MATH 423/502 Introduction to Commutative Algebra and Representation Theory

Detailed Course outline

• Tuesday Jan. 3 : Motivation; review of rings. A criterion for a ring to be local. Operations on ideals. (Reference: Atiyah-Macdonald, pp. 1-5).
• Thursday Jan. 5 : Nilradical and Jacobson radical. Extension and pullback of ideals. (Reference: pp.5-10 of Atiyah-Macdonald). Did Example on p.10 (behaviour of the prime ideals of Z under the ring extension to Z[i] and Fermat's theorem on the sums of squares) in detail. These details can be found in any book on algebraic Number Theory, or in Dummit and Foote, Proposition 18 in 8.3.


• Tuesday Jan. 10 : The notion of spectrum of a ring. Many examples. In A-M the material we covered appears in Exercises 15, 16, 18, 26. My main reference was Section 3.2 of The Rising Sea by Ravi Vakil. The picture for Spec Z[x] was a celebrated picture from Mumford's red book of varieties and schemes.
• Thursday Jan. 12 : Spectrum of a ring, continued. Functorial properties of the spectrum. More about Zariski topology. We covered the remaining Exercises about Spec from AM Chapter 1 (apart from those assigned as homework).


• Tuesday Jan. 17 and Thursday Jan. 19: Hilbert basis theorem. Grobner bases. Main references: Section 9.6 of Dummit and Foote, and Sections 1.6 -1.7 of "Singular introduction to commutative algebra" (in this book's terminology, we only consider global orderings).


• Tuesday Jan. 24 : Back to Atiyah-Macdonald, Chapter 2. Discussed modules: definitions, homomorphisms. Main examples: vector spaces, ideals; k[x]-modules -- vector spaces with a linear operator (if uncomfortable with this example, please take a look at Dummit and Foote, Sections 12.1-12.3)). defined the notion of a category, and discussed the Hom-functor.
  Lior Silberman's supplement on categories .


• Thursday Jan. 26 : Finitely generated modules; Nakayama's Lemma
  An illuminating discussion on Mathoverflow.


• Tuesday January 31 : The Hom-functor and exactness. Tensor product of modules. (Chapter 2 of AM).
  (We did not get to review classification of modules over PIDs; if you need a review, please read about it over the break, e.g. Section 12.1 of Dummit and Foote; if you are interetsed in seeing how it implies the existence of Jordan canonical form, see Section 12.2 -- this will not be used in our course though).
• Thursday Feb. 2 : (Sujatha) Extension and restriction of scalars. Tensor products and exact sequences. Chapter 2 of AM, continued.


• Tuesday February 7 (Sujatha) Finished Chapter 2 of AM and started Localization (AM, Chapter 3)
• February 9 : Localization, continued. Local properties. (finished AM, Chapter 3).


• Tuesday February 14 : Integral extensions (AM, Chapter 5). Geometric meaning of "integrally closed" -- normal varieties. Notes by Ravi Vakil. Also, take a look at Chapter 3 of "Singular".
• Thursday February 16 : Going up and going down theorems. (AM, Chapter 5). Discussed their geometric meaning. Proved the going-up theorem.


• Feb. 21-23: break.


• Tuesday Feb.28 : Going-down theorem: proof, and an example showing that the assumption that A is integrally closed is necessary (the example is taken from Matsumura's first book, "Commutative algebra").
• Thursday March 2: Noether normalization theorem. Noetherian rings. Zariski's Lemma -- proof by Artin and Tate. References: AM, Exercises 16-17, Chapter 6 (covered it quickly without dwelling on descending chains), Chapter 7 up to (not including) "primary decomposition in Noetherian rings" (which we will omit).


• Tuesday March 7: Hilbert's Nullstellensatz: weak and strong forms, various proofs. References: AM, Exercise 18 in Chapter 5 and Exercise 14 in Chapter 7. Some very nice notes: Notes by Mel Hochster ; Notes on Jacobson rings by Matt Emerton.
• Thursday March 9: Quick survey of dimension theory. Main reference: Eisenbud "Commutative algebra with a view towards Algebraic Geometry", Chapter 8.