Commutative Algebra

Class overview

This lecture course will provide Master students in Math with some basic concepts of Commutative Algebra and applications.

Organization:

+ 3 hours/ week (including exercise sessions)

+ Time of Lecture Course: 8:00-11:30  (Monday, Thursday)

+ Exercise Class: 8:00-11:30  (Monday, Thursday)

+ Room: Building L, Third Floor, Seminar Room 2 (Faculty of Math)

+ Written final examination (90-120 minutes)

Literature

[1] M. F. Atiyah, I. G. MacDonald, Introduction to Commutative Algebra, 1969.

[2] E. Kunz, Introduction to Commutative Algebra and Algebraic Geometry, Birkhäuser Boston, 1984.

[3]  R.Y. Sharp, Steps in commutative algebra, Cambridge University Press, 2000.

[4] D.A. Cox, J. Little, D. O'Shea, Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, Springer, 2015. 

[5] A. Bandini, P. Gianni, E. Sbarra, Commutative Algebra through Exercises, Springer, 2024 

[6] A. Simis, Commutative Algebra, De Gruyter, 2023.

Contents of the lecture course

1. Rings and Ideals: Covers the fundamental building blocks of commutative algebra.

   • Rings and ring homomorphisms

   • Ideals and quotient rings

   • Prime ideals and maximal ideals

   • Nilradical and Jacobson radical

   • Operations on ideals (sum, product, intersection)

   • Extension and contraction of ideals

2. Modules: Introduces the concept of modules, which are essential for studying rings.

   • Modules and module homomorphisms

   • Submodules and quotient modules

   • Exact sequences

   • Tensor product of modules

   • Finitely generated modules

3. Rings and Modules of Fractions: Explores the process of localization, a key technique in commutative algebra.

   • Localization of rings and modules

   • Properties of localization

4. Primary Decomposition: Discusses the generalization of the unique factorization of integers into prime powers.

   • Primary ideals

   • Primary decomposition theorem (Lasker-Noether theorem)

5. Integral Dependence and Valuations: Deals with integral extensions of rings.

   • Integral dependence

   • The Going-Up and Going-Down theorems

   • Valuation rings

6. Chain Conditions: Introduces the concepts of Noetherian and Artinian rings.

   • Ascending and descending chain conditions on ideals and modules

7. Noetherian Rings: Focuses on the properties of Noetherian rings, a particularly well-behaved class of rings.

   • Primary decomposition in Noetherian rings

8. Artin Rings: Explores the properties of Artinian rings, which satisfy the descending chain condition.

   • Characterization of Artinian rings

9. Discrete Valuation Rings and Dedekind Domains: Applies the preceding theory to study important types of integral domains.

   • Discrete valuation rings (DVRs)

   • Dedekind domains

   • Fractional ideals

10. Completions: Introduces the concept of completing a ring or module with respect to a topology.

    • Topologies and filtrations

    • Completions of rings and modules

11. Dimension Theory: Discusses the algebraic notion of dimension for rings, inspired by the geometric concept of dimension.

    • Hilbert functions

    • Krull dimension

    • Regular local rings

 Course Materials

• Lecture 1 and Exercises

• Lecture 2 and Exercises

• Lecture 3 and Exercises

• Lecture 4 and Exercises

• Lecture 5 and Exercises

• Lecture 6 and Exercises

• Lecture 7 and Exercises

• Lecture 8 and Exercises

• Lecture 9 and Exercises

• Lecture 10 and Exercises

Some Further Notes