|⁠ .

|$\textrm {PCoh}(A) \overset {\sim }{\longrightarrow } \textrm {PCoh}((A,A^{+})_{\blacksquare })_{\textrm {nc}}.$|

|$\textrm {Perf}(A) \overset {\sim }{\longrightarrow } \textrm {PCoh}((A,A^{+})_{\blacksquare })_{\textrm {nc}, \textrm {cpt}}$|⁠ ,

The following is an analogue of [ 1 , Proposition 5.38].

  Lemma 5.12. Let |$K$| be a complete non-archimedean field and |$A \to B$| be a flat map of affinoid |$K$| -algebras. Then for every extremally disconnected set |$S$| and every object |$M \in {\mathcal {D}}((A, A^{+})_{\blacksquare })$|⁠ , we have an equivalence $$ \begin{align*} &R\operatorname{\underline{Hom}}_{\underline{A}}((A, A^{+})_{\blacksquare}[S], M) \otimes_{(A, A^{+})_{\blacksquare}}^{{\mathbb{L}}} (B,B^{+})_{\blacksquare} \\ \simeq{} &R\operatorname{\underline{Hom}}_{\underline{B}}((B,B^{+})_{\blacksquare}[S], M \otimes_{(A, A^{+})_{\blacksquare}}^{{\mathbb{L}}} (B,B^{+})_{\blacksquare}). \end{align*}$$

  Proof. We take |$N_{B/A} \in {\mathcal {D}}(\underline {B})$| as in Theorem 4.10 . Then the lemma follows from the following computation: $$ \begin{align*} &R\operatorname{\underline{Hom}}_{\underline{A}}((A, A^{+})_{\blacksquare}[S], M) \otimes_{(A, A^{+})_{\blacksquare}}^{{\mathbb{L}}} (B,B^{+})_{\blacksquare} \\ \simeq{} &R\operatorname{\underline{Hom}}_{\underline{A}}(N_{B/A}, R\operatorname{\underline{Hom}}_{\underline{A}}((A, A^{+})_{\blacksquare}[S], M)) \\ \simeq{} &R\operatorname{\underline{Hom}}_{\underline{A}}(N_{B/A}\otimes_{(A,A^{+})_{\blacksquare}}^{{\mathbb{L}}} (A, A^{+})_{\blacksquare}[S], M) \\ \simeq{} &R\operatorname{\underline{Hom}}_{\underline{A}}((A, A^{+})_{\blacksquare}[S], R\operatorname{\underline{Hom}}_{\underline{A}}(N_{B/A}, M)) \\ \simeq{} &R\operatorname{\underline{Hom}}_{\underline{A}}((A, A^{+})_{\blacksquare}[S], M \otimes_{(A, A^{+})_{\blacksquare}}^{{\mathbb{L}}} (B,B^{+})_{\blacksquare}) \\ \simeq{} &R\operatorname{\underline{Hom}}_{\underline{B}}((B,B^{+})_{\blacksquare}[S], M \otimes_{(A, A^{+})_{\blacksquare}}^{{\mathbb{L}}} (B,B^{+})_{\blacksquare}). \end{align*}$$

The following is an analogue of [ 1 , Proposition 5.39].

Let |$K$| be a complete non-archimedean field and |$A \to B$| be a flat map of affinoid |$K$| -algebras. Then there exists a non-negative integer |$k$| such that for every integer |$i$| and every |$M \in {\mathcal {D}}^{\geq i}((A,A^{\circ })_{\blacksquare })$|⁠ , |$M\otimes _{(A,A^{\circ })_{\blacksquare }}^{{\mathbb {L}}} (B,B^{\circ })_{\blacksquare }$| lies in |${\mathcal {D}}^{\geq i-k}((A,A^{\circ })_{\blacksquare })$|⁠ .

By Lemma 4.7 , we may assume that |$B$| is equal to |$A\langle T \rangle $| or |$B$| is a perfect |$A$| -module (if |$B$| is a perfect |$A$| -module then |$B$| could not be flat over |$A$|⁠ , but it causes no problem). The latter case is trivial, so we may assume that |$B=A\langle T \rangle $|⁠ . In this case, the claim follows from Lemma 1.29 , Lemma 2.9 and that |$R\operatorname {\underline {Hom}}_{\underline {{\mathbb {Z}}}}(\underline {{\mathbb {Z}}[T]}, \underline {{\mathbb {Z}}})$| is a compact projective |${\mathbb {Z}}_{\blacksquare }$| -module.

