Math 6290 (Fall 2023):
Homological Algebra

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About this course

Welcome to Homological Algebra (Math 6290)! This is a one-semester graduate class on homological algebra and its applications. The core topics are abelian categories, chain complexes, derived functors, and spectral sequences. The two main applications will be to abelian categories of modules and sheaves of modules, with a goal of understanding Verdier duality. Time permitting, we will conclude with an introduction to derived categories.

Contact information

You may also contact me anonymously.

Office hours

My office is Room 204 in the Math Department. My office hours sometimes change, so I maintain a calendar showing the times I will be available. You can also make an appointment or drop in without an appointment.

Syllabus and textbook

There will be roughly 3 parts to the course. In the first third, we will learn generalities about abelian categories, chain complexes, and derived functors and apply them in the simplest example, modules over a ring. The goal here is to get to spectral sequences. The main text will be Weibel's:

C. Weibel, An introduction to homological algebra.

The middle third of the class will be about sheaf cohomology with a goal of getting to Verdier duality. Sheaves are discussed in Weibel's book, but we will need more depth, so we will us Kashiwara and Schapira:

M. Kashiwara and P. Schapira, Sheaves on manifolds.

The last third of the class will likely be about derived categories. We may discuss other topics, depend on interest from the class. Here are some possibilities: group cohomology, Lie algebra cohomology, André–Quillen cohomology, Grothendieck topologies, quasicoherent sheaves.

Prerequisites

You will need some comfort with algebra, particularly the theory of modules over a ring, and topology. For certain applications, additional background may be helpful.

Course goals

The goal is to become comfortable enough with homological algebra to read papers and textbooks that assume it. Students should leave the class able to read the homological aspects of Chapter III of Hartshorne's Algebraic geometry.

Homework and grading

Homework will be assigned weekly. To pass the course, students should do the reading assignments, participate actively in class, and submit homework regularly on Canvas. To receive an A, students should submit a significant number of homework problems every week.

Feedback

Special accommodations, CoViD-19, classroom behavior, and the honor code

The Office of Academic Affairs officially recommends a number of statements for course syllabi, all of which are supported in this class.

If you need special accommodation of any kind in this class, or are uncomfortable in the class for any reason, please contact me and I will do my best to remedy the situation. You may contact me in person, by e-mail, or anonymously.