Course Information

Course Name and Number

Selected Topics in Algebra: Derived categories in algebraic geometry II – Math 748


Spring 2022

Meeting Time and Location

Tuesdays and Thursdays 10:05-11:20 AM in Room 3006B of the Carolina Coliseum

Instructor Information

Office Hours

Tuesdays 12-3 or by appointment

Full Course Description

This is the second in a two-semester course. With much of the background of triangulated categories set, we focus on the structure of derived categories of (quasi-)coherent sheaves on schemes.


Math 738: Derived categories in algebraic geometry I. Fall 2021.

Learning Outcomes

After successful completion of this course, you will be able to:

  • Be familiar with seminal results on the structure of derived categories of coherent sheaves
  • Have a working knowledge of major tools and major questions in the area

Course Materials

The main resources for materials are

Supplemental course materials will be linked as necessary.

All course materials comply with copyright/fair use policies.

Course Requirements

Course Format

The course is a standard lecture course.

At the end of the class, a paper will be due in place of a final exam.

Course Communication

If you need to get in touch with me, the best method is via Zulip chat or email. Generally, I will reply within 24 hour.

If you are having trouble with this course or its material, you should contact me via Zulip chat or email to discuss the issues.

Announcements will be posted to this course whenever necessary. If there is any other information I think is important, I will send it to your preferred university email address. It is your responsibility to ensure that your email account works properly in order to receive email.

If you are unsure of your preferred email, check your account at For more information on setting your preferred university email, please see the Knowledge Base Article How To Change Your Primary University Email Address.


To participate in learning activities and complete assignments, you will need:

  • Access to a working computer that has a current operating system with updates installed with a modern web browser installed;

  • Reliable Internet access and a USC email account;

  • We are using the Microsoft Teams with the team COTEAM-BALLARMR-MATH-738-001-FALL-2021 run through your UofSC Microsoft account. To access the team for the first time on your desktop/laptop, use the join link included in your welcome email.

  • For chat, We will be using a service called Zulip to aid our learning. It exceeds Team's capabilities with regards to typsetting mathematics content. To access the SCAGNT Zulip chat for the first time on your desktop/laptop, use the join link included in your welcome email.

Minimal Technical Skills Needed

Minimal technical skills are needed in this course. All work in this course must be completed and submitted online. Therefore, you must have consistent and reliable access to a computer and the Internet. The minimal technical skills you have include the ability to:

  • Organize and save electronic files;

  • Check and use the course, GitHub Classroom, Zulip, and Microsoft Teams sites regularly;

  • Download and upload documents;

  • Locate and enter information with a browser;

  • Edit and compile TeX documents.

Technical Support

If you have problems with your computer, technology, IT-related questions, support, including Microsoft Teams, please contact the Division of Information Technology (DoIT) Service Desk at (803) 777-1800 or submit an online request through the Self-Service Portal or visit the Carolina Tech Zone. The Service Desk is open Monday – Friday from 8:00 AM – 6:00 PM (Eastern Daylight Time). The Thomas Cooper Library at USC has computers for you to use in case you encounter computer issues/problems.

If you have problems with Zulip or GitHub Classroom, contact me.

Course Assignments and Grading


In lieu of a finals, papers on more in depth topics are required. Contact me to arrange for a topic. Suggestions will be provided as we progress.

Evaluation and Grading Scale

All grades will be posted on Teams. A final letter grade will be assigned based on weighting below.

Assignment Weights

ComponentPercent of total

Grading Scale

Final total intervalsLetter Grade

Assignment Submission

Assignment submission will be through the GitHub Classroom for the course.

Academic Success


The Student Disability Resource Center (SDRC) empowers students to manage challenges and limitations imposed by disabilities. Students with disabilities are encouraged to contact me to discuss the logistics of any accommodations needed to fulfill course requirements (within the first week of the semester). In order to receive reasonable accommodations from me, you must be registered with the Student Disability Resource Center (1705 College Street Close-Hipp, Suite 102 Columbia, SC 29208, 803-777-6142). Any student with a documented disability should contact the SDRC to make arrangements for appropriate accommodations.

University Library Resources

University Libraries has access to books, articles, subject specific resources, citation help, and more. If you are not sure where to start, please Ask a Librarian! Assistance is available at 

Remember that if you use anything that is not your own writing or media (quotes from books, articles, interviews, websites, movies – everything) you must cite the source in MLA (or other appropriate and approved) format.

Teams and Technology

Teams and Technology. As a student in this course, you have access to support from the Division of Information Technology (DoIT) for Teams and computer issues. The service desk can be reached at 803-777-1800.

