MATH 423/502 Introduction to Commutative Algebra and Representation Theory.

MATH 423/502 Introduction to Commutative Algebra and Representation Theory


There is no required text, but the following books will be useful: Links to some online notes will also be posted.
Classes: Tue, Th 2-3:30pm in MATX1102.
My office: Math 217.
e-mail: gor at math dot ubc dot ca
Office Hours: for now, by appointment.


Final presentations: Thursday April 13

Final schedule:

Ideas for the final essay

The essay will be due April 27 (strict UBC deadline). Feel free to suggest your own topic (please discuss with me before committing to it). Here are some suggestions (some classical topics that fit very naturally in this course but we did not have time for):


There will bi-weekly homework assignments, generally due on Thursdays.
  • Problem set 1 (due Thursday January 12).
  • Problem set 2 (due Tuesday January 31). The scan of the exercise from DF.
  • Problem Set 3 (due Tuesday February 28 -- first class after the break).
  • Problem Set 4 (due approximately Thursday March 16, but this is not strict).
  • Review problems on linear algebra (not to hand in).
  • Problems on representation theory (Hand in ony if you want it to count as a final essay).
  • Two interesting representation theory problems. Do not hand in unless you really want to.

    Detailed Course outline

    Short descriptions of each lecture and relevant additional references will be posted here as we progress. The records for dates in the past reflect what happened; for the dates in the near future these are tentative plans, subject to change.
    • 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.

    Representation Theory (starting March 14)

    Some interesting links: Detailed outline:
    • March 14: started Representation theory. From now on the main reference is Serre, "Linear representations of finite groups". I hope to cover approximately Chapters 1-8. On Tuesday, defined representations, irreducible representations; Regular representation; some examples; existence of a complement to a sub-representation (Serre, Sections 1.1-1.3).
    • March 16: Decomposition into irreducible representations; Schur's Lemma; characters (Serre, Sections 1.4, 2.1-2.3).

    • March 21-23: NO CLASS!

    • March 28-31: Lectures on Tuesday and Thursday as usual, and the long make-up class, Friday March 31, 3-5pm.
      Tuesday and Thursday: Decomposition of the right regular representation; orthogonality of characters and matrix coefficients. (Serre: Sections 1.5, 2.1, 2.3-2.4, 2.5-2.6). However, the lectures were closer to Casselman's notes , Sections 3,4,6 and 7.
    • March 31: Long make-up lecture, in Math 126, 3-5pm. The space of central functions on G; an application -- dimension of representation has to divide the order of the group. The reference for this is Chapter 6 of Serre. Fourier analysis on abelian groups. References: Serre 3.1 and Chapter 4; Casselman's notes , Sections 8,9, 12-13. Another very useful reference: Notes on abelian groups by Keith Conrad.
      A copy of lecture notes .

    • April 4: Harmonic analysis on finite (and compact) groups -- statements. Induced representations (definitions). References for induced representations: Serre,Section 3.3, and Casselman's notes , Section 11. Please think of why the two definitions of an induced representation are equivalent.
    • April 6: Induced representations. Example: representations of the dihedral group. References: Serre, Chapters 5 and 7 and 8 (a selection of results from these chapters).