主题目录

    • University:   Massachusetts Institute of Technology

      Instructors:  Prof. Steven Kleiman

      Course Number:  18.705

      Level:  Graduate

      Course Description

      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.

  • 教学大纲(Syllabus)

    • Prerequisites

      Algebra I (18.701) and Algebra II (18.702)

  • 其他教学资源(Other Resources)

    • 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 #TOPICSREADINGS
      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