COMMUTATIVE ALGEBRA

The aim of this course is to deepen the study of coomutative ring theory, taking particular attention to polynomial rings and their quotients, with a view towards applications to allgebraic geometry and number theory.

The student will also improve his skill of making abrstact argomentations and will learn that a deep theoretical knowledge allows to develop signifinant applicative tools.

In the course the will be lectures and exercises, given at the blackboard by the lecturer, and class exercises. Ussually the lecturer alternates exercises and theoretical parts in the same day. As for class exercises, the lecturer gives some exercises to the students, that have to try to solve them working in small groups; the lecturer helps the students to find the proper way to appoach the exercises. 

Should teaching be carried out in mixed mode or remotely, it may be necessary to introduce changes with respect to previous statements, in line with the programme planned and outlined in the syllabus.

I. Rings and ideals. Definitions and first properites. Prime and maximal ideals. Local rings. Nilradical and Jacobson radical. Ideals operations.Homomorphisms.

II. Modules. Definitions and first properties. DIrect product and direct sum; free modules. Finite modules and Nakayama's lemma. Module homomorphisms. Algebras.

III. Frctions rings and modules. Definition and properties. Localization and local properties. Ideals in fraction rings.

IV. Noetherian rings. Affine varieties, affine K-algebras, correspondance betwwen algebra and algebraic-geometry concepts. Krull dimension. Noetherian rings and modules: definitions and first properties. Hilbert's basis theorem. Conditions for a sub-algebra to be finitely generated.

V. Artinian rings. Artinian rings and modules. Composition series. Length. A ring is artinian if and only if it is noetherian and zero-dimensional.

VI. Primary decomposition. Primary ideals and primary decomposition. Associated primes and their characterization. Zero-divisors. Unicity of isolated components. The noetherian case.

VII. Hilbert Nullstellensatz: weak and strong formulations.

VIII. Integral dependance. Definitions and first properties. Going Up theorem. Normal domains and Going Down theorem. Noether's normalization lemma.

IX. First steps in dimension theory. Chain of primes, height, dimension. Krull's principal ideal theorem.Krull's height theorem.Dimension for polynomial rings with coefficient in a field. Local rings. System of parameters. Embedding dimension. Regular local rings (only definition and geometric relevance).

1. M.F. Atiyah, I.G. Macdonald , Introduzione to commutative algebra.

2. E. Kunz, Introduction to commutative algebra and algebraic geometry, Birkauser 1985