Math 101
Modules, Linear Algebra, Groups

Last updated July 18, 2017 09:28:13 EDT

## Syllabus

The main focus of the course will be modules (projective, injective, free, torsion), applications to linear algebra including canonical forms, direct products, tensor products, coproducts, and categorical characterization in terms of universal mapping properites. We will also do some group theory towards the end of the course. This will involve chapters $10-12$ and $4 - 5$ of Dummit and Foote's text Abstract Algebra as well as supplementary material. Below is a tentative syllabus; the weekly syllabus contained on the Homework Assignments page will always be accurate.

Lectures Sections in Text Brief Description
Week 1 11.1, 11.2, 10.1 Linear Algebra (old and new perspectives), modules and examples
Week 2 10.2, 10.3, 10.5 Module isomorphism theorems, category theory, products and coproducts, Modules and exact sequences, Hom(D,-) and Hom(-,D) as functors
Week 3 10.3, 10.5 More on Hom(A, B), splitting SES, free, projective, injective modules
Week 4 10.4, 10.5 Tensor products, localization, more on exact sequeces
Week 5 10.5, 12.1 Extension of scalars, Flat modules, Modules over a PID
Week 6 12.1 - 12.2 Finitely generated modules over PIDs and canonical forms
Week 7 12.3, 4.1 - 4.3 Finish canonical forms, partitions, equivalence relations, cosets and consequences, direct products
Week 8 4.5, 4.6, 5.1, cyclic groups, Group Actions, Sylow theorems
Week 9 5.2, 5.4, 5.5 Symmetric and Alternating group, semidirect products

T. R. Shemanske
Last updated July 18, 2017 09:28:13 EDT