# Readings

The readings listed are taken from the three course 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.

SES # TOPICS READINGS
Rings and ideals
1 Introduction, examples, prime ideals R: Chapter 0
2 Maximal ideals, Zorn's lemma R: Chapters 1.4-1.9
3 Nilpotents, radical of an ideal, idempotents, local rings R: Chapters 1.10-2.3
Modules
4

Homomorphisms, generators, Cayley-Hamilton theorem, determinant trick, Nakayama's lemma

R: Chapters 2.4-2.8
5 Exact sequences, ascending chain condition, Noetherian rings R: Chapters 2.9-3.3
6 Hilbert basis theorem, Noetherian modules R: Chapters 3.4-3.6 and chapters 4.1-4.3
Integral dependence
7 Integral closure, Noether normalization R: Chapters 4.4-4.8
8 Proof of Noether normalization, weak Nullstellensatz

R: Chapters 4.9-5.2 and chapter 6.1

Handout: Proof of the refined version of the Noether normalization lemma (PDF)

Localization
9 Construction of S^{-1}A, basic properties R: Chapters 6.2-6.3
10 Ideals in A and S-1A, localization of modules R: Chapters 6.4-6.8
11 Exactness of localization R: Chapters 7.1-7.2
12 Support of a module SuppM, definition and properties of AssM R: Chapters 7.3-7.5
13 Relation between Supp and Ass, disassembling a module R: Chapters 7.6-7.9
Primary decomposition
14 Primary ideals, primary decomposition, uniqueness of primary decomposition R: Chapters 7.10-7.12
Dedekind domains
15 Definition of a DVR R: Chapter 7.13 and chapters 8.1-8.3
16 Main theorem on DVRs, general valuation rings R: Chapters 8.4-8.6
17 Serre's criterion of normality, Dedekind domains R: Chapters 8.7-8.9 and 9.3(e)-(f)
18 Fractional ideals AM: Chapter 9
19 Finiteness of normalization

AM: Chapter 9, pp. 96-98

R: Chapters 8.11-8.13

Dimension theory
20 Going up, lying over, going down, dimension of affine rings

AM: pp. 61-62

R: Chapter s8.11-8.13

21 Artin rings

AM: pp. 62-64 and 78

E: Chapter 13

22 Krull's principal ideal theorem, parameter ideals

AM: Chapter 8

E: Chapter 10

Tensor product
23 Tensor product of modules, restriction and extension of scalars, flatness

AM: pp. 24-27

E: Chapter 10

Length
24 Modules of finite length AM: pp. 24-31 and 39-40
25 Graded rings and modules, associated graded ring, Hilbert polynomials

AM: pp. 76-78

E: Chapter 2.4

26 Filtrations, Artin-Reese lemma, dimension and Hilbert-Samuel polynomials

AM: pp. 106-107, 111-112, and 116-121

E: Chapters 5.0-5.2 and chapter 12