The problems are exercises for your mathematical health; they provide a means for you to check your understanding of the material. They are meant to be neither difficult, nor tricky, nor involved. If you find that you are stuck on any problem, then review the relevant material; if you remain stuck, then discuss the problem with someone. Of course, you must still think through each problem on your own and write it up in your own words.
Normally the problem presented on Tuesday is due two days later. Although some problems have more than one solution, feel free to write up and pass in the solution presented; just think it through for yourself. You will learn more if you think about the problem before it is presented. Similarly, Thursday’s presentation normally treats a problem due that day. If you learn that you have made a mistake in your own solution, then you may request an extension so that you can correct your work. Although this practice may seem odd, remember what counts in the end: that you can solve the problem correctly and that you can explain the solution clearly.
Problems are to be handed in at the end of the indicated class session and will be graded in part on the quality of the write-up.
Most of the assignments listed are taken from the three textbooks:
R: Reid, Miles. Undergraduate Commutative Algebra: London Mathematical Society Student Texts. Cambridge, UK: Cambridge University Press, April 26, 1996. ISBN: 9780521458894.
AM: Atiyah, Michael, and Ian Macdonald. Introduction to Commutative Algebra. Reading, MA: Addison-Wesley, 1994. ISBN: 9780201407518.
E: Eisenbud, David. Commutative Algebra: With a View Toward Algebraic Geometry. New York, NY: Springer-Verlag, 1999. ISBN: 9780387942698.
SLK: Some of the problems are not taken from the course textbooks but can be found in the assignments handout. (PDF)
The [*] that appears before some problems indicates that the problem is to be presented in class.
| SES # | TOPICS | ASSIGNMENTS |
|---|---|---|
| Rings and ideals | ||
| 1 | Introduction, examples, prime ideals |
R: Problems 1.5, 1.6, 1.12(a) SLK: [*] 1 |
| 2 | Maximal ideals, Zorn’s lemma |
R: Problems [*] 1.18, 1.19 SLK: 2 |
| 3 | Nilpotents, radical of an ideal, idempotents, local rings | R: Problems 1.4, 1.11, [*] 1.10 |
| Modules | ||
| 4 |
Homomorphisms, generators, Cayley-Hamilton theorem, determinant trick, Nakayama’s lemma |
R: Problems 2.1, [*] 2.6 (and prove that every minimal generating set of M has the same number of elements), 2.8 (and prove that the idempotent is unique) |
| 5 | Exact sequences, ascending chain condition, Noetherian rings |
R: Problems [*] 2.9, 3.2, 3.5 (prove it, or give a counterexample) SLK: 3 |
| 6 | Hilbert basis theorem, Noetherian modules | R: Problems 3.3, 3.4, 3.7 (with p a prime), [*] 3.8 |
| Integral dependence | ||
| 7 | Integral closure, Noether normalization |
R: Problems 4.1(a), [*] 4.5 AM: Problem 5.6 |
| 8 | Proof of Noether normalization, weak Nullstellensatz |
R: Problems 4.9, [*] 4.10 AM: Problem 5.9 |
| Localization | ||
| 9 | Construction of S^{-1}A, basic properties | R: Problems: [*] 6.3(a), 6.1 (show that the ring A of problem 6.3(a) works), 6.5 (describe each subring A as a localization of Z) |
| 10 | Ideals in A and S_-1_A, localization of modules |
R: Problem [*] 6.13 SLK: 4, 5 |
| 11 | Exactness of localization |
AM: Problem [*] 3.1 SLK: 6, 7, 8 |
| 12 | Support of a module Supp_M_, definition and properties of Ass_M_ |
R: Problem [*] 7.2 (if true, prove it; if false, give a counterexample) SLK: 9 |
| 13 | Relation between Supp and Ass, disassembling a module | R: Problems 7.4, 7.6, [*] 7.7 |
| Primary decomposition | ||
| 14 | Primary ideals, primary decomposition, uniqueness of primary decomposition |
R: Problems 7.8, 7.10 SLK: [*] 10 |
| Dedekind domains | ||
| 15 | Definition of a DVR |
R: Problems 8.1, 8.2, 8.4 SLK: 11 |
| 16 | Main theorem on DVRs, general valuation rings |
R: Problem [*] 8.6 SLK: 12, 13 |
| 17 | Serre’s criterion of normality, Dedekind domains |
R: Problem 8.7 SLK: [*] 14, 15 |
| 18 | Fractional ideals |
AM: Problems 9.7, 9.8 SLK: [*] 16 |
| 19 | Finiteness of normalization | SLK: 17, 18, [*] 19, 20 |
| Dimension theory | ||
| 20 | Going up, lying over, going down, dimension of affine rings |
E: Problems 13.2, [*] 13.3 SLK: 21, 22, 23 |
| 21 | Artin rings |
E: Problem 9.4 SLK: [*] 24, 25, 26 |
| 22 | Krull’s principal ideal theorem, parameter ideals | SLK: 27, [*] 28, 29 |
| Tensor product | ||
| 23 | Tensor product of modules, restriction and extension of scalars, flatness | SLK: [*] 30, 31, 32 |
| Length | ||
| 24 | Modules of finite length | SLK: 33, [*] 34, 35 |
| 25 | Graded rings and modules, associated graded ring, Hilbert polynomials | |
| 26 | Filtrations, Artin-Reese lemma, dimension and Hilbert-Samuel polynomials | |
