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 unification which results, where the same underlying algebraic structures arise both in geometry and in number theory, has been one of the crowning glories of twentieth century mathematics and still plays an absolutely fundamental role in current work in both these fields.
One simple example of this unification will be familiar already to anyone who has noticed the strong parallels between the ring Z (a Euclidean Domain and hence also a Unique Factorization Domain) and the ring F[X] of polynomials over a field (which has both the same properties). More generally, the rings of algebraic integers which have been studied since the 19th century to solve problems in number theory have parallels in rings of functions on curves in geometry.
While self-contained, this course will also serve as a useful introduction to either algebraic geometry or algebraic number theory.

Module web page

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.

Topics: Gröbner bases, modules, localization, integral closure, primary decomposition, valuations and dimension.

Learning outcomes

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

• understand the Hilbert bases theorem, know several examples of Noetherian rings.
• be able to compute the Gröbner bases of an ideal of the polynomial ring; Spec of some rings; zero locus of ideals.
• understand the proof of Cayley-Hamilton theorem, and to be able to apply it to prove Nakayama Lemma, Tower’s law and other theorems in module theory.
• be able to determine algebraic integers via several methods in the theory of finite extensions.
• be able to compute the localisation, integral closure, dimension of rings.
• understand the proof of Hilbert Nullstellensatz, and to be able to apply it to compute the primary decomposition of ideals in a polynomial ring.

Indicative reading list

M.F. Atiyah, I.G. MacDonald, Introduction to Commutative Algebra. Addison-Wesley 1969; reprinted by Perseus 2000. [QA251.3.A8]
D. Eisenbud, Commutative algebra with a view toward algebraic geometry. Springer 1995. [QA251.3.E4]
M. Reid, Undergraduate Commutative Algebra. CUP 1995. [QA251.3.R3]
R.Y. Sharp, Steps in Commutative Algebra (2nd ed.) CUP 2000. [QA251.3.S4]
O. Zariski and P. Samuel, Commutative Algebra, (vols I and II). Springer 1975-6. [QA251.3.Z2]

Subject specific skills

The students will get their first knowledge in algebraic geometry and algebraic number theory. The students will train their programming skill in various assignments, especially computational problems for Gröbner bases.

Transferable skills

The students will acquire confidence in dealing with apparently complicated problems which nonetheless have a simple underlying solution.