Math 901 Fall 2021

Math 901: Algebra I

Lecture notes

Problem sets

• Problem set #1 (tex) due Wednesday, September 8
• Problem set #2 (tex) due Friday, September 24
• Problem set #3 (tex) due Wednesday, October 13
• Midterm (tex) due Monday, October 25
• Problem set #4 (tex)
• Problem set #5 (tex)
• Final (tex) (edited 12/7) due Wednesday, December 15 (edited 12/9 4:30pm)

Time and place:

MWF 11:30am–12:20pm Oldfather 304

Textbook:

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

• Algebra by Serge Lang
• Advanced Modern Algebra by Joseph Rotman
• An Introduction to Homological Algebra by Joseph Rotman
• Math 614 Lecture Notes by Mel Hochster

Topics:

In this class we will cover algebraic techniques and ideas that are useful in a wide range of mathematical areas. The four main themes are category theory, multilinear algebra, homological algebra, and representation theory of semisimple algebras and groups. In more detail, a likely path is:

• Categories and basic categorical notions
• Functors and natural transformations
• Rings and modules
• Module categories, and additive functors
• Hom functor
• Multilinear maps and tensor products
• Short exact sequences and exact functors
• Tensor functors and tensor algebras
• Exactnesses and Hom and tensor
• Projective, injective, and flat modules
• Group rings
• Simple modules
• Jordan-Holder theorem
• Semisimple rings
• Artin-Wedderburn
• Character theory
• Chain complexes and homology
• The long exact sequence of homology
• Free and injective resolutions
• Derived functors, Ext, and Tor
• Balancing of Ext and Tor