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Description
The purpose of the course is to further student鈥檚 understanding of algebraic objects, to provide a new (geometric) perspective on them and to provide necessary background for a course in algebraic geometry.
Modern commutative algebra was developed in the first half of the 20th century as a technical tool to study both number theory and algebraic geometry. It turned out that algebraic objects: rings, modules etc., can be thought of as geometric objects and geometric intuition is often very helpful in the study of these objects.听 Results are however proved formally.
The course starts with an example-based review of the basic notions: rings, ideals, modules, algebras, prime and maximal ideals, then introduces a fundamental tool in commutative algebra 鈥� localisation. 听The Nakayama lemma is proved. Then the notion of integral extensions of rings, which is absolutely essential in both number theory and algebraic geometry, is studied in some detail. The Noether normalisation theorem and its consequences (Zariski鈥檚 lemma, weak Nullstellensatz) are proved. Discrete Valuation rings and the Krull dimension of a ring are also studied. 听There is an emphasis on examples and on geometric understanding.
Module deliveries for 2024/25 academic year
Last updated
This module description was last updated on 8th April 2024.
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