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18.705

Commutative Algebra

3-0-9 units · Graduate

Exactness, direct limits, tensor products, Cayley-Hamilton theorem, integral dependence, localization, Cohen-Seidenberg theory, Noether normalization, Nullstellensatz, chain conditions, primary decomposition, length, Hilbert functions, dimension theory, completion, Dedekind domains.

Fall 2008

In this course students will learn about Noetherian rings and modules, Hilbert basis theorem, Cayley-Hamilton theorem, integral dependence, Noether normalization, the Nullstellensatz, localization, primary decomposition, DVRs, filtrations, length, Artin rings, Hilbert polynomials, tensor products, and dimension theory.

Readings

  • R: Reid, Miles. Undergraduate Commutative Algebra: London Mathematical Society Student Texts . Cambridge, UK: Cambridge University Press, April 26, 1996. ISBN: 9780521458894.
    Listed reading
  • AM: Atiyah, Michael, and Ian Macdonald. Introduction to Commutative Algebra . Reading, MA: Addison-Wesley, 1994. ISBN: 9780201407518.
    Listed reading
  • E: Eisenbud, David. Commutative Algebra: With a View Toward Algebraic Geometry . New York, NY: Springer-Verlag, 1999. ISBN: 9780387942698.
    Listed reading
  • SES #: 1 · TOPICS: Introduction, examples, prime ideals · READINGS: R: Chapter 0
    Reading assignment
  • SES #: 2 · TOPICS: Maximal ideals, Zorn’s lemma · READINGS: R: Chapters 1.4-1.9
    Reading assignment
  • SES #: 3 · TOPICS: Nilpotents, radical of an ideal, idempotents, local rings · READINGS: R: Chapters 1.10-2.3
    Reading assignment
  • SES #: 4 · TOPICS: Homomorphisms, generators, Cayley-Hamilton theorem, determinant trick, Nakayama’s lemma · READINGS: R: Chapters 2.4-2.8
    Reading assignment
  • SES #: 5 · TOPICS: Exact sequences, ascending chain condition, Noetherian rings · READINGS: R: Chapters 2.9-3.3
    Reading assignment
  • SES #: 6 · TOPICS: Hilbert basis theorem, Noetherian modules · READINGS: R: Chapters 3.4-3.6 and chapters 4.1-4.3
    Reading assignment
  • SES #: 7 · TOPICS: Integral closure, Noether normalization · READINGS: R: Chapters 4.4-4.8
    Reading assignment
  • R: Chapters 4.9-5.2 and chapter 6.1
    Listed reading
  • SES #: 9 · TOPICS: Construction of S^{-1}A, basic properties · READINGS: R: Chapters 6.2-6.3
    Reading assignment
  • SES #: 10 · TOPICS: Ideals in A and S ⁻¹ A , localization of modules · READINGS: R: Chapters 6.4-6.8
    Reading assignment
  • SES #: 11 · TOPICS: Exactness of localization · READINGS: R: Chapters 7.1-7.2
    Reading assignment
  • SES #: 12 · TOPICS: Support of a module SuppM , definition and properties of AssM · READINGS: R: Chapters 7.3-7.5
    Reading assignment
  • SES #: 13 · TOPICS: Relation between Supp and Ass, disassembling a module · READINGS: R: Chapters 7.6-7.9
    Reading assignment
  • SES #: 14 · TOPICS: Primary ideals, primary decomposition, uniqueness of primary decomposition · READINGS: R: Chapters 7.10-7.12
    Reading assignment
  • SES #: 15 · TOPICS: Definition of a DVR · READINGS: R: Chapter 7.13 and chapters 8.1-8.3
    Reading assignment
  • SES #: 16 · TOPICS: Main theorem on DVRs, general valuation rings · READINGS: R: Chapters 8.4-8.6
    Reading assignment
  • SES #: 17 · TOPICS: Serre’s criterion of normality, Dedekind domains · READINGS: R: Chapters 8.7-8.9 and 9.3(e)-(f)
    Reading assignment
  • SES #: 18 · TOPICS: Fractional ideals · READINGS: AM: Chapter 9
    Reading assignment
  • AM: Chapter 9, pp. 96-98
    Listed reading
  • R: Chapters 8.11-8.13
    Listed reading
  • R: Chapter s8.11-8.13
    Listed reading
  • E: Chapter 13
    Listed reading
  • AM: Chapter 8
    Listed reading
  • E: Chapter 10
    Listed reading
  • SES #: 24 · TOPICS: Modules of finite length · READINGS: AM: pp. 24-31 and 39-40
    Reading assignment
  • E: Chapter 2.4
    Listed reading
  • E: Chapters 5.0-5.2 and chapter 12
    Listed reading

Prerequisites & course information

Prereq: 18.702 G (Fall) 3-0-9 units

Sources