# MA4J8 Commutative Algebra II

**Lecturer: **Miles Reid

**Term(s): **Term 1

**Status for Mathematics students: **List A

**Commitment: **30 lectures

**Assessment: **85% by 3-hour examination, 15% coursework

**Formal registration prerequisites: **None

**Assumed knowledge: **Divisibility and ideals from MA249 Algebra II: Groups and Rings, the first half of the MA3G6 Commutative Algebra module or my book "Undergraduate Commutative Algebra".

**Useful background: **Material from, or an interest in:

- MA3A6 Algebraic Number Theory
- MA3G6 Commutative Algebra
- MA377 Rings and Modules
- MA3D5 Galois Theory
- MA3H6 Algebraic Topology with basic ideas on Homology

**Synergies: **The module runs in parallel with MA4A5 Algebraic Geometry and the overlap between the two may serve as useful repetition and reinforcement. The material of commutative algebra has many basic and more advanced links with Algebraic Geometry, Algebraic Number Theory and next term's MA4L7 Algebraic Curves.

**Content:
**

- Review of MA3G6 Commutative Algebra
- Completion and dimension of Noetherian local rings
- Regular local rings, free resolutions and projective dimension
- Regular sequences, Cohen-Macaulay and Gorenstein rings

**Aims:**

The module treats selected topics in commutative algebra, as a continuation of MA3G6 Commutative Algebra. Commutative algebra takes as its model 19th century work in arithmetic and algebraic geometry, as unified in the work of Dedekind and Weber, and later David Hilbert. In the modern era, commutative algebra is viewed as foundational for algebraic number theory, algebraic geometry (especially its scheme theoretic aspects) and computer algebra. Introductory texts on Algebraic Geometry commonly assume results from Commutative Algebra going beyond the first MA3G6 Commutative Algebra module, and my course will cover many of these results, together with topics in the algebra of commutative rings that are of independent interest.

**Objectives:
**By the end of the module the student should:

- Have developed a sophisticated command of many facets of a major branch of algebra with important applications across the whole of mathematics
- Be in a position to read standard texts and research papers on commutative algebra
- Be able to apply commutative algebra methods to problems in algebra, arithmetic and geometry, both on paper and in computer algebra
- Understand the dimension of rings and the relations between regular local rings and nonsingular points of algebraic varieties
- Know how to work with Cohen-Macaulay, Gorenstein and complete Intersection rings

**Website:
**Example sheets, assessed worksheets and additional material will be available from my webpage:

https://homepages.warwick.ac.uk/~masda/MA4J8/

I may produce notes on some topics, including colloquial “general interest” or “propaganda” items, and technical appendices.

**Books:
**M.F. Atiyah and I.G. MacDonald:

*Introduction to Commutative Algebra,*Warwick Library QA 251.3.A8, available as e-book

*David Eisenbud:*

*Commutative Algebra with a View Towards Algebraic Geometry,*Warwick Library QA 251.3.E4, available as e-book

*Hideyuki Matsumura:*

*Commutative Ring Theory,*Warwick Library QA 251.3.M2, available as e-book

*Miles Reid: Undergraduate Commutative Algebra, Warwick Library QA 251.3.R3, available as e-book*

*Marco Schlichting, Commutative algebra II (notes of the 2013 module by Florian Bouyer), get from:*

https://homepages.warwick.ac.uk/~masda/MA4J8/

## Additional Resources