# Math 210A: Modern Algebra (fall quarter)

Fall 2014

Tuesdays and Thursdays 2:15-3:30 in 380-W

About the course: According to the bulletin: basic commutative ring and module theory, tensor algebra, homological constructions, linear and multilinear algebra, introduction to representation theory. This course is intended to get across material important for graduate students embarking on a Ph.D. in mathematics. Others are welcome to attend the class, but should understand that the course will not be directed at them.

Prerequisites: complete fluency with the ideas of Math 120 and 121. In particular, I won't revisit the foundations of group theory, and will only briefy review the theory of rings.

Professor: Ravi Vakil (vakil@math), 383-Q.

Course Assistant: Evita Nestoridi (evnestor@math.stanford.edu), 381-D.

Office hours: Monday 2-3:30 pm (Evita), Tuesday 10:30 am-noon (Evita), Wednesday 10:30 am-noon (Ravi), Friday 9-10:30 am (Ravi). (For additional and changed office hours in week 10, see the syllabus below.)

Text: Lang's Algebra (revised third edition) and Dummit and Foote (3rd ed.). I see Dummit and Foote as a better text. Lang is useful as an encyclopedic reference. Don't feel obliged to buy a text, but you should definitely have access to books to make sure you understand the material.

Grading: Problem sets (70%), midterm (10%), final (20%).

Problem sets: Problem sets will be due every Thursday, unless otherwise noted. Please hand them in by noon to Evita Nestoridi's mailbox, or give them to me in class. No lates will be allowed. But to give everyone a chance to get sick, or have busy periods, the lowest homework grade will be dropped.

• Problem set 1 is due October 9.
• Problem set 2 is due October 16.
• Problem set 3 is due October 23.
• Problem set 4 (corrected version!) is due October 30. (The correction, now made: the horizontal arrows in problem 1 should point left. Thanks to those who caught this!)
• Problem set 5 is due November 20. (The correction, now made: in problem 4, the field must be algebraically closed, and one of the g's needs an "inverse". Thanks to Erik Bates and Adam Levine for pointing it out!)

Syllabus: The theme of the course will be modules over rings in different contexts. We will discuss basics of modules; linear algebra in the language of module theory (including the structure theorem for finitely generated modules over a PID); tensors (which will require an introduction to category theory); and homological algebra for modules. If time remains at the end of the class, we may discuss some representation theory or some of the theory of binary forms.

• Tues. Sept. 23: review of rings. Commtuative, 0-ring, unit, invertible, polynomial ring, formal power series, subring, center, category of rings, initial object and final object, integral domain, field, ideal, quotient by ideal, prime maximal principal ideals, Noetherian rings, euclidean algorithm, euclidean domain, greatest common divisor, principal ideal domain, associates.
• Thurs. Sept. 25: euclidean domain implies PID implies UFD. Localizaiton take 1. Gauss' Lemma. Eisenstein criterion. Quick review of modules.
• Tues. Sept. 30. Toward classification of finitely generated modules over PIDs. (Applications: classification of finitely generated abelian groups.) Noetherian modules. Rank of module over an integral domain. Statement of the "big theorem" on submodules of free modules over a PID, and how it implies the classification. Proof of uniqueness.
• Thurs. Oct. 2: Proof of the "big theorem". Using this to make some linear algebra (the linear endormorphisms of vector spaces) easy: minimal polynomial, characteristic polynomial, they have the same roots, rational canonical form, companion matrix, Cayley-Hamilton polynomial, Jordan blocks, Jordan canonical form (if the characteristic polynomial splits into linear factors, e.g. over an algebraically closed field), interesting matrices over non-algebraically closed fields, Smith normal form.
• Tues. Oct. 7: universal properties, localization of rings and modules (and properties thereof), tensor product (universal property and construction), determinant.
• Thurs. Oct. 9: more on universal properties and modules; symmetric and alternating forms, determinant.
• Tues. Oct. 14: properties of the category of modules over a ring; kernels, cokernels, etc. (and abelian categories, secretly). Complexes, exactness, homology. Long exact sequences in cohomology from short exact sequences of complexes. Construction of the Tor functors (take 1).
• Thurs. Oct. 16: Construction of the Tor functors (take 2). Left-derived functors of right-exact functors. "Enough projectives and injectives." Right-derived functors of left-exact functors.
• Tues. Oct. 21: the full statement of Yoneda's Lemma, Tor and flatness, calculations, Ext groups.
• Thurs. Oct. 23: spectral sequences and their care and feeding.
• Tues. Oct. 28: more linear algebra and spectral sequences.
• Thurs. Oct. 30: introduction to representation theory. The category of rpresentations of a group over a field. Group rings.
• Mon. Nov. 3: bonus office hours 2:15-4:15.
• Tues. Nov. 4: midterm. (Here is a practice midterm.)
• Thurs. Nov. 6: Zhiyuan Li discusses Wedderburn's theorem and some of its uses.
• Tues. Nov. 11 and Thurs. Nov. 13: Classifying representations. Character theory. Orthogonality relations.
• Tues. Nov. 18 and Thurs. Nov. 20: Finding all irreducible representations (for k algebraically closed, of characteristic not dividing the order of a group). First thoughts on removing algebraic closure hypotheses.
• Tues. Dec. 2: Evita's bonus office hours will be 12:30-2 and 3:30-5:30, not her usual office hours.
• Wed. Dec. 3: Evita's bonus office hours will be 2-5 pm. (Ravi's remain 10:30-12.)
• Thurs. Dec. 4: final exam. (Thursday morning I will be around, in meetings with graduate students, but you can ask me questions.)