Counseling Services

The University offers counseling and crisis services as well as outreach services and self-help.

Course Policies and Procedures

Attendance Policy and COVID Reporting

You are expected to participate actively in each course. If you anticipate an excused absence, you need to contact me in advance. You should submit a request in writing (email is acceptable) stating the dates of the anticipated absence, explaining the reason for absence, providing supporting documentation as required above, and including any request for make-up work. You should submit this request no later than the end of the second week of regularly scheduled classes in a full fall or spring semester term and within twice the length of the drop/add period for any other term.

If regularly attending class becomes difficult for any reason, please contact me to discuss the issue.

In accordance with university policy, a grade penalty of 10% may be imposed if a student has unexcused absensces exceeding 5% of the total number of courses. For this course, that means missing TWO courses without excuse.

All absences due to documented illness or quarantine will be excused, and no grade penalty will be assessed for missing classes for this reason. If you experience COVID-19 symptoms, please stay home, contact the COVID-19 Student Health Services (SHS) nurse line (803-576-8511), complete the COVID-19 Student Report Form and select the option allowing the Student Ombuds to contact your professors. When talking with the SHS nurse, be sure to ask for documentation of the consult as you will need this to document why you missed class. You will also use the COVID-19 Student Report Form if you have tested positive for COVID-19 or if you have been ordered to quarantine because of close contact with a person who was COVID-19 positive. In each of these situations you will be provided appropriate documentation that can be shared through the Student Report Form.

COVID Policies

As of 8/17/2021, UofSC requires face coverings, including in this class. For more information on this semester's COVID policies see the guidance from the Provost.

I strongly encourage getting vaccinated.

I encourage physical distancing. While not always possible, I will strive to keep everyone at least three feet apart, even when working in groups, which we will do regularly. If you feel uncomfortable with group work in my class, please come talk with me and I don’t mind letting you work independently at all.

I have been requested to keep a seating chart. I have been urged to keep this seating chart for contact tracing purposes and will do my best to abide by the University policy.

Questions You May Have

What if I get sick with COVID? Two things have to occur: Isolation: Students who have been diagnosed with COVID-19 are released from isolation when a medical professional has determined, based on the current CDC and DHEC guidelines, that they have recovered. Currently, these guidelines include being fever-free for at least 24 hours and at least 10 days from their first symptom or positive test if they are asymptomatic. Quarantine: Unvaccinated students who are deemed a close contact with a confirmed COVID-19 case will be quarantined for 7-14 days from their last contact with the infected individual. More specifically, students who test negative on day 5, 6, or 7 can leave quarantine after 7 full days; individuals who did not test but remain asymptomatic can leave after 10 full days. Individuals who are symptomatic or have other health concerns may be advised to remain under quarantine for 14 days. 

What is the attendance policy if I get COVID? In brief, I must provide make-up course work including content and assignments when students have excused absences which include (but are not limited to) being in quarantine or isolation, religious holidays, medical conditions related to pregnancy, and military duty. However, recorded classes and hybrid/online options are not required and should not be expected. All excused absences must have documentation. See syllabus for further attendance policies.

How will the Dr. Ballard know if I am absent due to quarantine or isolation? COVID-19 related absences must be document through the Student Ombudsman.  Students who have been diagnosed with COVID-19 or have been exposed and require quarantining should complete the COVID-19 Student Report Form and instructors should request this form in order to excuse the absence. 

Can I inquire about classmates condition with COVID? Sadly, not with me. These are health issues and the information is protected by state and federal law. If an individual student has questions about whether they should quarantine or believe that they have been in close contact, have them reach out to the COVID Phone Bank (803-576-8511).

Would we ever change to go online if too many people are sick? Only in the rare instance that 30% or more of students have documented excused absences may I take the course online. This is not to be expected and very complicated according to the current policies.

What if Dr. Ballard gets sick with COVID? I have been fully vaccinated and breakthrough infection symptoms most often resemble the common cold. In the event of a breakthrough infection, I will enter the self-isolation period and the course will switch modality to synchronous online temporarily.

In the rare circumstance I am unable to teach remotely, a substitute instructor will take over the course.

Academic Integrity

You are expected to practice the highest possible standards of academic integrity. Any deviation from this expectation will result in a minimum academic penalty of your failing the assignment, and will result in additional disciplinary measures. This includes improper citation of sources, using another student's work, and any other form of academic misrepresentation.

The first tenet of the Carolinian Creed is, "I will practice personal and academic integrity."

