## Math 901: Algebra I

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### Problem sets

### 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