Introductory description

Commutative Algebra is the study of commutative rings, and their modules and ideals. This theory has developed over the last 150 years not just as an area of algebra considered for its own sake, but as a tool in the study of two enormously important branches of mathematics: algebraic geometry and algebraic number theory.

The resulting unification, where the same underlying algebraic structures arise both in geometry and in number theory, is one of the crowning glories of modern mathematics and plays a fundamental role in current work. A simple instance of this unification will already be familiar to anyone who has seen the strong parallel between the ring of integers Z and the polynomial ring k[X] over a field k: both are integral domains having division-with-remainder, and therefore a Euclidean algorithm. In either case, unique factorisation into primes and the treatement of ideals and modules in terms of localisation at primes follows by the same kind of arguments. The ring of integers of an algebraic number field used since the 19th century to solve problems in number theory runs in parallel with the geometry of algebraic curves and their function theory.

This module is a second course in the subject (following MA3G6), and will bring students to the frontiers of research in Algebra, Number Theory and Algebraic Geometry.

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

Review of basics of commutative algebra.
Factorisation, discrete valuations rings and normal rings.
The Artin-Rees lemma for completions and power series methods.
Dimension theory, the Hilbert and Hilbert-Samuel function.
Systems of parameters and multiplicity.
Regular sequences, the Koszul complex, depth, Cohen-Macaulay and Gorenstein rings.
Characterisation of regular local rings.

Learning outcomes

By the end of the module, students should be able to:

• Develop a sophisticated command of the many facets of a major branch of algebra with important applications across the whole of mathematics.
• Apply advanced notions of commutative algebra such as localisation, completion, dimension, extension, regular sequences, Hilbert and Hilbert-Samuel series, to rings and their modules, both in theoretical arguments and in practical applications.

Indicative reading list

M.F. Atiyah, I.G. MacDonald, Introduction to Commutative Algebra. Addison-Wesley 1969; reprinted by Perseus 2000.
M. Reid, Undergraduate Commutative Algebra. CUP 1995.
D. Eisenbud, Commutative algebra with a view toward algebraic geometry. Springer 1995.
H. Matsumura, Commutative ring theory, CUP 1986.

Subject specific skills

Working knowledge of commutative algebra.
Ability to apply commutative algebra methods to a variety of problems. including those in Algebraic Geometry and Number Theory.

Transferable skills

Ability to translate scientific ideas into mathematical language.
Ability to communicate complex ideas and mathematical results clearly.
Ability to analyse and solve abstract mathematical problems.