MATH 423/502 Introduction to Commutative Algebra and Representation Theory
There is no required text, but the following books will be useful:
- Atiyah and Macdonald ``Introduction to commutative algebra''
- D.S. Dummit, R.M. Foote, `` Abstract Algebra''.
- ``A Singular introduction to commutative algebra'' by Greuel and
Pfister (available online at the library website)
- J.-P. Serre, `` Linear Representations of Finite groups'' (for the
Representation Theory part).
- S. Bosch, ``Algebraic geometry and Commutative Algebra'' (available
online at the library website)
- D. Eisenbud ``Commutative Algebra with a view towards Algebraic Geometry''.
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
- The final presentations will be on Thursday April 13, 1-5pm --
please see the schedule below.
- Please sign up for a Piazza forum. The
Final presentations: Thursday April 13
- 1pm -1:50pm, Alex talking about direct and inverse limits (in MATX
- 2pm-2:50pm, Nicolas talking about homological algebra (in MATX 1118)
- 2:50pm-3:10pm -- coffee break (in MATX 1118)
- 3:10pm - 4pm Brian will talk about representation theory (in MATH
- 4:10pm - 5pm Javier will talk about Cohen-Macaulay rings. (in MATH
- 5pm -5:50pm Nina will talk about primary decompositions, and Dedekind
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
course but we did not have time for):
- Flatness, Ext and Tor.
- Cohen-Macaulay rings and smoothness.
- Direct and inverse limits. Completions and graded rings.
- Dedekind domains.
- How to compute normalizations algorithmically (Chapter 3 of
"Singular Introduction" book).
- Primary decomposition and irreducible varieties.
- Representation theory of the symmetric group S_n.
There will bi-weekly homework assignments, generally due on Thursdays.
Problem set 1 (due Thursday January
Problem set 2 (due Tuesday January
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.
- Here are some resources
if you you need resources for using TeX (optional).
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
assigned as homework).
- Tuesday Jan. 17 and Thursday
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
- 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.
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:
Mel Hochster ;
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:
- 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
- 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;
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.
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
- April 6:
Induced representations. Example: representations of the dihedral group.
References: Serre, Chapters 5 and 7 and 8 (a selection of results from