### Math 2520: Algebra, Spring 2020MWF 1-1:50pm Kassar 105

 Instructor: Melody Chan office: 311 Kassar House course website: http://www.math.brown.edu/~mtchan/2020Spring_2520.html office hours, Thursdays 4-5pm and Fridays 2-3pm, Kassar 311

Course description

This is a second semester graduate algebra course. The topics are commutative algebra, homological algebra, and category theory, as time permits. Broadly speaking, I am interested in teaching you the kind of algebra that supports the study of algebraic geometry and algebraic topology, and I will foreground the relationship between algebra and geometry.

This course follows Math 2510, which is a prerequisite. I will by default assume you have taken Math 2510 last semester with Prof. Goodwillie. If you have taken Math 2510 in a different year, or have taken a roughly equivalent course elsewhere, that is fine too, just let me know, so that I am aware of what topics you have seen.

Knowing topology at least at the undergraduate level (Math 1410) is also highly recommended. Please ask me any questions about whether this course is suited for you.

I will expect a good amount of work outside class on the problem sets.

Topics

Category theory: categories and functors review. Initial and final. Universal properties. Adjoints. Limits, colimits. Natural transformations, equivalence of categories, representability, Yoneda's lemma.

Commutative algebra: Rings, Spec, modules. Extension and contraction. Nakayama's Lemma. Tensor-hom adjunction. Exactness. Flatness. Noetherian rings. Primary decomposition. Integral dependence, normal rings, going up. Hilbert basis theorem. Nullstellensatz. A little representation theory, and other topics time permitting, weighed against how much homological algebra we decide to do.

Homological algebra: abelian categories, chain complexes, derived functors, and other topics time permitting.

Recommended textbook

Atiyah and MacDonald, Introduction to commutative algebra.
This book is available at the bookstore for \$85.95 new, \$64.46 used.
If cost is an issue, let me know.

Supplementary textbooks

We will use some secondary sources, which I will make an effort to ensure are freely available online.
Eisenbud Commutative algebra with a view towards algebraic geometry
Vakil Foundations Of Algebraic Geometry
Riehl Category theory in context

Weekly problem sets posted here, typically due Mondays, at the beginning of class in hard copy (LaTeX strongly preferred) and stapled. No late problem sets will be accepted, other than very extenuating circumstances which should be presented to me. Your lowest problem set grade will be dropped.
The first problem set is due Monday 1/27.
The problem sets are the most important component of the course.

Optional 15 minute in-class presentation and short writeup, in the second half of the course. This is optional but recommended. Your presentation grade replaces 1.5 lowest problem set grades, in addition to the one free drop.

Problem sets

All problem sets in one PDF

Schedule

Reading period schedule: The last day of class will be Wednesday, April 29.

Collaboration

You are welcome and encouraged to collaborate with other students in the class on your homework. I suggest that you think carefully about each problem on your own first. You are required to write up your solutions separately and write the names of the students with whom you worked on the assignment. (You may only use the Internet as a general reference, at the level of generality of Wikipedia.)

How much time will this class take?

Roughly speaking, you should expect to spend twelve hours every week outside of class, including attending office hours, reviewing class material, preparing your class presentation, and doing problem sets. In addition to three hours of class every week, I estimate a total of 15*13 = 195 hours of time spent on this class.

Accommodations for students with disabilities

Any student with a documented disability is welcome to contact me as early in the semester as possible so that we may arrange reasonable accommodations. As part of this process, please be in touch with Student and Employee Accessibility Services by calling 401-863-9588 or online at
http://brown.edu/Student_Services/Office_of_Student_Life/seas/index.html