Graduate Algebra 506, Spring 2019

Instructor: Julia Pevtsova

Place: Padelford Hall, C36

Time: 9:30-10:20, MWF

Office Hours: Monday, 11:30-1pm or by appointment

Teaching Assistant: Varodom (Smart) Thep

Office: ART 336/333

Office hours: Tuesday 11:30-1:00 pm, starting April 9

Course Description. This is the third quarter of the first-year algebra sequence. Here is the Syllabus including recommended texts.

Grading system. Grades will be determined based on homework assignments, a midterm and a final as follows:

· Homework 40%

· Midterm 20%

· Final 40%

Schedule: NO CLASS April 12 and 15.

Midterm is on Monday, May 6^th, in class, 9:00-10:20am

A two sided sheet of notes is allowed, but no other electronic or written help.

The midterm will have four problems; 100% will correspond to solving all four. The results will be curved.

List of topics. Rep theory: Maschke’s theorem, irreducible representations of finite groups, character theory. The material of the first chapter of Serre’s book except that the compact Lie groups were optional. Commutative algebra: prime and maximal ideals, nilpotent and Jacobson radicals, operations with ideals, including radicals. Nakayama lemma – various formulations and consequences. Noetherian topological spaces, irreducible components. Spec R, Zariski topology, closed points, irreducible sets and principal open sets in Spec R.

Final exam is TAKE HOME. It will posted here on Monday, June 10^th, by 5PM. Due Wednesday, June 12^th, by 10:30AM. You could bring it to my office or send a scanned or typed copy by e-mail. No late exam will be accepted. The exam is individual; your class notes are allowed but no other material to be used. If you use your computer/other device to type the exam and to consult electronic notes, make sure it is in offline mode and no books or other sources except for the notes are available to you. The exam is for two hours, please don’t spend more than four.

FINAL exam (take home; due June 12 by 10:30 am). tex

Homework. Assignments will be posted on this website on a weekly basis. You are encouraged to tex your homework especially if you did not take a calligraphy course in the past. The homework will be due in class on Wednesday morning. No late homework is accepted. E-mailing homework before the deadline is fine.

Worksheets. This is a special homework assignment. It counts towards the total homework grade. The format is different from the regular homework. The worksheet is designed as an independent study (or review) of a particular topic. You’ll get a short written introduction the topic with all the proofs missing. You’ll need to ``fill in the blanks”, that is, supply the proofs. Once the worksheet is graded and returned to you, it should be added to your notes. You may attach your proofs to the original worksheet, or download the tex file and add the proofs right where they belong so that you get a nice and continuous exposition. The material from the worksheets will be used later in the course and relied on in exams in the same way as the material presented in lecture.

Textbooks.

1. Representation Theory:

o Linear Representations of Finite Groups, J.-P. Serre (main reference)

o A Course on Finite Group Representation Theory, P. Webb

o Representation Theory, W. Fulton, J. Harris

2. Commutative algebra:

o Introduction to Commutative Algebra, M. Atiyah and I. Macdonald (main reference)

o Commutative algebra with a view towards algebraic Geometry, D. Eisenbud

o ``Undergraduate commutative algebra” and “Undergraduate algebraic geometry”, M. Reid

3. Homological algebra:

o An introduction to homological algebra, C. Weibel

4. General references:

o Abstract Algebra, D. Dummit and R. Foote

o Algebra, S. Lang

Notes. Max Götzler has graciously agreed to share the notes he is typing up for the course. They can be found here.

Practice problems. Here is the list of RT related problems from past prelim exams (Thank you, Martin!)

Homework 0, due Wednesday, April 3. OPTIONAL. tex

This homework contains suggested reading for the first unit of the Spring quarter:

J.-P. Serre, “Linear Representations of Finite Groups”, Chapter 1.

Supplemental: P. Webb, “A Course on Finite Group Representation Theory”.

Homework 1 (idempotents), due Wednesday, April 10. tex

Homework 2 (nilpotent elements and ideals, Noetherian and Artinian conditions), due Wednesday, April 24. tex

Homework 3 (Spec), due Wednesday, May 1. tex

Homework 4 (Spec), due Wednesday, May 8. tex

Worksheet on Artinian rings – Homework 5, due Wednesday, May 15. tex

Homework 6 (algebraic sets), due Friday May 24. tex

Homework 7(Hom and tensor), due Friday, May 31. tex

Homework 8 (homological algebra), due Friday June 7. tex

MyDefn file for compiling tex files