The material for Math210B will be divided between commutative algebra, with Dimension Theory being an important topic, and group representation theory. In addition to the material mentioned in the syllabus we will discuss primary decomposition in commutative rings, and Mackey theory.
In addition to Lang's Algebra, the syllabus mentions some texts, including Jacobson, Basic Algebra II. Jacobson's book is excellent and inexpensive but we will mostly stick to sources that are available on-line through the Stanford Libraries, so I will refer to Lang when possible instead of Jacobson
An important topic this quarter will be the dimension theory of commutative rings, and for this Matsumoto's book Commutative Ring Theory is given as a reference. We also recommend Atiyah and Macdonald Commutative Algebra as a reference for the dimension theory and other aspects of commutative algebra. This book is also available on-line through the Library.
During the first week we will discuss integral ring extensions and transcendence degree of field extensions. So please read Lang, Sections VII.1, VIII.1. In the second week we will discuss Noether's Normalization Theorem so you may also want to look at Section VIII.2.
This week we will prove the Nullstellensatz on Monday, then review the relationship between affine algebraic geometry and commutative algebra.
This week we will start the dimension theory, including the primary decomposition theorem and the Hilbert polynomial. See Dimension I and Dimension II linked below, with Atiyah and Macdonald as an alternate sourse.
This week and next we will consider some facts from commutative algebra, the "Going up" and "Going down" theorems that concern the behavior of prime ideals in integral extensions. The significance of this topic is that one of several notions of the dimension of a variety, the Krull dimension of a commutative ring, is the length $d$ of a maximal chain of prime ideals:
$$\mathfrak{p}_0 \subset \mathfrak{p}_1 \subset \cdots \subset \mathfrak{p}_d$$So we want to understand how such chains behave in ring extensions. Supplementing Lang's Algebra, I recommend Chapter 5 in the book of Atiyah and Macdonald, which you can access on-line through the Stanford Libraries.
Leading into the topic of Group Representation Theory, I recommend reading at least the first 4 sections of Chapter XVII (Semisimplicity) in Lang's Algebra, and start reading Chapter XVIII. In addition to Lang's Algebra, you might want to consult other sources. Unfortunately some books I'd like to recommend are not available on-line through the Stanford Libraries, so I won't list them here. I don't really recommend Dummit and Foote for this topic, but it does have lots of good problems and is useful as long as it is not your primary reference. Here are some notes of mine:
From the departmental web page:
The material for Math210B will be divided between commutative algebra, with Dimension Theory being an important topic, and group representation theory.
Homeworks will be submitted on Gradescope, which you may access through Canvas. Usually homeworks will be due on Wednesdays. Solutions will be posted here after the due date.
| Homework | Latex | Due date | Solutions |
| Homework 1 (pdf) | (latex) | Tuesday, Jan 17, 2023 | HW1 Solutions |
| Homework 2 (pdf) | (latex) | Tuesday, Jan 24, 2023 | HW2 Solutions |
| Homework 3 (pdf) | (latex) | Tuesday, Jan 31, 2023 | HW3 Solutions |
| Homework 4 (pdf) | (latex) | Wednesday, Feb 8, 2023 | HW4 Solutions |
| Homework 5 (pdf) | (latex) | Tuesday, Feb 14, 2023 | HW5 Solutions |
| Homework 6 (pdf) | (latex) | Thursday, Feb 23, 2023 | HW6 Solutions |
| Homework 7 (pdf) | (latex) | Thursday, Mar 2, 2023 | HW7 Solutions |
| Last Homework 8 (pdf) | (latex) | Thursday, Mar 9, 2023 | HW8 Solutions |