Below are some websites for you to visit to learn more about University policies:


Using the words or ideas of another as if they were one’s own is a serious form of academic dishonesty. If another person’s complete sentence, syntax, key words, or the specific or unique ideas and information are used, one must give that person credit through proper citation. You should in particular cite any resources, person, text, or otherwise, you used to assist in preparation of your work. Copying proofs or problem solutions is strictly forbidden.

Group Work

Group work should be performed in safe manner. Remote work will certainly form a larger component of a career going forward. You are encouraged to take advantage of Microsoft Teams video conferencing and Zulip chat abilities to aid in collaboration.

Class Conduct

Professionalism will be expected at all times, but most especially with your interactions online and in person. Because the university classroom is a place designed for the free exchange of ideas, we must show respect for one another in all circumstances. We will show respect for one another by exhibiting patience and courtesy in our exchanges. Appropriate language and restraint from verbal attacks upon those whose perspectives differ from your own is a minimum requirement. Courtesy and kindness is the norm for those who participate in the class.

Mistakes, in particular during the running phase, are expected and natural. Mistakes are how learning happens. All students should recognize and respect the bravery of a student presenting a proof or solution. If you ever feel uncomfortable beyond the intellectual challenge of the course, please contact me.

Teams is a way for you to share your ideas and learning with your colleagues in this class. We do this as colleagues in learning, and the online space is meant to be a safe and respectful environment for us to conduct these discussions.

Some general netiquette rules:

  • Treat one another with respect. It will be expected that we will not attack one another personally for holding different opinions.

  • Do not use all CAPITAL LETTERS in emails or discussion board postings. This is considered "shouting" and is seen as impolite or aggressive.

  • Begin emails with a proper salutation (Examples: Dr. Name; Ms. Name; Hello Professor Name; Good afternoon Mr. Name). Starting an email without a salutation or a simple "Hey" is not appropriate.

  • When sending an email, please include a detailed subject line. Additionally, make sure you reference the course number (Ex. ENGL 287) in the message and sign the mail with your name.

  • Use proper grammar, spelling, punctuation, and capitalization. Text messaging language is not acceptable.

  • Use good taste when communicating. Profanity should be avoided.

  • Re-Read, think, and edit your message before you click "Send/Submit/Post."

  • Please remember when posting to be respectful and courteous to your colleagues, and limit your communication to topics of this course and the assignments.

Late Work/Make-up Policy

All assignments due by the deadline as posted on the course schedule. Late work is not accepted and not eligible for revision.

Please plan accordingly, and complete these assignments in advance of their deadlines to ensure any unanticipated circumstances do not result in a missed assignment. User error does not qualify you for any kind of makeup or retake opportunity.

Completing and submitting the assignments by the due date is the sole responsibility of you. If you fail to submit the assignment or test by the due date, then your score for that assignment will be recorded as "zero."

You will be allowed to access the assignments an unlimited number of times until the due date/time. If you are concerned about missing a deadline, post your assignment the day before the deadline.

Be Careful: The clock on your computer may be different than the clock in Teams. If the clock is different by one second, you will be locked out of the assignment. Plan accordingly.

Incomplete Grades

The grade of Incomplete will be granted only in accordance with university policy.

Diversity and Inclusion

The university is committed to a campus environment that is inclusive, safe, and respectful for all persons, and one that fully embraces the Carolinian Creed: “I will discourage bigotry, while striving to learn from differences in people, ideas and opinions.” Likewise, the Student Code of Conduct stresses, “The University of South Carolina strives to maintain an educational community that fosters the development of students who are ethical, civil and responsible persons.”

To that end, all course activities will be conducted in an atmosphere of friendly participation and interaction among colleagues, recognizing and appreciating the unique experiences, background, and point of view each student brings. You are expected at all times to apply the highest academic standards to this course and to treat others with dignity and respect.

Title IX and Gendered Identity

This course affirms equality and respect for all gendered identities and expressions. Please don’t hesitate to correct me regarding your preferred gender pronoun and/or name if different from what is indicated on the official class roster. Likewise, I am committed to nurturing an environment free from discrimination and harassment. Consistent with Title IX policy, please be aware that I as a responsible employee am obligated to report information that you provide to me about a situation involving sexual harassment or assault.

Expectations of the Instructor

I am expected to facilitate learning, answer questions appropriately, be fair and objective in grading, provide timely and useful feedback on assignments and treat you as I would like to be treated.

Copyright/Fair Use Statement

I will cite and/or reference any materials that I use in this course that I do not create.

Anything that appears on this website is copyright © 2022 Matthew Ballard and is distributed by an MIT license.

Course materials that do not appear on this website are copyright © 2022 Matthew Ballard and all rights are reserved. In particular, you may not distribute any of these course materials in any fashion.