Math 2520: Algebra, Spring 2019

Instructor: Melody Chan
office: 311 Kassar House course website: http://www.math.brown.edu/~mtchan/2019Spring_2520.html (or scan the QR code at the right) office hours, Wednesdays 3:304:30, Fridays 10:3011:30 
Course description
This is a second semester graduate algebra course. The topics are commutative algebra, some homological algebra, and a little 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.
In principle, this course follows Math 2510. But as long as you are willing to work hard, I will not require Math 2510 as a prerequisite. A very solid grounding in undergraduate abstract algebra (at the level of Math 1530) is definitely required, and Math 1410 (Topology) will also be useful background knowledge. I will expect a good amount of work outside class on the problem sets. If you are unsure about whether to take this course, please discuss it with me.
Topics
Rings, modules, Hom and tensor, exactness, localization, integral dependence, valuations, Noetherian conditions, and further topics as time permits.
Recommended textbook
Atiyah and MacDonald, Introduction to commutative algebra.
This book is available at the bookstore for $82.95 new, $62.21 used.
If cost is an issue, let me know.
Supplementary textbooks
We may also use some secondary sources, which I will make an effort to ensure are freely available online.
A likely possibility is Chapter 1 of Vakil's Foundations Of Algebraic Geometry.
Grading
Grading: 100% homework. Optional 15 minute inclass presentation and short writeup drops 1.5 lowest homeworks.
Problem sets
Problem sets are the most important part of the course. With the exception of the first problem set due Monday 1/28, problem sets will be assigned and due every Friday. Problem sets are due at the beginning of class, hard copy (LaTeX strongly preferred) and stapled. No late problem sets will be accepted. However, your lowest score will be dropped.
All problem sets in one PDF
Student presentations
Schedule
Week 1, Jan 23: Introduction. Categories and functors. (Reference: Vakil Chapter 1.2)
Week 2, Jan 28: Initial and final. Universal properties. Adjoints. Limits, colimits. (Reference: Vakil Chapter 1.3, 1.4, 1.5)
Week 3, Feb 4: Natural transformations, equivalence of categories, representability, Yoneda's lemma. Rings, ideals, prime, maximal. Existence of maximal ideals. Radical ideals. Various operations on ideals. Spec R and the Zariski topology.
Week 4, Feb 11: Extension and contraction. Modules. Basic operations. Colon, annihilator. Free, finitely generated modules.
Week 5, Feb 18: Nakayama's Lemma. Tensor products, adjointness of tensor and hom, extension and restriction of scalars.
Week 6, Feb 25: Tensor products of algebras. Exact sequences.
Week 7, Mar 4: Flatness, criteria for flatness. Snake lemma. Local properties.
Week 8, Mar 11: Noetherian rings and primary decomposition.
Week 9, Mar 18: Integral dependence, normal rings, going up.
Week 10, Apr 1: Noether normalization, Nullstellensatz. DVRs.
Week 11, Apr 8: DVRs. Topological groups
Week 12, Apr 15: Completions, filtrations, graded rings/modules, ArtinRees.
Week 13, Apr 22: Krull intersection theorem. Dimension: Poincare series of graded modules
Week 14, Apr 29: Hilbert functions and polynomials; various notions of dimension are equal. CohenMacaulay condition.
Reading period schedule: I plan to hold class as usual up through Friday May 3.
Collaboration
You are welcome to collaborate with other students in the class on your homework, although 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 4018639588 or online at
http://brown.edu/Student_Services/Office_of_Student_Life/seas/index.html