Math 902 Spring 2022

Math 902: Algebra II

Lecture notes (through 4/27)

Problem sets

• Problem set #1 (tex) due Friday, February 4
• Problem set #2 (tex) due Wednesday, February 23
• Problem set #3 (tex) due Wednesday, March 9
• Midterm (tex) due Friday, April 1
• Problem set #4 (tex) due Monday, April 18
• Problem set #5 (tex) Problem #2 optional/bonus due Wednesday, April 27
• Final (tex) due Wednesday, May 4

Time and place:

MWF 11:30am–12:20pm Burnett 232

Textbook:

I will be following and posting my own notes for the class. Recommended supplementary texts include

• Math 614 Lecture Notes by Mel Hochster
• Introduction to Commutative Algebra by Atiyah and Macdonald
• Introduction to Commutative Algebra and Algebraic Geometry by Ernst Kunz

Topics:

In this class we will cover introductory theory of commutative rings and modules. In more detail, a likely path is:

• Finite generation of algebras and modules
• Integral extensions
• Commutative Noetherian rings and Hilbert's Basis Theorem
• Finite generation of rings of invariants
• Graded rings
• Affine varieties
• Zariski topology
• Irreducible varieties
• Transcendence degree
• Nullstellensatz
• (Anti)equivalence between varieties and reduced affine algebras
• Spectrum of a ring
• Local rings
• Nakayama's Lemma
• Support of a module
• Associated primes
• Primary decomposition
• Krull dimension and height
• Cohen-Seidenberg theorems
• Noether normalization
• Dimension of affine algebras and varieties
• Krull's height theorem
• Systems of parameters
• Hilbert functions
• Regular local rings
• Normal rings