Math 200C: Algebra III (Spring 2016)

Course description

This is the third quarter of the three-part graduate algebra sequence. In this quarter, we will cover primarily commutative algebra.

Instructor:	Alina Bucur (alina at math dot ucsd dot edu), APM 7151
Lectures:	MWF 9-10am, APM 5402
Office hours:	M 2-3pm if you cannot make this time, email me for an appointment

TA:	François Thilmany (fthilman at ucsd dot edu)
Office hours:	WF 3-4pm

Books:	M. Atiyah and I.G. MacDonald, Introduction to Commutative Algebra (the main textbook) The bookstore carries it, but you might find better priced copies elsewhere. Online shopping is your friend. Another good and more verbose reference, especially for people interested in arithmetic/algebraic geometry, is Eisenbud's Commutative Algebra: with a View Toward Algebraic Geometry.
Prerequisites:	MATH 200A and 200B.
Homework:	Homework will be assigned weekly (most weeks). The assigned problems will be posted on the class website. You are free to discuss the homework problems in general with the professor, the TA, or each other, but your final write-up of the problems must be your work alone. If you actually submit solutions that are not your own work, for example directly copying from an online solution bank, you will not get credit and I will have to report you for academic dishonesty.
Exams:	None.
Grading:	Your grade will be based entirely on homework. I will drop your lowest score assignment.

Announcements

Since the qual is on May 24, there will be no hw due on May 20. Instead, I will assign a longer homework that will be due on Wednesday June 1. This will be the last hw of the term.
The first part will be posted as usual on F 5/13 (so if you feel so inclined, you can start working on it). The pdf will be updated later with the rest of the problems.
On M 5/16, office hours will be 11am-12pm (instead of the usual 2-3pm).
F 5/20 and M 5/23 we will do review for the qual in lecture. Please come armed with questions!
François has put together some problems to help you with qual prep. Click here.
Special office hours for the qual:
- F 5/20 3-5pm (François)
- M 5/23 1-3pm (Alina)

Lecture summaries

Lecture 1: Categories: definition, examples, universal objects, products, coproducts, monomorphisms, epimorphisms.
Lecture 2: Additive and abelian categories, kernels, cokernels, images. Functors: defintion, examples.
Lecture 3: Exact/left exact/right exact functors. Examples. Chapter 1, Sections 1-4 of A-M: rings, ideals, quotient rings, subrings; zero-divisors, nilpotents, idempotents; prime and maximal ideals; local rings: definition and characterizations.
Lecture 4: Nilradical and Jacobson radical: definition, characterization, properties. Opreations on ideals: finite sums, finite products, finite intersectiosn. Distributivity. Chinese remainder theorem.
Lecture 5: Properties of prime ideals. Ideal quotient: definition, properties, examples. Radical of an ideal: definition, properties, examples. Behavior of ideals with respect to a ring homorphism: extenstion and contraction (definition, properties, examples). The category of modules over a ring: objects and morphisms. Examples.
Lecture 6: The category of R-modules: subobjects, kernels, cokernels, images, arbitrary sums and products, 0 object. The sum of submodules of a module. Isomorphism theorems for modules. Quotient ideal of two submodules, annihilators. If I ⊆ Ann_RM, then M is an R/I-module. Snake lemma.
Lecture 7: Tensor product of modules: definition, examples, universality property.
Lecture 8: Tensor product of modules: properties. Tensor product and Hom are adjoint. Tensor product is right exact.
Lecture 9: Finitely generated modules. Nakayama's lemma. Finitely generated modules over a local ring.
Lecture 10: Free modules and exactness of Hom and tensor product. Projective modules: definition, equivalent characterizations.
Lecture 11: Flat modules: definition, characterizations, properties, examples. Direct and inverse limits.
Lecture 12: Restriction and extention of scalars. Tensor product of algebras. The tensor product of a field extension with a separable finite field extension splits into a direct sum of fields.
Lecture 13: S^-1R: definition, examples, universal property. S^-1M: definition.
Lecture 14: Localization is an exact functor. Tensoring with S^-1R is the same as localizing. Local properties: M=0, injectivity.
Lecture 15: Flatness is a local property. Extended and contracted ideals in rings of fractions.
Lecture 16: More localization. Integral elements over a ring: definition and characterization. (The missing part of the last result about localization in pdf.)
Lecture 17: Integral closure and integrally closed rings. Integrality is transitive, commutes with quotient and localizations. Behavior of prime ideals in an integral extension of rings.
Lecture 18: Going-up theorem. Integral closures commute with localization. Being integrally closed is a local property. Integral elements over an ideal.
Lecture 19: Going-down theorem. Integral closures and Galois extensions. Valuation rings: defintion, examples, properties.
Lecture 20: Construction of a valuation ring in a field. The integral closure as intersection of valuation rings. Nullstellensatz.
Lecture 21: Ascending and descending chain conditions. Noetherian and Artinian R-modules: definition, examples.
Lecture 22: Going-down theorem. Valuation rings: definition, examples, properties.
Lecture 23: More about valuation rings. The integral closure as intersection of valuation rings.
Lecture 24: Hilbert's nullstellensatz. Noetherian and Artinian modules: defintion, examples. A module is noetherian iff all its submodules are finitely generated.
Lecture 25: Artinian and noetherian rings. Composition series.
Lecture 26: Length of a module: definiton, additive function. Noetherian rings. Rings of integers in number fields: defintion, they are noetherian. Localization of noetheian rings.
Lecture 27: Recap for qual: 200A.
Lecture 28: Recap for qual: 200B.
Lecture 29: Hilbert basis theorem, a field extension that is finitely generated as a k-algebra is a finite algebraic extension. Artinian rings: every prime ideal is maximal. Every Artinian ring is semilocal.
Lecture 30: Dimension of a ring. Artinian rings are noetherian rings of dimension 0.
Lecture 31: Structure theorem for Artinian rings. Discrete valuations: definition, examples. Discrete valuation rings.
Lecture 32: Characterizations of discrete valuation rings. Dedekind domains. Every ring of integers in a number field is a Dedekind domain. Zeta functions.

Homework

due Fridays by 5pm in your TA's mailbox on the 7th floor of APM
If you do not have access to the mailroom, ask the front desk to put in the mailbox.

HW1 due F 4/8 (pdf)
HW2 due ~~F 4/15~~ M 4/18 by 5pm (pdf)
HW3 due F 4/22 (pdf)
HW4 due F 4/29 (pdf)
HW5 due F 5/6 (pdf)
HW6 due F 5/13 (pdf)
HW7 due W 6/1 (part A, part B): the two parts will be graded separately, each counting as one hw.