Math 250B: Graduate Algebra II

IMPORTANT NOTE: Starting on 1/27/2015, lectures will be held in 3113 Etcheverry.

Basic information:

• Faculty: Adam Topaz.
• Lecture: TR 5:00-6:30pm in 3113 Etcheverry.
• Office Hours: Tuesday 4-5pm, Wednesday 1-2pm, and by appointment.
• Office Number: 889 Evans.

Literature:

• Algebra, by S. Lang.
• Commutative Algebra: with a view toward algebraic geometry, by D. Eisenbud.
• ... other sources will be discussed in class.

Note: Personal electronic copies of these texts are freely available to Berkeley students via springer link.

List of Topics:

The emphasis will be on commutative algebra required for algebraic geometry and number theory. Other topics will be discussed, especially some general homological algebra and group cohomology.

Tentative List of Topics:

• Basics of category theory: categories; functors; natural transformation; the Yoneda lemma; projective/injective resolutions and basics of derived functors; $\operatorname{Ext}$ and $\operatorname{Tor}$ (as needed).
• Basics of group cohomology: homogeneous and inhomogeneous cocycles; functorial properties; $\operatorname{H}^2$ and group extensions; cohomology of cyclic groups; group homology.
• Commutative rings: localization; $\operatorname{Spec}$ of a ring; associated primes and primary decomposition; Nakayama's Lemma; Noether normalization; Hilbert's Nullstellensatz; flat and etale morphisms; dimension theory; regular rings; modules of differentials; complete and Henselian rings; possibly other topics.
• Valuation rings: DVR's and general valuation rings; refinement and coarsening; connections with Galois theory and algebraic geometry.

Notes

Below are some lecture notes from the class. NOTE: These are mostly informal, and you should read them with caution. You should assume that every sentence has typos and mistakes. Also, many proofs are omitted, especially ones that involve commutative diagrams. Don't use these notes as a reference!

Category Theory and Homological Algebra:

• Basic Definitions
• The Yoneda Lemma
• Basic Homological Algebra
• Derived Functors
• Group Cohomology
• Tensor Products

Homework

Homework assignments will appear below.

• Homework 1
• Homework 2
• Homework 3
• Homework 4
• Homework 5
• Homework 6 -- Erratum
• Homework 7
• Homework 8 -- Section 2 is optional, but I highly recommend that you think about it!

Grading

Final grades will be based on homework, including a final longer (somewhat cumulative) homework assignment.