By Lemma 5.11 , in order to prove descent for perfect complexes and pseudo-coherent complexes, it is enough to prove descent for nuclear objects, pseudo-coherent objects, and compact objects.

|$\textrm {PCoh}((A,A^{\circ })_{\blacksquare }) \overset {\sim }{\longrightarrow } \varprojlim _{[n] \in \Delta } \textrm {PCoh}((B^{(n+1)/A}, (B^{(n+1)/A})^{\circ })_{\blacksquare })$|⁠ .

|${\mathcal {D}}((A,A^{\circ })_{\blacksquare })_{\textrm {nc}} \overset {\sim }{\longrightarrow } \varprojlim _{[n] \in \Delta } {\mathcal {D}}((B^{(n+1)/A}, (B^{(n+1)/A})^{\circ })_{\blacksquare })_{\textrm {nc}}$|⁠ .

|${\mathcal {D}}((A,A^{\circ })_{\blacksquare })_{\textrm {cpt}} \overset {\sim }{\longrightarrow } \varprojlim _{[n] \in \Delta } {\mathcal {D}}((B^{(n+1)n/A}, (B^{(n+1)/A})^{\circ })_{\blacksquare })_{\textrm {cpt}}$|⁠ .

The same argument as in [ 1 , Theorem 5.42] works well to prove (2) by using Lemma 5.12 instead of [ 1 , Proposition 5.38].

For |$M \in \textrm {PCoh}((A,A^{\circ })_{\blacksquare })$|⁠ , |$M \otimes _{(A,A^{\circ })_{\blacksquare }}^{{\mathbb {L}}} (B^{(m+1)/A},(B^{(m+1)/A})^{\circ })_{\blacksquare }$| is pseudo-coherent for every non-negative integer |$m$|⁠ .

For |$M \in {\mathcal {D}}((A,A^{\circ })_{\blacksquare })$|⁠ , if |$M \otimes _{(A,A^{\circ })_{\blacksquare }}^{{\mathbb {L}}} (B^{(m+1)/A},(B^{(m+1)/A})^{\circ })_{\blacksquare }$| is pseudo-coherent for every non-negative integer |$m$|⁠ , then |$M$| is pseudo-coherent.

For |$M \in {\mathcal {D}}((A,A^{\circ })_{\blacksquare })_{\textrm {cpt}}$|⁠ , |$M \otimes _{(A,A^{\circ })_{\blacksquare }}^{{\mathbb {L}}} (B^{(m+1)/A},(B^{(m+1)/A})^{\circ })_{\blacksquare }$| is compact for every non-negative integer |$m$|⁠ .

For |$M \in {\mathcal {D}}((A,A^{\circ })_{\blacksquare })$|⁠ , if |$M \otimes _{(A,A^{\circ })_{\blacksquare }}^{{\mathbb {L}}} (B^{(m+1)/A},(B^{(m+1)/A})^{\circ })_{\blacksquare }$| is compact for every non-negative integer |$m$|⁠ , then |$M$| is compact.

  Theorem 5.15. Let |$K$| be a complete non-archimedean field and |$A \to B$| be a faithfully flat map of affinoid |$K$| -algebras. Let |$B^{n/A}$| denote the |$n$| -fold completed tensor product of |$B$| over |$A$|⁠ . Then we have the following equivalence of |$\infty $| -categories: $$ \begin{align*} &{\mathcal{C}}(A) \overset{\sim}{\longrightarrow} \varprojlim_{[n] \in \Delta} {\mathcal{C}}(B^{(n+1)/A}),\end{align*}$$ where |${\mathcal {C}}$| is one of the following |$\infty $| -categories: |$\textrm {PCoh}$|⁠ , |$\textrm {PCoh}^{\leq m}$|⁠ , |$\textrm {Perf}$|⁠ , |$\textrm {Perf}^{[a,b]}$|⁠ , and |$\textrm {FP}$|⁠ .

The descent for |$\textrm {PCoh}$| and |$\textrm {Perf}$| follows from Theorem 4.18 , Lemma 5.11 , Theorem 5.14 . To prove the descent for |$\textrm {PCoh}^{\leq m}$|⁠ , it is enough to show that for |$M \in \textrm {PCoh}(A)$| if |$M \otimes _{A}^{{\mathbb {L}}} B$| belongs to |$\textrm {PCoh}^{\leq m}(B)$| then |$M$| also belongs to |$\textrm {PCoh}^{\leq m}(A)$|⁠ , where we implicitly use Lemma 5.10 and Lemma 5.11 . This follows from the fact that |$A \to B$| is faithfully flat. Finally, to prove the descent for |$\textrm {Perf}^{[a,b]}$|⁠ , it is enough to show that for |$M \in \textrm {Perf}(A)$| if |$M \otimes _{A}^{{\mathbb {L}}} B$| belongs to |$\textrm {Perf}^{[a,b]}(B)$| then |$M$| also belongs to |$\textrm {Perf}^{[a,b]}(A)$|⁠ . Since |$A\to B$| is faithfully flat, we have that for |$N\in {\mathcal {D}}(A)$| and |$i\in {\mathbb {Z}}$|⁠ , |$H^{i}(N)=0$| is equivalent to |$H^{i}(N\otimes _{A}^{{\mathbb {L}}}B)=0$|⁠ . The claim easily follows from it.

This paper is based on the author’s master thesis. The author is grateful to the adviser Yoichi Mieda for his support during the studies of the author. In addition, the author is grateful to Ko Aoki for his answers in questions about higher algebra and to Lucas Mann for his comments on this paper. Finally, the author would like to thank the referee for several corrections.

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