Math 210A, Fall 2016

No class Monday, December 5.We will have class as usual on December 7 and 9, which is the last day of classes. I will not have office hours this week (Dec 5-9) but will be in my office all of next week.

We will not follow it linearly or very closely. We will cover (among other things) the following topics: structure theory of finitely-generated modules over a principal ideal domain, and applications, including the canonical forms of linear algebra; basic commutative algebra, including localization; some category theory; multilinear algebra; homological algebra including ext, tor and derived functors; some multilinear algebra, in particular quadratic and symplectic spaces.

Grade will be based on homeworks.

I am happy to have homework by email to bump@math.stanford.edu. Please put Math210 or Math210A in the subject heading. Also if you do this, please send a pdf file in 12 point font. You can add this to the beginning of your latex file:

\documentclass[12pt]{article}

Readings in Lang

Week of September 26
Ideas of Category Theory
Universal Properties
Product and Coproduct
initial and terminal objects
Tensor Product of two modules
 Chapter I, Sec.11; Chapter 3, Sec.1-3; Chapter 16, Sec. 1
Week of October 3
Free and Projective Modules
Rank of a Free Module over Commutative Ring
 Chapter 2, Sec.5; Chapter 3, Sec.4,5.
Week of October 10
Noetherian Rings
Hilbert Basis Theorem
Principal Ideal Domains
Torsion and Torsion-Free Modules
Localization
 Chapter 10, Sec.1; Chapt.4, Sec. 4; Chapter 2, Sec.4,5; Chapter 3, Sec.7. The Hilbert Basis Theorem is Theorem IV.4.1.
Week of October 17
Right exactness of tensor product.
Flatness.
Localization is flat.
The Snake Lemma.
Extension of the Base Ring.
 Chapter 3, Section 9. Chapter 16, Sections 2, 3, 4.

Week of October 24
Ext and Tor
Homological Algebra
Chapter 16, Sec.3 (continued).
Chapter 20 Sec.1-6
Another recommended text:
Hilton and Stammbach
A Course in Homological Algebra
Chapters 1-4 (available online through Library)
Week of October 31
Derived Functors
Integral Dependence
The Extension Theorem
Chapter 20 (continued)
Chapter 7 Sections 1 and 3
The "Extension Theorem" is Corollary 3.3
on page 348. It implies the Nullstellensatz.
Week of November 7
Integral Dependence (continued)
Nakayama's Lemma
The Extension Theorem
The Nullstellensatz
Chapter 7 Sections 1 and 3
(continued)
Chapter 9 Sections 1 and 2
Chapter 10 Section 4
Week of November 14
The Nullstellensatz (continued)
Affine algebraic geometry
Spec of a ring
Multilinear Algebra
Chapter 9 Sections 1, 2 and 5
Chapter 16 Section 6,7,8
Chapter 19 Section 1
After Thanksgiving
Group Cohomology
Extensions of Modules and Groups
$Ext$ and module extensions
$H^2$ and group extensions
Exercises 1-5 and 27 in Chapter 20
(See posted solutions)
Hilton and Stammbach Chapter 3