Math 200c Spring 2011

Announcements:

The quarter is now over. Please contact me to make an appointment if you want to see and talk over your qual exam. A blank copy of the qual exam is posted below the homework, in case you want to think more about problems you missed. Otherwise have a good summer!

Instructors:

Professor:

Teaching Assistants:

Course description:

This is the third quarter of the three-part graduate algebra sequence. We will cover tensor products, an introduction to algebraic geometry, an introduction to noncommutative rings, and an introduction to representations of finite groups. While all of the topics covered this quarter are fundamental to further study in algebra, the qualifying exam will be based only on the material covered in the first two quarters of Math 200 and the very beginning of this quarter only (tensor products and maybe a bit more.)

Textbook:

D. Dummit, R. M. Foote, Abstract Algebra, 3rd edition.

Grading:

I will assign grades considering your record on homework and exams as a whole. Roughly, your grade is likely to correspond to your result on the qual (if you take it): A = PhD Pass, A- = Provisional PhD Pass, B+/B = Master's Pass, C = not a passing grade on the qual. With excellent homework your final grade might be better than your qual result, and with little or no homework your final grade might be worse. If you do not plan to take the qual, the default will be to base your final grade on the homework assignments. This course is graded seriously, so you should not register for the course if you are not prepared to hand in the homework.

Exams:

The only exam will be the qualifying exam which will be in late May, which substitutes for a final exam. Please see Lois Stewart to sign up for the qual if you plan to take it.

Homework:

The homework will be due roughly every two weeks, with the first homework set not due until week 2. Copying your homework answers from another student or from an online source is academically dishonest, and is almost sure to lead to a poor qual result.

What we plan to cover:

The topics we plan to cover are for the most part found in the textbook in some form. The relevant sections are 10.4, 10.5, Chapter 15, and Chapter 18-19. Time permitting, we may cover a few other sections in the text. Wedderburn's classification theorem on semisimple algebras is not proved in the text; so that topic will be based primarily on course notes.

Lecture Summaries

• Lecture 1 (3/28/11): Section 10.4 Bilinear maps. Definition of a tensor product of modules over a commutative ring. Existence and uniqueness of the tensor product.
• Lecture 2 (3/30/11): Section 10.4 Examples of tensor product. The basis of a tensor product over a field. Extension of scalars.
• Lecture 3 (4/1/11): Section 10.4 Examples of extension of scalars. The tensor product of two algebras and some examples.
• Lecture 4 (4/4/11): Section 10.5 Short exact sequences and extensions of modules. Characterizations of split exact sequences. -- tensor_R N acts on homomorphisms and is a right exact operation.
• Lecture 5 (4/6/11): Section 10.5 Proof of right exactness of tensor. Left exactness of Hom( --, M) and Hom(M, --). Projective and modules. Free modules are projective. Exact sequences of the form 0 -> M -> N -> P -> 0 with P projective split.
• Lecture 6 (4/8/11): Section 10.5 A module is projective if and only if it is a direct summand of a free module. Injective modules and examples. Baer criterion (no proof) and proof that divisible modules over a PID are injective. Hom(--, E) is exact if and only if E is injective.
• Lecture 7 (4/11/11): Section 10.5, 15.1. Flat modules. Projective modules are flat. Finitely generated k-algebras. Affine n-space A^n. Introduction to the correspondence between ideals of the polynomial ring k[x_1, ..., x_n] and subsets of A^n.
• Lecture 8 (4/13/11): Section 15.1. More basic properties of the correspondence between ideals of the polynomial ring and subsets of affine space. The Zariski topology.
• No lecture 4/15/11
• Lecture 9 (4/18/11): Section 15.1. Morphisms (regular maps) between affine closed subsets. Regular maps V to W are in bijective correspondence with k-algebra homomorphisms k[W] to k[V] between their coordinate rings.
• Lecture 10 (4/20/11): Section 15.2: Irreducible closed subsets. Every closed subset of affine space is (uniquely) a finite union of irreducible closed subsets.
• Lecture 11 (4/22/11): Section 15.3: Integral extensions of rings.
• Lecture 12 (4/25/11): Section 15.3. Noether normalization: every finitely generated k-algebra is integral over a subring isomorphic to a polynomial ring. The Weak Nullstellensatz: if k is algebraically closed, all maximal ideals of k[x_1, ... , x_n] are of the form (x_1 - a_1, ... , x_n - a_n).
• Lecture 13 (4/27/11): Completion of the proof of the Nullstellensatz. Examples.
• Lecture 14 (4/29/11): Introduction to noncommutative rings and modules. Semisimple rings: rings such R that R is a direct sum of simple modules. Schur's lemma. Endomorphism rings. The endomorphism ring of a direct sum.
• Lecture 15 (5/02/11): More on the endomorphism ring of a direct sum. First version of Wedderburn's theorem: Semisimple rings are isomorphic to products of matrix rings over division rings.
• Lecture 16 (5/04/11): Structure of semisimple modules: submodules and factor modules of semisimple modules are semisimple. The Jacobson radical and its first properties.
• Lecture 17 (5/06/11): More on the Jacobson radical. Artinian rings with zero Jacobson radical are semisimple.
• Lecture 18 (5/09/11): The Jacobson radical of an Artinian ring is nilpotent. Prime and semiprime ideals. Artinian semiprime rings are semisimple.
• Lecture 19 (5/11/11): Artinian rings are Noetherian. Nakayama's Lemma. Introduction to representation theory, definitions and first examples.
• Lecture 20 (5/13/11): Representation theory. Correspondence between representations of G and modules over the group algebra of G.
• Lecture 21 (5/16/11): Proof of Maschke's theorem. The center of the group algebra CG. The number of matrix rings in an expression of CG as a product of matrix rings is equal to the number of conjugacy classes of G.
• No lecture 5/18, 5/20, 5/23, 5/25
• Lecture 22 (5/27/11): Characters of representations. Character tables. Orthogonality of irreducible characters.
• Memorial Day (no class) (5/30/11):
• Lecture 23 (6/1/11): Burnside's theorem: finite groups whose order is divisible by at most two primes are solvable. Beginning of proof.
• Lecture 24 (6/3/11): Conclusion of the proof of Burnside's theorem.

Homework

• Homework 1: Homework 1 due Friday 4/8/11 (extended to Monday 4/11/11)

• Homework 2: Homework 2 due Friday 4/22/11

• Homework 3: Homework 3 due Friday 5/6/11

• Homework 4: Homework 4 due Friday 6/3/11

• Algebra qualifying exam administered May 19, 2011: Spring 2011 qual