## Math 902: Algebra II

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### Problem sets

### Time and place:

MWF 11:30am–12:20pm Burnett 232

### Textbook:

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

*Math 614 Lecture Notes* by Mel Hochster
*Introduction to Commutative Algebra* by Atiyah and Macdonald
*Introduction to Commutative Algebra and Algebraic Geometry* by Ernst Kunz

### Topics:

In this class we will cover introductory theory of commutative rings and modules. In more detail, a likely path is:

- Finite generation of algebras and modules
- Integral extensions
- Commutative Noetherian rings and Hilbert's Basis Theorem
- Finite generation of rings of invariants
- Graded rings
- Affine varieties
- Zariski topology
- Irreducible varieties
- Transcendence degree
- Nullstellensatz
- (Anti)equivalence between varieties and reduced affine algebras
- Spectrum of a ring
- Local rings
- Nakayama's Lemma
- Support of a module
- Associated primes
- Primary decomposition
- Krull dimension and height
- Cohen-Seidenberg theorems
- Noether normalization
- Dimension of affine algebras and varieties
- Krull's height theorem
- Systems of parameters
- Hilbert functions
- Regular local rings
- Normal rings